Modelling of Rabies Transmission Dynamics Using …downloads.hindawi.com/journals/jam/2017/2451237.pdfexplored three control methods: vaccination, vaccination pulsefertilitycontrol,andculling.Anordinarydifferential

Research ArticleModelling of Rabies Transmission Dynamics Using OptimalControl Analysis

Joshua Kiddy K Asamoah1 Francis T Oduro1 Ebenezer Bonyah2 and Baba Seidu3

1Department of Mathematics Kwame Nkrumah University of Science and Technology Kumasi Ghana2Department of Statistics and Mathematics Kumasi Technical University Kumasi Ghana3Department of Mathematics University for Development Studies Navrongo Ghana

Correspondence should be addressed to Joshua Kiddy K Asamoah jasamoahaimsedugh

Received 1 January 2017 Revised 13 March 2017 Accepted 19 April 2017 Published 16 July 2017

Academic Editor Zhen Jin

Copyright copy 2017 Joshua Kiddy K Asamoah et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We examine an optimal way of eradicating rabies transmission from dogs into the human population using preexposureprophylaxis (vaccination) and postexposure prophylaxis (treatment) due to public education We obtain the disease-freeequilibrium the endemic equilibrium the stability and the sensitivity analysis of the optimal control model Using the Latinhypercube sampling (LHS) the forward-backward sweep scheme and the fourth-order Range-Kutta numerical method predictthat the global alliance for rabies controlrsquos aim of working to eliminate deaths from canine rabies by 2030 is attainable throughmassvaccination of susceptible dogs and continuous use of pre- and postexposure prophylaxis in humans

1 Introduction

Rabies is an infection that mostly affects the brain of aninfected animal or individual caused by viruses belonging tothe genus Lyssavirus of the family Rhabdoviridae and orderMononegavirales [1 2] This disease has become a globalthreat and it is also estimated that rabies occurs in morethan 150 countries and territories [2] Raccoons skunks batsand foxes are the main animals that transmit the virus inthe United States [2] In Asia Africa and Latin America itis known that dogs are the main source of transmission ofthe rabies virus into the human population [2] When therabies virus enters the human body or that of an animal theinfection (virus) moves rapidly along the neural pathways tothe central nervous system from there the virus continuesto spread to other organs and causes injury by interruptingvarious nerves [2] The symptoms of rabies are quite similarto those of encephalitis (see [3]) Due tomovement of dogs inhomes or the surroundings the risk of not being infected bya rabid dog can never be guaranteed Rabies is a major healthproblem in many populations dense with dogs especiallyin areas where there are less or no preventive measures

(vaccination and treatment) for dogs and humans Treatmentafter exposure to the rabies virus is known as postexposureprophylaxis (PEP) and vaccination before exposure to theinfection is known as preexposure prophylaxis

The study of optimal control analysis in maximizing orminimizing a said target was introduced by Pontryagin andhis collaborators around 1950They developed the key idea ofintroducing the adjoint function to a differential equation byforming an objective functional [4] and since then there hasbeen a considerable study of infectious disease using optimalcontrol analysis (see [4ndash12])

Research published by Aubert [13] on the advancementof the expense of wildlife rabies in France incorporatedvarious variablesThey follow immunization of domestic ani-mals the reinforcement of epidemiological reconnaissancesystem and the bolster given to indicative research laborato-ries the costs connected with outbreaks of rabies the clinicalperception of those mammals which had bitten humansthe preventive immunization and postexposure treatment ofpeople A significant percentage (72) of the cost was thepreventive immunization of local animals In France as inother European nations in which the red fox (Vulpes) is the

HindawiJournal of Applied MathematicsVolume 2017 Article ID 2451237 23 pageshttpsdoiorg10115520172451237

2 Journal of Applied Mathematics

speciesmost affected two primary procedures for controllingrabies were assessed in [13] at the repository level to bespecific fox termination and the oral immunization of foxesThe consolidated costs and advantages of both systems werelooked at and included either the expenses of fox separationor the cost of oral immunizationThe total yearly costs of bothtechniques stayed practically identical until the fourth yearafter which the oral immunization methodology turned outto be more cost effectiveThis estimate was made in 1988 andreadjusted in 1993 and affirmed by ex-postinvestigation fiveyears later Accordingly it was presumed that fox terminationbrought about a transient diminishment in the event ofthe infection while oral immunization turned out to beequipped for wiping out rabies even in circumstances inwhich fox population was growing Anderson and May [14]formulated a mathematical model based on each time stepdynamic which was calculated independently in every cellLater Bohrer et al [15] published a paper on the viability ofdifferent rabies spatial immunization designs in a simulatedhost population

The research presented by Bohrer [15] stated that indesert environments where host population size varies overtime nonuniform spreading of oral rabies vaccination mayunder certain circumstances be more effective than the com-monly used uniform spread The viability of a nonarbitraryspread of the immunization depends to some extent onthe dispersal behavior of the carriers The outcomes likewiseexhibit that in a warm domain in a few high-density regionsencompassed by populations with densities below the criticalthreshold for the spread of the disease the rabies infectioncan persist

Levin et al [16] also presented a model for the immuneresponses to rabies virus in bats Coyne et al [17] proposedan SEIR model which was also used in a study predictingthe local dynamics of rabies among raccoons in the UnitedStates Childs et al [18] also researched rabies epidemics inraccoons with a seasonal birth pulse using optimal controlof an SEIRSmodel which describes the population dynamicsHampson et al [19] also noted that rabies epidemic cycleshave a period of 3ndash6 years in dog populations in Africa sothey built a susceptible exposed infectious and vaccinatemodel with an intervention response variable which showedsignificant synchrony

Carroll et al [20] also used compartmental modelsto describe rabies epidemiology in dog populations andexplored three control methods vaccination vaccinationpulse fertility control and culling An ordinary differentialequation model was used to characterize the transmissiondynamics of rabies between humans and dogs by [21 22]The work by Zinsstag et al [23] further extended the existingmodels on rabies transmission between dogs to include dog-to-human transmission and concluded that human postex-posure prophylaxis (PEP) with a dog vaccination campaignwas the more cost effective in controlling the disease inthe long run Furthermore Ding et al [24] formulated anepidemic model for rabies in raccoons with discrete timeand spatial features Their goal was to analyze the strategiesfor optimal distribution of vaccine baits to minimize thespread of the disease and the cost of carrying out the control

Smith and Cheeseman [25] show that culling could be moreeffective than vaccination given the same efficacy of controlbut Tchuenche and Bauch suggest that culling could becounterproductive for some parameter values (see [26])

Thework in [27 28] also presented amathematical modelof rabies transmission in dogs and from the dog population tothe human population in ChinaTheir study did not considerthe use optimal control analysis to the study of the rabiesvirus in dogs and from the dog population to the humanpopulation Furthermore the insightful work ofWiraningsihet al [29] studied the stability analysis of a rabies model withvaccination effect and culling in dogs where they introducedpostexposure prophylaxis to a rabies transmission modelbut the paper did not consider the noneffectiveness of thepre- and postprophylaxis on the susceptible humans andexposed humans and that of the dog population and theuse of optimal control analysis Therefore motivated by theresearch predictions of the global alliance of rabies control[30] and the workmention above we seek to adjust themodelpresented in [27ndash29] by formulating an optimal controlmodel so as to ascertain an optimal way of controlling rabiestransmission in dogs and from the dog population to thehuman population taking into account the noneffectiveness(failure) of vaccination and treatment

The paper is petition as follows Section 2 contains themodel formulation mathematical assumptions the mathe-matical flowchart and the model equations Section 3 con-tains themodel analysis invariant region equilibriumpointsbasic reproduction number R0 and the stability analysis ofthe equilibria In Section 4 we present the parameter valuesleading to numerical values of the basic reproduction numberR0 the herd immunity threshold and sensitivity analysisusing Latin hypercube sampling (LHS) and some numericalplots Section 5 contains the objective functional and theoptimality system of the model Finally Sections 6 and 7contain discussion and conclusion respectively

2 Model Formulation

We present two subpopulation transmission models of rabiesvirus in dogs and that of the human population (see Figure 1)based on the work presented in [27ndash29] The dog populationhas a total of four compartments The compartments repre-sent the susceptible dogs 119878119863(119905) exposed dogs 119864119863 infecteddogs 119868119863(119905) and partially immune dogs 119877119863(119905)Thus the totaldog population is119873119863(119905) = 119878119863(119905) + 119864119863(119905) + 119868119863(119905) + 119877119863(119905) Thehuman population also has four compartments representingsusceptible humans 119878119867(119905) exposed humans 119864119867(119905) infectedhumans 119868119867(119905) and partially immune humans 119877119867(119905) Thusthe total human population is 119873119867(119905) = 119878119867(119905) + 119864119867(119905) +119868119867(119905) + 119877119867(119905) It is assumed that there is no human to humantransmission of the rabies virus in the human submodel (see[29]) In the dog submodel it is assumed that there is a directtransmission of the rabies virus fromone dog to the other andfrom the infected dog compartment to the susceptible humanpopulation It is further assumed that the susceptible dogpopulation 119878119863(119905) is increased by recruitment at a rate 119860119863and 119861119867 is the birth or immigration rate into the susceptiblehuman population 119878119867(119905) It is assumed that the transmission

Journal of Applied Mathematics 3

Human population dynamics

Dog population dynamics

]DSD

RD

mDRD

ID

(D + mD) ID

(1 minus D) DED

DED

ED

(mD + CD)ED

(1 minus ]D) DDSDID

DED

mDSD

AD SD

DRD

RH

mHRH

IH

(H + mH) IH

(1 minus H) HHEH

HEH

]HSH

mHEH

EH

HRH

(1 minus ]H) DHSHID

HHEH

SH

mHSH

BH

Figure 1 Optimal control model of rabies transmission dynamics

and contact rate of the rabid dog into the dog compartmentis 120573119863119863 Suppose that ]119863 represents the control strategy due topublic education and vaccination in the dogs compartmentthen the transmission dynamics become (1 minus ]119863)120573119863119863119878119863119868119863where (1minus ]119863) is the noneffectiveness (failure) of the vaccineIt is also assumed that the contact rate of infectious dogsto the human population is 120573119863119867 Similarly administratingvaccination to the susceptible humans the progression rateof the susceptible humans to the exposed stage becomes(1 minus ]119867)120573119863119867119878119867119868119863 where ]119867 is the preexposure prophylaxis(vaccination) (1 minus ]119867) represents the failure of the preex-posure prophylaxis in the human compartment Further-more administrating postexposure prophylaxis (treatment)to affected humans at the rate 120588119867 decreases the progressionrate of the rabies virus at the exposed class to the infectiousclass as (1 minus 120588119867)120575119867120574119867119864119867 where (1 minus 120588119867) is the failure rate ofthe postexposure prophylaxis and 120575119867120574119867 represents the rate atwhich exposed humans progress to the infected compartment[27] The rate of losing immunity in both compartments isrepresented by 120572119863 and 120572119867 respectively

The exposed humans without clinical rabies that moveback to the susceptible population are denoted by the rate120575119867120576119867 The natural death rate of dogs is 119898119863 and 119898119867denotes the mortality rate of humans (natural death rate) 120583119863represents the death rate associated with rabies infection indogs and 120583119867 represents the disease induce death in humansThe rate at which exposed dogs die due to culling is 119862119863and 120575120576119863 represents the rate at which exposed dogs withoutclinical rabies move back to the susceptible dog compart-ment Subsequently using the idea presented in [29] weassumed that the exposed dogs are treated or quarantined bytheir owners at the rate 120588119863 this implies that (1 minus 120588119863)120575120574119863119864119863is the progression rate of the exposed dogs to the infectiouscompartment where (1 minus 120588119863) is the failure of the treatmentor quarantined strategy and 120575120574119863119864119863 denotes those exposeddogs that develop clinical rabies [27] Figure 1 shows themathematical dynamics of the rabies virus in both com-partments

From Figure 1 transmission flowchart and assumptionsgive the disease pathways as

119889119878119863119889119905 = 119860119863 minus (1 minus ]119863) 120573119863119863119878119863119868119863 minus (119898119863 + ]119863) 119878119863 + 120575120576119863119864119863 + 120572119863119877119863


119889119864119863119889119905 = (1 minus ]119863) 120573119863119863119878119863119868119863 minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 119864119863119889119868119863119889119905 = (1 minus 120588119863) 120575120574119863119864119863 minus (119898119863 + 120583119863) 119868119863119889119877119863119889119905 = ]119863119878119863 + 120588119863119864119863 minus (119898119863 + 120572119863) 119877119863119889119878119867119889119905 = 119861119867 minus (1 minus ]119867) 120573119863119867119878119867119868119863 minus (119898119867 + ]119867) 119878119867 + 120575119867120576119867119864119867 + 120572119867119877119867119889119864119867119889119905 = (1 minus ]119867) 120573119863119867119878119867119868119863 minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 119864119867119889119868119867119889119905 = (1 minus 120588119867) 120575119867120574119867119864119867 minus (119898119867 + 120583119867) 119868119867119889119877119867119889119905 = ]119867119878119867 + 120588119867119864119867 minus (119898119867 + 120572119867) 119877119867with 119878119863 (0) gt 0 119864119863 (0) ge 0 119868119863 (0) ge 0 119877119863 (0) ge 0 119878119867 (0) gt 0 119864119867 (0) ge 0 119868119867 (0) gt 0 119877119867 (0) gt 0

(1)

3 Model Analysis

Model system (1) will be studied in a biological feasible regionas outlined below Model system (1) is basically divided intotwo regions thusΩ = Ω119863 times Ω119867Lemma 1 The solution set 119878119863 119864119863 119868119863 119877119863 119878119867 119864119867 119868119867 119877119867 isin1198778+ of model system (1) is contained in the feasible region ΩProof Suppose 119878119863 119864119863 119868119863 119877119863 119878119867 119864119867 119868119867 119877119867 isin 1198778+ for all119905 gt 0 We want to show that the region Ω is positivelyinvariant so that it becomes sufficient to look at the dynamicsof model system (1) given that

119873119863 (119905) = 119878119863 (119905) + 119864119863 (119905) + 119868119863 (119905) + 119877119863 (119905) (2)119873119867 (119905) = 119878119867 (119905) + 119864119867 (119905) + 119868119867 (119905) + 119877119867 (119905) (3)

where119873119863(119905) is the total population of dogs at any time (119905) and119873119867(119905) is total population of humans at any time (119905)Equation (2) gives

119889119873119863119889119905 = 119860119863 minus (119878119863 + 119864119863 + 119868119863 + 119877119863)119898119863 minus 120583119863119868119863minus 119862119863119864119863 (4)

which yields

119889119873119863119889119905 = 119860119863 minus 119898119863119873119863 minus 120583119863119868119863 minus 119862119863119864119863 (5)

Similarly (3) gives

119889119873119867119889119905 = 119861119867 minus 119898119867119873119867 minus 120583119867119868119867 (6)

Now assuming that there are no disease induced deathrate and culling effect in the dogsrsquo compartment it impliesthat (5) and (6) become

119889119873119863119889119905 = 119860119863 minus 119898119863119873119863119889119873119867119889119905 = 119861119863 minus 119898119867119873119867 (7)

Suppose 119889119873119863119889119905 le 0 119889119873119867119889119905 le 0119873119863 le 119860119863119898119863 and119873119867 le119861119867119898119867 and then imposing the theorem proposed in [32] ondifferential inequality results in 0 le 119873D le 119860119863119898119863 and 0 le119873119867 le 119861119867119898119867Therefore (7) becomes

119889119873119863119889119905 le 119860119863 minus 119898119863119873119863 (8)

119889119873119867119889119905 le 119861119863 minus 119898119867119873119867 (9)

Solve (8) and (9) using the integrating factor (IF)methodThus 119889119910119889119905 + 119901(119905)119910 = 119876 119868119865 = 119890int119901(119905)119889119905 After some algebraicmanipulation the feasible solution of the dogsrsquo population inmodel system (1) is in the region

Ω119863 = (119878119863 119864119863 119868119863 119877119863) isin R4+ 119873119863 le 119860119863119898119863 (10)

Similarly the human population follows suit and from (9)this implies that the feasible solution of the humanpopulationof model system (1) is in the region

Ω119867 = (119878119867 119864119867 119868119867 119877119867) isin R4+ 119873119867 le 119861119867119898119867 (11)


Therefore the feasible solutions are contained inΩThusΩ =Ω119863 times Ω119867 From the standard comparison theorem used ondifferential inequality in [33] it implies that

119873119863 (119905) le 119873119863 (0) 119890minus(119898119863)119905 + 119860119863119898119863 (1 minus 119890minus(119898119863)119905) 119873119867 (119905) le 119873119867 (0) 119890minus(119898119867)119905 + 119861119867119898119867 (1 minus 119890minus(119898119867)119905) (12)

Hence the total dog population size119873119863(119905) rarr 119860119863119898119863 as119905 rarr infin Similarly the total human population size119873119867(119905) rarr119861119867119898119867 as 119905 rarr infinThismeans that the infected state variables(119864119863 119868119863 119864119867 119868119867) of the two populations tend to zero as timegoes to infinityTherefore the regionΩ is pulling (attracting)all the solutions in R8+This gives the feasible solution set ofmodel system (1) as

((((((((

119878119863119864119863119868119863119877119863119878119867119864119867119868119867119877119867

))))))))

isin R8+ |

((((((((((((((((((

119878119863 gt 0119864119863 ge 0119868119863 ge 0119868119863 ge 0119877119863 ge 0119878119867 gt 0119864119867 ge 0119868119867 ge 0119877119867 ge 0119873119863 le 119860119863119898119863119873119867 le 119861119867119898119867

))))))))))))))))))

(13)

Hence (1) ismathematically well posed and epidemiolog-ically meaningful

31 Disease-Free Equilibrium E0 Suppose there is no infec-tion of rabies in both compartments then (119864119863 = 0 119868119863 =0 119864119867 = 0 119868119867 = 0) Incorporating this into (1) leads to119860119863 minus (119898119863 + ]119863) 119878119863 + 120572119863119877119863 = 0

]119863119878119863 minus (119898119863 + 120572119863) 119877119863 = 0119861119867 minus (119898119867 + ]119867) 119878119867 + 120572119867119877119867 = 0]119867119878119867 minus (119898119867 + 120572119867) 119877119867 = 0

(14)

After some algebraic manipulation of (14) the disease-free equilibriumpoint becomesE0 = (1198780119863 1198640119863 1198680119863 1198770119863 1198780119867 11986401198671198680119867 1198770119867) with

E0 = ( 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 0 0119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 0 0119861119867]119867119898119867 (119898119867 + 120572119867 + ]119867)) (15)

32 Basic Reproduction Number R0 Here the basic repro-duction number (R0) measures the average number of newinfections produced by one infected dog in a completelysusceptible (dog and human) population (see also [34]) Nowtaking119864119863 119868119863 119864119867 and 119868119867 as our infected compartments gives

1198911 = (1 minus ]119863) 120573119863119863119878119863119868119863minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 1198641198631198912 = (1 minus 120588119863) 120575120574119863119864119863 minus (119898119863 + 120583119863) 1198681198631198913 = (1 minus ]119867) 120573119863119867119878119867119868119863minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 1198641198671198914 = (1 minus 120588119867) 120575119867120574119867119864119867 minus (119898119867 + 120583119867) 119868119867(16)

where 1198911 = 119889119864119863119889119905 1198912 = 119889119868119863119889119905 1198913 = 119889119864119867119889119905 and 1198914 =119889119868119867119889119905Now using the next generation matrix operator 119866 =119865119881minus1 and the Jacobian matrix

119869 =((((((((

1205971198911120597119864119863 1205971198911120597119868119863 1205971198911120597119864119867 12059711989111205971198681198671205971198912120597119864119863 1205971198912120597119868119863 1205971198912120597119864119867 12059711989121205971198681198671205971198913120597119864119863 1205971198913120597119868119863 1205971198913120597119864119867 12059711989131205971198681198671205971198914120597119864119863 1205971198914120597119868119863 1205971198914120597119864119867 1205971198914120597119868119867

))))))))

(17)

as described in [34] results in

119869

=(((

minus((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (1 minus ]119863) 120573119863119863119878119863 0 0(1 minus 120588119863) 120575120574119863 minus (119898119863 + 120583119863) 0 00 (1 minus ]119867) 120573119863119867119878119867 minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 00 0 (1 minus 120588119867) 120575119867120574119867 minus (119898119867 + 120583119867))))

(18)


Using the fact that 119869 = 119865 minus 119881 gives 119865 and 119881 evaluated atE0 as

119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((

((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 0 0 0minus (1 minus 120588119863) 120575120574119863 (119898119863 + 120583119863) 0 00 0 ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 00 0 minus (1 minus 120588119867) 120575119867120574119867 (119898119867 + 120583119867))))

(19)

where the element in matrix 119865 constitutes the new infectionterms while that of matrix 119881 constitutes the new trans-fer of infection terms from one compartment to another

Now splitting matrix 119881 into four 2 times 2 submatrices andfinding its corresponding inverses result in 119866 = 119865119881minus1 givenby

119866

=(((((

(1 minus 120588119863) (1 minus ]119863) 120575120574119863120573119863119863119860119863 (119898119863 + 120572119863)((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (119898119863 + 120583119863)119898119863 (119898119863 + ]119863 + 120572119863) (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 0(1 minus 120588119863) 120575120574119863 (1 minus ]119867) 120573119863119867119861119867 (119898119867 + 120572119867)((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (119898119863 + 120583119863)119898119867 (119898119867 + ]119867 + 120572119867) (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867(119898119863 + ]119863)119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0)))))

(20)

Letting

119886 = (1 minus 120588119863) (1 minus ]119863) 120575120574119863120573119863119863119860119863 (119898119863 + 120572119863)((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (119898119863 + 120583119863)119898119863 (119898119863 + ]119863 + 120572119863) 119887 = (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 119888 = (1 minus 120588119863) (1 minus ]119867) 120575120574119863120573119863119867 (119898119867 + 120572119867)((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (119898119863 + 120583119863)119898119867 (119898119867 + ]119867 + 120572119867) 119889 = (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867(119898119863 + ]119863)119898119867 (119898119867 + ]119867 + 120572119867)

(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)

Finding the matrix determinant of (22) and denoting it by 119863give the expression119863 = |119866minusI120582| where I is the identitymatrixof a 4 times 4matrix thus

119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 minus 120582 119887 0 00 minus120582 0 0119888 119889 minus120582 00 0 0 minus120582

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)

This gives a characteristic equation of the form 1205823(119886 minus120582) = 0 solving the characteristic polynomial results in thefollowing eigenvalues 120582119894 = [0 0 0 119886] The basic reproduc-tion number R0 is the spectral radius (largest eigenvalue)120588(119865119881minus1) also defined as the dominant eigenvalue of 119865119881minus1

Therefore

R0 = (1 minus 120588119863) (1 minus ]119863) 120575120574119863120573119863119863119860119863 (119898119863 + 120572119863)((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) (119898119863 + 120583119863)119898119863 (119898119863 + ]119863 + 120572119863) (24)

Remark 2 R0 contains the secondary infection produced bythe infectious compartment of dogs (in the presence of preex-posure prophylaxis (vaccination) postexposure prophylaxis(treatmentquarantine) and culling of exposed dogs) WhenR0 lt 1 the infection gradually leaves the dog compartmentbut when R0 gt 1 the rabies virus remains in the dog

compartments for a longer time thereby increasing the rateat which the susceptible dogs and humans get infected by arabid dog

33 Endemic Equilibrium E1 The endemic equilibrium isgiven as

119878lowast119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863)R0 119864lowast119863 = (119898119863 + 120583119863)(1 minus 120588119863) 120575120574119863 119868lowast119863119868lowast119863 = [(1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120574119863] (119898119863 + 120583119863)119898119863 (119898119863 + ]119863 + 120572119863) (R0 minus 1)(119898119863 + 120572119863) (1 minus ]119863) 120573119863119863 [(1 minus 120588119863) 120575120574119863 + 119898119863 + 119862119863] + 119898119863 (1 minus ]119863) 120573119863119863120588119863 119877lowast119863 = 119860119863]119863 (1 minus ]119863) 120573119863119863 (1 minus 120588119863) 120575120574119863 (119898119863 + 120572119863) + (1 minus ]119863) 120573119863119863120588119863 (119898119863 + 120583119863) 119868lowast119863119898119863 (119898119863 + ]119863 + 120572119863)R0 (1 minus ]119863) 120573119863119863 (1 minus 120588119863) 120575120574119863 (119898119863 + 120572119863) 119878lowast119867 = 119861119867 (119898119867 + 120572119867) + [120575119867120576119867 + 120572119867120588119867] 119864lowast119867[(1 minus ]119867) (119898119867 + 120572119867) 120573119863119867119868lowast119863 + 119898119867 (119898119867 + 120572119867 + ]119867)] 119864lowast119867= (1 minus ]119867) 119861119867 (119898119867 + 120572119867) 120573119863119867119868lowast119863(119898119867 + 120572119867) [(1 minus ]119867) 120573119863119867119868lowast119863 ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867) + (119898119867 + ]119867) ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867)] minus (1 minus ]119867) 120573119863119867119868lowast119863120572119867120588119867 119868lowast119867 = (1 minus 120588119867) 120575119867120574119867119898119867 + 120583119867 119864lowast119867119877lowast119867 = 119861119867]119867 (119898119867 + ]119867) + [(]119867120575119867120576119867 + ]119867120572119867120588119867) + 120588119867 (1 minus ]119867) (119898119867 + 120572119867) 120573119863119867119868lowast119863 + 120588119867119898119867 (119898119867 + 120572119867 + ]119867)] 119864lowast119867[(1 minus ]119867) (119898119867 + 120572119867)2 120573119863119867119868lowast119863 + (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867)]

(25)

Note that if R0 = 1 it results in the disease-free equi-librium if R0 gt 1 then there exists a unique endemic

equilibrium if R0 lt 1 then there exist two endemic equi-libriums


34 Stability Analysis of E0 Linearizing (1) at E0 and sub-tracting eigenvalue 120582 along the main diagonal yield

J (E0) =((((((((((((

1198871 minus 120582 1198877 1198861 120572119863 0 0 0 00 1198862 minus 120582 1198863 0 0 0 0 00 11988710 1198872 minus 120582 0 0 0 0 0]119863 120588119863 0 1198873 minus 120582 0 0 0 00 0 1198864 0 1198874 minus 120582 1198875 0 1205721198670 0 1198865 0 0 1198866 minus 120582 0 00 0 0 0 0 1198879 1198876 minus 120582 00 0 0 0 ]119867 120588119867 0 1198878 minus 120582

))))))))))))

(26)

where

1198861 = minus (1 minus ]119863) 120573119863119863 (119898119863 + 120572119863) 119860119863119898119863 (119898119863 + ]119863 + 120572119863) 1198862 = minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 1198863 = (1 minus ]119863) 120573119863119863 (119898119863 + 120572119863) 119860119863119898119863 (119898119863 + ]119863 + 120572119863) 1198864 = minus (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867)119898119867 (119898119867 + ]119867 + 120572119867) 1198865 = (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867)119898119867 (119898119867 + ]119867 + 120572119867) 1198866 = minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 1198871 = minus (119898119863 + ]119863) 1198872 = minus (119898119863 + 120583119863) 1198873 = minus (119898119863 + 120572119863) 1198874 = minus (]119867 + 119898119867) 1198875 = 1205751198671205761198671198876 = minus (119898119867 + 120583119867) 1198877 = 1205751205761198631198878 = minus (119898119867 + 120572119867) 1198879 = (1 minus 120588119867) 12057511986712057411986711988710 = (1 minus 120588119863) 120575120574119863

(27)

Simplifying matrix J(E0) gives(1198876 minus 120582) (1198866 minus 120582) (1198874 minus 120582) (1198878 minus 120582)sdot [1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614] = 0 (28)

where11988611 = (minus1198872 minus 1198862 minus 1198871 minus 1198873) 11988612 = ]119863120572119863 + 11988621198873 + 11988621198871 + 11988721198873 + 11988721198871 + 11988731198871 + 11988721198862minus (1 minus 120588119863) 120575120574119863119886311988613 = minus1198862]119863120572119863 minus 1198872]119863120572119863 + 1198863 (1 minus 120588119863) 1205751205741198631198872 + 1198863 (1minus 120588119863) 1205751205741198631198871 minus 119886211988721198873 minus 119887211988621198871 minus 119886211988731198871 minus 11988721198873119887111988614 = (1198871119887211988731198862 + (1 minus 120588119863) 120575120574119863119886311988731198871+ (1 minus 120588119863) 1205751205741198631198863]119863 + ]11986312057211986311988721198862)

(29)

From (28) the four characteristic factors that are negativeare 1205821 = 11988761205822 = 11988661205823 = 11988741205824 = 1198878

(30)

where 1198866 = minus((1 minus 120588119867)120575119867120574119867 + 119898119867 + 120588119867 + 120575119867 + 120575119867120576119867)1198876 = minus(119898119867 + 120583119867) 1198874 = minus(]119867 + 119898119867) and 1198878 = minus(119898119867 + 120572119867)The other four characteristic factors can be obtained using theRouth-Hurwitz criterion Routh-Hurwitz stability criterion isa test to ascertain the nature of the eigenvalues If the rootsof the polynomial are all positive then the polynomial has anegative real part [35 36] The remaining four characteristiceigenvalues are obtained as follows

1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)

Hence simplifying the coefficient of the above character-istic polynomial in (31) yields

11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)


11988612 = ]119863120572119863 + ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)sdot [(119898119863 + 120572119863) + (119898119863 + ]119863)] + (119898119863 + 120583119863)sdot [(119898119863 + 120572119863) + (119898119863 + ]119863)] + (119898119863 + 120572119863)sdot (119898119863 + ]119863) + ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 119862119863)sdot (119898119863 + 120583119863) (1 minusR0) 11988613 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) ]119863120572119863+ (119898119863 + 120583119863) ]119863120572119863 + (119898119863 + 120583119863) (119898119863 + 120572119863)sdot (119898119863 + ]119863)+ ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)sdot (119898119863 + 120572119863) (119898119863 + 120583119863) [1 minus R0 (119898119863 + 120583119863)119898119863 + 120572119863 ]+ ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)sdot (119898119863 + 120572119863) (119898119863 + ]119863) [1 minus R0119898 + 120572] 11988614 = ]119863120572119863 (119898119863 + ]119863)sdot ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (1 minus 120588119863) 120575120574119863 (1 minus ]119863) 120573119863119863 (119898119863 + 120572119863) 119860119863]119863119898119863 (119898119863 + 119898119863 + ]119863 + 120572119863)+ (119898119863 + ]119863) (119898119863 + 120583119863) (119898119863 + 120572119863)sdot ((1 minus 120588119863) 120575119863120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)sdot (1 minusR0)

(32)

Therefore from the Routh-Hurwitz criterion of orderfour it implies that the conditions 11988611 gt 0 11988612 gt 0 11988613 gt0 11988614 gt 0 and 119886111198861211988613 gt 119886213 + 11988621111988614 are satisfied ifR0 lt 1 Hence the disease-free equilibrium E0 is locallyasymptotically stable whenR0 lt 1 (see [37])341 Global Stability of E0

Theorem 3 The disease-free equilibrium E0 of model (1) isglobally asymptotically stable ifR0 le 1 and unstable ifR0 gt 1Proof LetV be a Lyapunov function with positive constantsK1K2K3 andK4 such that

V = (119878119863 minus 1198780119863 minus 1198780119863 ln 1198781198631198780119863) +K1119864119863 +K2119868119863+ (119877119863 minus 1198770119863 minus 1198770119863 ln 1198771198631198770119863)

+ (119878119867 minus 1198780119867 minus 1198780119867 ln 1198781198671198780119867) +K3119864119867 +K4119868119867+ (119877119867 minus 1198770119867 minus 1198770119867 ln 1198771198671198770119867)

(33)

Taken the derivative of the Lyapunov function withrespect to time gives

119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)

Plugging (1) into (34) results in

119889V119889119905 = (1 minus 1198780119863119878119863) [119860119863 minus (1 minus ]119863) 120573119863119863119878119863119868119863minus (119898119863 + ]119863) 119878119863 + 120575120576119863119864119863 + 120572119877119863]+K1 [(1 minus ]119863) 120573119863119863119878119863119868119863minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 119864119863]+K2 [(1 minus 120588119863) 120575120574119863119864119863 minus (119898119863 + 120583119863) 119868119863] + (1minus 1198770119863119877119863) []119863119878119863 + 120588119863119864119863 minus (119898119863 + 120572119863) 119877119863] + (1minus 1198780119867119878119867) [119861119867 minus (1 minus ]119867) 120573119863119867119878119867119868119863 minus (119898119867 + ]119867) 119878119867+ 120575119867120576119867119864119867 + 120572119867119877119867] +K3 [(1 minus ]119867) 120573119863119867119878119867119868119863minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 119864119867]+K4 [(1 minus 120588119867) 120575119867120574119867119864119867 minus (119898119867 + 120583119867) 119868119867] + (1minus 1198770119867119877119867) []119867119878119867 + 120588119867119864119867 minus (119898119867 + 120572119867) 119877119867]

(35)

Now after forming the Lyapunov function V on thespace of the eight state variables thus (119878119863 119864119863 119868119863 119877119863119878119867 119864119867 119868119867 119877119867) and introducing the idea from [37] it isclear that if 119864119863(119905) 119868119863(119905) 119864119867(119905) and 119868119867(119905) at the disease-freeequilibrium are globally stable (thus 119864119863 = 0 119868119863 = 0 119864119867 = 0and 119868119867 = 0) then 119878119863(119905) rarr 119860119863(119898119863+120572119863)119898119863(119898119863+120572119863+]119863)119877119863(119905) rarr 119860119863]119863119898119863(119898119863 + 120572119863 + ]119863) 119878119867(119905) rarr 119861119867(119898119867 +120572119867)119898119867(119898119867 + 120572119867 + 119899119906119867) and 119877119867(119905) rarr 119861119867]119867119898119867(119898119867 +120572119867 + 119867 + ]119867) as 119905 rarr infin


Therefore it can be assumed that

119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)

(see [38]) and replacing it into (35) yields

119889V119889119905 le K1 [(1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119868119863minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 119864119863]+K2 [(1 minus 120588119863) 120575120574119863119864119863 minus (119898119863 + 120583119863) 119868119863]+K3 [(1 minus ]119867) 120573119863119867119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 119868119889minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 119864119867]+K4 [(1 minus 120588119867) 120575119867120574119867119864119867 minus (119898119867 + 120583119867) 119868119867]

(37)

This implies that

119889V119889119905 le [K1 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863)minusK2 (119898119863 + 120583119863)+ K3 (1 minus ]119867) 120573119863119867119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) ] 119868119863+ [K2 (1 minus 120588119863) 120575120574119863minusK1 ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)] 119864119863+ [K4 (1 minus 120588119867) 120575119867120574119867minusK3 ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120575119867120576119867)] 119864119867minusK4 (119898119867 + 120583119867)

(38)

Equating the coefficient of 119868119863 119864119863 119868119867 and 119864119867 in (38) tozero gives

K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)

and we obtain119889V119889119905 le ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)sdot (119898119863 + 120583119863) (R0 minus 1) 119868119863le 0 if R0 le 1

(40)

Additionally 119889V119889119905 = 0 if and only if 119868119863 = 0 Thereforefor 119864119863 = 119868119863 = 119864119867 = 119868119867 = 0 it shows that 119878119863(119905) rarr 119860119863(119898119863 +120572119863)119898119863(119898119863 +120572119863 + ]119863) 119877119863(119905) rarr 119860119863]119863119898119863(119898119863 +120572119863 + ]119863)119878119867(119905) rarr 119861119867(119898119867 + 120572119867)119898119867(119898119867 + 120572119867 + ]119867) and 119877119867(119905) rarr119861119867]119867119898119867(119898119867 + 120572119867 + ]119867) as 119905 rarr infin Hence the largestcompact invariant set in (119878119863 119864119863 119868119863 119877119863 119878119867 119864119867 119868119867 119877119867) isinΩ 119889V119889119905 le 0 is the singleton set E0 Therefore from LaSallersquos invariance principle we conclude that E0 is globallyasymptotically stable inΩ ifR0 le 1 (see also [38 39])35 Global Stability of Endemic Equilibrium E1

Theorem 4 The endemic equilibrium E1 of model (1) isglobally asymptotically stable wheneverR0 gt 1Proof Suppose R0 gt 1 then the existence of the endemicequilibrium point is assured Using the common quadraticLyapunov function

119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)

as illustrated in [40] we consider a Lyapunov function withthe following candidate

V (119878119863 119864119863 119868119863 119877119863 119878119867 119864119867 119868119867 119877119867) = 12 [(119878119863 minus 119878lowast119863)+ (119864119863 minus 119864lowast119863) + (119868119863 minus 119868lowast119863) + (119877119863 minus 119877lowast119863)]2+ 12 [(119878119867 minus 119878lowast119867) + (119864119867 minus 119864lowast119867) + (119868119867 minus 119868lowast119867)+ (119877119867 minus 119877lowast119867)]2

(42)

Now differentiating (42) along the solution curve of (1)gives

119889V119889119905 = [(119878119863 minus 119878lowast119863) + (119864119863 minus 119864lowast119863) + (119868119863 minus 119868lowast119863)+ (119877119863 minus 119877lowast119863)] 119889 (119878119863 + 119864119863 + 119868119863 + 119877119863)119889119905+ [(119878119867 minus 119878lowast119867) + (119864119867 minus 119864lowast119867) + (119868119867 minus 119868lowast119867)+ (119877119867 minus 119877lowast119867)] 119889 (119878119867 + 119864119867 + 119868119867 + 119877119867)119889119905

(43)


From (1) it implies that 119889(119878119863 + 119864119863 + 119868119863 + 119877119863)119889119905 = 119860119863 minus119898119863(119878119863+119864119863+119868119863+119877119863) minus119862119863119864119863minus120583119863119868119863 and 119889(119878119867+119864119867+119868119867+119877119867)119889119905 = 119861 minus 119898(119878119867 + 119864119867 + 119868119867 + 119877119867) minus 120583119867119868119867 which whenplugged into (43) gives

119889V119889119905 = [(119878119863 minus 119878lowast119863) + (119864119863 minus 119864lowast119863) + (119868119863 minus 119868lowast119863)+ (119877119863 minus 119877lowast119863)] (119860119863 minus 119898119863 (119878119863 + 119864119863 + 119868119863 + 119877119863)minus 119862119863119864119863 minus 120583119863119868119863) + [(119878119867 minus 119878lowast119867) + (119864119867 minus 119864lowast119867)+ (119868119867 minus 119868lowast119867) + (119877119867 minus 119877lowast119867)] (119861119867minus 119898 (119878119867 + 119864119867 + 119868119867 + 119877119867) minus 120583119867119868119867) (44)

Now assuming

119860119863 = 119898119863 (119878lowast119863 + 119864lowast119863 + 119868lowast119863 + 119877lowast119863) + 119862119863119864lowast119863 + 120583119863119868lowast119863119861119867 = 119898119867 (119878lowast119867 + 119864lowast119867 + 119868lowast119867 + 119877lowast119867) + 120583119867119868lowast119867 (45)

and substituting it into (44) we have

119889V119889119905 = [(119878119863 minus 119878lowast119863) + (119864119863 minus 119864lowast119863) + (119868119863 minus 119868lowast119863) + (119877119863minus 119877lowast119863)] [119898119863 (119878lowast119863 + 119864lowast119863 + 119868lowast119863 + 119877lowast119863) + 119862119863119864lowast119863 + 120583119863119868lowast119863minus 119898119863 (119878119863 + 119864119863 + 119868119863 + 119877119863) minus 119862119863119864119863 minus 120583119863119868119863]+ [(119878119867 minus 119878lowast119867) + (119864119867 minus 119864lowast119867) + (119868119867 minus 119868lowast119867) + (119877119867minus 119877lowast119867)] [119898119867 (119878lowast119867 + 119864lowast119867 + 119868lowast119867 + 119877lowast119867) + 120583119867119868lowast119867minus 119898 (119878119867 + 119864119867 + 119868119867 + 119877119867) minus 120583119867119868119867] 119889V119889119905 = [(119878119863 minus 119878lowast119863) + (119864119863 minus 119864lowast119863) + (119868119863 minus 119868lowast119863) + (119877119863minus 119877lowast119863)] [(minus119898119863 (119878119863 minus 119878lowast119863) minus 119898119863 (119864119863 minus 119864lowast119863)minus 119898119863 (119868119863 minus 119868lowast119863) minus 119898119863 (119877119863 minus 119877lowast119863) minus 119862119863 (119864119863 minus 119864lowast119863)minus 120583119863 (119868119863 minus 119868lowast119863))] + [(119878119867 minus 119878lowast119867) + (119864119867 minus 119864lowast119867)+ (119868119867 minus 119868lowast119867) + (119877119867 minus 119877lowast119867)] [(minus119898119867 (119878119867 minus 119878lowast119867)minus 119898119867 (119864119867 minus 119864lowast119867) minus 119898119867 (119868119867 minus 119868lowast119867)minus 119898119867 (119877119867 minus 119877lowast119867) minus 120583119867 (119868119867 minus 119868lowast119867))]

(46)

This also implies that

119889V119889119905 = minus119898119863 (119878119863 minus 119878lowast119863)2 minus (119862119863 + 119898119863) (119864119863 minus 119864lowast119863)2minus (119898119863 + 120583119863) (119868119863 minus 119868lowast119863)2 minus 119898119863 (119877119863 minus 119877lowast119863)2 minus (2119898119863+ 119862119863) (119878119863 minus 119878lowast119863) (119864119863 minus 119864lowast119863) minus (2119898119863 + 120583119863) (119878119863minus 119878lowast119863) (119868119863 minus 119868lowast119863) minus (2119898119863 + 120583119863 + 119862119863) (119864119863 minus 119864lowast119863) (119868119863minus 119868lowast119863) minus 2119898119863 (119877119863 minus 119877lowast119863) (119868119863 minus 119868lowast119863) minus (2119898119863 + 120583119863

+ 119862119863) (119877119863 minus 119877lowast119863) (119868119863 minus 119868lowast119863) minus 119898119867 (119878119867 minus 119878lowast119867)2minus 119898119867 (119864119867 minus 119864lowast119867)2 minus (119898119867 minus 120583119867) (119868119867 minus 119868lowast119867)2minus 119898119867 (119877119867 minus 119877lowast119867)2 minus 2119898119867 (119878119867 minus 119878lowast119867) (119864119867 minus 119864lowast119867)minus (2119898119867 minus 120583119867) (119878119867 minus 119878lowast119867) (119868119867 minus 119868lowast119867) minus (2119898119867 + 120583119867)sdot (119864119867 minus 119864lowast119867) (119868119867 minus 119868lowast119867)minus 119898119867 [(119868119867 minus 119868lowast119867) (119877119867 minus 119877lowast119867)+ (119878119867 minus 119878lowast119867) (119877119867 minus 119877lowast119867)] (47)

This shows that 119889V119889119905 is negative and 119889V119889119905 = 0 if andonly if 119878119863 = 119878lowast119863 119864119863 = 119864lowast119863 119868119863 = 119868lowast119863 119877119863 = 119877lowast119863 119878119867 =119878lowast119867 119864119867 = 119864lowast119867 119868119867 = 119868lowast119867 119877119867 = 119877lowast119867 Additionally everysolution of (1) with the initial conditions approaches E1 as119905 rarr infin (see [38 39]) therefore the largest compact invariantset in (119878119863 119864119863 119868119863 119877119863 119878119867 119864119867 119868119867 119877119867) isin Ω 119889V119889119905 le 0is the singleton set E1 Therefore from Lasallersquos invariantprinciple [41] it implies that the endemic equilibrium E1 isglobally asymptotically stable inΩ wheneverR0 gt 14 Numerical Analysis

Considering the parameter values in Table 1 we will ascertainthe numerical importance of our analysis

41 Different Scenarios of the Basic Reproduction NumberR0 We shall denote R0 without pre- and postexposureprophylaxis (treatment) as Rlowast0 and R0 without preexpo-sure prophylaxis and culling as Rlowastlowast0 and the R0 withoutpostexposure prophylaxis (treatment) and culling as Rlowastlowastlowast0 Therefore using the parameter values in Table 1 Rlowast0 R

lowastlowast0

andRlowastlowastlowast0 are given as follows

Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0

= (1 minus 120588119863) 120575120574119863120573119863119863119860119863((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863) (119898119863 + 120583119863)119898119863 Rlowastlowast0 = 2181

Rlowastlowastlowast0

= (1 minus ]119863) 120575120574119863120573119863119863119860119863 (119898119863 + 120572119863)(120575120574119863 + 119898119863 + 120575120576119863) (119898119863 + 120583119863)119898119863 (119898119863 + ]119863 + 120572119863) Rlowastlowastlowast0 = 1914

(48)

Therefore from the above calculations it indicates that thebest way in reducing or minimizing the rabies virus in thedogs compartment is to use more of preexposure prophylaxis(vaccination)


Table 1 Parameter values

Parameter Description Standard value Source119860119863 Recruitment rate of dogs 3 times 106119910minus1 [27]120572119863 Loss of immunity in dogs 1119910minus1 [27]119862119863 Death rate of dogs due to culling 03119910minus1 Assumed119898119863 Natural death rate of dogs 0056119910minus1 [27]120583119863 Disease induced mortality in dogs 1119910minus1 [27]]119863 Preexposure prophylaxis for dogs 025119910minus1 Assumed120588119863 Postexposure prophylaxis for dogs 02119910minus1 [27]120573119863119863 Transmission rate in dogs 158 times 10minus7119910minus1 [27]120574119863 Latency period in dogs (2376) 119910minus1 [27]120575120576119863 Rate of no clinical rabies 04119910minus1 [27]119861119867 Birth rate (humans) 00314119910minus1 [31]120573119863119867 Transmission rate (dog-humans) 229 times 10minus12119910minus1 [27]120572119867 Loss of immunity (humans) 1119910minus1 [27]119898119867 Natural death rate (humans) 00074119910minus1 [31]120583119867 Disease induced mortality (humans) 1119910minus1 [27]]119867 Preexposure prophylaxis for humans 054119910minus1 Assumed120588119867 Postexposure prophylaxis for humans 01119910minus1 [27]120574119867 Latency rate (humans) (16) 119910minus1 [27]120574119867120576119867 Rate of no clinical rabies (humans) 24119910minus1 [27]

411 Herd Immunity Threshold 1198671 Therefore from theabove numerical values we are motivated to know thenumber of humans or dogs that should be vaccinated whenRlowast0 = 3027

1198671 fl 1 minus 1Rlowast0

= 066 (49)

This shows that if Rlowast0 = 3027 then 66 of individualsand dogs should receive vaccination

42 Sensitivity Analysis To determine parameters that con-tribute most to the rabies transmission we used two sensi-tivity analysis approach the normalised forward sensitivityindex as presented in [37] and the Latin hypercube samplingas described in [42] To determine the dependence of param-eters inR0 using a sampling size 119899 = 1000 the partial rankcorrection coefficients (PRCC) value of the ten parameters inR0 are shown in Figure 2(a)The longer the bar in Figure 2(a)suggests that the statistical influence of those parameters tochanges in R0 is high Also using the normalised forwardsensitivity index gives the following values and the nature oftheir signs in Table 2 based on the parameter value given inTable 1The plus sign orminus sign signifies that the influenceis positive or negative respectively [42]

Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0

= 120597R0120597120575120576119863 120575120576119863R0= 120575120576119863((1 minus 120588119863) 120575120574119863 minus 120575120576119863 minus 119862119863 minus 119898119863 minus 120588119863)= minus161

Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)

Therefore from Table 2 it shows that an addition or areduction in the values of 120573119863119863 120572119863 120575120574119863 and 119860119863 will havean increase or a decrease in the spread of the rabies virus For


02010 03 04minus02minus03 minus01minus04

PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD

(a) LHS plot for the parameters inR0

AD = 4 times 106AD = 5 times 106

AD = 3 times 106AD = 1 times 106AD = 2 times 106

AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H

(b) Effect of varying recruitment rate on the infected humans

0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H

(c) Effect of an increase inR0 on the infected humans

0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106

ID (0) = 4 times 106ID (0) = 2 times 106ID (0) = 3 times 106

ID (0) = 1 times 106

Infe

cted

hum

ansI

H

(d) Effect of varying the initial infected dog population size on the infectedhumans

Figure 2 The graphical representation of some parameters inR0 and the effect of varying some initial state values on the model

example Γ120573119863R0

= 1 indicates that increasing or reducing thetransmission rate by 5 may increase or reduce the numberof secondary infection by 5 The negative sign in Table 2will have a reduction in the basic reproduction numberR0when the values of those parameters are increased and areduction in the values of 120588119863 ]119863 120583119863 119898119863 and 120575120576119863 will leadto an increase in the number of secondary infections

TheLatin hypercube sampling (LHS) in Figure 2(a) showsthat 120583119863 119862119863 120572119863 and 120575120574119863 have a minimal influence on therate at which the rabies virus is spread The Latin hypercubesampling (LHS) plots for the ten parameters inR0 show thatculling of exposed dogs does not actuallyminimize the spread

of rabies as compared to vaccination of susceptible dogsFigure 2(a) also shows that the most influential parameter inspreading the infection is 120573119863119863 followed by 119860119863 Figure 2(c)shows that an increase in the basic reproduction numberwill contribute to a high level of secondary infection inthe human population Similarly Figure 2(a) shows thatvaccination of dogs ]119863 is themost effective way of controllingthe rabies virus in the dog population as compared to thetreatmentquarantine of exposed dogs 120588119863 Figure 3(a) givesthe contour nature of ]119863 and 120588119863 which shows a more sat-urated effect on the basic reproduction number Figure 3(b)shows that 120573119863119863 and 120572119863 have a positive relation with the basic


Table 2 Sensitivity signs ofR0 to the parameters in (24)

Parameter Description Sensitivity sign120573119863119863 Transmission rate of dogs +ve119860119863 Recruitment rate of dogs +ve120583119863 Disease induce death rate of dogs minusve120575120576119863 Rate of no clinical rabies minusve119862119863 Culling of exposed dogs minusve120572119863 Loss of immunity in dogs +ve119898119863 Natural death rate of dogs minusve120575120574119863 Rate at which exposed dogs become infective (infective rate) +ve120588119863 Postexposure prophylaxis (treatmentquarantined) minusve]119863 Preexposure prophylaxis (vaccination) minusve

0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D

(a) The contour plot of ]119863 and 120588119863 toR0

10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107

(b) The 3D plot ofR0 to 120572119863 and 120573119863119863

02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD

(c) The 3D plot ofR0 to119898119863 and 120583119863

500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D

(d) The 3D plot ofR0 to 120588119863 and ]119863

Figure 3 The graphical representation of some parameters inR0

reproduction numberR0 Therefore an increase in 120573119863119863 and120572119863 will have a direct increase in the spread of the rabies virusFigure 2(b) indicates that with a high number of recruitmentof dogs into the susceptible dogrsquos compartment will havea corresponding high increase in the number of infectedhumans Figure 2(d) demonstrates that a high number ofinfected dogs in the compartment will lead to an increase

in the number of infected humans Figure 3(c) shows thata high increase in the number of disease induce death rateand natural death rate will have a negative reflection onR0 biologically we would not recommend this approach inminimizing the spread of the disease since an increase inboth 120583119863 and 119898119863 may result in a high rate of the disease inthe human population even though 120583119863 and 119898119863 naturally


reduce the number of susceptible and infected dogs inthe population Finally Figure 3(d) shows the 3D plot ofFigure 3(a)

5 Objective Functional

Given that 119910(119905) isin 119884 isin R119899 is a state variable of model system(1) and 119906(119905) isin 119880 isin R119899 are the control variables at any time (119905)with 119905(0) le 119905 le 119905(119891) then an optimal control problem con-sists of finding a piecewise continuous control 119906(119905) and itscorresponding state 119910(119905) This optimizes the cost functional119869[119910(119905) 119906(119905)] using Pontryaginrsquos maximum principle [43]Therefore we set the following likelihood control strate-gies

(1) 1199061 = ]119863 is the control effort aimed at increasing theimmunity of susceptible dogs (preexposed prophy-laxis)

(2) 1199062 = 120588119863 is the control effort aimed at treating theexposed dogs (postexposed prophylaxis)

(3) 1199063 = ]119867 is the control effort aimed at increasingthe immunity of susceptible humans (preexposureprophylaxis)

(4) 1199064 = 120588119867 is the control effort aimed at treating theexposed humans (postexposed prophylaxis)

Our goal is to seek optimal controls such as ]lowast119863 120588lowast119863 ]lowast119867and 120588lowast119867 that minimize the objective functional

119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)

Therefore (51) is subject to

119889119878119889119889119905 = 119860119863 minus (1 minus ]119863) 120573119863119863119878119863119868119863 minus (119898119863 + ]119863) 119878119863 + 120575120576119863119864119863 + 120572119863119877119863119889119864119889119889119905 = (1 minus ]119863) 120573119863119863119878119863119868119863 minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 119864119863119889119868119863119889119905 = (1 minus 120588119863) 120575120574119863119864119863 minus (119898119863 + 120583119863) 119868119863119889119877119863119889119905 = ]119863119878119863 + 120588119863119864119863 minus (119898119863 + 120572119863) 119877119863119889119878119867119889119905 = 119861119867 minus (1 minus ]119867) 120573119863119867119878119867119868119889 minus (119898119867 + ]119867) 119878119867 + 120575119867120576119867119864119867 + 120572119867119877119867119889119864119867119889119905 = (1 minus ]119867) 120573119863119867119878119867119868119889 minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 119864119867119889119868119867119889119905 = (1 minus 120588119867) 120575119867120574119867119864119867 minus (119898119867 + 120583119867) 119868119867119889119877119867119889119905 = ]119867119878119867 + 120588119867119864119867 minus (119898119867 + 120572119867) 119877119867 119878119863 gt 0 119864119863 ge 0 119868119863 ge 0 119877119863 ge 0 119878119867 gt 0 119864119867 ge 0 119868119867 ge 0 119877119867 ge 0

(52)

From (51) the quantities 1198601 and 1198602 denote the weightconstants of the exposed classes and1198603 and1198604 are the weightof the infectious classes respectively 1198611 1198612 1198613 1198614 are theweight constants for the dog and human controls 1198611]211986311986121205882119863 1198613]2119867 11986141205882119867 describe the cost associated with rabiesvaccination and treatmentThe square of the control variablesshows the severity of the side effects of the vaccination andtreatment Employing Pontryaginrsquos maximum principle weform the Hamiltonian equation with state variables 119878119863 = 119878lowast119863

119864119863 = 119864lowast119863 119868119863 = 119868lowast119863 119877lowast119863 and 119878119867 = 119878lowast119867 119864119867 = 119864lowast119867 119868119867 =119868lowast119867 119877lowast119867 as119867 = 1198601119864lowast119863 + 1198602119864lowast119867 + 1198603119868lowast119863 + 1198604119868lowast119867 + 11986112 ]2119863 + 11986122

sdot 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867 + 1205821 [119860119863minus (1 minus ]119863) 120573119863119863119878lowast119863119868lowast119863 minus (119898119863 + ]119863) 119878lowast119863 + 120575120576119864lowast119863


+ 120572119863119877lowast119863] + 1205822 [(1 minus ]119863) 120573119863119863119878lowast119863119868lowast119863minus ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) 119864lowast119863]+ 1205823 [(1 minus 120588119863) 120575120574119863119864lowast119863 minus (119898119863 + 120583119863) 119868lowast119863]+ 1205824 []119863119878lowast119863 + 120588119863119864lowast119863 minus (119898119863 + 120572119863) 119877lowast119863] + 1205825 [119861119867minus (1 minus ]119867) 120573119863119867119878lowast119867119868lowast119863 minus (119898119867 + ]119867) 119878lowast119867 + 120575119867120576119867119864lowast119867+ 120572119867119877lowast119867] + 1205826 [(1 minus ]119867) 120573119863119867119878lowast119867119868lowast119863minus ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867) 119864lowast119867]+ 1205827 [(1 minus 120588119867) 120575119867120574119867119864lowast119867 minus (119898119867 + 120583119867) 119868lowast119867]+ 1205828 []119867119878lowast119867 + 120588119867119864lowast119867 minus (119898119867 + 120572119867) 119877lowast119867] (53)

Considering the existence of adjoint functions 120582119894 119894 =1 2 8 satisfying1198891205821119889119905 = minus 120597119867120597119878lowast119863= 1205821 ((1 minus ]119863) 120573119863119863119868lowast119863 + 119898119863 + ]119863)minus 1205822 (1 minus ]119863) 120573119863119863119868lowast119863 minus 1205824]1198631198891205822119889119905 = minus 120597119867120597119864lowast119863= 1205822 ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)minus 1205821120575120576119863 minus 1205823 (1 minus 120588119863) 120575120574119863 minus 1205824120588119863 minus 11986011198891205823119889119905 = minus 120597119867120597119868lowast119863= 1205823 (119898119863 + 120583119863) + 1205821 (1 minus ]119863) 120573119863119863119878lowast119863+ 1205825 (1 minus ]119867) 120573119863119867119878lowast119867 minus 1205822 (1 minus ]119863) 120573119863119878lowast119863minus 1205826 (1 minus ]119867) 120573119863119867119878lowast119867 minus 11986031198891205824119889119905 = minus 120597119867120597119877lowast119863 = 1205824 (119898119863 + 120572119863) minus 12058211205721198631198891205825119889119905 = minus 120597119867120597119878lowast119867= 1205825 ((1 minus ]119867) 120573119889119867119868lowast119863 + 119898119867 + ]119867)minus 1205826 (1 minus ]119867) 120573119863119867119868lowast119863 minus 1205828]1198671198891205826119889119905 = minus 120597119867120597119864lowast119867= 1205826 ((1 minus 120588119867) 120575119867120574119867 + 119898119867 + 120588119867 + 120575119867120576119867)minus 1205825120575119867120576119867 minus 1205827 (1 minus 120588119867) 120575119867120574119867 minus 1205828120588119867 minus 1198602

1198891205827119889119905 = minus 120597119867120597119868lowast119867 = 1205827 (119898119867 + 120583119867) minus 11986041198891205828119889119905 = minus 120597119867120597119877lowast119867 = 1205828 (119898119867 + 120572119867) minus 1205825120572119867

(54)

with transversality condition 120582119894(119905119891) = 0 for 119894 = 1 8 forthe control set 119906119894 hence we have

120597119867120597119906119894 = 0 where 119894 = 1 2 3 4120597119867120597]119863 10038161003816100381610038161003816100381610038161003816]119863=]lowast119863 fl 1198611]lowast119863 minus 1205821119878lowast119863 + 1205824119878lowast119863 + 1205821120573119863119863119878lowast119863119868lowast119863

minus 1205822120573119863119878lowast119863119868lowast119863 = 0]lowast119863 = (1205821119878lowast119863 minus 1205824119878lowast119863) + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611

120597119867120597120588119863 10038161003816100381610038161003816100381610038161003816120588119863=120588lowast119863 fl 1198612120588lowast119863 minus 1205821119864lowast119863 + 1205824119864lowast119863 + 1205822119864119863120575120574119863119864lowast119863minus 1205823120575120574119863119864lowast119863 = 0

120588lowast119863 = (1205822119864lowast119863 minus 1205824119864lowast119863) + (1205823 minus 1205822) 120575120574119863119864lowast1198631198612 120597119867120597]119867 10038161003816100381610038161003816100381610038161003816]119867=]lowast119867 fl 1198613]lowast119867 minus 1205825119878119867 + 1205828119878119867 + 1205825120573119863119867119878119867119868119863


120597119867120597120588119867 10038161003816100381610038161003816100381610038161003816120588119867=120588lowast119867 fl 1198614120588lowast minus 1205826119864lowast119867 + 1205828119864lowast119867 + 1205826120575119867120574119867119864lowast119867minus 1205827120575119867120574119867119864lowast119867 = 0

120588lowast119867 = (1205826119864lowast119867 minus 1205828119864lowast119867) + (1205827 minus 1205826) 120575119867120574119867119864lowast1198671198614

(55)

Now using an appropriate variation argument and takingthe bounds into account the optimal control strategies aregiven as

]lowast119863 = minmax(0 (1205821 minus 1205824) 119878lowast119863 + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611 ) ]119863max (56)

120588lowast119863 = minmax(0 (1205822 minus 1205824) 119864lowast119863 + (1205823 minus 1205822) 120575120574119863119864lowast1198631198612 ) 120588119863max (57)




Optimality System Substituting the representation of the opti-mal vaccination and treatment control with correspondingadjoint function we have the optimality system as

119889119878119863119889119905 = 119860119863 minus (1 minusminmax(0(1205821 minus 1205824) 119878lowast119863 + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611 ) ]119863max)sdot 120573119863119863119878119863119868119863 minus 119898119863119878119863 minusminmax(0(1205821 minus 1205824) 119878lowast119863 + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611 ) ]119863max119878119863+ 120575120576119863119864119863 + 120572119863119877119863119889119864119863119889119905 = (1 minusminmax(0(1205821 minus 1205824) 119878lowast119863 + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611 ) ]119863max)sdot 120573119863119863119878119863119868119863 minus ((1minusminmax(0 (1205822 minus 1205824) 119864lowast119863 + (1205823 minus 1205822) 120575120574119863119864lowast1198631198612 ) 120588max)120575120574119863 + 119898119863 + 120575120576119863 + 119862119863)119864119863 minusminmax(0(1205822 minus 1205824) 119864lowast119863 + (1205823 minus 1205822) 120575120574119863119864lowast1198631198612 ) 120588119863max119864119863

119889119868119863119889119905 = 120575120574119863119864119863 minus (119898119863 + 120583119863) 119868119863119889119877119863119889119905 = minmax(0(1205821 minus 1205824) 119878lowast119863 + (1205822 minus 1205821) 120573119863119863119868lowast119863119878lowast1198631198611 ) ]119863max119878119863minus (119898119863 + 120572119863) 119877119863 +minmax(0(1205822 minus 1205824) 119864lowast119863 + (1205823 minus 1205822) 120575120574119863119864lowast1198631198612 ) 120588119863max119864119863

119889119878119867119889119905 = 119861119867 minus (1 minusminmax(0(1205825 minus 1205828) 119878lowast119867 + (1205826 minus 1205825) 120573119863119867119878lowast119867119868lowast1198631198613 ) ]119867max)sdot 120573119863119867119878119867119868119863 minus 119898119867119878119867 minusminmax(0(1205825 minus 1205828) 119878lowast119867 + (1205826 minus 1205825) 120573119863119867119878lowast119867119868lowast1198631198613 ) ]119867max119878119867+ 120575119867120576119867119864119867 + 120572119867119877119867119889119864119867119889119905 = (1 minusminmax(0(1205826 minus 1205828) 119864lowast119867 + (1205827 minus 1205826) 120575119867120574119867119864lowast1198671198614 ) 120588119867max)sdot 120573119863119867119878119867119868119863 minus (120575119867120574119867 + 119898119867 + 120575119867120576119867) 119864119867minusminmax(0(1205826 minus 1205828) 119864lowast119867 + (1205827 minus 1205826) 120575119867120574119867119864lowast1198671198614 ) 120588119867max119864119867


1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)

51 Numerical Simulations of the Optimality System Todetermine the control strategies ]119863 120588119863 ]119867 and 120588119867 as givenin the objective functional we began an iteration of themodeluntil convergence is achieved The results of the simulationof the control strategies are displayed below We considerequal weights of (1198601 = 1 1198602 = 1 1198603 = 1 1198604 = 1) for bothexposed and infected classes We varied the cost associatedwith the objective functional which indicate that with lowcost of vaccination the rate at which individuals will seek forvaccination of their susceptible dogs will increase and thiscould result in low transmission of rabies in a heterogeneouspopulation We consider the various cost of preexposureprophylaxis and postexposure prophylaxis to be (1198611 = 1


0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e

Postexposure prophylaxis

Preexposure prophylaxisfor humans (]H)

for humans (H)Postexposure prophylaxis

Preexposure prophylaxisfor dogs (]D)

for dogs (treatment) (D)

Figure 4 The simulation effect of the controls

0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

Figure 5 The trajectories of the model with and without pre- and postexposure prophylaxis on exposed humans and that of the exposeddogs

1198612 = 4 1198613 = 1 1198614 = 4) We found that the optimal timein controlling the infection using preexposure prophylaxis indogs is much better than using postexposure prophylaxis indogs as shown by the trajectories of the red line and blue linein Figure 4 respectively The blue line in Figure 4 indicatesthat applying postexposure prophylaxiswill considerably takea longer time in controlling of rabies in dogs The green linein Figure 4 signifies that preexposure prophylaxis in humans

increases the immunity levels of humans and hence reducesthe rate at which individuals move to the infected stage Fig-ures 5 and 6 show the effect of using only one control strategyon themodelTherefore Figure 5(a) shows that applying onlypostexposure prophylaxis (treatment or quarantine) of dogshas a low positive impact on the model Figure 5(b) showsthat sticking to the use of pre- and postexposure prophylaxisin human without administering pre- and postexposure


0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0

Figure 6 The trajectories of the model with and without pre- and postexposure prophylaxis on infected humans and that of the infecteddogs

prophylaxis in the dog population will result in a high of therabies infection in the human population Figure 6(a) alsoshows that combining pre- and postexposure prophylaxis(vaccination and treatmentquarantine) in the dog com-partment will reduce the spread of the rabies virus therebyreducing the using of pre- and postexposure prophylaxis(vaccination and treatment) in humans Figure 6(b) indicatesthat a rapid use of pre- and postexposure prophylaxis in thehuman population will reduce the number of rabies deaths inthe human population Figure 7 shows the simulation effectsof applying both controls on the model Figure 7(a) showsthat with the use of the optimal control strategies the rate ofthe infection in the susceptible dogs will reduce significantlyFigures 7(b) and 7(c) show that there is a proportionaldecrease in the number of exposed and infected dogs whenthe control measures are applied Similarly Figures 7(e) and7(f) show a significant decrease in the number of infectedand exposed humans when the control measures are appliedFigure 7(d) shows that there is a proportional increase in thenumber of recovered dogs when the control measures areapplied Finally Figures 8(a)ndash8(h) show the simulation effectof corresponding adjoint functions

6 Discussion

The numerical simulations of the resulting optimality systemshow that during the case where it is more expensive tovaccinate than treatment more resources should be investedin treating affected individuals until the disease prevalencebegins to fall This option however does not reduce thenumber of individuals expose to the disease quickly enoughthus resulting in an overall increase in the infected humanpopulation On the other hand if it is more expensive to

treat than to vaccinate then more susceptible dogs shouldbe vaccinated so as to lower the rate at which newborn dogsget infected Nevertheless in the case where both measuresare equally expensive the simulation shows that the optimalway to drive the epidemic towards eradication within anyspecified period is to use more preexposure prophylaxis inboth compartments

7 Conclusion

We studied an optimal control model of rabies transmissiondynamics in dogs and the best way of reducing death rateof rabies in humans The stability analysis shows that thedisease-free equilibrium is locally and globally asymptoticallystable We also obtained an optimal control solution for themodel which predicts that the optimal way of eliminatingdeaths from canine rabies as projected by the global alliancefor rabies control [30] is using more of preexposure pro-phylaxis in both dogs and humans and public educationhowever the results show that the effective and optimalconsideration of preexposure prophylaxis and postexposureprophylaxis in humans without an optimal use of vaccinationin the dog population is not beneficial if total elimination ofthe disease is desirable in Africa and Asia Any combinationstrategy which involves vaccination in the dogsrsquo populationgives a better result and hence it may be beneficial ineliminating the disease in Asia Africa and Latin America

Disclosure

The authors fully acknowledge that this paper was developedas a result of the first authorrsquos thesis work submitted to theDepartment of Mathematics Kwame Nkrumah University of


1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)

Without optimal controlWith optimal control

(a) Susceptible dogs with and without control

0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)


(b) Exposed dogs with and without control

1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)


(c) Infected dogs with and without control

05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)


(d) Recovered dogs with and without control

1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)


(e) Infected humans with and without control

500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)


(f) Exposed humans with and without control

Figure 7 The trajectories of the model with and without optimal control on individual compartments


0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1

(a) The cost function 1205821 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2

(b) The cost function 1205822 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3

(c) The cost function 1205823 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3

(d) The cost function 1205824 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8

(e) The cost function 1205825 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6

(f) The cost function 1205826 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7

(g) The cost function 1205827 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8

(h) The cost function 1205828 for 1198601 = 1198602 = 1198603 = 1198604 = 1 and 1198611 =1 1198612 = 4 1198613 = 1 1198614 = 4

Figure 8 The trajectories of the model with and without optimal control on individual compartments and corresponding adjoint function

Science and Technology (see httpirknustedughxmluihandle12345678910053)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first authorrsquos work (thesis) was partly supported by theGovernment of Ghana Annual Research Grant for Postgrad-uate Studies

References

[1] D T S Hayman N Johnson D L Horton et al ldquoEvolutionaryhistory of rabies in Ghanardquo PLoS Neglected Tropical Diseasesvol 5 no 4 Article ID e1001 2011

[2] ldquoRabies fact sheetrdquo 2016 httpwwwwhointmediacentrefact-sheetsfs099en

[3] C E Rupprecht D Briggs C M Brown R Franka S LKatz H D Kerr et al Use of A Reduced (4-Dose) VaccineSchedule for Postexposure Prophylaxis to Prevent Human RabiesRecommendations of The Advisory Committee on ImmunizationPractices Department of Health and Human Services Centersfor Disease Control and Prevention 2010

[4] R L M Neilan Optimal control applied to population anddiseasemodels [PhD thesis] University of Tennessee KnoxvilleTenn USA 2009

[5] A O Isere J E Osemwenkhae and D Okuonghae ldquoOptimalcontrol model for the outbreak of cholera in Nigeriardquo AfricanJournal of Mathematics and Computer Science Research vol 7no 2 pp 24ndash30 2014

[6] H S Rodrigues ldquoOptimal control and numerical optimiza-tion applied to epidemiological modelsrdquo 2014 httpsarxivorgabs14017390context=math

[7] A A LashariMathematical Modeling and Optimal Control of AVector National University of Modern Languages IslamabadPakistan 2012

[8] AAbdelrazecModeling Dynamics andOptimal Control ofWestNile Virus with Seasonality York University Toronto Canada2014

[9] D Greenhalgh ldquoAge-structured models and optimal control inmathematical equidemiology a surveyrdquo 2010

[10] A StashkoTheEffects of Prevention andTreatment Interventionsin A Microeconomic Model of HIV Transmission Duke Univer-sity Durham NC USA 2012

[11] B Seidu and O D Makinde ldquoOptimal control of HIVAIDSin the workplace in the presence of careless individualsrdquoComputational and Mathematical Methods in Medicine vol2014 Article ID 831506 19 pages 2014

[12] S D D Njankou and F Nyabadza ldquoAn optimal control modelfor Ebola virus diseaserdquo Journal of Biological Systems vol 24 no1 pp 29ndash49 2016

[13] M Aubert ldquoCosts and benefits of rabies control in wildlife inFrancerdquo Revue Scientifique et Technique de lrsquoOIE vol 18 no 2pp 533ndash543 1999

[14] R M Anderson and R M May ldquoPopulation biology ofinfectious diseasesrdquo in Proceedings of the Dahlem WorkshopSpringer Berlin Germany March 1982

[15] G Bohrer S Shem-Tov E Summer K Or and D SaltzldquoThe effectiveness of various rabies spatial vaccination patternsin a simulated host population with clumped distributionrdquoEcological Modelling vol 152 no 2-3 pp 205ndash211 2002

[16] S A Levin T G Hallam and L J Gross Eds Applied Mathe-matical Ecology vol 18 of Biomathematics Springer BerlinGermany 2012

[17] M J Coyne G Smith and F E McAllister ldquoMathematic modelfor the population biology of rabies in raccoons in the mid-Atlantic statesrdquo American journal of veterinary research vol 50no 12 pp 2148ndash2154 1989

[18] J E Childs A T Curns M E Dey et al ldquoPredicting the localdynamics of epizootic rabies among raccoons in the United


Statesrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 97 no 25 pp 13666ndash13671 2000

[19] K Hampson J Dushoff J Bingham G Bruckner Y H Ali andA Dobson ldquoSynchronous cycles of domestic dog rabies in sub-Saharan Africa and the impact of control effortsrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 104 no 18 pp 7717ndash7722 2007

[20] M J Carroll A Singer G C Smith D P Cowan andGMasseildquoThe use of immunocontraception to improve rabies eradi-cation in urban dog populationsrdquoWildlife Research vol 37 no8 pp 676ndash687 2010

[21] XWang and J Lou ldquoTwodynamicmodels about rabies betweendogs and humanrdquo Journal of Biological Systems vol 16 no 4 pp519ndash529 2008

[22] W Yang and J Lou ldquoThe dynamics of an interactional modelof rabies transmitted between human and dogsrdquo Bollettino dellaUnione Matematica Italiana Serie 9 vol 2 no 3 pp 591ndash6052009

[23] J Zinsstag S Durr M A Penny et al ldquoTransmission dynamicsand economics of rabies control in dogs and humans in anAfrican cityrdquo Proceedings of the National Academy of Sciencesof the United States of America vol 106 no 35 pp 14996ndash150012009

[24] W Ding L J Gross K Langston S Lenhart and L A RealldquoRabies in raccoons optimal control for a discrete time modelon a spatial gridrdquo Journal of Biological Dynamics vol 1 no 4pp 379ndash393 2007

[25] G C Smith and C L Cheeseman ldquoA mathematical model forthe control of diseases in wildlife populations culling vaccina-tion and fertility controlrdquo Ecological Modelling vol 150 no 1-2pp 45ndash53 2002

[26] J M Tchuenche and C T Bauch ldquoCan culling to preventmonkeypox infection be counter-productive Scenarios from atheoretical modelrdquo Journal of Biological Systems vol 20 no 3pp 259ndash283 2012

[27] J Zhang Z Jin G-Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 p e20891 2011

[28] O A Grace ldquoModelling of the spread of rabies with pre-expo-sure vaccination of humansrdquo Mathematical Theory and Model-ing vol 4 no 8 2014

[29] E D Wiraningsih F Agusto L Aryati et al ldquoStability analysisof rabies model with vaccination effect and culling in dogsrdquoApplied Mathematical Sciences vol 9 no 77-80 pp 3805ndash38172015

[30] ldquoGlobal alliance for rabies controlrdquo 2016 httpsrabiesallianceorg

[31] CIA world factbook and other sources 2016 httpwwwtheodoracomwfbcurrentghanaghana_peoplehtml

[32] G Birkhoff and G Rota Ordinary differential equations Ginnamp Co New York NY USA 1962

[33] V Lakshmikantham S Leela and S Kaul ldquoComparison prin-ciple for impulsive differential equations with variable timesand stability theoryrdquo Nonlinear Analysis Theory Methods ampApplications vol 22 no 4 pp 499ndash503 1994

[34] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio 1198770 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[35] E C Pielou An introduction to Mathematical Ecology Wiley-InterScience Hoboken NJ USA 1969

[36] R C May ldquoStable limit cycles in pre-predator population Acommentrdquo Science p 1074 1973

[37] M Martcheva An Introduction to Mathematical Epidemiologyvol 61 of Texts in AppliedMathematics Springer New York NYUSA 2015

[38] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2012

[39] S D Hove-Musekwa F Nyabadza and H Mambili-Mam-boundou ldquoModelling hospitalization home-based care andindividual withdrawal for people living with HIVAIDS in highprevalence settingsrdquo Bulletin of Mathematical Biology vol 73no 12 pp 2888ndash2915 2011

[40] C V De Leon ldquoConstructions of Lyapunov functions for clas-sics SIS SIR and SIRS epidemic model with variable popu-lation sizerdquo Foro-Red-Mat Revista Electronica De ContenidoMatematico vol 26 no 5 article 1 2009

[41] J LaSalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the CBMS-NSF Regional Conference Series in Applied Mathe-matics 25 SIAM Philadelphia Pa USA 1976

[42] T Zhang KWang X Zhang and Z Jin ldquoModeling and analyz-ing the transmission dynamics of HBV epidemic in XinjiangChinardquo PLoS ONE vol 10 no 9 Article ID e138765 2015

[43] O Sharomi and T Malik ldquoOptimal control in epidemiologyrdquoAnnals of Operations Research vol 251 no 1-2 pp 55ndash71 2017

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Stochastic AnalysisInternational Journal of


speciesmost affected two primary procedures for controllingrabies were assessed in [13] at the repository level to bespecific fox termination and the oral immunization of foxesThe consolidated costs and advantages of both systems werelooked at and included either the expenses of fox separationor the cost of oral immunizationThe total yearly costs of bothtechniques stayed practically identical until the fourth yearafter which the oral immunization methodology turned outto be more cost effectiveThis estimate was made in 1988 andreadjusted in 1993 and affirmed by ex-postinvestigation fiveyears later Accordingly it was presumed that fox terminationbrought about a transient diminishment in the event ofthe infection while oral immunization turned out to beequipped for wiping out rabies even in circumstances inwhich fox population was growing Anderson and May [14]formulated a mathematical model based on each time stepdynamic which was calculated independently in every cellLater Bohrer et al [15] published a paper on the viability ofdifferent rabies spatial immunization designs in a simulatedhost population

The research presented by Bohrer [15] stated that indesert environments where host population size varies overtime nonuniform spreading of oral rabies vaccination mayunder certain circumstances be more effective than the com-monly used uniform spread The viability of a nonarbitraryspread of the immunization depends to some extent onthe dispersal behavior of the carriers The outcomes likewiseexhibit that in a warm domain in a few high-density regionsencompassed by populations with densities below the criticalthreshold for the spread of the disease the rabies infectioncan persist

Levin et al [16] also presented a model for the immuneresponses to rabies virus in bats Coyne et al [17] proposedan SEIR model which was also used in a study predictingthe local dynamics of rabies among raccoons in the UnitedStates Childs et al [18] also researched rabies epidemics inraccoons with a seasonal birth pulse using optimal controlof an SEIRSmodel which describes the population dynamicsHampson et al [19] also noted that rabies epidemic cycleshave a period of 3ndash6 years in dog populations in Africa sothey built a susceptible exposed infectious and vaccinatemodel with an intervention response variable which showedsignificant synchrony

Carroll et al [20] also used compartmental modelsto describe rabies epidemiology in dog populations andexplored three control methods vaccination vaccinationpulse fertility control and culling An ordinary differentialequation model was used to characterize the transmissiondynamics of rabies between humans and dogs by [21 22]The work by Zinsstag et al [23] further extended the existingmodels on rabies transmission between dogs to include dog-to-human transmission and concluded that human postex-posure prophylaxis (PEP) with a dog vaccination campaignwas the more cost effective in controlling the disease inthe long run Furthermore Ding et al [24] formulated anepidemic model for rabies in raccoons with discrete timeand spatial features Their goal was to analyze the strategiesfor optimal distribution of vaccine baits to minimize thespread of the disease and the cost of carrying out the control

Smith and Cheeseman [25] show that culling could be moreeffective than vaccination given the same efficacy of controlbut Tchuenche and Bauch suggest that culling could becounterproductive for some parameter values (see [26])

Thework in [27 28] also presented amathematical modelof rabies transmission in dogs and from the dog population tothe human population in ChinaTheir study did not considerthe use optimal control analysis to the study of the rabiesvirus in dogs and from the dog population to the humanpopulation Furthermore the insightful work ofWiraningsihet al [29] studied the stability analysis of a rabies model withvaccination effect and culling in dogs where they introducedpostexposure prophylaxis to a rabies transmission modelbut the paper did not consider the noneffectiveness of thepre- and postprophylaxis on the susceptible humans andexposed humans and that of the dog population and theuse of optimal control analysis Therefore motivated by theresearch predictions of the global alliance of rabies control[30] and the workmention above we seek to adjust themodelpresented in [27ndash29] by formulating an optimal controlmodel so as to ascertain an optimal way of controlling rabiestransmission in dogs and from the dog population to thehuman population taking into account the noneffectiveness(failure) of vaccination and treatment

The paper is petition as follows Section 2 contains themodel formulation mathematical assumptions the mathe-matical flowchart and the model equations Section 3 con-tains themodel analysis invariant region equilibriumpointsbasic reproduction number R0 and the stability analysis ofthe equilibria In Section 4 we present the parameter valuesleading to numerical values of the basic reproduction numberR0 the herd immunity threshold and sensitivity analysisusing Latin hypercube sampling (LHS) and some numericalplots Section 5 contains the objective functional and theoptimality system of the model Finally Sections 6 and 7contain discussion and conclusion respectively

2 Model Formulation

We present two subpopulation transmission models of rabiesvirus in dogs and that of the human population (see Figure 1)based on the work presented in [27ndash29] The dog populationhas a total of four compartments The compartments repre-sent the susceptible dogs 119878119863(119905) exposed dogs 119864119863 infecteddogs 119868119863(119905) and partially immune dogs 119877119863(119905)Thus the totaldog population is119873119863(119905) = 119878119863(119905) + 119864119863(119905) + 119868119863(119905) + 119877119863(119905) Thehuman population also has four compartments representingsusceptible humans 119878119867(119905) exposed humans 119864119867(119905) infectedhumans 119868119867(119905) and partially immune humans 119877119867(119905) Thusthe total human population is 119873119867(119905) = 119878119867(119905) + 119864119867(119905) +119868119867(119905) + 119877119867(119905) It is assumed that there is no human to humantransmission of the rabies virus in the human submodel (see[29]) In the dog submodel it is assumed that there is a directtransmission of the rabies virus fromone dog to the other andfrom the infected dog compartment to the susceptible humanpopulation It is further assumed that the susceptible dogpopulation 119878119863(119905) is increased by recruitment at a rate 119860119863and 119861119867 is the birth or immigration rate into the susceptiblehuman population 119878119867(119905) It is assumed that the transmission




]DSD

RD

mDRD

ID

(D + mD) ID

(1 minus D) DED

DED

ED

(mD + CD)ED

(1 minus ]D) DDSDID

DED

mDSD

AD SD

DRD

RH

mHRH

IH

(H + mH) IH

(1 minus H) HHEH

HEH

]HSH

mHEH

EH

HRH

(1 minus ]H) DHSHID

HHEH

SH

mHSH

BH








(1)

3 Model Analysis


119873119863 (119905) = 119878119863 (119905) + 119864119863 (119905) + 119868119863 (119905) + 119877119863 (119905) (2)119873119867 (119905) = 119878119867 (119905) + 119864119867 (119905) + 119868119867 (119905) + 119877119867 (119905) (3)



which yields


Similarly (3) gives

119889119873119867119889119905 = 119861119867 minus 119898119867119873119867 minus 120583119867119868119867 (6)


119889119873119863119889119905 = 119860119863 minus 119898119863119873119863119889119873119867119889119905 = 119861119863 minus 119898119867119873119867 (7)


119889119873119863119889119905 le 119860119863 minus 119898119863119873119863 (8)

119889119873119867119889119905 le 119861119863 minus 119898119867119873119867 (9)


Ω119863 = (119878119863 119864119863 119868119863 119877119863) isin R4+ 119873119863 le 119860119863119898119863 (10)


Ω119867 = (119878119867 119864119867 119868119867 119877119867) isin R4+ 119873119867 le 119861119867119898119867 (11)





((((((((

119878119863119864119863119868119863119877119863119878119867119864119867119868119867119877119867

))))))))

isin R8+ |

((((((((((((((((((


))))))))))))))))))

(13)




(14)


E0 = ( 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 0 0119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 0 0119861119867]119867119898119867 (119898119867 + 120572119867 + ]119867)) (15)




119869 =((((((((

1205971198911120597119864119863 1205971198911120597119868119863 1205971198911120597119864119867 12059711989111205971198681198671205971198912120597119864119863 1205971198912120597119868119863 1205971198912120597119864119867 12059711989121205971198681198671205971198913120597119864119863 1205971198913120597119868119863 1205971198913120597119864119867 12059711989131205971198681198671205971198914120597119864119863 1205971198914120597119868119863 1205971198914120597119864119867 1205971198914120597119868119867

))))))))

(17)


119869

=(((


(18)



119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((


(19)



119866

=(((((


(20)

Letting


(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







]DSD

RD

mDRD

ID

(D + mD) ID

(1 minus D) DED

DED

ED

(mD + CD)ED

(1 minus ]D) DDSDID

DED

mDSD

AD SD

DRD

RH

mHRH

IH

(H + mH) IH

(1 minus H) HHEH

HEH

]HSH

mHEH

EH

HRH

(1 minus ]H) DHSHID

HHEH

SH

mHSH

BH








(1)

3 Model Analysis


119873119863 (119905) = 119878119863 (119905) + 119864119863 (119905) + 119868119863 (119905) + 119877119863 (119905) (2)119873119867 (119905) = 119878119867 (119905) + 119864119867 (119905) + 119868119867 (119905) + 119877119867 (119905) (3)



which yields


Similarly (3) gives

119889119873119867119889119905 = 119861119867 minus 119898119867119873119867 minus 120583119867119868119867 (6)


119889119873119863119889119905 = 119860119863 minus 119898119863119873119863119889119873119867119889119905 = 119861119863 minus 119898119867119873119867 (7)


119889119873119863119889119905 le 119860119863 minus 119898119863119873119863 (8)

119889119873119867119889119905 le 119861119863 minus 119898119867119873119867 (9)


Ω119863 = (119878119863 119864119863 119868119863 119877119863) isin R4+ 119873119863 le 119860119863119898119863 (10)


Ω119867 = (119878119867 119864119867 119868119867 119877119867) isin R4+ 119873119867 le 119861119867119898119867 (11)





((((((((

119878119863119864119863119868119863119877119863119878119867119864119867119868119867119877119867

))))))))

isin R8+ |

((((((((((((((((((


))))))))))))))))))

(13)




(14)


E0 = ( 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 0 0119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 0 0119861119867]119867119898119867 (119898119867 + 120572119867 + ]119867)) (15)




119869 =((((((((

1205971198911120597119864119863 1205971198911120597119868119863 1205971198911120597119864119867 12059711989111205971198681198671205971198912120597119864119863 1205971198912120597119868119863 1205971198912120597119864119867 12059711989121205971198681198671205971198913120597119864119863 1205971198913120597119868119863 1205971198913120597119864119867 12059711989131205971198681198671205971198914120597119864119863 1205971198914120597119868119863 1205971198914120597119864119867 1205971198914120597119868119867

))))))))

(17)


119869

=(((


(18)



119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((


(19)



119866

=(((((


(20)

Letting


(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

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35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(1)

3 Model Analysis


119873119863 (119905) = 119878119863 (119905) + 119864119863 (119905) + 119868119863 (119905) + 119877119863 (119905) (2)119873119867 (119905) = 119878119867 (119905) + 119864119867 (119905) + 119868119867 (119905) + 119877119867 (119905) (3)



which yields


Similarly (3) gives

119889119873119867119889119905 = 119861119867 minus 119898119867119873119867 minus 120583119867119868119867 (6)


119889119873119863119889119905 = 119860119863 minus 119898119863119873119863119889119873119867119889119905 = 119861119863 minus 119898119867119873119867 (7)


119889119873119863119889119905 le 119860119863 minus 119898119863119873119863 (8)

119889119873119867119889119905 le 119861119863 minus 119898119867119873119867 (9)


Ω119863 = (119878119863 119864119863 119868119863 119877119863) isin R4+ 119873119863 le 119860119863119898119863 (10)


Ω119867 = (119878119867 119864119867 119868119867 119877119867) isin R4+ 119873119867 le 119861119867119898119867 (11)





((((((((

119878119863119864119863119868119863119877119863119878119867119864119867119868119867119877119867

))))))))

isin R8+ |

((((((((((((((((((


))))))))))))))))))

(13)




(14)


E0 = ( 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 0 0119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 0 0119861119867]119867119898119867 (119898119867 + 120572119867 + ]119867)) (15)




119869 =((((((((

1205971198911120597119864119863 1205971198911120597119868119863 1205971198911120597119864119867 12059711989111205971198681198671205971198912120597119864119863 1205971198912120597119868119863 1205971198912120597119864119867 12059711989121205971198681198671205971198913120597119864119863 1205971198913120597119868119863 1205971198913120597119864119867 12059711989131205971198681198671205971198914120597119864119863 1205971198914120597119868119863 1205971198914120597119864119867 1205971198914120597119868119867

))))))))

(17)


119869

=(((


(18)



119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((


(19)



119866

=(((((


(20)

Letting


(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

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D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

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4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

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4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

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08

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int f

unct

ion

020 40 60 800

Time (years)

2


05

1

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Adjo

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unct

ion

20 40 60 800Time (years)

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2

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8

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Adjo

int f

unct

ion

20 40 60 800Time (years)

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times10minus3


1

2

3

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5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








((((((((

119878119863119864119863119868119863119877119863119878119867119864119867119868119867119877119867

))))))))

isin R8+ |

((((((((((((((((((


))))))))))))))))))

(13)




(14)


E0 = ( 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 0 0119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + ]119867) 0 0119861119867]119867119898119867 (119898119867 + 120572119867 + ]119867)) (15)




119869 =((((((((

1205971198911120597119864119863 1205971198911120597119868119863 1205971198911120597119864119867 12059711989111205971198681198671205971198912120597119864119863 1205971198912120597119868119863 1205971198912120597119864119867 12059711989121205971198681198671205971198913120597119864119863 1205971198913120597119868119863 1205971198913120597119864119867 12059711989131205971198681198671205971198914120597119864119863 1205971198914120597119868119863 1205971198914120597119864119867 1205971198914120597119868119867

))))))))

(17)


119869

=(((


(18)



119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((


(19)



119866

=(((((


(20)

Letting


(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119865 (E0) =(((((

0 (1 minus ]119863) 120573119863119863119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + ]119863 + 120572119863) 0 00 0 0 00 (1 minus ]119867) 120573119863119867 (119898119867 + 120572119867) 119861119867119898119867 (119898119867 + ]119867 + 120572119867) 0 00 0 0 0

)))))

119881(E0) =(((


(19)



119866

=(((((


(20)

Letting


(21)


implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





implies

119866 =(((

119886 119887 0 00 0 0 0119888 119889 0 00 0 0 0)))

(22)


119863 =10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (23)


Therefore






(25)





J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






J (E0) =((((((((((((


))))))))))))

(26)

where


(27)



(29)


(30)


1205824 + 119886111205823 + 119886121205822 + 11988613120582 + 11988614 = 0 (31)


11988611 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863)+ (119898119863 + 120583119863) + (119898119863 + 120572119863) + (119898119863 + ]119863)



(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(32)





(33)


119889V119889119905 = (1 minus 1198780119863119878119863) 119889119878119863119889119905 +K1119889119864119863119889119905 +K2

119889119868119863119889119905+ (1 minus 1198770119863119877119863) 119889119877119863119889119905 + (1 minus 1198780119867119878119867) 119889119878119867119889119905+K3

119889119864119867119889119905 +K4119889119868119867119889119905 + (1 minus 1198770119867119877119867) 119889119877119867119889119905

(34)



(35)




119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119878119863 le 1198780119863 = 119860119863 (119898119863 + 120572119863)119898119863 (119898119863 + 120572119863 + ]119863) 119877119863 le 1198770119863 = 119860119863]119863119898119863 (119898119863 + 120572119863 + ]119863) 119878119867 le 1198780119867 = 119861119867 (119898119867 + 120572119867)119898119867 (119898119867 + 120572119867 + 119899119906119867) 119877119867 le 1198770119867 = 119861119867]119867119898119867 (119898119867 + 120572119867 + 119867 + ]119867)

(36)



(37)

This implies that


(38)


K4 = K3 = 0K2 = ((1 minus 120588119863) 120575120574119863 + 119898119863 + 120588119863 + 120575120576119863 + 119862119863) K1 = (1 minus 120588119863) 120575120574119863

(39)


(40)



119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (41)



(42)



(43)




Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Now assuming




(46)







lowastlowast0


Rlowast0 = 120573119863119863119860119863120575120574119863(120575120574119863 + 119898119863 + 120575120576119863 + 119862119863) times (119898119863 + 120583119863)119898119863

Rlowast0 = 3027

Rlowastlowast0




(48)







= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









= 066 (49)



Γ120573119863119863R0

= 120597R0120597120573119863 120573119863119863R0= 1

Γ119860119863R0

= 120597R0120597119860119863 119860119863R0 = 1Γ120583119863R0

= 120597R0120597120583119863 120583119863R0

= minus120583119863(119898119863 + 120583119863) = minus095

Γ120575120576119863R0


Γ119862119863R0


Γ120572119863R0

= 120597R0120597120572119863 120572119863R0 = 028Γ119898119863R0

= 120597R0120597119898119863 119898119863R0 = minus164Γ120575120574119863R0

= 120597R0120597120575120574119863 120575120574119863R0 = 133Γ120588119863R0

= 120597R0120597120588119863 120588119863R0

= minus05Γ]119863R0

= 120597R0120597]119863 ]119863R0

= minus052(50)




PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






PRCC values

ＬＢＩD

ＨＯD

delta gamＧD

AD

mD

ＦＪＢD

delta varepsiloＨD

CD

ＧＯD

＜ＮD




AD = 2 times 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 20 30 40 50 60 70 800Time (years)

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

5000

6000

20 40 60 80705030100Time (years)

R0 = 212

R0 = 318

R0 = 064

R0 = 084

R0 = 106

Infe

cted

hum

ansI

H


0

1000

2000

3000

4000

6000

8000

5000

7000

5 10 15 20 25 30 35 400Time (years)

ID (0) = 5 times 106ID (0) = 6 times 106


ID (0) = 1 times 106

Infe

cted

hum

ansI

H



example Γ120573119863R0







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

100

200

0

2

minus400

minus300

minus200

minus100

0

100

200

1

15

2

25

3

35

4

45

5

D

minus100

minus200

minus300

3 40 1

]D


10

5

01

05

0 02

46

9

8

7

6

5

4

3

2

times107

R0

D

DDtimes 10

minus7

times107


02

015

01

005

01

05

0 12

34

5

014

012

01

008

006

004

002

R0

D

mD


500

0

minus500

minus10006

4

2

0 01

23

4

200

100

0

minus100

minus200

minus300

minus400

R0

D

]D














119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of













119869 = min int1199051198911199050[1198601119864119863 + 1198602119864119867 + 1198603119868119863 + 1198604119868119867 + 11986112 ]2119863 + 11986122 1205882119863 + 11986132 ]2119867 + 11986142 1205882119867] 119889119905 (51)



(52)








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








(54)









(55)












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of












1198891205821119889119905 1198891205822119889119905 1198891205823119889119905 1198891205824119889119905 1198891205825119889119905 1198891205826119889119905 1198891205827119889119905 1198891205828119889119905 with 120582119894 (119905119891) = 0 119894 = 1 2 3 4 5 6 7 8

(60)



0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 10 20 30 40 50 60 70 80Time (years)

02

03

04

05

06

07

08

Con

trol p

rofil

e







0

05

1

15

2

25

3

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Expo

sed

dogs

ED

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

500

1000

1500

2000

2500

3000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Expo

sed

hum

ansE

H

(b) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

4

2

6

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

times106

Infe

cted

dog

sID

(a) A plot of ]119863 = 120588119863 = 0 and ]119867 = 120588119867 = 0

0

1000

2000

3000

4000

5000

6000

10 20 30 40 50 60 70 800Time (years)

]D = D = ]H = H = 0

]D = D = 0 ]H = H = 0

Infe

cted

hum

ansI

H

(b) A plot of ] = 120588 = 0 and ]119867 = 120588119867 = 0



6 Discussion



7 Conclusion


Disclosure



1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





1

15

2

25

3

35

4

45

5Su

scep

tible

dog

s

times107

20 40 60 800Time (years)



0

05

1

15

2

3

25

35

4

Expo

sed

dogs

times106

20 40 60 800Time (years)



1

2

3

4

5

6

7

8

0

Infe

cted

dog

s

times106

20 40 60 800Time (years)



05

1

15

0

2Re

cove

red

dogs

times107

20 40 60 800Time (years)



1000

0

2000

3000

4000

5000

6000

7000

8000

Infe

cted

hum

ans

20 40 60 800Time (years)



500

1000

1500

2000

2500

3000

3500

4000

0

Expo

sed

hum

ans

20 40 60 800Time (years)





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

0002

0004

0006

0008

001Ad

join

t fun

ctio

n

20 40 60 800Time (years)

1


02

04

06

08

1

Adjo

int f

unct

ion

020 40 60 800

Time (years)

2


05

1

15

2

25

3

35

4

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

4

times10minus3


1

2

3

4

5

6

7

8

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

5

times10minus8


005

01

015

02

025

03

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

6


Figure 8 Continued


02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





02

04

06

08

1

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

7


1

2

3

4

5

6

0

Adjo

int f

unct

ion

20 40 60 800Time (years)

8

times10minus8






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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