Stellenbosch University ://core.ac.uk/download/pdf/37401067.pdf · 2020. 1. 7. · Lungu y ersit (Univ of ana) Botsw Co-Promoter: Prof. John e v Hargro y ersit (Univ of bh) Stellenosc

Mathemati al modelling on intera tionbetween malaria parasites and the hostimmune systembyTheresia MarijaniDissertation presented in full ful�lment of thea ademi requirements for the degree ofDo tor of Philosophy in Mathemati sat the Stellenbos h University

Promoter: Prof. Edward M. Lungu (University of Botswana)Co-Promoter: Prof. John Hargrove (University of Stellenbos h)Mar h 2012

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Stellenbosch University SUNScholar Repository

https://core.ac.uk/display/37401067?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1

De larationI, the undersigned, hereby de lare that the work ontained in this dissertation is my ownoriginal work and has not previously, in its entirety or in part, been submitted at anyuniversity for a degree.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Theresia Marijani Date

Copyright ©2012 Stellenbos h UniversityAll rights reserved

Stellenbosch University http://scholar.sun.ac.za

Abstra tMalaria is a deadly tropi al disease aused by protozoa of the genus plasmodium. Themalaria parasite life y le involves three y les namely the sporogony (mosquito stages),exo-erythro yti s hizogony (human liver stages), and the erythro yti s hizogony (humanblood stage). We onsider a mathemati al model for malaria involving, sus eptible redblood ells, latent infe ted red blood ells, a tive infe ted red blood ells, intra ellularparasites, extra ellular parasites and e�e tor ells. We extend the model to in lude all thethree stages of the malaria life y le. The e�e t of treatment on the prognosis of malaria isalso introdu ed in these models. The models are analysed mathemati ally and numeri ally.One of the question addressed in our study is: what repli ative hara teristi s o�er theparasite opportunities to evade the host immune system? The results showed that thelonger it takes to produ e the parasites, the higher the han e that an infe ted red blood ell will be identi�ed and apoptosised by the e�e tor ells. Our sensitivity analysis resultsshow that poor parametri estimation has serious impli ations on the prognosis of thedisease. Treatment results suggest that a high drug e� a y an stop the development ofthe disease. The study has revealed that the parasite repli ative hara teristi s enable theparasite to evade the immune response during the red blood stage malaria. Firstly, we havefound that the parasite has a strategy of infe ting older red blood ells as a strategy toevade immune surveillan e. Se ondly, we dis overed that the administration of an e�e tivedrug an prevent malaria in all stages despite the urrent belief that only a malaria va ine an reliably prote t against all stages malaria infe tion. We re ommend treatment to beused in areas where anti-malarial drugs do not show resistan e to the parasites. We alsore ommend that individuals with malaria or showing some symptoms should be treatedfor both malaria and hroni infe tions. i


OpsommingMalaria is 'n dodelike tropiese siekte wat veroorsaak word deur die protoso ë van die genusPlasmodium. Die malariaparasiet lewensiklus bestaan uit drie siklusse naamlik die sporo-gony (muskiet stadiums), exo-erythro yti s hizogony (menslike lewer stadiums), en dieerythro yti s hizogony (menslike bloed stadium). Ons kyk na 'n wiskundige model virmalaria, vatbaar rooibloedselle, latente besmet rooi bloedselle, aktief besmette rooibloed-selle, intrasellul êre parasiete, ekstrasellulêre parasiete en e�ektor selle. Ons brei die modelom al die drie fases van die lewensiklus malaria te sluit. Die e�ek van behandeling opdie voorspelling van malaria is ook in hierdie modelle ingevoer. Die modelle is wiskundigen numeries ontleed. Een van die vraag in ons studie is: watter repli ative eienskappebied die parasiet geleenthede om die gasheer se immuunrespons te ontduik? Die resultatehet getoon dat hoe langer dit neem om die parasiete te produseer, hoe groter die kansdat besmette rooibloedselle sal geïdenti�seer word en deur die e�ektor selle apoptosised.Ons sensitiwiteitsanalise resultate toon dat die arme parametriese beraming het ernstigeimplikasies vir die voorspelling van die siekte. Behandeling resultate dui daarop dat 'n hoëdwelm doeltre�endheid kan die ontwikkeling van die siekte stop. Die studie het getoon datdie parasiet repli ative eienskappe die parasiet in staat stel om die immuunreaksie tydensdie rooi bloed stadium malaria te ontduik. Eerstens, het ons gevind dat die parasiet het 'nstrategie van besmet om ouer rooi bloed selle as' n strategie om immuun toesig te ontduik.Tweedens, het ons ontdek dat die administrasie van 'n doeltre�ende middel kan malariain alle stadiums voorkom ten spyte van die huidige oortuiging dat slegs' n malaria-entstofbetroubaar kan beskerm teen alle stadia malaria infeksie. Ons raai dat behandeling gebruikword in gebiede waar die anti-malaria medisyne nie weerstand toon aan die parasiete. Onsbeveel ook aan dat individue met malaria of wat sekere simptome het, behandel moet wordvir beide malaria en hroniese malaria infeksies.ii


Dedi ationI dedi ate this dissertation to the greater glory of God, to his gra e and power, to the peoplewho su�ers with malaria, to my beloved parents Mr. and Mrs. Marijani.

iii


A knowledgmentsI would like thank almighty God for his gra e, inspiration, strength and guidan e, to Godbe the glory!I would like to thank my beloved parents for their en ouragement and prayer through allthe time.I would like give thank my promoter Prof. Edward Lungu, for his valuable guidan eand supervision. He has been an ex ellent supervisor, a wise mentor, an inspiring andmotivational tea her available in moments of need. His help during the time in need is realappre iated. My sin ere thanks go to Mrs. Elizabeth Lungu.I deeply appre iate my o-promoter Prof. John Hargrove for his valuable suggestions andinsightful omments on this dissertation.I real appre iate the dire tion and help by the Dire tor of South Afri an Centre of Epidemi-ologi al Modelling and Analysis (SACEMA) Dr. Alex Welte, resear h manager LynnemoreS heepers, assistant dire tor training Dr. Gavin Hit h o k, administrator Natalie Roman,Dr. Rashid Ouifki, all the sta� at SACEMA, University of Stellenbos h, International o� eat Stellenbos h University. University of Botswana, mathemati s department at Universityof Botswana, O� e of International Edu ation and Partnerships at University of Botswanaduring these years of my study. My spe ial thanks to them for all o� ial assistan e theygave me and the personal on ern they showed me.I would like to thank all my fellow students at SACEMA and at University of Botswanafor their help and support in personally and a ademi ally.My spe ial thanks go to Huba Bosho� and Tebogo Magetse for their help and supportiv


vduring the time of ex hanging my studies from Stellenbos h University to Botswana Uni-versity.This thesis is supported by SACEMA and OWSD (Organization for Women in S ien e forthe Developing World). International o� e at Stellenbos h University, O� e of Interna-tional Edu ation and Partnerships at University of Botswana. I would like to thank allthese sponsors.To my family members, my sisters, my brothers' in-law, my brother, sister's in-law, thankyou for being patient with me during this time of my study, espe ially for my nephews(Godwin and Crispin) and my nie es (Winlove, Imma ulata and Agnes). Thank you somu h my family for your support, en ouragement and prayers I real appre iate.Last, but not least, my deepest a knowledgement to all my friends, Angelina Lutambi,Asha Kalula, Sara Mkango, Rose Kibe hu, Doreen Mbabazi, Joseph Ssebuliba, AmaniLusekelo, Maggie Goosen, Boitumelo Mogaleemang, Malebogo, Christina Meela and FaziaDu Plessis thanks for their help on reading and omments.


GlossaryAbbreviation MeaningRBCs Red Blood CellsLVCs Liver CellsMGCs Midgut CellsCD in CD4 Cluster of Di�erentiation 4CD in CD8 Cluster of Di�erentiation 8HIV Human immunode� ien y virusAIDS A quired Immunode� ien y SyndromeODE Ordinary Di�erential EquationT in T- ell ThymusACT Artemisin-based Combination TherapyWHO World Health OrganizationDDT Di hloro-Diphenyl-Tri hloroethaneUNICEF United Nations Children's FundSIV Simian Immunode� ien y VirusCDC Centers for Disease Control & Prevention

vi


ContentsAbstra t iOpsomming iiDedi ation iiiA knowledgements ivGlossary vi1 Introdu tion 11.1 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Obje tives of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Organization of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Literature review 83 Mathemati al tools 163.1 The de�nition and omputation of R0 . . . . . . . . . . . . . . . . . . . . . 163.2 The Routh-Hurwitz riterion . . . . . . . . . . . . . . . . . . . . . . . . . . 18vii


Contents viii3.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 A within host model of blood stage malaria 224.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.2 Mathemati al analysis of the model . . . . . . . . . . . . . . . . . . 324.2.3 A within host treatment model of blood stage malaria . . . . . . . . 364.2.4 Simulations of the within host model of blood stage malaria . . . . 374.2.5 Simulations of the within host treatment model of blood stage malaria 394.3 Results of the within host model of blood stage malaria . . . . . . . . . . . 414.4 Results of within host treatment model of blood stage malaria . . . . . . . 514.5 Dis ussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 A within host treatment model with three stages of malaria life y le 565.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.1 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.2 Mathemati al Analysis of the model . . . . . . . . . . . . . . . . . 655.2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.1 Dynami s of the system before treatment . . . . . . . . . . . . . . . 735.3.2 Treatment strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


Contents ix5.4 Dis ussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 Con lusion 866.1 Limitations and re ommendations . . . . . . . . . . . . . . . . . . . . . . . 876.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Appendix 88A Parameters values and initial variables used in simulations 88


List of Figures1.1 A diagram showing malaria endemi area in Afri a. . . . . . . . . . . . . . 22.1 A diagram showing ampli�ed relationship between HIV and malaria . . . . 92.2 Malaria life y les, opied from Parasite image library [10℄ . . . . . . . . . 103.1 Representation of F and V . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1 A diagrammati representation of within host malaria model. . . . . . . . . 254.2 A diagram showing sensitivity of various parameters on the reprodu tionnumber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Shows a diagram of parasite-free equilibrium with R02 = 0.8327. . . . . . . 424.4 A diagram of parasite-present equilibrium with R02 = 1.3165. . . . . . . . . 434.5 A diagram showing population of intra ellular parasites for n1 < 16 theparasite-present equilibrium ases and n1 ≥ 16 the parasite-free equilibrium ases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 A diagram showing population of extra ellular parasites for n1 < 16 theparasite-present equilibrium ases and n1 ≥ 16 the parasite-free equilibrium ases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7 Represents relative impa t of the two parasite produ tion me hanisms 10∗P1and (P2) for n1 = 12 and R02 = 1.1277. . . . . . . . . . . . . . . . . . . . . 46x


List of �gures xi4.8 A diagram showing the evolution of RBCs with time. . . . . . . . . . . . . 474.9 A diagram showing the evolution of a tive infe ted RBCs with time. . . . . 474.10 Contour plots for n1 = 12 and n1 = 15 and n1 = 16. . . . . . . . . . . . . 484.11 Shows the population of a tively infe ted RBCs at for n1 = 24. . . . . . . . 484.12 Shows the population of latently infe ted RBCs at for n1 = 24. . . . . . . . 494.13 Diagram showing the population of a tively infe ted RBCs population fordi�erent values of m and ktp. . . . . . . . . . . . . . . . . . . . . . . . . . . 494.14 Diagram showing the population of sus eptible RBCs population for di�er-ent values of m and ktp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.15 A diagram shows malaria without treatment. . . . . . . . . . . . . . . . . . 524.16 Diagram showing one type of drug in treatment of malaria after 32 days andǫ1 = 0 =⇒ R04 = 1.3723, ǫ1 = 0.4 =⇒ R04 = 1.2114, ǫ1 = 0.6 =⇒ R04 =

1.0932, ǫ1 = 0.95 =⇒ R04 = 0.5074. . . . . . . . . . . . . . . . . . . . . . . 535.1 Diagram shows sensitivity analysis of R07. . . . . . . . . . . . . . . . . . . 725.2 Diagram shows the parasite-free equilibrium (DFE) at liver stage with R07 =

0.0063. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Diagram shows the parasite-free equilibrium (DFE) at blood stage R07 =

0.0063. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4 Diagram shows the parasite-free equilibrium (DFE) at mosquito stage R07 =

0.0063. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5 Diagram shows the parasite-present equilibrium point (EEP) at liver stageR07 = 4.7265. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.6 Diagram shows the parasite-present equilibrium point (EEP) at blood stageR07 = 4.7265. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


List of �gures xii5.7 Diagram shows the parasite-present equilibrium point (EEP) at mosquitostage R07 = 4.7265. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.8 Shows the ontour plot of R07 as a fun tion of an average number of s hizontsrelease from an infe ted liver ells that die naturally (n1) and the rate ofloss of s hizonts inside liver ells that are killed by e�e tor ells (ktp). . . . 795.9 Shows the ontour plot of R07 as a fun tion of the natural death of aninfe ted liver ells (µil) and natural death of sus eptible midgut ells (µmc). 805.10 Shows the ontour plot of R07 as a fun tion of the growth rate due to infe -tion of RBCs (kr) the rate of killing of merozoites by e�e tor ells(k7). . . . 815.11 Diagram shows an appli ation of treatment after 30 days at liver stage andǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒ R06 =

2.9200, ǫ1 = 0.99 =⇒ R06 = 0.7491. . . . . . . . . . . . . . . . . . . . . . . 825.12 Diagram shows an appli ation of treatment after 30 days at blood stage andǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒ R06 =

2.9200, ǫ1 = 0.99 =⇒ R06 = 0.7491. . . . . . . . . . . . . . . . . . . . . . . 835.13 Diagram shows an appli ation of treatment after 30 days at mosquito stageand ǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒

R06 = 2.9200, ǫ1 = 0.99 =⇒ R06 = 0.7491. . . . . . . . . . . . . . . . . . . 84


List of Tables1.1 The table that shows the drug resistan e for anti-malarial drug . . . . . . . 52.1 The table shows plasmodium spe ies and hara teristi s. . . . . . . . . . . 103.1 The Routh-Hurwitz table showing the hara teristi equation . . . . . . . 184.1 The table with the variables, des riptions and units. . . . . . . . . . . . . 254.2 The table that shows parameters and their des riptions. . . . . . . . . . . 264.3 The table that shows the parameter values of the model. . . . . . . . . . . 374.4 The table that shows the parameter values of the model. . . . . . . . . . . 405.1 The table des ribing the variables and units of variables . . . . . . . . . . 585.2 The table des ribing the parameters and units of parameters . . . . . . . . 595.3 The table des ribing the parameters and units of parameters . . . . . . . . 605.4 The table with the parameters, values and sour e . . . . . . . . . . . . . . 705.5 The table with the parameters, values and sour e . . . . . . . . . . . . . . 71A.1 The table that shows the initial variables that used in FIG. (4.3,4.4) . . . 88A.2 The table with parameters values used in FIG. (4.3) . . . . . . . . . . . . . 88xiii


List of �gures xivA.3 The table with parameters values used in FIG. (4.4) . . . . . . . . . . . . . 89A.4 The table that shows the initial variables that used in FIG. (4.5,4.6) . . . 89A.5 The table with parameters values used in FIG. (4.5,4.6) . . . . . . . . . . . 89A.6 The table that shows the initial variables that used in FIG. (4.7) . . . . . . 89A.7 The table with parameters values used in FIG. (4.7) . . . . . . . . . . . . . 89A.8 The table that shows the initial variables that used in FIG. (4.8,4.9, 4.10) . 89A.9 The table with parameters values used in FIG. (4.8,4.9,4.10) . . . . . . . . 90A.10 The table that shows the initial variables that used in FIG.(4.11,4.12, 4.13,4.14) 90A.11 The table with parameters values used in FIG. (4.11,4.12, 4.13,4.14) . . . 90A.12 The table that shows the initial values used in FIG. (4.15,4.16) . . . . . . . 90A.13 The table with parameters and values used in FIG. (4.15,4.16) . . . . . . . 90A.14 The table that shows the initial values used in FIG. (5.2,5.3,5.4) . . . . . 90A.15 The table with parameters and values used in FIG. (5.2,5.3,5.4) . . . . . . 91A.16 The table that shows the initial values used in FIG. (5.5,5.6,5.7) . . . . . 91A.17 The table with parameters and values used in FIG.(5.8,5.9, 5.10,5.5,5.6,5.7,5.11,5.12, 5.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


Chapter 1Introdu tion1.1 Ba kgroundMalaria is a mosquito borne infe tious disease aused by protozoan of genus plasmod-ium [73℄. The four spe ies that an infe t humans are; Plasmodium fal iparum, whi h auses severe disease and possibly death [41, 92℄ if not diagnosed and treated promptly,the other three plasmodium vivax, plasmodium ovale and plasmodium malariae gener-ally ause milder disease that is rarely fatal. [50℄. The parasite was dis overed in 1880 byCharles Laveran [45℄, who was working in the military hospital in Constantine, Algeria. Heobserved the parasites in a blood smear taken from a patient who had just died of malaria.But Laveran linked the ause of malaria with the monkeys [8℄. In 1902 Sir Ronald Ross dis- overed the malaria parasite in the gastrointestinal tra t of the anopheles mosquito. Thisled to the realization that malaria was transmitted by the Anopheles mosquito [98℄. Thisstudy [98℄ laid the foundation for ombating the disease su h as the use of the pesti ideDi hloro-Diphenyl -Tri hloroethane (DDT) for the ontrol of mosquitoes during world warII [63℄ and the treatment drug hloroquine in 1950.

1


Chapter 1. Introdu tion 2

FIG. 1.1. A diagram showing malaria endemi area in Afri a.Malaria remains a burden in terms of morbidity and mortality for developing ountries(FIG. 1.1) with tropi al and subtropi al limates. Malaria is prevalent in these regionsbe ause of heavy rainfall, warm onsistent temperatures and high humidity onditions thatare ondu ive to the development of the larvae. Temperature determines ve tor survival,in ubation period and transmission while heavy rainfall auses the stagnant waters in whi htheir larvae mature and provide mosquito with the environment needed for ontinuous


Chapter 1. Introdu tion 3breeding [23, 77℄.It is estimated that half of the world's population, over 3.3 billion lives in malaria endemi areas. There are about 300 to 500 million ases of lini al malaria reported [24, 67, 80, 96℄,resulting in 1.5 to 2.7 million deaths annually. sub-Sahara Afri a is the region with thehighest infe tion rate [96℄. In this region alone, the disease kills at least one million peopleea h year and is responsible for as many as half of the deaths in Afri an hildren underthe age of 5 [96℄, and a ounts for 20% of all hildhood deaths globally. Malaria de reasesthe gross domesti produ t by as mu h as 1.3% in ountries with high disease rate [70, 96℄.This ontributes to poverty and underdevelopment in most sub-Sahara ountries.The Malaria parasites are introdu ed into the human blood stream after a bite by a femaleanopheles mosquito. Parasites in the form of sporozoites, enter the liver where they divideseveral times before maturing into s hizonts whi h rupture and release merozoites whi h ompleting the initial parasite repli ation in the liver (exo-erythro yti s hizogony). Dur-ing this initial stage, two spe ies, namely, plasmodium vivax and plasmodium ovale anremain dormant (hypnozoites) in the liver and ause repla e by invading the blood streamweeks, or even years later. After this initial repli ation in the liver, the merozoites enterthe blood stage where they undergo asexual multipli ation in the erythro ytes (erythro- yti s hizogony). Mature merozoites in the blood stream are apable of invading the redblood ell. At this point the symptoms of disease will start to manifest, in the form offever, heada he, vomiting, hills, weakness and sweating. These symptoms are intermittentdepending on the immunity of the host. Some merozoites di�erentiate and develop intosexual forms of the parasite, alled male and female gameto ytes, that ir ulate in thebloodstream [2, 11, 41, 45, 101℄.When a mosquito bites an infe ted human, it ingests the gameto ytes whi h initiate par-asite multipli ation repli ation in the mosquito known as the sporogoni y le. In themosquito's stoma h, male and female gametes fuse to form diploid zygotes whi h developinto ookinetes that invade the mosquito midgut wall and form oo ysts. The oo ysts grow,rupture, and release thousands of sporozoites. Sporozoites invade the mosquito salivaryglands from where they an be inje ted into human hosts to ontinue the infe tion pro esswhen the mosquito takes a blood meal [45, 50℄. Malaria an be prevented [41, 46, 92℄mainly through awareness of risk; People at risk in lude those who have no or those with


Chapter 1. Introdu tion 4low immunity to malaria like young hildren, pregnant woman and visitors that travel totropi al areas from malaria free areas. Malaria an be prevented by preventing bites frominfe ted mosquitoes. This an be a hieved by wearing long lothes that over as mu h of theskin as possible, treating exposed parts of the body with inse t repellent, using inse ti ide-impregnated bed nets while sleeping and spraying indoors with inse ti ide around sleepingareas. People visiting malaria endemi areas are advised to take anti-malarial drugs be-fore entering these area. It is advisable for all residents of malaria endemi areas to bediagnosed for malaria routinely and be treated if they test positive.Despite the intense resear h and number of lini al trial, urrently there is no e�e tiveva ine [23, 64℄ and there has been very little su ess in produ ing su h va ines [64℄.A urate diagnosis of malaria is an integral part of treatment of malaria patients and pre-vention of further spread of malaria in the ommunity. Treating malaria depends on manyfa tors in luding disease severity, the spe ies of malaria parasite ausing the infe tion andthe part of the world in whi h the infe tion was a quired. The use of a simple, inexpen-sive and rapid diagnosis tests for malaria may be of in reasing importan e as ountriesin Afri a shift from low- ost anti-malarial to more expensive drugs Artemisin-based Com-bination Therapy (ACT). ACT is the most e�e tive strategy for plasmodium fal iparuminfe tion re ommended by WHO in the fa e of wide spread drug resistan e [63, 96℄.Parasites have be ome resistan e due to usual anti-malarial drug like hloroquine whi hwas the drug of hoi e to treat malaria for de ades following World War II; it was stoppedafter parasites be ame resistant to it. Also the mosquito has be ame resistant to mostinse ti ide [6, 54℄, DDT whi h was a very e�e tive ve tor ontrol pesti ide was stoppedbe ause the mosquito be ame resistant to it [63℄. Malaria an be managed with properdiagnosis and prompt treatment. Early diagnosis and prompt treatment are the prin iplete hni al omponents of the global strategy to ontrol malaria and is highly dependenton the e� a y, safety, availability, a�ordability and a eptability of anti-malarial drugs[91℄. An e�e tive anti-malarial drug not only redu es mortality and morbidity of malariabut also redu es the risk of drug resistan e. Due to drug resistan e of parasites towards,available anti-malarial drugs for various regions are given below;


Chapter 1. Introdu tion 5TABLE. 1.1. The table that shows the drug resistan e for anti-malarial drugAnti-malarial drug Pla es that shows drug resistan eChloroquine Plasmodium fal iparum spe ies all areasof the world ex ept the following:North Afri a; the Middle East(though ases have been reportedin Oman, Yemenand Iran);Haiti; Domini anRepubli ; rural areas of Mexi o;and Central Ameri a,north and westof the Panama anal [86℄.Fansidar South East Asia;the Indiansub- ontinent; the Amazon basin;many ountries in Afri a southof the Sahara; and O eania [90℄.Me�oquine South East Asia espe ially in Thailand;parts of Afri a and South Ameri a;the Middle East; and O eania [90℄.Quinine South East Asia;parts of Afri a;Brazil; and O eania [90℄.Halofantrine Thailand and shows ross resistan ewith me�oquine, fansidar andsulfadoxine-pyrimethamine [90℄.1.2 Statement of the problemIn-host mathemati al models are important and ne essary to enhan e our understandingof the dynami s of the Malaria parasites [58℄. Su h models an also be used to give aninsight into the e�e tiveness of treatment drug and other intervention strategies. In thisstudy, we investigate the dynami s of the malaria parasite during the red blood y le, thenextend the model to look at all the stages of the malaria life y le, namely; liver stage, redblood ell stage and mosquito stage. These models, whi h also in lude the e�e tor ellswill be used to address the following questions:(i) What repli ative hara teristi s o�er the parasite opportunities to evade the hostimmune system?(ii) Can we �nd the riti al e�e tor ell killing rates whi h must be maintained (or ex- eeded) to ensure that the parasite does not establish itself within the host?


Chapter 1. Introdu tion 6(iii) What are the e�e t of therapy on the prognosis of the malaria disease?(iv) What are these models' ontribution to the publi health?1.3 Obje tives of the studyMalaria remains one of the world's worst problem. It is shown that more people are lini ally ill with malaria than any other disease [96℄. A ording to WHO estimates, in2008 alone, there o urred 190− 311 million lini al ases of malaria [11, 96℄. Hopes thatmalaria might be eradi ated have proved impossible to realize. In many tropi al areas,the threat of epidemi malaria is in reasing and the ontrol measures are be oming lesse�e tive. For this malaria asso iated burden to be redu ed, we ondu t a study whoseobje tives are summarized as follows;� To understand biologi al pro esses that enable the parasites to evade the immuneresponse.� To gaining insights into the dynami s between the malaria parasites and the immunesystem� To investigate the e�e t of anti-malarial therapy on the prognosis of the disease.� To ontribution poli y re ommendations on management of malaria.1.4 Organization of the workThis work is organized as follows: Chapter 2 reviews various studies about malaria. Chap-ter 3 provides mathemati al and numeri al tools ne essary for this study. Chapter 4presents the model with the following lasses; sus eptible RBCs, latent RBCs, a tive in-fe ted RBCs, intra ellular parasites, extra ellular parasites and e�e tor ells. Also presentsthe e�e ts of treatment. The model is analysed mathemati ally and numeri ally, and theresults are dis ussed. Chapter 5 presents the treatment model with three stages of themalaria life y le, namely; (i) the liver stage whi h omprises of; sus eptible liver ells


Chapter 1. Introdu tion 7(LVCs), infe ted liver ells (LVCs), sporozoites and s hizonts inside the liver ells (LVCs),(ii) the blood stage whi h onsist of; sus eptible red blood ells (RBCs), infe ted RBCs,merozoites, trophozoites, and s hizonts inside the red blood ells (RBCs), (iii) the mosquitostage whi h has; sus eptible midgut ells (MGCs), infe ted midgut ells (MGCs), gameto- ytes. The model also has an e�e tor ell lass. The mathemati al and numeri al simula-tions are done, the results are also dis ussed. Chapter 6 on ludes what has been dis ussedin the results and suggests limitations, re ommendations, and possibilities of future work.This is followed by the appendix.


Chapter 2Literature reviewIt is over one hundred years sin e malaria was re ognized as a disease in humans. Initially,malaria was known as mal′aria implying that the disease was aused by bad or spoiledair. The fever lini al symptoms were related to swaps and low lying water. In 1880, Lav-eran dis overed parasites inside red blood ells of a si k person and mistakenly related thedisease to monkeys following his earlier study whi h identi�ed the same parasites inside amonkey's red blood ells [8℄.The study by Ross (see [8℄ and the referen es therein) on malaria has be ome the basis ofepidemiologi al studies of malaria in luding ve tor spread, treatment et . and these haveled to immunologi al studies of the disease [8, 41℄. To demonstrate the seriousness of themalaria epidermi , we have analysed available re ords for the trend. Re ords regardingmalaria lini al ases and mortality before the year 1950 are not available. The treatmentdrug hloroquine was dis overed in 1950 and at almost the same time the pesti ide DDTwas dis overed. From available re ords depi ted by FIG. 2.1, it is evident that malaria lini al ases were in de line between 1950 and 1975 due to e�e tive ve tor ontrol pro-grams and the e�e tiveness of the treatment drug hloroquine [66, 76℄. Over a long periodof administration of hloroquine, the parasites developed resistan e to the drug [6, 54℄ and urrently, hloroquine is not re ommended for treatment of malaria in sub-Sahara Afri a[66℄. The trend for malaria lini al ases has been on the in rease sin e 1975 (FIG. 2.1).There appears to be a link between malaria and HIV/AIDS (FIG. 2.1). During the late1980's, HIV/AIDS be ame an epidemi in most sub-Sahara Afri a. Coin identally, when8


Chapter 2. Literature review 9

FIG. 2.1. A diagram showing ampli�ed relationship between HIV and malariaHIV/AIDS exploded malaria lini al ases also started to in rease rapidly.We want to exer ise aution about the link mentioned above and to note that the rapidin rease in malaria lini al ases during the late 1980's ould also be related to other fa -tors su h as the ollapse of the ve tor ontrol programs following the deterioration in moste onomies in sub-Sahara Afri a, but it may also be due to weakened immune responses inpatients o-infe ted with HIV and malaria.There are four spe ies of plasmodium that are known to infe t humans namely, Plasmodiumfal iparum, Plasmodium malariae, Plasmodium vivax, and Plasmodium ovale. The dis-tribution of the various malaria parasites are indi ated in the [TABLE 2.1℄. Plasmodiumfal iparum, whi h auses more deaths in humans, is found mainly in tropi al and sub-tropi al areas of the world whi h in lude sub-Sahara Afri a and most of the poor regionsof Asia. These parasites develop through a y le depi ted in (FIG.2.2) and dis ussed in de-tail in [26, 50℄. Clearly from this �gure, the malaria life y le is very omplex and involvesstages within the mosquito and the human host. If malaria is to be ontrolled and/or


Chapter 2. Literature review 10TABLE. 2.1. The table shows plasmodium spe ies and hara teristi s.Spe ies Global distribution Re ur Type of RBCs Infe tionP. fal iparum Tropi al and sub-tropi al worldwide Re rudes en e All Severe anaemiaP. malariae Worldwide Re rudes en e Older RBCs Milder diseaseP. ovale Afri a Relapse Young RBCs Normal infe tionP. vivax Asia, Latin Ameri asome part of Afri a Relapse Young RBCs Normal infe tion

FIG. 2.2. Malaria life y les, opied from Parasite image library [10℄.


Chapter 2. Literature review 11eradi ated, it is important to understand the fa tors that in�uen e malaria pathogenesis[1, 31℄. It is evident from (FIG.2.2) that the important questions for the mathemati alstudy of malaria are both epidemiologi al and immunologi al and span several levels ofbiologi al organization.Knowing the fa tors involved in pathogenesis, however, is only the �rst step towards quan-tifying how ea h fa tor in�uen es the result [51℄. As a sequel to the study by Andersonet al. [1℄, several studies Hetzel and Anderson [31℄, Antia et al. [2℄, et ., have studiedthe within-host ellular dynami s of blood stage malaria. Hetzel and Anderson [31℄ inves-tigated the properties of a mathemati al model of blood stage infe tion of malaria. Theanalysis [31℄, done in the absen e of the host immune response to demonstrate the rela-tionship between host and parasites parameters, led to the determination of parametersne essary for the su essful invasion and persisten e of the parasites. The parameters inHetzel and Anderson [31℄ are used in this study as a �rst approximation in our model tounderstand the role of treatment and innate immune response to malaria pathogenesis.A more re ent study by Antia et al. [2℄ has onsidered a ute malaria infe tions with aview to determine the dynami s of parasite and anaemia during a ute primary malaria in-fe tions and why some strains of malaria rea h higher densities and ause greater anaemiathan others. While most studies agree that spe i� immunity does not play a major role inthe initial dynami s of pathogenesis, there is onsiderable ontroversy over whi h fa torsdrive the dynami s shortly after infe tion [1, 30, 31, 55℄.Despite these numerous studies on malaria pathogenesis, the relative signi� an e of dif-ferent fa tors in�uen ing malaria pathogenesis ited in [1, 2, 31℄, the development of thedisease is far from lear [60℄. So far, many studies fo us on the innate immune response[31, 34℄ to understand the RBC-parasites intera tion and a few on adaptive immune re-sponse. However the fa t that resear hers have su essfully ategorized the immune re-sponses does not imply understanding of these pro esses. For example, Stevenson and Riley[82℄ have dis ussed how several adaptive immune response omponents like ma rophagesand natural killer ells are involved in the innate immune response and on luded that theintera tion of those ells remains spe ulative and on�i ting experimental data has openeda way for mathemati al models of pathogenesis to be used as tools to a hieve deeper un-derstanding of this pro ess [31℄.


Chapter 2. Literature review 12Several studies on the innate immune response to malaria infe tion have been formulated[13, 65, 84, 89℄. A study by Su et. al [84℄ on the syn hronization of parasite repli ation indi�erent red blood ells, onsiders an age-stru tural human malaria infe tion of red blood ells. The numeri al simulation results in this study show that syn hronization with regu-lar periodi os illation o urs when the repli ation rates in rease. A more re ent study byNiger and Gumel [65℄ has investigated the innate immune response to malaria infe tion andthe e�e t of imperfe t va ines. The simulation results [65℄ show that a va ine e� a y ofat least 87% is ne essary to eliminate Infe ted Red Blood Cells (IRBCs )in vivo.The study by M Queen and M Kenzie [55℄ onsiders the sus eptibility of red blood ellsand the dynami s of malaria infe tion. The authors [55℄ assume a predator-prey type re-lationship between a population of repli ating parasites and a replenishing population ofred blood ells. The study explores the hypothesis that some malaria-parasite spe ies thatinfe t humans su h as Plasmodium malariae and Plasmodium vivax have preferen e forparti ular age lasses of red blood ells. Our study onsiders the infe tion of red blood ellsby Plasmodium fal iparum whi h has di�erent infe tion hara teristi s [34, 55℄. We wantto investigate whether Plasmodium fal iparum too has a tenden y for age sele tion likethe other malaria spe ies, noting that the di�eren e in infe tion hara teristi s betweenPlasmodium fal iparum and the spe ies onsidered by M Queen and M Kenzie [55℄ o ursprimarily in the range of parameters used [34℄.A review by Engwerda and Good [23℄ onsidered the intera tion between malaria parasitesand the host immune system and revealed the potential for designing and implementingnew va ine and drug programs through understanding of ell-immune, ell-parasite inter-a tion. This study [23℄ in luded the adaptive response whi h is not part of this study butprovides insight into the ell-parasite dynami s whi h has guided our study.A review by Mideo et al. [58℄ re ommended the use of mathemati al models as a tool forre�ning knowledge of within-host pro esses and has suggested why under ertain ir um-stan es mathemati al models may be better than experimentation. Generally, however,Mideo et al. [58℄ re ommends the use of both approa hes sin e together these approa hespossess the potential for informing the design of intervention and health poli y for address-ing lingering questions about the basi biology of malaria whi h should guide the modelformulation. The malaria parasite repli ation is a very omplex pro ess and involves three


Chapter 2. Literature review 13stages as shown in (FIG. 2.2). This ompli ated repli ation y le has impli ations regard-ing eradi ation of malaria whi h is too expensive and probably unrealisti for poor resour esub-Sahara Afri an ountries. A treatment or va ination strategy suggested by mathe-mati al models may ontribute to utting ost of intervention programs.This study will attempt to quantitatively predi t the pattern of pathogenesis as a fun tionsome underlying within-host regulatory fa ts. However, as the biology of malaria pathogen-esis is very omplex and involves many fa tors, this study will in lude only those biologi alpro ess that we believe an plausibly explain the dynami of malaria pathogenesis.M Kenzie and Bossert [34, 53, 55℄ modelled malaria pathogenesis using a system of ou-pled di�erential equations involving only uninfe ted RBCs, infe ted RBCs and merozoites.These studies omitted the detailed biology su h as how the merozoites are repli ated asex-ually and on entrated on the basi infe tion dynami s that lead to lini al malaria. Theaim in those studies was to understand pathogenesis and not how the immune systemresponds to infe tion. However, the study by Hoshen et al. [34℄ gives useful insights and on lusions that have guided our study. The question one ask is whether a simpli�ed modelsu h as Hoshen et al. [34℄ an yield reliable information to guide planning and poli y. Fur-thermore, one wonders whether these simple models an reliably estimate the severity ofthe disease?Some larger models [61℄ have in luded more biologi al pro ess su h as innate and adaptiveimmune responses. However, su h studies have la ked lini ally determined parameter val-ues to alibrate and validate their models. The obje tive behind su h large models is todetermine parasite repli ation me hanisms but in the absen e of biologi ally determinedlaws it has proved di� ult to justify results from su h models. It is known that whenmerozoites infe t red blood ells, to initiate the asexual repli ation of the merozoite pop-ulation [55, 61℄, not all infe ted red blood ells ontribute to the population of merozoitessin e some of them are apoptosed by the natural killer ells. Adjustment to re�e t this fa thas been done in an ad ho manner [40℄. The di�eren es in a ounting for the repli ationlaws for malaria, TB and HIV [25, 55℄ are indi ative of the absen e of lear biologi alunderstanding of the pro esses. Some authors have assumed very simple laws [34, 55℄ tokeep the models parsimonious. The la k of biologi al fa ts on this subje t has a�e ted our


Chapter 2. Literature review 14de�nition of the parasite repli ation law whi h we admit has no biologi al basis.These omplex mathemati al models on malaria pathogenesis have onsidered the develop-ment of erebral malaria in hildren and adult travelers living in non endemi malaria areasand have on luded that severe malaria is an immune-mediated disease [4℄. This study [4℄ onsidered the role of innate and adaptive immune responses in terms of (i) prote tionfrom lini al malaria and (ii) their potential role in immunopathology and the subsequen edevelopment of lini al immunity. Another study [3℄ has determined the potential on-tribution of innate immune responses to the early pro-in�ammatory ytokine response toPlasmodium fal iparum malaria. The study examined the kineti s and ellular sour es ofinterferon-gamma produ tion in response to infe tion of red blood ells. The study on- ludes that early interferon-gamma response ould redu e red blood ells infe tion.There is onsiderable ontroversy over whi h fa tors drive the malaria pathogenesis shortlyafter infe tion [1, 30, 31, 55℄. Some of the assumptions made to explain the di�eren esin the initial dynami s of malaria strains in lude virulen e evolution [2, 14, 27, 49℄, (ii)red blood ell age spe i� infe tion strategy [55℄ and (iii) innate or early spe i� immuneresponses to regulate the initial dynami s of infe tion and anaemia [21, 29℄. Antia et al.[2℄, however, have given two reasons why it is di� ult to as ertain the ontribution of thesefa tors to the dynami s of a ute infe tion namely (i) limited data on the dynami s of theparasite and loss of RBCs following infe tion of humans with human malaria parasites and(ii) the dynami s of the infe tion ould involve many intera ting populations.Several studies have investigated the potential for a va ine as the best strategy to ombatmalaria targeting the liver stage for a va ine [23, 33, 59℄. A ording to [23℄, this stageposes many obsta les to anti-infe tion va ines and drugs. These in lude: (i) the liverstage malaria parasites have distin t metabolism whi h helps them to evade anti-malarialdrugs (ii) Plasmodium malaria parasites an lie dormant in the liver and relapse to bloodinfe tion after months or even years. A ording to Morrow and Moss [63℄, liver stage para-sites annot be targeted by any li ensed drug ex ept primaquine whi h is fatal to pregnantwomen and diabeti individuals.Our hypothesis in this study follows from the observation made by M Queen and M Kenzie[55℄. We ask the question whether Plasmodium fal iparum targets older red blood ells asa strategy for a elerating parasite repli ation. Although we have not developed an age-


Chapter 2. Literature review 15stru tured model, we have manipulated the natural death term in the infe ted red blood ell population to a hieve our goal. It is suggested in [39, 104℄ that CD8 ells may notfun tion optimally in individuals su�ering from hroni illnesses, we have investigate thiss enario and a ordingly have made a re ommendation regarding the treatment of malariafor individuals su�ering from hroni infe tions.Despite a large area in resear h on malaria pathogenesis (within-host me hanisms throughwhi h plasmodium parasite auses disease) [5, 23, 55, 58℄ many questions remain unan-swered. Issues in pathogenesis need to be explored to develop better treatment [58, 60℄. Itis known that most of the drugs a t best against repli ating pathogens in ombination withe�e tive immunologi al responses [5℄. There is a need also to better understand e�e tor ell me hanism in the development of immunity to malaria [1℄.


Chapter 3Mathemati al tools3.1 The de�nition and omputation of R0De�nition 3.1.1 The basi reprodu tion number (R0), sometimes alled the basi repro-du tive rate or the basi reprodu tive ratio of an infe tion is the average number of se -ondary ases aused by an infe ted individual introdu ed in a ompletely sus eptible popu-lation [20℄.For the ase of a single infe ted ompartment, R0 is simply the produ t of the infe tionrate and the mean duration of the infe tion [93℄. But for models with several infe ted ompartments, this de�nition is not su� ient. That is a more sophisti ated te hnique isrequired as reviewed below. In this thesis, we only give a brief overview of the al ulationof the basi reprodu tion number R0, using the next generation method dis ussed byDiekmann [20℄ and van den Driess he et. al [93℄. Let x = (x1, . . . , xn)

t with xi ≥ 0, bethe number of individuals in ea h ompartment (i = 1, . . . , n). We sort the ompartmentsso that the �rst m ompartments orrespond to infe ted individuals and then, de�ne theparasite free equilibria as x0 = {x ≥ 0 | xi = 0, i = 1, . . . , m}.For the omputation of R0, it is important to distinguish new infe tions from all other hanges in the population. As illustrated in �gure (3.1), we let: Fi (x) be the rate ofappearan e of new infe tions in ompartment i, V+i be the rate of transfer of individualsinto ompartment i and V−

i (x) be the rate of transfer of individuals out of ompartmenti. 16


Chapter 3. A mathemati al tools 17PSfrag repla ementsFi (x)

V+i (x)

V−i (x)FIG. 3.1. Representation of F and V

Fi (x) ,V+i (x) , and V−

i (x) are di�erentiable ontinuous fun tions. In general, we anexpress a system of di�erential equations of the form x = fi (x) in the form that a ountsfor in�ow and out�ow (3.1) as:x (t) = Fi (x)− Vi (x) , andVi(x) = V−

i (x)− V+i (x), for i = 1, ..., n.Let the fun tions Fi (x) ,V

+i (x) , and V−

i (x) satisfy the following onditions.(A1) : If x ≥ 0, then Fi,V+i ,V

−i ≥ 0.(A2) : If xi = 0, then V−

i = 0. In parti ular, if x ∈ x0, then V−i = 0, for i = 1, ...., m.(A3) : If i > m,Fi = 0.(A4) : If x ∈ x0, then Fi (x) = 0, and V+

i (x) = 0.(A5) : If f (x) = 0 , then the eigenvalues of Df (x0) have negative real parts and x0 is theparasite free equilibrium.The onditions listed above allow us to partition the matrix DF(x) as shown in Remark1.Remark 1 DF (x0) is the derivative [∂fi∂xj

] evaluated at DFE. The onditions (A1 − A5)allow us to partition the matrix DF(x0) and DV(x0) as;DF(x0) =

(F 00 0

) and DV(x0) =

(V 0J3 J4

)

,where F and V are m×m matri es de�ned as F =

[∂F〉

∂xj

(x0)

]

, and V =

[∂V〉

∂xj

(x0)

]

. Sin eF is non-negative, V is a non-singular matrix and all eigenvalues of J4 have positive realparts.


Chapter 3. A mathemati al tools 18If f(x) satis�es (A1 −A5), then the reprodu tive ratio is de�ned as R0 = η(FV−1

), whereη is the spe tral radius [93℄.When R0 < 1, the infe tion dies out, ex ept for a model whi h exhibits a ba kwardbifur ation. For R0 > 1, the infe tion spreads in a population. Thus R0 is a thresholdparameter su h that when R0 < 1, the population remains healthy be ause the diseasefails to establish itself. The stability of the disease free equilibrium point (maintaining the onditions for absen e of disease in a population) is dis ussed by Diekmann ([20℄), and vanden Driess he [93℄. The stability of parasite free equilibrium point is stated in theorem3.1.2.Theorem 3.1.2 The parasite free equilibrium point is lo ally asymptoti ally stable forR0 < 1 and unstable for R0 > 1.The proof of this theorem is given in several studies [16, 43, 44℄.3.2 The Routh-Hurwitz riterionLet

anSn + an−1S

n−1 + · · ·+ a1S0 + a0 = 0, (3.1)be a hara teristi equation of a given Ja obian matrix. The Routh-Hurwitz table [36, 62℄for the hara teristi equation (3.1) of degree n an be determined as illustrated in TABLE3.1. TABLE. 3.1. The Routh-Hurwitz table showing the hara teristi equation

Sn an an−2 an−4 an−6

Sn−1 an−1 an−3 an−5 an−7

Sn−2 b1 =an−1an−2 − anan−3

an−1b2 =

an−1an−4 − anan−5

an−1b3 =

an−1an−6 − anan−7

an−1

Sn−3 b1an−3 − b2an−1

b1

b1an−5 − b3an−1

b1... ... ... ... ...S0


Chapter 3. A mathemati al tools 19This table an be used as follows:� If there are sign hanges in the �rst olumn, then the eigenvalues have positive realparts.� The number of sign hanges in the �rst olumn is equal to the number of positivereal roots of the hara teristi equation.� If there exists a mixture of positive and negative signs, then the given system isunstable.� If there are no sign hanges in the �rst olumn, then all eigenvalues are either positiveor negative. If there are negative signs only, then the eigenvalues have negative realparts and the system is stable.3.3 Sensitivity analysisThe parameter values and assumptions of any model are subje t to hanges and errors.Sensitivity analysis is a te hnique for establishing the signi� an e of a parameter and howit impa ts the dynami s of the model. An independent variable will impa t a parti ulardependent variable if the variable is a di�erentiable fun tion of that parameter [12, 74℄.Sensitivity analysis is a very useful tool for hara terizing the un ertainty asso iated witha parameter with regard to model on lusions. Its importan e is part and par el of goodmodelling pra ti e and requires a modeller to provide an evaluation of on�den e in themodel results. Furthermore, it validates the relevan e of the inputs by determining theoutput of the model [74, 99℄.Un ertainty analysis may be used to asses the variability in the out ome variable thatis due to the un ertainty in estimating the input values. Sensitivity analysis an extendun ertainty analysis by identifying important parameters that yield reliable predi tions [7℄.An alternative sensitivity analysis design, for a K parameter model, is to �x the values


Chapter 3. A mathemati al tools 20of K − 1 parameters and then vary only the value of the Kth parameter over a spe i�edrange. This sensitivity analysis design has the advantage that it is simple and qui k, butsu�ers from major disadvantages. That is only one parameter may be varied at a time,only a small region of a K−dimension parameter spa e an be explored and values of K−1parameters have to be estimated [7℄.Latin Hyper ube Sampling (LHS) is the type of strati�ed Monte Carlo sampling and maybe viewed as an extension of the Latin Square sampling. In LHS, the un ertainty estimationfor ea h input parameter is modelled by treating ea h input parameter as a random variable.It is an extremely e� ient sampling design be ause is used only on e in the analysis. Aninput ve tor is generated for ea h omputer simulation of the deterministi model and themodel is then run N times [7, 74℄.A distribution fun tion for ea h of the out ome variable an be dire tly derived be auseof the probability sele tion te hnique. LHS enables the results of a deterministi model tobe interpreted within a statisti al framework. The distribution may be hara terized bysimple des riptive statisti s. Sensitivity analysis may be then be performed by al ulatingthe partial rank orrelation oe� ients (PRCC) for ea h input parameter and ea h out omevariable [7, 75℄.The LHS/PRCC te hnique involves seven steps [7℄:(a) De�ning the probability distribution fun tion for parameters and state variables. Amathemati al model ontains a ertain number of parameters and state variables, theestimated values for all or only a subset of these will be un ertain.(b) Cal ulating the number of simulations (N). The LHS design involves sampling withoutrepla ement. Therefore if only k draws are made (where k equals the number ofun ertain variables), the kth draw would be predetermined. Hen e the lower limit tothe value of N (where N equals the number of simulations) should be at least (k + 1).( ) Dividing the range of ea h of the K parameters into N equal probable intervals. Therange of ea h parameter is divided into N non-overlaping equiprobable intervals (whereN is the number of simulations) and ea h interval is sequentially assigned a samplingindex.


Chapter 3. A mathemati al tools 21(d) Creating the LHS table. The LHS design involves random sampling without repla e-ment and every equiprobable interval of ea h input variable is sampled on e.(e) Sampling the values of the input parameters and performing the N simulations. TheLHS table is used to generate for example 100 by 30 input matrix.(f) Analysing model out omes of un ertainty analysis. The results of the simulation runsof the model onsist of N observations of ea h out ome variable. Distribution fun tionsof ea h out ome variable an be dire tly derived and hara terized by simple des riptivestatisti s.(g) Analysing the model out ome with respe t to the N observations of ea h out omevariable whi h may be used to assess the sensitivity of the out ome variables and toestimate the un ertainty of the input parameters.


Chapter 4A within host model of blood stagemalaria4.1 Introdu tionMalaria parasites are transmitted from a mosquito to a human host. Upon entering thehuman host, extra ellular malaria sporozoites must �rst take up residen e in the liverbefore initiating red blood ell infe tion. In the liver, the sporozoites undergo spe ta ularphenotypi hanges prior to multipli ation [37, 69℄. The sporozoites mature into s hizontswhi h rapture and release merozoites [11℄, whi h are fully ompetent to infe t red blood ells and instigate the pathology asso iated with malaria [83℄.Several studies on the innate immune response to malaria infe tion have been formulated[13, 65, 84, 89℄. A study by Su et. al [84℄ on the syn hronization of parasite repli ationin di�erent red blood ells, onsiders an age-stru tural human malaria infe tion of redblood ells. The numeri al simulation results in that study showed that syn hronizationwith regular periodi os illation o urs when the repli ation rates in rease. A more re entstudy by Niger and Gumel [65℄ has investigated the innate immune response to malariainfe tion and the e�e t of imperfe t va ines assuming a s enario of parasite life y les.The simulation results [65℄ showed that a va ine e� a y of at least 87% is ne essary toeliminate Infe ted Red Blood Cells (IRBCs )in vivo.As a sequel to these studies [5, 55, 65℄, we onsider two stages of the parasite y le namely (i)22


Chapter 4. A within host model of blood stage malaria 23stage 1: A short stage when merozoites are released from the liver to initiate the red blood ell infe tion. It is estimated that a primary infusion of 104 to 105 parasites are releasedinto the blood [55℄. (ii) stage 2: The red blood stage during whi h asexual multipli ationof merozoites and infe tion of red blood ells by the merozoites o urs on urrently. Ourmodel does not in lude the early stages of the exo-erythro yti y le whi h is known to bea reservoir for the erythro yti stage [59℄. Despite this, the results of this model an still ontribute towards understanding the repli ative dynami s of the parasite and hen e thedevelopment of lini al malaria.In-host mathemati al models are important and ne essary to enhan e our understandingof the dynami s of Malaria pathogenesis [58℄. Su h models an also be used to give insightinto the e�e tiveness of drug treatment and other intervention strategies [65℄. In thisChapter, we investigate the dynami s of the malaria parasite during the red blood y le.Our model in ludes the red blood ells, extra ellular parasites, intra ellular parasites ande�e tor ells. This model di�ers from the models in the earlier studies [5, 34, 55, 65℄ in thatwe introdu e a lass of intra ellular parasites in the pathogenesis pro ess. We believe it isimportant to in lude this pro ess in the dynami s so that intervention strategies an betargeted at di�erent stages of the repli ation pro ess as is the ase in HIV/AIDS treatment[38℄. Our study addresses the following questions: (i) what repli ative hara teristi s o�erthe parasite opportunities to evade the host immune system? (ii) A signi� ant numberof individuals in sub-Sahara Afri a are o-infe ted with viral (for example HIV, simianimmunode� ien y virus (SIV)), ba terial and parasiti infe tions other than malaria [38, 42,88, 104℄, it is important to investigate how su h individuals respond to a malaria infe tion.Studies by Kalia et el [39℄ and Zhang et el [104℄ have shown that in hosts su�ering from hroni infe tions, CD8 T- ell fun tions are ompromised and are dysfun tional . To giveinsight into what happens when CD8 T- ell responses have been altered and impaired, weshall onsider a hypotheti al situation where a host is infe ted with the malaria parasitewhi h would lear if CD8 T- ell responses were normal. We shall then alter the CD8 T- ell response parameters m and ktp at time t < tc, where, tc is the time of learan e ofthe malaria infe tion, and investigate the prognosis of the malaria disease by �nding the riti al e�e tor ell killing rate whi h must be maintained (or ex eeded) to ensure that theparasite does not establish itself within the host?Malaria an be managed with proper diagnosis and prompt treatment . Early diagnosis and


Chapter 4. A within host model of blood stage malaria 24prompt treatment are the prin iple te hni al omponents of the global strategy to ontrolmalaria. This strategy is highly dependent on the drug e� a y. E�e tive anti-malarialdrugs not only redu e mortality and morbidity of malaria but also redu e the risk of drugresistan e of the parasites toward available anti-malarial drugs [90℄.Our model in ludes the repli ation of the parasite within the infe ted red blood ells.While the pro ess of entry into the red blood ells by the merozoites has been extensivelystudied, the repli ation pro ess of the parasite within an infe ted red blood ell is not wellunderstood. In this study, we have assumed a repli ation law for intra ellular parasitesimilar to that of the tuber ulosis ba teria in ma rophages [25℄. This assumption maynot be an a urate representation of the malaria parasite repli ation law, however, wehave utilized the available data [18℄ to ensure that parasite repli ation in a red blood ell produ es between 8 to 32 merozoites. Using this information, we have determinedthe rate of bursting, kb, for infe ted red blood ells taking the ell's arrying apa ityto be 32. Furthermore, we have assumed that the release of the merozoites is throughbursting of the infe ted red blood ell as is the ase for ba teria in ma rophages [25℄. Wehave made this assumption be ause very little is known about the a tual parasite releaseme hanism involved. We believe, however, that this study will stimulate experimentalbiologists into investigating the reprodu tive law for intra ellular malaria parasites andthe release me hanism of parasites from infe ted red blood ells.We also extend the model to investigate the e� a y level required to lear the parasites.We introdu e treatment with a drug of onstant e� a y ǫ1 targeting the infe tion terms[71℄.4.2 Methodology4.2.1 Model formulationThe system of equations des ribing the dynami s of in-host malaria is given below: Themodel represents the human blood stage of the malaria disease alled the erythro yti y le.


Chapter 4. A within host model of blood stage malaria 25SUSCEPTIBLE LATENT ACTIVE

INTRACELLULAR EXTRACELLULAR

RBCs RBCs

PARASITES

EFFECTOR

RBCs

Interaction,

Surce ofRBCs

Growth of

Growth of Intracellular parasite

Movement

effector cells

Source of

effector cells

PARASITES

Source extracellular

parasites

PSfrag repla ements µrl

αkRPe

(1− α)kRPe

γRl

µraµr

k∗n∗RPe

n1µraPiktpNEPe

k∗n∗RPek11NRaG

µpe

kbRaG

mERa

FIG. 4.1. A diagrammati representation of within host malaria model.TABLE. 4.1. The table with the variables, des riptions and units.Variables Des riptions UnitsR Sus eptible red blood ell ell/mlRl Latent red blood ell ell/mlRa A tivated red blood ell ell/mlPi Intra ellular parasites ell/mlPe Extra ellular parasites ell/mlE E�e tor ell ell/mlRed blood ells (RBCs)

R = Sr − µrR− kRPe. (4.1)


Chapter 4. A within host model of blood stage malaria 26TABLE. 4.2. The table that shows parameters and their des riptions.Parameter Des riptions UnitsSr Constant sour es of red blood ell cell/ml.dayµr Natural death rates of sus eptible RBCs day−1

k Infe tion rate ml/cell.dayα Proportion of RBCs dimensionlessγ Rate of a tivation day−1

µrl Natural death rate of latent RBCs day−1

m Rate of killing a tivated RBCs by e�e tor ell ml/cell.daykb Rate of bursting day−1

µra Natural death rates of a tivated RBCs day−1

N Number of parasites that �ll the RBCs dimensionlesskpi Rate of growth of intra ellular parasites day−1

k11 Rate of loss due to burst of a tivated ell day−1

n1 A threshold number of intra ellular parasites releasedas a results of the natural death of an a tivated RBC dimensionlessktp Rate of loss of extra ellular parasites that are killedby e�e tor ells ml/cell.dayn∗k∗ Threshold number as a results of gain due to infe tionof sus eptible RBC by extra ellular parasites ml/cell.dayµpe Natural death rate of extra ellular parasites day−1

ωe Growth rate of e�e tor ells day−1

re Carrying apa ity of e�e tor ell cell/mlSpe Sour e of extra ellular parasites cell/ml.dayDuring the human blood stage, the sporozoites inje ted into the human host by the femaleanopheles mosquito enter the blood stream and infe t sus eptible red blood ells (RBCs).The dynami s of the sus eptible red blood ell (RBC) population are given in (4.1). Theterms in this equation have the following meaning: The �rst term represents a onstantnatural sour e for the red blood ell population. The se ond term represents natural deathof the sus eptible red blood ells at a onstant rate µr and the third term representsinfe tion of RBCs by extra ellular parasites (merozoites) at a onstant rate k. The newlyinfe ted red blood ells may be ome latently infe ted with the malaria parasite, a statewhi h inhibits parasite repli ation, or the RBCs may be ome a tively infe ted, meaningthat parasite repli ation persists in them. The rate of hange for the latently infe ted redblood ell population, Rl, is given by equation (4.2)

Rl = αkRPe − (γ + µrl)Rl. (4.2)


Chapter 4. A within host model of blood stage malaria 27The terms in this equation have the following meaning: The �rst term represents a propor-tion of infe ted RBCs that have be ome latently infe ted, and the se ond term representslosses due to a tivation at a onstant rate γ and due to natural death at a onstant rate µrl.The a tively infe ted red blood ell population evolves a ording to the following equation(4.3) below:Ra = (1− α)kRPe + γRl −mERa − kbRa

(Pi

2

Pi2 + (NRa)2

)

− µraRa. (4.3)The �rst term in equation (4.3) represents the proportion of sus eptible red blood ellsthat be ome a tively infe ted, the se ond term represents gain due to a tivation of latentlyinfe ted red blood ells, Rl, the third term represents the removal of a tivated infe tedRBCs due to killing by e�e tor ells. When the merozoites infe t red blood ells, theystart to repli ate within the infe ted red blood ells. This pro ess an go on until thenumber of parasites within the infe ted red blood ell rea hes 32 [55℄ ausing it to burst.The fourth term measures an e�e tive number of infe ted red blood ells that burst torelease intra ellular parasite. The bursting rate in this model is assumed to be dependenton the densities of intra ellular parasites and infe ted red blood ells [25, 28℄. This burstinglaw has been used for pathogens su h as TB [25℄. To the best of our knowledge this has notbeen used for malaria and is not supported by any literature. Not all infe ted red blood ells burst to release parasite. There is an e�e tive number of infe ted red blood ells thatburst to release parasite into the blood stream. The fa tor;P 2i

P 2i + (NRa)2

,measures the proportion of infe ted red blood ells that burst to release parasite. Wehave hosen this ratio so that the repli ation pro ess has an upper bound.This law has revealed orre t repli ative dynami s for the TB ba teria [25℄ andis adopted in this study. The �fth term a ounts for natural death of infe ted red blood ells at a onstant rate µra.ParasitesIntra ellular parasites repli ate inside an infe ted red blood ell. It is assumed in this studythat the intra ellular parasite population grows a ording to a law similar to the growth of


Chapter 4. A within host model of blood stage malaria 28TB ba teria in ma rophages [25℄. In the absen e of experimentally or lini ally determinedgrowth law for the intra ellular malaria parasite, we assume a growth law of intra ellularparasites inside an infe ted red blood ell of the form.Pi = kpiPi

(

1−Pi

2

Pi2 + (NRa)2

)

+ k∗n∗RPe − k11NRa

(Pi

2

Pi2 + (NRa)2

)

−n1µraPi. (4.4)In equation (4.4), the �rst term represents the growth of intra ellular parasites. The se ondterm represents gain of Pi due to infe tion of sus eptible RBCs by extra ellular parasites(merozoites), Pe, at a threshold n∗k∗, the third term represents an e�e tive number ofintra ellular parasites lost due to bursting of a tivated RBCs and the forth term representsloss of intra ellular parasites due to natural death of an infe ted red blood ell, Ra, wheren1 denotes a threshold number of intra ellular parasites released. Upon bursting of ana tively infe ted red blood ell, it releases the merozoites into the blood stream to ontinuethe parasite y le.

kpiPi

(

1−P 2i

P 2i + (NRa)2

)

= kpiPi

((NRa)

2

P 2i + (NRa)2

)

.This term has the following hara teristi s;limRa→0

kpiPi

((NRa)

2

P 2i + (NRa)2

)

= 0.There is no growth of intra ellular parasites.lim

Ra→∞kpiPi

1(

Pi

NRa

)2

+ 1

= kpiPi.In this ase the intra ellular in reases exponentially. As Pi → ∞ the number of burstinginfe ted red blood ells de reases and the loss of infe ted red blood ells may beexponential. The author is not aware of the repli ation law for malaria parasite hen e,the repli ation law hosen here is similar to that for infe tion of ma rophages by TBba teria [25℄. This is at best an approximation whi h needs further investigation. The�fth term represents loss due to natural death at a onstant rate µra.Upon bursting of the a tively infe ted red blood ells, the merozoites are released into the


Chapter 4. A within host model of blood stage malaria 29blood stream to ontinue the parasite y le. The rate of hange of the merozoite populationis des ribed by equation by (4.5):Pe = SpePe + k11NRa

(Pi

2

Pi2 + (NRa)2

)

+ n1µraPi − ktpNEPe

−k∗n∗RPe − µpePe. (4.5)The �rst term in (4.5) represents the amount of extra ellular parasites present in theblood stream, the se ond and third terms in (4.5) represent gains of extra ellular parasitepopulation due to bursting of a tivated RBCs and natural death of a tivated infe ted redblood ells, Pi, the forth term is the loss due to killing of extra ellular parasites by e�e tor ells, the �fth term is loss due to infe tion of sus eptible RBCs by merozoites and the sixthterm is the natural death of merozoites at a onstant rate µpe.E�e tor ellsE�e tor ells are fully di�erentiated stru tural lympho yte ells that are spe ialized ininitiating and e�e ting an immune response. This population in ludes CD4+ T- ells, andCD8+ T- ells and is assumed to grow logisti ally [9℄ as shown in equation (4.6)E = ωe

(

1−E

re

)

E, (4.6)where ωe denotes the onstant growth rate of this population and re is the arrying a-pa ity of the e�e tor ells per millilitre of blood. We have simpli�ed the dynami s of thispopulation but we believe we have aptured the biologi al role of these ells.Positivity of the solutionsLemma 1 Let R(0) ≥ 0, Rl(0) ≥ 0, Ra(0) ≥ 0 Pi(0) ≥ 0, Pe(0) ≥ 0 and E(0) ≥ 0. Then,the solution (R(t), Rl(t), Ra(t), Pi(t), Pe(t), E(t)) are all non negative for all time t > 0 inthe regionΓ6 = (R,Rl, Ra, Pi, Pe, E)ǫ R6

+. (4.7)Proof 1 Consider the sus eptible RBCs dynami s given by equation (4.1):dR

dt= Sr − µrR− kRPe.


Chapter 4. A within host model of blood stage malaria 30Multiplying equation (4.1) by the integration fa tor e(k∫ t

0Pe(s)ds+µrt), and rearranging, weobtain

d

dt

(

R(t)e(k∫ t

0Pe(s)ds+µrt)

)

= Sre(k

∫ t

0Pe(s)ds+µrt),

R(t1) = e(−k∫ t10

Pe(s)ds−µrt)︸︷︷︸

≥0

R(0)︸︷︷︸︸︷︷︸

≥0

+Sr ×

∫ t1

0

e(k∫ t0Pe(s)ds+µrt)

︸︷︷︸

≥0

ds

≥ 0, As t −→ ∞Now, onsider the dynami s of the latently infe ted RBCs;dRl

dt= αkRPe − (γ + µrl)Rl.Multiplying equation (4.2)the integrating fa tor e(γ+µrl)t, we obtain;

d

dt

(Rl

(e(γ+µrl)t

))= αkR(t)Pe(t)e

(γ+µrl)t,

Rl(t1) = e(−(γ+µrl)t1)

(

Rl(0) + αk

∫ t1

0

Rl(s)Pe(s)e(γ+µrl)sds

)

≥ 0 As t1 −→ ∞.The equation for a tivated infe ted RBCs gives;dRa

dt= (1− α)kRPe + γRl −mERa − kbRa

(Pi

2

Pi2 + (NRa)2

)

− µraRa,

≥ (1− α)kRPe + γRl − (mE + µra + kb)Ra,

= (1− α)kRPe + γRl − (mE + µra + kb)Ra.Multiplying through by the integrating fa tor e(m∫ t

0E(s)ds+(µra+kb)t), gives:

d

dt

(

em∫ t0E(s)ds+(µra+kb)t

)

Ra ≥ (1− α)kR(t)Pe(t)em

∫ t0E(s)ds+(µra+kb)t.

Ra(t1) ≥ Rae(−m(

∫ t10

E(s)ds+(µra+kb)t1))

+ (1− α)ke(−(m∫ t10

E(s)ds+(µra+kb)t1))

×

∫ t1

0

Ra(v)Pe(v)e(m

∫ v0E(p)dp+(µra+kb)v)dv,

≥ 0 As t1 −→ ∞.


Chapter 4. A within host model of blood stage malaria 31A similar analysis for the intra ellular parasite gives;dPi

dt= kpiPi

(

1−Pi

2

Pi2 + (NRa)2

)


(Pi

2

Pi2 + (NRa)2

)

−n1µraPi,

≥ k∗n∗RPe − k11NRa

P 2i

P 2i + (NRa)2

− n1µraPi,

≥ k∗n∗RPe − k11NRa − n1µraPi.

= k∗n∗RPe − (k11 + n1µraPi) .Multiplying through by the integrating fa tor is e(k11+n1µra)t, we obtain;d

dt

(Pi(t)e

(k11+n1µra)t)

≥ k∗n∗R(t)Pe(t)e(k11+n1µra)t

Pi(t1) ≥ Pi(0)e(−(k11+n1µra)t1)

+k∗n∗e(−(k11+n1µra)t1)

∫ t1

0

R(s)Pe(s)e(k11+n1µra)ds,

≥ 0 As t −→ ∞.Lastly,d

dt

(

Pe(t)e(ktpN

∫ t

0E(s)ds+k∗n∗

∫ t

0R(s)ds+µpet)

)

≥ n1µraPi(t)e(ktpN

∫ t

0E(s)ds+k∗n∗

∫ t

0R(s)ds+µpet)

Pe(t1) ≥ Pe(0)e−ktp

∫ t10

E(s)ds−k∗n∗

∫ t

0R(s)ds−µpet

+n1µrae(−ktp

∫ t10

E(s)ds−k∗n∗

∫ t0R(s)ds−µpet)

×

∫ t1

0

Pi(t)e(ktpN

∫ t10

E(s)ds+k∗n∗

∫ t0R(s)ds+µpet)dt,

≥ 0 As t −→ ∞.The e�e tor ells equation is logisti , and so its solution isE =

re1 + Ce−ωet

As t −→ ∞ E −→ rethen E(0) ≥ 0.The model equation (4.1) - (4.6) is mathemati ally and epidemiologi ally well-posed andwe pro eed to onsider the dynami s of the �ow generated by it in Γ6.


Chapter 4. A within host model of blood stage malaria 324.2.2 Mathemati al analysis of the modelLo al Stability of the parasite-free equilibrium pointThe system of equations (4.1) - (4.6) has equilibrium points namely the parasite-free andthe parasite-present equilibrium points. The parasite-free equilibrium point, obtained bysetting the infe ted states and the parasite states to zero, that is, Rl = Ra = Pi = Pe = 0,is given by;x02 =

(Sr

µr

, 0, 0, 0, 0, re

)

. (4.8)The model equations (4.1) to (4.6), depending on parameter values, an pro ess either aunique parasite-present equilibrium point or multiple parasite-present equilibrium pointx03 = (R∗, R∗

l , R∗a, P

∗i , P

∗e , E

∗) ,but analyti al determination of su h points is too umbersome for a large model. Ournumeri al simulation, however, will demonstrate the existen e and stability of these points.We have determined the model reprodu tion number by rearranging the system (4.1) to(4.6) as in the previous model (see [93℄ for details). We have:Rl = αkRPe − (γ + µrl)Rl,

Ra = (1− α)kRPe + γRl −mERa − kbRa

(Pi

2

Pi2+(NRa)2

)

− µraRa,

Pi = kpiPi

(

1− Pi2

Pi2+(NRa)2

)


(Pi

2

Pi2+(NRa)2

)

−n1µraPi,

Pe = SpePe + k11NRa

(Pi

2

Pi2+(NRa)2

)

+ n1µraPi − ktpNEPe − k∗n∗RPe

−µpePe,

R = Sr − µrR− kRPe,

E = ωe

(

1− Ere

)

E.

(4.9)New infe tions o ur in the four infe ted lasses namely the lass of intra ellular parasites,extra ellular parasites, latently infe ted red blood ells and a tively infe ted red blood ells. The new infe tions are given in the matrix below:


Chapter 4. A within host model of blood stage malaria 33F =

αkRPe

(1− α)kRPe

k∗n∗RPe

k11NRa

(Pi

2

Pi2+(NRa)2

)

+ n1µraPi

.The Ja obian DF|(x02) of F is given byF = J(F )|(x02) =

0 0 0 αkSr

µr

0 0 0 (1−α)kSr

µr

0 0 0 k∗n∗Sr

µr

0 0 n1µra 0

.The other transitions among the states are given byV =

(γ + µrl)Rl

−γRl +mERa + kbRa

(Pi

2

Pi2+(NRa)2

)

+ µraRa

−kpiPi

(

1− Pi2

Pi2+(NRa)2

)

+ k11NRa

(Pi

2

Pi2+(NRa)2

)

+ n1µraPi

−Spe + ktpNEPe + k∗n∗RPe + µpePe

,and the asso iated Ja obian isV = J(V )|(x02) =

(γ + µrl) 0 0 0

−γ mre + µra 0 0

0 0 −kpi + n1µra 0

0 0 0 ktpNre +k∗n∗Sr

µr+ µpe − Spe

.The produ t FV−1 is given byFV−1 =

0 0 0 kαSr

µr(Nktpre+µpe+k∗n∗Sr

µr−Spe)

0 0 0 k(1−α)Sr


µr−Spe)

0 0 0 k∗n∗Sr


µr−Spe)

0 0 n1µra

n1µra−kpi0

.

The reprodu tion number is de�ned as the largest absolute eigenvalue of the matrix FV−1


Chapter 4. A within host model of blood stage malaria 34and is given byR02 =

√

k∗n∗n1Srµra

(n1µra − kpi) (µpeµr + k∗n∗Sr +Nktpreµr − Speµr)

=

√(n1µra

(n1µra − kpi)

)(k∗n∗Sr

(µpeµr + k∗n∗Sr +Nktpreµr − Speµr)

)

=√R∗

02,

(4.10)whereR∗

02 = RomRop (4.11)Rom =

n1µra

n1µra − kpi, n1µra − kpi > 0. (4.12)

Rop =k∗n∗Sr

(µpeµr + k∗n∗Sr +Nktpreµr − Speµr). (4.13)The number Rop is positive if the inequality

µpeµr +Nktpreµr − Speµr ≥ 0, n1µra − kpi ≥ 0, (4.14)is satis�ed. This simpli�es to µpe + Nktpre ≥ Spe. We an summarize this information asfollows:1. The positivity of the number R∗02 requires that n1µra−kpi > 0 and µpe+Nktpre ≥ Spe.2. The numberR∗

02 is a produ t of two numbers, Rop andRom, representing two pro essesof the red blood y le, the infe tion of red blood ells by extra ellular parasites andthe asexual repli ation of parasites within an infe ted red blood ell respe tively.This produ t des ribes a host-ve tor nature of the pro ess whereby an extra ellularparasite must infe t a sus eptible red blood ell �rst, multiply asexually within theinfe ted red blood ell before the parasites are released to ontinue the red blood y le.Hen e, from the numbers (1, 2), we on lude that, based on the model reprodu tion num-ber, lini al malaria is aused mainly by the asexual reprodu tion of the parasite ( Rom > 1). A ording to our model, although a high number of sus eptible red blood ells may beinfe ted by extra ellular parasites, this pro ess alone does not generate enough se ondary


Chapter 4. A within host model of blood stage malaria 35infe tions (Rop ≤ 1) to ause lini al malaria but the high number of merozoites releasedinto the blood stream by the e� ient asexual pro ess is responsible for rapid depletion ofthe red blood ell population resulting in anaemia. One may ask the following questions" For what values of n1 is the model reprodu tion number R02 greater than one and forwhat numbers n1 is it less than one? Can we �nd the threshold n∗1 whi h determines theprognosis of the disease? We address these questions in detail in the se tion (4.2.4) onsimulation. We an state the following stability theorem for the parasite-free equilibriumpoint.Theorem 4.2.1 The parasite-free equilibrium of the system (4.1) - (4.6) is lo ally stableif µpeµr +Nktpreµr − Speµr ≥ 0, n1µra − kpi ≥ 0, and R∗

02 < 1.From the lo al stability ondition we have al ulated the threshold killing rate of merozoitesby the e�e tor ells and is given byktp =

kpik∗n∗Sr

Nreµr (n1µra − kpi)−

µpe

Nre.If ktp > ktp, the immune ells manage to ontrol the parasite population. In malariaendemi areas of the developing ountries where many individuals su�er from hroni in-fe tions and their e�e tor ell responses an be altered and impaired, the e�e tor ellskilling rate for some patients may be below ktp as a results of ell fatigue [94℄. This sit-uation may be responsible for the re urren e of malaria. Note that the threshold killingrate, m, of a tively infe ted red blood ells by e�e tor ells annot be determined fromthe stability onditions of the parasite-free equilibrium point. We investigate the impa tof the parameter m in se tion (4.2.4) on simulation.


Chapter 4. A within host model of blood stage malaria 364.2.3 A within host treatment model of blood stage malariaTreatment with a drug of onstant e� a y (ǫ1) an be in orporated in the model (4.1) -(4.6) by inserting the fa tor (1− ǫ1) next to the infe tion terms as shown below:R = Sr − µrR− k(1− ǫ1)RPe,

Rl = αkR(1− ǫ1)Pe − (γ + µrl)Rl,

Ra = (1− α)kR(1− ǫ1)Pe + γRl −mERa − kbRa

(Pi

2

Pi2+(NRa)2

)

− µraRa,

Pi = kpiPi

(

1− Pi2

Pi2+(NRa)2

)

+ k∗n∗R(1− ǫ1)Pe − k11NRa

(Pi

2

Pi2+(NRa)2

)

−n1µraPi,

Pe = SpePe + k11NRa

(Pi

2

Pi2+(NRa)2

)

+ n1µraPi − ktp

(

1 + ǫ11+ǫ1

)

NEPe

−k∗n∗R(1− ǫ1)Pe − µpePe,

E = ωe

(

1− Ere

)

E.

(4.15)The reprodu tion number al ulated as in [93℄ is given by

R04 =√

k∗n∗n1(1−ǫ1)Srµra

(n1µra−kpi)(H3µr)

=

√(

n1µra

(n1µra−kpi)

)(k∗n∗(1−ǫ1)Sr

(H3µr)

)

=√R∗

04.whereH3 = ktp

(

1 +ǫ1

1 + ǫ1

)

Nre +k∗n∗(1− ǫ1)Sr

µr

+ µpe − Spe. (4.16)andR∗

04 =

(n1µra

n1µra − kpi

)(k∗n∗(1− ǫ1)Sr

(H3µr)

)

. (4.17)The number R∗04 onsists of two parts, ( n1µra

(n1µra−kpi)

) and (k∗n∗(1−ǫ1)Sr

H3µr

)

. These parts de-�ne the infe tion pro ess whi h starts with the infe tion of red blood ells (the part(

k∗n∗(1−ǫ1)Sr

H3µr

)) and the asexual reprodu tion of the parasite (( n1µra

(n1µra−kpi)

)).Note that; (k∗n∗(1− ǫ1)Sr

H3µ

)

< 1,

(n1µra

(n1µra − kpi)

)

> 1. (4.18)The e� a y parameter ǫ1 ranges between 0 and 1. As ǫ1 −→ 1, R∗04 −→ 0, and as ǫ1 −→ 0,

R∗04 = R∗

02. We want to determine, in the se tion on simulation, the least e� a y levelǫ∗1 < 1 for whi h the disease lears. We an summarize the lo al stability of the parasite-


Chapter 4. A within host model of blood stage malaria 37free equilibrium point in the following theorem:Theorem 4.2.2 The model (4.15) is stable for R∗04 < 1, implying that treatment main-tains the stability of the model.4.2.4 Simulations of the within host model of blood stage malariaSensitivity and un ertainty analysis of R02 and parameter estimationThe model (4.1) - (4.6) requires several input parameter values, whi h an be divided intothree sets, namely (i) those that have been measured lini ally or experimentally [TABLE4.3℄, (ii) those that have been estimated by other resear hers [TABLE 4.3℄ and (iii) thosethat have been estimated by us using the parameters in (i) and (ii) above. Sin e a number ofparameters are not known [TABLE 4.3℄, we begin by investigating the level of un ertaintiesTABLE. 4.3. The table that shows the parameter values of the model.Parameter Value Sour e

Sr 2.5 ∗ 108 < Sr < 2.5 ∗ 109 Estimated [89, 101℄µr 0.01 [101℄k 2 ∗ 10−9 [89℄µrl 0.022 < µrl < 0.01 Estimated [5, 101℄µra 0.014 < µra < 0.02 [101℄N 32 [55℄n1 12 < n1 < 32 Estimated [55℄n∗k∗ 10−8 Estimatedk11 0.01 Estimatedkb 0.05 < kb < 0.4 Estimatedµpe 0.0208 [89℄γ 0.0001 < γ < 0.04 Estimatedα 0.2 Estimatedm 10−8 [5, 89℄kpi 0.08745 Estimatedktp 0.01 Estimatedre 4000 < re < 15000 [100℄ωe 0.04 EstimatedSpe 0 < Spe < 50 Estimated [78℄in the model parameters and their sensitivity using Latin Hyper ube Sampling Te hniques


Chapter 4. A within host model of blood stage malaria 38

1 2 3 4 5 6 7 8 9 10

−15

−10

−5

0

5

10

15

Sr

µr

µra

N

kpi

n1

ktp

re

µpe

k*n*

Parameters

Se

nsi

tivity

ind

ice

s

FIG. 4.2. A diagram showing sensitivity of various parameters on the reprodu tion number.[32℄. The partial rank orrelation oe� ients plotted in FIG. 4.2 show the impa t of thevarious parameters on the reprodu tion number R02. We an see that the reprodu tionnumber is highly positively orrelated to kpi, the rate of growth of intra ellular parasites,and highly negatively orrelated to µra, the natural death rate of intra ellular parasitesand n1 another intra ellular parasite related parameter. The range of the parameter n1is known and is given in [55℄. The parameter µra is also known and is given in [101℄.The parameter kpi has not been estimated lini ally or experimentally and is estimatedin this study using the parameters from M Queen and M Kenzie [55℄ and the referen estherein. Using the values in [TABLE 4.3℄, we have found using an iterative pro edure that


Chapter 4. A within host model of blood stage malaria 39n1 = 16 is the threshold value for whi h R02 = 1. Using this value of n1 and the parametervalues in [TABLE 4.3℄, we have estimated the value of the growth rate of parasites to bekpi = 0.08745.We want to use the parameters in [TABLE 4.3℄ to study the following hypotheti al prob-lems: (a) First, from the model equations (4.1) - (4.6), we want to determine the number,n1, of merozoites in an infe ted red blood ell at the time of its natural death for whi hR02 < 1, and for whi h R02 > 1. In other words, we want to use the threshold valuen∗1 = 16 (R02 = 1), to determine the prognosis of the malaria disease. (b) Se ondly,while a ute infe tions usually result in e�e tive immune responses, hroni infe tions areasso iated with suboptimal e�e tor ell responses [94℄. To illustrate this, we onsider ahypotheti al situation where a host has malaria infe tion whi h would lear if CD8 T- ellresponses were normal (that is R02 < 1). We then alter the CD8 T- ell response param-eters m and ktp at time t < tc, where tc is the time of learan e of the malaria infe tion,and investigate the prognosis of the disease. (iii) The malaria extra ellular parasites aregenerated either by bursting of infe ted RBCs or by natural death of infe ted red blood ells. We investigate whi h of these repli ation me hanisms is responsible for the develop-ment of lini al malaria.4.2.5 Simulations of the within host treatment model of bloodstage malariaThe models equations (4.15) require several input parameter values, whi h an be dividedinto three sets, namely (i) those that have been measured lini ally or experimentally[TABLE 4.4℄, ( ii) those that have been estimated by other resear hers [TABLE 4.4℄, and(iii) those that have been estimated by us using the parameters in (i) and (ii) above. Weused MATLAB routines ODE45, ODE15s et , to analyse our system and the sensitivityanalysis for estimation of parameter on [TABLE 4.4℄ was done in se tion 4.2.4 and is notrepeated here. We have in luded [TABLE 4.4℄ here to make referen ing easier. The timeused from simulations is zero to 60 days, whi h, lini ally, is the time for an infe ted personto shows symptoms of malaria. In this example, we have started treatment on 32 day after


Chapter 4. A within host model of blood stage malaria 40TABLE. 4.4. The table that shows the parameter values of the model.Parameter Value Units Sour eSr 2.5 ∗ 107.2 cell/ml.day Estimated [5, 55℄µr 0.01 day−1 [101℄k 2 ∗ 10−9.5 ml/cell.day [5, 89℄µrl 0.008 day−1 Estimated [5, 101℄µra 0.014 day−1 [101℄N 32 dimensionless [55℄n1 12 dimensionless Estimated [55℄n∗k∗ 10−8 ml/cell.day Estimatedk11 0.01 day−1 Estimatedkb 0.4 day−1 Estimatedµpe 0.0208 day−1 [89℄γ 0.0001 day−1 Estimatedα 0.2 dimensionless Estimatedm 10−8 ml/cell.day [5℄kpi 0.08745 day−1 Estimatedktp 0.0009 ml/cell.day Estimatedµpe 0.0208 day−1 [89℄re 880 cell/ml [100℄ωe 0.05 day−1 EstimatedSpe 20 cell/ml.day Estimated [78℄ǫ1 0 < ǫ1 < 0.95 dimensionless Estimatedinitial infe tion. This is be ause, the average time for lini al symptoms to appear isbetween (8− 25) days but it an take longer depending on the immune system of the host[48℄.


Chapter 4. A within host model of blood stage malaria 414.3 Results of the within host model of blood stagemalariaDynami s of the systemFIG. (4.3) shows the time plots for the various lasses of human ells and parasite lassesfor R02 = 0.8327 As expe ted the sus eptible red blood ell drops initially but settles at astead state level whi h is high enough to sustain life. All the infe ted states tend to zeroand the disease does not establish itself. FIG. 4.4 shows a time plot for all the human ell populations and the parasite populations for R02 = 1.3165. It is lear that the diseaseestablishes itself. The level of red blood ells falls rapidly below a level su� ient to supportlife.Figures 4.5 and 4.6 show the evolution of the intra ellular (FIG. 4.5) and extra ellular(FIG. 4.6) parasites for various values of the parameter n1 ( 8 ≤ n1 ≤ 32) and the orre-sponding reprodu tion numbers. These �gures, surprisingly, show that the repli ation ofintra ellular and extra ellular parasites in the host is fastest for n1 = 8. At this level ofrepli ation, ea h merozoite released from an infe ted red blood ell generates on average1.6679 se ondary infe tions. The repli ation of extra ellular and intra ellular parasitesde reases as n1 in reases. Likewise, the reprodu tion number de reases as n1 in reases(FIG. 4.5 and FIG. 4.6). We an see from these plots that the parasite persists for n1 < 16possibly resulting in lini al malaria but fails to invade the host for n1 ≥ 16 provided thee�e tor ell fun tions are normal. The fa t that the parasite persists for small values n1suggests that the in rease in the merozoite population within the host does not depend onthe bursting of the infe ted red blood ells (�lled to ell apa ity, that is, N = 32) but onthe natural death of infe ted red blood ells. FIG. 4.7, whi h is typi al for other valuesof n1, illustrates the relative impa t of the two merozoite repli ation me hanisms namely,repli ation by bursting of infe ted red blood ells, P1, for N = 32 and repli ation by naturaldeath of infe ted red blood ells, P2, for the ase n1 = 12. It is lear from (FIG. 4.7) thatrepli ation by natural death of infe ted red blood ells ontributes signi� antly more thanrepli ation from bursting of infe ted red blood ells thereby on�rming our earlier asser-tion. We are led to on lude that the merozoites may have preferen e for infe ting oldersus eptible red blood ells that do not have a long time left to live. Figures (4.8 and 4.9)



0 100 200 300 400 5003

4

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 100 200 300 400 5000

2

4x 10

4

Time (days)

La

te

nt R

BC

s

0 100 200 300 400 5000

5

10x 10

4

Time (days)

Activ

e R

BC

s

0 100 200 300 400 5000

5

10x 10

4

Time (days)In

te

rce

llu

lar p

ara

site

s

0 100 200 300 400 5000

500

Time (days)Extra

ce

llu

lar p

ara

site

s

0 100 200 300 400 5000

1000

2000

Time (days)

Effe

cto

r ce

lls

FIG. 4.3. Shows a diagram of parasite-free equilibrium with R02 = 0.8327.show the evolution of red blood ell and infe ted red blood ell populations respe tively.The population of red blood ells de lines rapidly while the infe ted red blood ell popu-lation in reases to a peak before going down. The peak of the infe tion, measured by the



0 100 200 300 400 5000

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 100 200 300 400 5000

5

10x 10

8

Time (days)

La

te

nt R

BC

s

0 100 200 300 400 5000

5

10x 10

8

Time (days)

Activ

e R

BC

s

0 100 200 300 400 5000

1

2x 10

10

Time (days)In

te

rce

llu

lar p

ara

site

s

0 100 200 300 400 5000

1

2x 10

8

Time (days)Extra

ce

llu

lar p

ara

site

s

0 100 200 300 400 5000

500

1000

Time (days)

Effe

cto

r ce

lls

FIG. 4.4. A diagram of parasite-present equilibrium with R02 = 1.3165.level of infe ted red blood ells, o urs approximately 37 days after the infe tion started,whi h agrees with lini al data [91, 96℄. For n1 < 16 the infe tion never lears and thered ell steady state level is only 25%− 40% (FIG. 4.10) of the parasite-free level,a whi h



0 50 100 150 200 250 3000

1

2

3

4

5x 10

11

Time (days)

Intracellular parasites

A plot of intracellular parasites vs time

n1=12, R

0=1.1277

n1=16, R

0=1.0001

n1=24, R

0=0.9079

n1=32, R

0=0.8704

n1=8, R

0=1.6679

FIG. 4.5. A diagram showing population of intra ellular parasites for n1 < 16 the parasite-present equilibrium ases and n1 ≥ 16 the parasite-free equilibrium ases.is too low to sustain life. Clearly, for n1 < 16 treatment with malaria drugs is the onlymeans of ontrolling the malaria parasite. The results for n1 ≥ 16 show that the parasite lears from the host as illustrated in FIG. 4.11 and FIG. 4.12. These �gures show that thea tivated and latently infe ted red blood ell populations be ome extin t. For n1 = 16,FIG. 4.10 shows that after the infe tion has leared the red ell steady state level is 80%of the parasite-free level whi h is high enough to sustain life. However, for individualswith hroni illnesses [22, 72, 81, 104℄, the e�e tor ell fun tions an be ompromised. Toillustrate this, we onsider a malaria infe tion whi h under normal e�e tor ell fun tionwould lead to a parasite-free equilibrium state and investigate the e�e t of redu ing therole of e�e tor ells by de reasing the parameters ktp and m. Taking the state variables at



0 50 100 150 200 250 3000

1

2

3

4

5

6x 10

7

Time (days)

Extracellular parasites

A plot of extracellular parasites vs time

n1=32, R

0=0.8704

n1=24, R

0=0.9079

n1=16, R

0=1.0001

n1=12, R

0=1.1277

n1=8, R

0=1.6679

FIG. 4.6. A diagram showing population of extra ellular parasites for n1 < 16 the parasite-present equilibrium ases and n1 ≥ 16 the parasite-free equilibrium ases.time t = 350 and de reasing the parameters ktp and m, we noti e that the population ofa tivated red blood ells (FIG. 4.13) immediately in reases while the sus eptible red ellpopulation (FIG. 4.14) de lines. We on lude that if the role of the e�e tor ells is redu edeven for the ase n1 = 24 the host would develop malaria.



0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

70

Time (days)

Re

lative

im

pa

ct

A plot of relative impact vs time

P2

10P1

FIG. 4.7. Represents relative impa t of the two parasite produ tion me hanisms 10 ∗ P1and (P2) for n1 = 12 and R02 = 1.1277.



0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

A plot of susceptible RBCs vs time

FIG. 4.8. A diagram showing the evolution of RBCs with time.

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5x 10

9

X: 37.28Y: 2.242e+009

Time (days)

Active

RB

Cs

A plot of active infected RBCs vs time

FIG. 4.9. A diagram showing the evolution of a tive infe ted RBCs with time.



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 109

0

5

10

15x 10

8

Susceptible red blood cell

Active

in

fecte

d r

ed

blo

od

ce

ll

Contour plot of active infected RBCs vs susceptible RBCs

n1=15

n1=12

n1=16

FIG. 4.10. Contour plots for n1 = 12 and n1 = 15 and n1 = 16.

0 100 200 300 400 5000

2

4

6

8

10x 10

4

Time (days)

Activ

e in

fect

ed R

BCs


FIG. 4.11. Shows the population of a tively infe ted RBCs at for n1 = 24.



0 100 200 300 400 5000

0.5

1

1.5

2

2.5x 10

4

Time (days)

Late

nt in

ecte

d R

BC

s

A plot of latent infected RBCs vs time

FIG. 4.12. Shows the population of latently infe ted RBCs at for n1 = 24.

300 310 320 330 340 350 360 3700

0.5

1

1.5

2

2.5

3

3.5

4x 10

9

Time (days)

Active infected RBCs


m=10−8, ktp

=9*10−4, R0=0.7422

m=10−9,ktp

=9*10−5,R0=1.0862

FIG. 4.13. Diagram showing the population of a tively infe ted RBCs population fordi�erent values of m and ktp.



300 310 320 330 340 350 360 3700

1

2

3

4

5x 10

9

Time (days)

Susceptible RBCs

A plot of susceptible RBCs vs time

m=10−8, ktp

=9*10−4, R0=0.7422

m=10−9, ktp

=9*10−5, R0=1.0862

FIG. 4.14. Diagram showing the population of sus eptible RBCs population for di�erentvalues of m and ktp.


Chapter 4. A within host model of blood stage malaria 514.4 Results of within host treatment model of bloodstage malariaFIG. (4.15) shows that sus eptible red blood ell population de reases between day 20and day 50 in the absen e of treatment. The latently and a tively infe ted red blood ellpopulation in rease during the same period.The FIG. 4.16 shows plots when treatment is administered for drug e� a ies ranging from(ǫ1 = 0) to (ǫ1 = 0.95). the �gure shows that the depletion of the sus eptible red blood ell population de reases with in reasing e� a y. On the other hand the latently anda tively infe ted red blood ell populations de rease with in reasing e� a y. Intra ellularparasites de rease with in reasing e� a y while extra ellular parasites in rease and showthe same hara teristi as time in reases. the reprodu tion number is a de reasing fun tionof e� a y from R04 = 1.3723 for ǫ1 = 0 to R04 = 0.5074 for ǫ1 = 0.95. .



0 10 20 30 40 502

3

4

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 10 20 30 40 500

1

2

3x 10

8

Time (days)

La

ten

t R

BC

s

0 10 20 30 40 500

1

2x 10

9

Time (days)

Active

RB

Cs

0 10 20 30 40 500

5

10

15x 10

9

Time (days)

Inte

rce

llu

lar p

ara

site

s

0 10 20 30 40 500

5

10x 10

7

Time (days)

Extr

ace

llu

lar p

ara

site

s

0 10 20 30 40 50200

400

600

800

Time (days)

Effe

cto

r c

ells

FIG. 4.15. A diagram shows malaria without treatment.



0 10 20 30 40 50 602

3

4

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 10 20 30 40 50 600

2

4x 10

8

Time (days)

La

ten

t R

BC

s

0 10 20 30 40 50 600

1

2x 10

9

Time (days)

Active

RB

Cs

0 10 20 30 40 50 600

1

2x 10

10

Time (days)

Inte

rce

llu

lar p

ara

site

s

0 10 20 30 40 50 600

5

10x 10

7

Time (days)

Extr

ace

llu

lar p

ara

site

s

0 10 20 30 40 50 600

500

1000

Time (days)

Eff

ecto

r c

ells

ε1=0 ε

1=0.4, ε

1=0.6 ε

1=0.95

75% of initial value of RBCs

FIG. 4.16. Diagram showing one type of drug in treatment of malaria after 32 days andǫ1 = 0 =⇒ R04 = 1.3723, ǫ1 = 0.4 =⇒ R04 = 1.2114, ǫ1 = 0.6 =⇒ R04 = 1.0932,ǫ1 = 0.95 =⇒ R04 = 0.5074.


Chapter 4. A within host model of blood stage malaria 544.5 Dis ussionSensitivity and un ertainty analysis has shown that the reprodu tion number is highlysensitive to the rate of growth of intra ellular parasites (kpi). An overestimate or under-estimate of this parameter has serious impli ations regarding the prognosis of the disease.Therefore, there is need for a lini al or experimental determination of this parameter inparti ular but also a number of other parameters, su h as rate of loss of extra ellular par-asites that are killed by e�e tor ells (ktp), threshold number as a results of gain due toinfe tion of sus eptible RBC by extra ellular parasites (n∗k∗), rate of loss due to burst ofa tivated ell (k11), rate of bursting (kb), rate of a tivation (γ), proportion of RBCs (α),to enhan e the understanding of malaria.We have found that parasites are repli ated in two ways: (i) from naturally dying infe tedred blood ells and (ii) from bursting of infe ted red blood ells �lled to arrying apa ityN. Of the two me hanisms, we have found that natural death of infe ted red blood ells ontributes more to the population of extra ellular parasites. Interestingly, the number ofparasites in a dying infe ted red blood ell need not be high for lini al malaria to develop.In fa t the larger the number of parasites in a dying infe ted ell the smaller the han esof lini al malaria developing. Furthermore, we have found the following link between athreshold number of intra ellular parasites released as a results of the natural death of ana tivated RBC (n1) and the reprodu tion number R02 : For n1 < 16, R02 > 1 and forn1 ≥ 16, R02 ≤ 1. This result suggests that the larger n1 is, the longer it takes to produ ethe parasites and the higher the han e of an infe ted red blood ell being identi�ed andapoptosised by the e�e tor ells. This has led us to on lude that in order to minimize thepossibility of infe ted red blood ells being dete ted and eliminated by the immune ells,the parasite has a strategy of infe ting older red blood ells whose life expe tan y is mu hshorter than the younger ells [55℄ and in this way avoid the infe ted red blood ell beingdete ted and apoptosised.The reprodu tion number (4.10) is a produ t of two reprodu tion numbers Rop and Romthat reveal the host-ve tor infe tion nature of the pro ess during the erythro yti stages.Of the two pro esses namely infe tion of sus eptible red blood ells by extra ellular para-sites and asexual produ tion of parasites, it is the asexual repli ation pro ess, measured byRom, whi h is responsible for the pathology of lini al malaria as the reprodu tive rate forthis pro ess is greater than one per merozoite. To prevent lini al malaria from develop-ing, it is important to administer highly e� a ious drugs to stop the erythro yti stage ofthe parasite y le. However, the fa t that the major sour e of parasite produ tion is fromnaturally dying infe ted red blood ells has impli ations regarding the ontrol of malaria aseven treatment with highly e� a ious drugs ould generate drug resistant malaria strainsif some infe ted red blood ells died before the drug hemodynami e�e ts are ompleted.We believe that only a malaria va ine an reliably prote t against blood stage malariainfe tion.


Chapter 4. A within host model of blood stage malaria 55We have also shown that hroni infe tions an transform manageable malaria into a morea tive disease. The re ommendation from our study is that in malaria endemi areas, indi-viduals with malaria or showing malaria symptoms should be tested for hroni infe tionsand those who test positive for any hroni infe tion should be treated for both malaria andthe hroni infe tion.We have investigated the e�e ts of treatment on parasite depletion of sus eptible red blood ells by introdu ing a drug of e� a y ǫ1. As a �rst step, we determined the reprodu tionnumber whi h is now dependent on the drug e� a y. Taking ǫ = 0 in the formula forR0 gives the reprodu tion number obtained in the model without treatment. Clearly, thee�e t of the drug is apparent on the population sizes of red ells. It is found that the pop-ulation of sus eptible red blood ells in reases with in reasing e� a y (FIG. 4.16). Theinfe ted red blood ell populations on the other hand de rease with in reasing e� a y.A drug of e� a y of at least 0.6 is apable of learing the parasites (FIG. 4.16). Mostmalaria treatment drugs are of higher e� a y than 0.6. It is possible to ombat malariawith treatment drugs but the administration of drugs should be done at health entres orhospitals to ensure that patients omplete their treatment. This will go a long way towardsredu ing the rate of drug resistan e to urrent malaria drugs.


Chapter 5A within host treatment model withthree stages of malaria life y le5.1 Introdu tionMalaria remains one of the most serious global health problems and it is one of the leading auses of death and illness in hildren and adults in tropi al ountries [6, 91, 95℄. Malaria ontrol requires an integrated approa h, su h as prevention whi h in ludes ve tor ontroland treatment with e�e tive anti-malarial drugs. The on e a�ordable and widely availableanti-malarial treatment drug, hloroquine, that was used from the 1950's to the 1970'sis now totally ine�e tive [6, 102, 103℄. The use of ACTs is the best urrent treatmentstrategy; unfortunately the implementation of this treatment strategy has lagged behinddue to various fa tors su h as high ost of the drugs [95℄.The malaria life y le (FIG. 2.2), involves two hosts namely an infe ted female anophelesmosquito whi h ino ulates sporozoites into the blood ir ulation of a human host duringa blood meal, and the human host. Initial repli ation of the parasite starts in the liverwith the sporozoites infe ting and invading liver ells (hepato ytes). The sporozoites thenmature into s hizonts whi h rupture to release merozoites into the blood stream. In theblood stream, the merozoites infe t the red blood ells (RBCs) within whi h they undergoasexual multipli ation (erythro yti s hizogony). This stage is responsible for the devel-opment of lini al malaria [2, 11, 41, 45, 101℄.The sporogoni y le (mosquito y le) starts when the gameto ytes (mi ro-gametes andma ro-gametes) are ingested by a female anopheles mosquito during a blood meal. Whilein the mosquito stoma h, fertilization o urs and to form zygotes. The zygotes trans-form into ookinetes whi h invade the midgut wall of the mosquito where they transforminto oo ysts. Oo ysts grow, rupture and release sporozoites whi h invade mosquito sali-56


Chapter 5. A within host model with three stages of malaria life y le 57vary glands. When the mosquito bites a sus eptible human, the mosquito life y le starts[11, 45, 50℄. Mathemati al models have been valuable de ision-making tools for va i-nation and treatment strategies against infe tious diseases. Many mathemati al modelsof malaria parasite pathogenesis and its transmission ontinue to be onstru ted by re-sear hers [52, 55�57, 65, 84℄. Most of these models fo us on within-host dynami s of bloodstage malaria or ell-mediated immune response to pathogenesis. Very few resear hers,among them Miranda [87℄, have onsidered the within-mosquito parasite repli ation y le.The study by Iggidr et. al [35℄ onsiders a malaria within host model with k stages of agefor the parasitized red blood ells and n strains for the parasite. A study by Niger andGumel [65℄ has investigated the innate immune response to malaria infe tion and the e�e tof imperfe t va ines assuming a s enario of parasite life y les. The simulation results [65℄showed that a va ine e� a y of at least 87% is ne essary to eliminate Infe ted Red BloodCells (IRBCs )in vivo. Our model will in lude three stages of the parasite life y le namely:the exo-erythro yti s hizogony, whi h in ludes sus eptible liver ells (Sl), infe ted liver ells (Il), Sporozoites (P ) and S hizonts (Cl), the erythro yti s hizogony, whi h in ludessus eptible red blood ells (R), infe ted red blood ells (Ri), merozoites (Mr), trophozoites(T ), and s hizonts (Cr) and the sporogoni y le, in lude sus eptible midgut ells (Smc),infe ted midgut ells (Imc) and gameto ytes (G). We shall also introdu e treatment witha drug of onstant e� a y ǫ1, and the immune response represented by the e�e tor ells.Despite vast lini al and experimental knowledge of the immunology of the malaria para-site life y le [11, 26, 45, 50℄, development of the va ine for malaria is still elusive. Thedevelopment of the va ine is as mu h immunologi al as it is epidemiologi al. The epi-demiology of malaria guides the pro esses of development of new va ines and treatments[79℄ and implementation of the va ine and treatment trials. Needless to say, the e�e tiveand a�ordable drug that will treat all the three stages of the malaria y le will have majorimpa t on global publi health.In this hapter, we formulate a model based on the s hemata of the parasite life y lepresented on the Center for Disease Control Website (FIG. 2.2).


Chapter 5. A within host model with three stages of malaria life y le 585.2 Methodology5.2.1 Model developmentWe shall use the following variablesTABLE. 5.1. The table des ribing the variables and units of variablesVariables Des riptions UnitsSl Sus eptible liver ells ell/mlIl Infe ted liver ells ell/mlP Sporozoites ell/mlCl S hizonts inside liver ells ell/mlR Sus eptible red blood ell ell/mlRi A tivated red blood ell ell/mlMr Merozoites ell/mlT Trophozoites ell/mlCr S hizonts inside red blood ells ell/mlSmc Sus eptible midgut ells ell/mlIl Infe ted liver ells ell/mlG Gameto ytes ell/mlE E�e tor ells ell/ml


Chapter 5. A within host model with three stages of malaria life y le 59TABLE. 5.2. The table des ribing the parameters and units of parametersParameter Des riptions UnitsΠl Constant sour e of liver ells cell/ml.dayβl Infe tion rate ml/cell.dayµsl Natural death rate of sus eptible liver ells day−1

kl Rate of bursting of infe ted liver ells day−1

N Cell arrying apa ity dimensionlessm Rate of killing of infe ted ell by e�e tor ells ml/cell.dayµil Natural death of an infe ted liver ells day−1

Πp Sour e of sporozoites day−1

βp Infe tion rate ml/cell.dayαp Rate of gain/loss of sporozoites day−1

k12 Rate of bursting of infe ted midgut ells day−1

n3 Average number of sporozoites release froman infe ted midgut ells that dies naturally dimensionlessµimc Natural death of an infe ted midgut ells day−1

µp Natural death of sporozoites day−1

kcl Growth rate of s hizonts day−1

ktp Rate of loss of s hizonts inside the liver ellsthat are killed by e�e tor ells ml/cell.dayn1 Average number of s hizonts release froman infe ted liver ells that dies naturally dimensionlessµcl Natural death of s hizonts inside the liver ells day−1

Sr Constant sour e of red blood ells (RBCs) cell/ml.dayk Infe tion rate ml/cell.day


Chapter 5. A within host model with three stages of malaria life y le 60

TABLE. 5.3. The table des ribing the parameters and units of parametersParameter Des riptions Unitsµr Natural death of sus eptible RBCs day−1

kb Rate of loss of infe ted RBCsdue to bursting of infe ted RBCs day−1

µri Natural death of infe ted RBCs day−1

kr Growth rate due to infe tion of RBCs day−1

k7 Rate of killing of merozoites by e�e tor ells ml/cell.dayk13 Rate of loss of merozoites day−1

µmr Natural death of merozoites day−1

γ Proportion of merozoites dimensionlessω Rate of gain/loss of trophozoites day−1

µt Natural death of trophozoites day−1

k11 Rate of loss of s hizonts due tobursting of infe ted RBCs day−1

µcr Natural death of s hizonts inside RBCs day−1

Πmc Constant sour e of midgut ells cell/ml.dayβmc Infe tion rate ml/cell.dayµmc Natural death of sus eptible midgut ells day−1

µimc Natural death of infe ted midgut ells day−1

µg Natural death of gameto ytes day−1

ωe Growth rate of e�e tor ells day−1

re Carrying apa ity of e�e tor ell cell/ml


Chapter 5. A within host model with three stages of malaria life y le 61Liver stage (exo-erythro yti s hizogony)From FIG. 2.2, it is easy to see that, the model for the liver stages (exo-erythro yti s hizogony) must in lude sus eptible liver ells (Sl), infe ted liver ells (Il), Sporozoites(P ) and S hizonts (Cl). We also introdu e treatment with a drug of onstant e� a y ǫ1.The dynami s for the sus eptible liver ells is given by:Sl = Πl − βl(1− ǫ1)SlP − µslSl. (5.1)The �rst term of equation (5.1) is a sour e for the sus eptible liver ells, the se ond termrepresents loss due to infe tion of sus eptible liver ells by sporozoites [85℄ and the thirdterm represents natural death of sus eptible liver ell at onstant rate µsl . The rate of hange of the population of infe ted liver ells is given by:

Il = βl(1− ǫ1)SlP − klIl

(Cl

2

Cl2 + (NIl)2

)

−mEIl − µilIl. (5.2)The �rst term in equation (5.2) is due to infe tion of liver ells, the se ond term representsloss due to bursting of infe ted liver ells whi h is assumed to be dependent on the densitiesof gameto ytes and infe ted liver ells [28℄, the third and fourth terms represent losses duelysing of infe ted ells by e�e tor ells and natural death at a onstant rate µil respe tively.We have assumed a bursting rate similar to that of ba teria in infe ted ma rophages [25, 28℄,an assumption whi h has not been on�rmed lini ally or experimentally for malaria.P = ΠpP − βpSlP − αpP + k12Imc

(G2

G2 + (NImc)2

)

+n3µimcImc − µpP. (5.3)Sporozoites enter the sus eptible liver ells immediately after being ino ulated into theblood stream by the mosquito (FIG. 2.2). Within the liver ells the sporozoites developinto s hizonts [26, 45℄. In equation (5.3) the �rst term represents a sour e of sporozoites,the se ond term is loss of sporozoites due to infe tion of liver ells, the third term is lossdue to maturity of sporozoites into s hizonts, the fourth term measures the number ofinfe ted midgut ells that burst to release gameto ytes. The �fth term is the gain due tonatural death of midgut ells, and the sixth term is the natural death of the sporozoites.Cl = αpP + kclSl

(

1−Cl

2

Cl2 + (NIl)2

)

− ktpNECl − n1µclCl. (5.4)Equation (5.4) represents the evolution of liver s hizonts. The �rst term of the equationrepresents the maturation of sporozoites into the liver s hizonts [50℄, the se ond term isgrowth of s hizonts by an assumed law [25, 26, 28℄. The third term is the loss of s hizontslysis by e�e tor ells, the forth term is the loss due to death of infe ted liver ell.


Chapter 5. A within host model with three stages of malaria life y le 62Blood stages (erythro yti s hizogony)Erythro yti s hizogony [45℄ is the blood stage of the malaria infe tion des ribed in anearlier hapter and it involves sus eptible red blood ells (R), infe ted red blood ells (Ri),merozoites (Mr), trophozoites (T ), and s hizonts (Cr). S hizonts here are a result of theasexual reprodu tion during the blood stage of the infe tion. The red blood ell populationevolves a ording to the equation below:R = Sr − k(1− ǫ1)RMr − µrR. (5.5)The terms in equation 5.5 an be explained as follows: The �rst term represents a sour eof health red blood ells from bone marrow [55℄ and the se ond term represents infe tion ofhealthy red blood ells by merozoites. The third term represents natural death of healthyred blood ell at a onstant rate µr. An infe ted red blood ell is an in ubator of theparasite and as su h is targeted by the e�e tor ells as explained in equation (5.6). Thedynami s if the infe ted red ells is given by:

Ri = k(1− ǫ)RMr −mERi − kbRi

(Cr

2

Cr2 + (NRi)2

)

− µriRi, (5.6)The �rst term of equation (5.6) represents gain by the population of infe ted red blood ells due to infe tion of sus eptible red blood ells by merozoites [26℄, the se ond term isloss as a results of the immune response whi h triggers the e�e tor ells to remove theinfe ted red blood ells assumed here to be at a onstant rate m, the third term is lossdue to bursting of infe ted red blood ells. The fourth term represents natural death ofinfe ted red blood ell at a onstant rate µri. Within an infe ted red blood ell, we assumea law for the evolution of the merozoites population given in equation (5.7) below:Mr = n1µilCl + krMr

(

1−Mr

2

Mr2 + (NRi)2

)

− k7EMr

−k13Mr − µmrMr, (5.7)In equation (5.7) whi h represents the merozoite dynami s, the �rst term represents mero-zoites released from dying infe ted liver ells, the se ond term is logisti growth of themerozoite population, the third term is loss due lysing of merozoites by e�e tor ells, thefourth term is loss as merozoites mature into trophozoites and the �fth term is the natu-ral death of merozoites at onstant rate µmr. The rate of hange of the trophozoites ands hizonts populations are des ribed by equation (5.8) and (5.9)T = γk13Mr − ωT − µtT. (5.8)In equation (5.8) whi h represents the trophozoites dynami s, the �rst term represents aproportion of trophozoites that mature into merozoites, the se ond term is loss as tropho-


Chapter 5. A within host model with three stages of malaria life y le 63zoites mature into s hizonts and the third term is the natural death of trophozoites at a onstant rate µt.Cr = ωT − k11NRi

(Cr

2

Cr2 + (NRi)2

)

− µcrCr. (5.9)The mature trophozoites asexually divide to form s hizonts [26℄. In equation (5.9) whi hrepresents the evolution of blood s hizonts, the �rst term represents gain due to maturityof trophozoite, the se ond term is loss due to bursting of infe ted red blood ells, and thethird term is the natural death of s hizonts at a onstant rate µcr.Mosquito stages (sporogony)The mosquito stages (sporogoni ) onsist of three lasses namely, sus eptible midgut ells(Smc), infe ted midgut ells (Imc) and gameto ytes (G). The dynami s of the midgut ellsis given by:˙Smc = Πmc − βmcGSmc − µmcSmc. (5.10)The �rst term of equation (5.10) represents a onstant sour e of midgut ells, the se ondterm represents infe tion of sus eptible midgut ells by gameto ytes at a onstant rate βmc,and the third term is natural death of sus eptible midgut ells at a onstant rate µmc.˙Imc = βmcGSmc − n3µimcImc. (5.11)The �rst term of equation (5.11) represents a gain due infe tion of sus eptible midgut ellsby gameto ytes at a onstant rate βmc, the se ond term represents loss of infe ted midgut ells due to death of infe ted mid gut ells. The dynami s of the gameto ytes are explainedby equation (5.12) below:

G = (1− γ)k13Mr − k12Imc

(G2

G2 + (NImc)2

)

− µgG. (5.12)Gameto ytes are taken by the sus eptible mosquito as part of a blood meal. The gameto- ytes dynami s are represented by equation (5.12). The �rst term represents a proportionof merozoites that be ome gameto ytes, the se ond term is loss due to bursting of infe tedmidgut ells, and third term is natural death of gameto ytes at onstant rate µg.


Chapter 5. A within host model with three stages of malaria life y le 64E�e tor ellsThe e�e tor ells in lude CD4+ T- ells, and CD8+ T- ells. We have simpli�ed the dy-nami s of this group of ells but we hope their biologi al role is adequately represented.E = ωe

(

1−E

re

)

E. (5.13)In the equation (5.13), ωe denotes the onstant growth rate of this population and re isthe arrying apa ity of the e�e tor ells per millilitre of blood.Positivity of solutionsLemma 2 Let Sl(0) ≥ 0, Il(0) ≥ 0, P (0) ≥ 0, Cl(0) ≥ 0, R(0) ≥ 0, Ri(0) ≥0, Mr(0) ≥ 0, T (0) ≥ 0. Cr(0) ≥ 0, Smc(0) ≥ 0, Imc(0) ≥ 0, G(0) ≥ 0, E(0) ≥ 0.Then, the state variables in the solution(Sl(t), Il(t), P (t), Cl(t), R(t), Ri(t), Mr(t), T (t), Cr(t), Smc(t), Imc(t), G(t), E(t)) areall non negative for all time t > 0 in the region.

Γe = (Sl, Il, P, Cl, R, Ri, Mr, T, Cr, Smc, Imc, G, E) ǫ R13+ . (5.14)Proof 2 Consider the sus eptible liver ells;

dSl

dt= Πl − (βl(1− ǫ1)P − µsl)Sl.Multiplying both sides by the integration fa tor (

eβl(1−ǫ1)∫ t0P (s)ds+µslt

)

, we obtain:d

dt

(

Sl(t)eβl(1−ǫ1)

∫ t

0P (s)ds+µslt

)

= Πle(βl(1−ǫ1)

∫ t0P (s)ds+µslt)

Sl(t1) = Sl(0)e(−βl(1−ǫ1)

∫ t10

P (s)ds−µslt)

+Πle(−βl(1−ǫ1)

∫ t1

0P (s)ds−µslt)

∫ t1

0

e(βl(1−ǫ1)∫ t

0P (s)ds+µslt)dt

≥ 0.At t = 0, Sl(t) = Sl(0) ≥ 0.As t1 −→ ∞, Sl(t) ≥ 0.


Chapter 5. A within host model with three stages of malaria life y le 65Similarly, we an show thatIl(t1) −→ 0, P (t1) −→ 0, Cl(t1) −→ 0, R(t1) −→ 0

Ri(t1) −→ 0, Mr(t1) −→ 0, T (t1) −→ 0, Cr(t1) −→ 0

Smc(t1) −→ 0, Imc(t1) −→ 0, G(t1) −→ 0, R(t1) −→ 0

E(t1) −→ 0

As t1 −→ ∞. The system (5.1) to (5.13) is mathemati ally and immunologi ally well-posedand we pro eed to onsider the dynami s of the �ow generated by it in Γe.5.2.2 Mathemati al Analysis of the modelThe analysis that follows involves deriving the equilibrium points, the reprodu tion number,and establishing the stability of the equilibrium points.Stability of parasite-free equilibrium and the reprodu tion numberWe derive the parasite-free equilibrium points by putting the variables for infe ted statesequal to zero, giving;x06 = (Sl, Il, P, Cl, R, Ri,Mr, T, Cr, Smc, Imc, G, E)

=

(Πl

µsl

, 0, 0, 0,Sr

µr

, 0, 0, 0, 0,Πmc

µmc

, 0, 0, re

)

.The model equations (5.1 - 5.13), depending on parameter values, an possess a uniqueparasite present equilibrium point or multiple parasite-present equilibrium pointsx07 = (Sl, Il, P, Cl, R, Ri,Mr, T, Cr, Smc, Imc, G, E)

= (S∗l , I

∗l , P

∗, C∗l , R

∗, R∗i ,M

∗r , T

∗, C∗r , S

∗mc, I

∗mc, G

∗, E∗) ,whi h algebrai ally are too ompli ated to al ulate for a large model. Our numeri alsimulation, however, will demonstrate the existen e of these points. The reprodu tion


Chapter 5. A within host model with three stages of malaria life y le 66number is omputed by onsidering the infe tious states only as in the previous hapters:Il = βl(1− ǫ1)SlP − klIl

(Cl

2

Cl2+(NIl)2

)

−mEIl − µilIl,

P = ΠpP − βpSlP − αpP + k12Imc

(G2

G2+(NImc)2

)

+n3µimcImc − µpP,

Cl = αpP + kclSl

(

1− Cl2

Cl2+(NIl)2

)

− ktpNECl − n1µclCl,

Ri = k(1− ǫ1)RMr −mERi − kbRi

(Cr

2

Cr2+(NRi)2

)

− µriRi,

Mr = n1µilCl + krMr

(

1− Mr2

Mr2+(NRi)2

)

− k7EMr

−k13Mr − µmrMr,

T = γk13Mr − ωT − µtT,

Cr = ωT − k11NRi

(Cr

2

Cr2+(NRi)2

)

− µcrCr,

˙Imc = βmcGSmc − n3µimcImc,

G = (1− γ)k13Mr − k12Imc

(G2

G2+(NImc)2

)

− µgG.

(5.15)From the model(5.15), the new infe tions are given by

F =

βl(1− ǫ1)SlP

k12ImcG2

G2+(NImc)2+ n3µimcImc

αpPk(1− ǫ1)RMr

n1µilCl

γk13µr

ωTβmcGSmc

(1− γ)k13Mr

,

and its Ja obian matrix at the parasite-free equilibrium is given byF = DF|(x06) =

0 βl(1−ǫ1)Πl

µsl0 0 0 0 0 0 0

0 0 0 0 0 0 0 n3µimc 00 αp 0 0 0 0 0 0 0

0 0 0 0 k(1−ǫ1)Sr

µr0 0 0 0

0 0 n1µil 0 0 0 0 0 00 0 0 0 γk13 0 0 0 00 0 0 0 0 ω 0 0 0

0 0 0 0 0 0 0 0 βmcΠmc

µmc

0 0 0 0 (1− γ)k13 0 0 0 0

.


Chapter 5. A within host model with three stages of malaria life y le 67The other transitions among the states are given byV =

klIl

(C2

l

C2

l+(NIl)2

)

+mEIl + µilIl

−ΠpP + βpSlP + αpP + µpP

−kclSl

(

1−C2

l

C2

l+(NIl)2

)

+ ktpNECl + n1µclCl

mERi + kbRi

(C2

r

C2r+(NRi)2

)

+ µriRi

−krMr

(

1− M2r

M2r+(NRi)2

)

+ k7EMr + k13Mr + µmrMr

ωT + µtT

k11NRi

(C2

r

C2r+(NRi)2

)

+ µcrCr

n3µimcImc

k12Imc

(G2

G2+(NImc)2

)

,

with its Ja obian matrix at the parasite-free equilibrium given byV = DV|(x06) =

Q1 0 0 0 0 0 0 0 00 Q2 0 0 0 0 0 0 00 0 Q3 0 0 0 0 0 00 0 0 Q4 0 0 0 0 00 0 0 0 Q5 0 0 0 00 0 0 0 0 Q6 0 0 00 0 0 0 0 0 µcr 0 00 0 0 0 0 0 0 n3µimc 00 0 0 0 0 0 0 0 µg

.

WhereQ1 = mre + µil, Q2 =

(Πlβp

µsl

+ αp + µp − Πp

)

,

Q3 = Nktpre + n1µcl, Q4 = mre + µri,

Q5 = k13 − kr + µmr + k7re, Q6 = ω + µt.


Chapter 5. A within host model with three stages of malaria life y le 68The produ t FV−1 is given byFV−1 =

0 Πl(1−ǫ1)βl

Q2

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 00 αp

Q2

Q3 0 0 0 0 0 0

0 0 0 0 k(1−ǫ1)Sr

µrQ5

0 0 0 0

0 0 n1µil

Q3

0 0 0 0 0 0

0 0 0 0 γk13Q5

0 0 0 0

0 0 0 0 0 ωω+µt

0 0 0

0 0 0 0 0 0 0 0 Πmcβmc

µgµmc

0 0 0 0 (1−γ)k13Q5

0 0 0 0

,

The largest eigenvalue of FV−1 is alled the reprodu tion number and is given by;R06 = 5

√(Πmc(1− ǫ1)βmc

µgµmc

)(Πlβl

Q2

)(k(1− ǫ1)Sr

µrQ5

)(n1µil

Q3

)(ω

ω + µt

) (5.16)R∗

06 =

(Πmc(1− ǫ1)βmc

µgµmc

)(Πlβl

Q2

)(k(1− ǫ1)Sr

µrQ5

)(n1µil

Q3

)(ω

ω + µt

) (5.17)Sin e ǫ1 ranges between 0 and 1. We see that as ǫ1 −→ 1, R∗06 −→ 0 and as ǫ1 −→ 0,

R∗06 = R∗

07 =

(Πmcβmc

µgµmc

)(Πlβl

Q2

)(kSr

µrQ5

)(n1µil

Q3

)(ω

ω + µt

)

, (5.18)whereR07 =

5

√

R07∗ = 5

√(Πmcβmc

µgµmc

)(Πlβl

Q2

)(kSr

µrQ5

)(n1µil

Q3

)(ω

ω + µt

)

. (5.19)We want to determine, in the se tion on simulation, the least e� a y ǫ∗1 < 1 for whi h thedisease lears. The reprodu tion numbers R06 and R07 are positive ifk13 + µmr + k7re > kr and Πlβp

µsl+ αp + µp > ΠpWe an explain R∗

06 in terms of the three stages of the malaria life y le as follows; The ontribution to the reprodu tion number R06 by ea h stage is explained below: R0L =(Πlβl

Q2

) de�nes the reprodu tive ratio for the liver stage. R0B =(

kSr

µrQ5

)(n1µil

Q3

)(ω

ω+µt

)de�nes the reprodu tion ratio for the red blood stage and R0M =(

Πmcβmc

µgµmc

) de�nes thereprodu tion ratio for the mosquito stage. The �fth root of the reprodu tion number R06indi ates the path the parasite undergoes through from the mosquito through the liver,the blood stream before starting another y le in the mosquito. We an state the following


Chapter 5. A within host model with three stages of malaria life y le 69stability theorem for the parasite-free equilibrium point.Theorem 5.2.1 The parasite-free equilibrium is lo ally asymptoti ally stable if R06 < 1,otherwise it is unstable.


Chapter 5. A within host model with three stages of malaria life y le 705.2.3 SimulationsThe model equations (5.1) to (5.13) require several input parameter values, whi h an be di-vided into three sets, namely (i) those that have been measured lini ally or experimentally[TABLE 5.4℄ and [TABLE 5.5℄, (ii) those that have been estimated by other resear hers[TABLE 5.4℄ and [TABLE 5.5℄, (iii) those that have been estimated by us [TABLE 5.4℄and [TABLE 5.5℄.TABLE. 5.4. The table with the parameters, values and sour eParameter Value/range of values Sour eΠl 9.9 ∗ 105 < Πl < 2 ∗ 1011 Estimated [15, 19, 68℄βl 4 ∗ 10(−10) [19℄µsl 0.004 < µsl < 0.01 Estimated[19℄kl 0.2 EstimatedN 32 [55℄m 10(−8) [5℄µil 0.01 < µil < 0.1 [19℄Πp 0 < Πp < 105 Estimated [47, 87℄βp 4 ∗ 10(−10) < βp < 4 ∗ 10(−9) Estimatedαp 0.2 < αp < 0.5 Estimatedk12 0.2 < k12 < 0.3 Estimatedn3 5 < n3 < 30 Estimatedµimc 0.01 < µimc < 0.2 Estimatedµp 0.001 < µp < 0.01 Estimatedkcl 0.02 Estimatedktp 0.0001 < ktp < 0.25 Estimatedn1 12 < n1 < 32 [55℄µcl 0.01 EstimatedSr 2.5 ∗ 108 < Sr < 2.5 ∗ 109 Estimated [89, 101℄k 2 ∗ 10(−9.25) < k < 2 ∗ 10(−8) Estimated [89℄Sensitivity analysisFIG. 5.1 shows how some parameters are orrelated to R07, and how sensitive R07 is to hanges in these parameters. Knowledge of how these parameters a�e t R07 helps todetermine whether the severity of the disease will be overestimated or underestimated asthese parameters vary. If a parameter is positively orrelated to R07 then in reasing thatparameter overestimates the severity of the disease. On the other hand, if a parameteris negatively orrelated to R07 then hanges in that parameter ould underestimate the


Chapter 5. A within host model with three stages of malaria life y le 71TABLE. 5.5. The table with the parameters, values and sour eParameter Value/range of values Sour eµr 0.01 [101℄kb 0.4 Estimatedµri 0.2 [101℄kr 0.001 < kr < 0.01 Estimatedk7 10(−8) [5, 89℄k13 0.6 Estimatedµmr 0.0208 < µmr < 0.1 Estimated [89℄γ 0.003 < γ < 0.4 Estimatedω 10(−4) < ω < 0.01 Estimatedµt 0.01 Estimatedk11 0.02 Estimatedµcr 0.2 EstimatedΠmc 2.4 ∗ 102 < Πmc < 2.4 ∗ 104 [17℄βmc 10(−8) < βmc < 10(−7) Estimatedµmc 10(−10) < µmc < 0.025 Estimatedµimc 0.01 < µimc < 0.2 Estimatedµg 0.04 < µg < 0.4 Estimated [89℄ωe 0.04 Estimatedre 2.1 ∗ 105 [97, 100℄severity of the disease.The range of the parameter n1 is known and is given in M Queen and M Kenzie [55℄. Theparameter µsl is signi� antly positively orrelated to R07. This is an important parameterestimated in [19℄. Parameters βl and µil are given in [19℄ and parameter Πmc is given in[17℄. The parameters Πl, Sr, k, whi h have not been estimated lini ally or experimentallyare estimated in [15, 19, 68, 89, 101℄ respe tively. We an see from the sensitivity analysisthat the reprodu tion number R07 is highly positively orrelated to ω, rate of loss orgain of trophozoites, βmc, infe tion rate and kr, growth rate due to infe tion of RBCs.Parameters k7,N, re, µr, are given/estimated in [5, 55, 89, 97, 100, 101℄ respe tively. Theparameters µg and µmr have not been determined lini ally or experimentally but areestimated in [89℄. We an also see that, the reprodu tion number is negatively orrelatedto ktp, µp, µcl, µmc, αp, k13, βp, µt. There is a need to determine these parameters either lini ally or experimentally.The time used in our simulations is from t = 0 days to 60 days, whi h, lini ally, is thetime for an infe ted person to show symptoms of malaria. In this example, we have startedtreatment on 30th day after the initial infe tion. This is be ause, the average time for lini al symptoms to appear is estimated to be between (8− 25) days although this period



0 5 10 15 20 25−1.5

−1

−0.5

0

0.5

1

1.5

n1

µil S

rµ

slΠ

mc k

kr

βmc ω β

l

πl

µmr

k7

N ktp r

e µg

βp

µp

µcl

µmc

k13 µ

rµ

t

αp

Parameters

Se

nsitiv

ity In

dic

es

FIG. 5.1. Diagram shows sensitivity analysis of R07. an be mu h longer depending on the immune system of the host [48℄.


Chapter 5. A within host model with three stages of malaria life y le 735.3 ResultsThe numeri al simulation of the model equations (5.1) to (5.13) onsiders how hanges inthe parameters a�e t the dynami s of the disease and the role of treatment.5.3.1 Dynami s of the system before treatment

0 100 200 300 400 500 6002

4

6

8

10x 10

10

Time (days)

Su

sce

ptib

le L

VC

s

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5x 10

5

Time (days)

Infe

cte

d L

VC

s

0 100 200 300 400 500 6000

100

200

300

Time (days)

Sp

oro

zo

ite

s

0 100 200 300 400 500 6000

1

2

3

4x 10

5

Time (days)

Sch

izo

nts

in

LV

Cs

FIG. 5.2. Diagram shows the parasite-free equilibrium (DFE) at liver stage with R07 =0.0063.FIG. 5.2 shows the time plots for the various lasses during the liver stage. Sus eptibleliver ells (LVCs) drop initially but eventually settle at parasite-free level. All the infe tedLVCs, sporozoites and s hizonts inside LVCs tend to zero and the disease does not establishitself.FIG. 5.3 shows the time plots for the various lasses during the blood stage of the disease.The sus eptible RBCs drop initially but eventually settle at parasite-free level whi h is



0 100 200 300 400 500 6003

4

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 100 200 300 400 500 6000

2

4x 10

6

Time (days)

In

fe

cte

d R

BC

s

0 100 200 300 400 500 6000

1

2x 10

5

Time (days)

Me

ro

zo

ite

s

0 100 200 300 400 500 6000

5

10x 10

5

Time (days)

Tro

po

zo

ite

s

0 100 200 300 400 500 6000

200

400

Time (days)

Sch

izo

nts in

R

BC

s

0 100 200 300 400 500 6000

2

4x 10

5

Time (days)

Effe

cto

r ce

lls

FIG. 5.3. Diagram shows the parasite-free equilibrium (DFE) at blood stage R07 = 0.0063.high enough to sustain the life. All the infe ted lasses tend to zero. The e�e tor ellsin rease logisti ally.FIG. 5.4 shows the various lasses during the mosquito stage of the repli ation y le. The



0 200 400 6008

9

10x 10

5

Time (days)

Su

sce

ptib

le M

GC

s

0 200 400 6000

2

4x 10

4

Time (days)In

fecte

d M

GC

s

0 100 200 300 400 500 6000

5

10

15x 10

4

Time (days)

Ga

me

tocyte

s

FIG. 5.4. Diagram shows the parasite-free equilibrium (DFE) at mosquito stage R07 =0.0063.sus eptible MGCs drop initially and settle at parasite-free level. All the infe ted lassestend to zero.



0 100 200 300 400 500 6001

2

3

4x 10

10

Time (days)

Su

sce

ptib

le

L

VC

s

0 100 200 300 400 500 6000

500

Time (days)In

fe

cte

d L

VC

s

0 100 200 300 400 500 6000

10

20

Time (days)

Sp

oro

zo

ite

s

0 100 200 300 400 500 6000

2

4x 10

5

Time (days)

Sch

izo

nts in

L

VC

s

FIG. 5.5. Diagram shows the parasite-present equilibrium point (EEP) at liver stageR07 = 4.7265.FIG. 5.5, FIG. 5.6 and FIG. 5.7 are time plot of various lass during the liver, blood stageand mosquito stages respe tively. It is lear that the disease establishes itself.



0 100 200 300 400 500 600

4.6

4.8

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 100 200 300 400 500 6000

2

4x 10

7

Time (days)

In

fe

cte

d R

BC

s

0 100 200 300 400 500 6000

5

10x 10

4

Time (days)

Me

ro

zo

ite

s

0 100 200 300 400 500 6000

5000

Time (days)

Tro

po

zo

ite

s

0 100 200 300 400 500 6000

200

400

Time (days)Sch

izo

nts in

R

BC

S

0 100 200 300 400 500 6000

500

1000

Time (days)

Effe

cto

r ce

lls

FIG. 5.6. Diagram shows the parasite-present equilibrium point (EEP) at blood stageR07 = 4.7265.



0 200 400 6000.5

1

1.5x 10

5

Time (days)

Su

sce

ptib

le M

GC

s

0 200 400 6000

500

1000

Time (days)In

fecte

d M

GC

s

0 100 200 300 400 500 6000

5

10x 10

5

Time (days)

Ga

me

tocyte

s

FIG. 5.7. Diagram shows the parasite-present equilibrium point (EEP) at mosquito stageR07 = 4.7265.FIG. 5.8 shows the variation of ktp and n1 with R06. As ktp in reases it redu es the re-produ tion number R07 in agreement the results of our sensitivity analysis. On the otherhand, in reasing µil in reases R07 FIG. 5.9.In reasing killing rate of merozoites by e�e tor ells, redu es the disease manifestation inthe blood as illustrated in the FIG. 5.10. Also in reasing the growth rate of merozoites,in reases the severity of the disease manifested by a sharp de line in population of RBCs.


Chapter 5. A within host model with three stages of malaria life y le 790.18

0.18

0.2

0.2

0.2

0.2

0.22

0.22

0.22

0.22

0.24

0.240.24

0.260.26

0.26

0.280.28

0.28

0.3

n1

ktp

12 14 16 18 20 22 24 26 28 30 32

0.5

1

1.5

2x 10

−3

FIG. 5.8. Shows the ontour plot of R07 as a fun tion of an average number of s hizontsrelease from an infe ted liver ells that die naturally (n1) and the rate of loss of s hizontsinside liver ells that are killed by e�e tor ells (ktp).


Chapter 5. A within host model with three stages of malaria life y le 800.4

0.4

0.45

0.45

0.5

0.5

0.5

0.5

0.55

0.55

0.55

0.55

0.55

0.6

0.6

0.6

0.6

0.65

0.65

0.65

0.65

0.7

0.7

0.7

0.75

0.750.75

0.80.8

Natural death of an infected LVCs (µil)

N

atu

ra

l d

ea

th

o

f su

sce

ptib

le M

GC

s ( µ

mc)

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.021

2

3

4

5

6

7

8

9

10x 10

−6

FIG. 5.9. Shows the ontour plot of R07 as a fun tion of the natural death of an infe tedliver ells (µil) and natural death of sus eptible midgut ells (µmc).



0.297

0.297

0.2975

0.2975

0.2975

0.298

0.298

0.298

0.2985

0.2985

Growth rate due to infection of RBCs (kr)

Ra

te o

f killin

g o

f m

ero

zo

ite

s b

y e

ffe

cto

r c

ells (

k7 )

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

1

2

3

4

5

6

7

8

9x 10

−6

FIG. 5.10. Shows the ontour plot of R07 as a fun tion of the growth rate due to infe tionof RBCs (kr) the rate of killing of merozoites by e�e tor ells(k7).


Chapter 5. A within host model with three stages of malaria life y le 825.3.2 Treatment strategy

0 10 20 30 40 50 602.4

2.6

2.8

3

3.2x 10

10

Time (days)

Su

sce

ptib

le L

VC

s

0 10 20 30 40 50 600

50

100

150

Time (days)

Infe

cte

d L

VC

s

0 10 20 30 40 50 600

1

2

3

4

Time (days)

Sp

oro

zo

ite

s

0 10 20 30 40 50 600

0.5

1

1.5

2x 10

5

Time (days)

Sch

izo

nts

in

sid

e L

VC

s

ε1=0, ε

1=0.3, ε

1=0.7 ε

1=0.99

FIG. 5.11. Diagram shows an appli ation of treatment after 30 days at liver stage andǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒ R06 = 2.9200,ǫ1 = 0.99 =⇒ R06 = 0.7491.FIG. 5.11 shows time plot of various lasses during the liver stage as treatment is admin-istered for drug e� a y ranging from ǫ1 = 0 to ǫ1 = 0.99. The sus eptible liver ell popu-lations do not hange due to learan e of the pathogen as the drug e� a y in reases. Theinfe ted liver ell populations, Sporozoites and s hizonts de rease with in reasing e� a y.The reprodu tion number is a de reasing fun tion of e� a y, for example, R06 = 4.0980for ǫ1 = 0 and R06 = 0.7491 for ǫ1 = 0.99.FIG. 5.12 shows time plot of various lasses during the blood stage of the disease in the



0 10 20 30 40 50 604.9

4.95

5x 10

9

Time (days)

Su

sce

ptib

le R

BC

s

0 10 20 30 40 50 600

1

2x 10

7

Time (days)

In

fe

cte

d R

BC

s

0 10 20 30 40 50 600

2

4x 10

4

Time (days)

Me

ro

zo

ite

s

0 10 20 30 40 50 600

500

1000

Time (days)

Tro

po

zo

ite

s

0 10 20 30 40 50 600

2040

Time (days)

Sch

izo

nts in

sid

e R

BC

s

0 10 20 30 40 50 600

500

1000

Time (days)

Effe

cto

r ce

lls

ε1=0, ε

1=0.3, ε

1=0.7, ε

1=0.99

FIG. 5.12. Diagram shows an appli ation of treatment after 30 days at blood stage andǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒ R06 = 2.9200,ǫ1 = 0.99 =⇒ R06 = 0.7491.presen e of treatment. The sus eptible RBCs population in reases with in reasing treat-ment e� a y. The infe ted lasses de rease with in reasing treatment e� a y.



0 10 20 30 40 50 601

1.05

1.1

1.15x 10

5

Time (days)

Su

sce

ptib

le M

GC

s

0 10 20 30 40 50 600

50

100

150

200

250

Time (days)In

fecte

d M

GC

s

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5x 10

5

Time (days)

Ga

me

tocyte

s

ε1=0, ε

1=0.3, ε

1=0.7, ε

1=0.99.

FIG. 5.13. Diagram shows an appli ation of treatment after 30 days at mosquito stageand ǫ1 = 0 =⇒ R06 = 4.7265, ǫ1 = 0.3 =⇒ R06 = 4.0980, ǫ1 = 0.7 =⇒ R06 = 2.9200,ǫ1 = 0.99 =⇒ R06 = 0.7491.


Chapter 5. A within host model with three stages of malaria life y le 855.4 Dis ussionWe investigated the intera tion between the malaria parasites and the immune ells forall the three stages of malaria life y le. The model has been studied and results werepresented. The e�e t of treatment was dis ussed using numeri al results with reasonable hoi es of parameter values. Therapy whi h was represented by drug e� a y ǫ1 and tar-geted to kill sporozoites, merozoites and gemoto ytes.The model investigated the intera tion between the malaria parasites and the immune ellsfor all three stages of the malaria life y le. The three- stage malaria model presents many hallenges in that there are many parameters that are not known lini ally. However, us-ing the known parameters, we have performed a sensitivity analysis ([7℄) whi h has helpedto determine the magnitude and range of our unknown parameters. This analysis hasshown that the reprodu tion number R07 is positively orrelated to the rate of gain/lossof trophozoites (ω), infe tion rate (βmc), and growth rate due to infe tion of RBCs (kr),and is negatively orrelated to rate of loss of s hizonts inside the liver ells that are killedby e�e tor ells (ktp), natural death of sporozoites (µp), natural death of s hizonts insidethe liver ells (µcl), natural death of sus eptible midgut ells (µmc), rate of gain/loss ofsporozoites (αp), rate of loss of merozoites (k13), infe tion rate (βp) and natural death oftrophozoites (µt). Overestimation of rate of gain/loss of trophozoites (ω), βmc, and growthrate due to infe tion of RBCs (kr) or underestimation of rate of loss of s hizonts insidethe liver ells that are killed by e�e tor ells (ktp), µp, µcl, µmc, αp, k13, βp and µt an haveserious impli ations regarding the prognosis of the disease. There is a need for lini al orexperimental determination of these parameters.The ontour plot results (FIG. 5.8, FIG. 5.9 and FIG. 5.10) have shown how hanges insome parameters a�e ted the reprodu tion number. The results are in agreement withthe sensitivity and un ertainty analysis results. The graphs orresponding to R07 < 1 andthose orresponding to R07 > 1 are in agreement with the predi tions of (Theorem 5.2.1).As in the previous hapter on treatment, treatment had the e�e t of slowing down thedepletion of sus eptible ells and learan e the parasite populations. Our results showedthat when using therapy with an e� a y of ǫ1 = 0.99, the infe ted liver ells, and infe tedred blood ells take 10 days to revert to a parasite-free equilibrium state R06 = 0.7494.As in the previous Chapter 4, we re ommend a hange in the manner treatment is ad-ministered. Spe i� ally, malaria treatment should be administered at health entres liketuber ulosis and in the presen e of healthy workers. We believe that only a malaria va ine an reliably prote t against all stages of malaria infe tion.


Chapter 6Con lusionMathemati al models of intera tions of malaria parasites and the host immune system havebeen presented in this study. We dis ussed the mathemati al and numeri al analysis ofthese models. We then introdu ed treatment e� a y (ǫ1) to enable us explore the e�e t oftreatment on the pathogenesis of malaria. The results obtained from the models presentedin Chapters 4, and 5 are as follows:1. The model of the blood stage in Chapter 4, revealed that parasite repli ative hara -teristi s allow the parasite to evade the immune response during the red blood stageof malaria infe tion. We found that, the larger a threshold number of intra ellularparasites released as a results of the natural death of an a tivated RBC (n1) is, thelonger it takes to produ e the parasites and the higher the han e of an infe ted redblood ell being identi�ed and apoptosised by the e�e tor ells. Hen e we on ludedthat in order to minimize the possibility of infe ted red blood ells being dete tedand eliminated by the immune ells, the parasite infe ts older red blood ells whoselife expe tan y is mu h shorter than the younger ells thereby avoiding the infe tedred blood ell from being dete ted and apoptosised.2. The e�e t of drug e� a y at blood stage y le in Chapter 4 showed that, a high druge� a y of ǫ1 = 0.95 ould stop the development of the disease. Sin e most malariatreatment drugs are of higher e� a y than 0.6, it is possible to ombat malaria withtreatment drugs but the administration of drugs should be done at health entres orhospitals to ensure that patients omplete their treatment.3. The model with three stages of malaria life y le in Chapter 5, showed that, as thedrug e� a y in reases up to ǫ1 = 0.99 and R06 = 0.7494, the populations of theinfe ted LVC and infe ted RBC approa h the parasite-free equilibrium after 10 days.Sporozoites, s hizonts and gameto ytes took more than 10 days to approa h the dis-ease free equilibrium with an e� a y of ǫ1 = 0.99. We found that, treatment had the86


Chapter 6. Con lusion 87e�e t slowing down the depletion of sus eptible ells and learing the parasite popu-lations. This showed that an e�e tive drug treatment ould stop the development ofdisease.6.1 Limitations and re ommendationsDue to the detailed nature of the models developed in this study, some of the parametersused have not been lini ally or experimentally determined before. This has been thebiggest hallenge in our assessment. The models studied in this thesis la ked data setsthat would be �tted on them so as to validate the predi tions of the observed outputs.We therefore re ommend that, the lini ians or experimentalists estimate the unknownand important parameters su h as: rate of loss of s hizonts inside the liver ells that arekilled by e�e tor ells (ktp), threshold number as a results of gain due to infe tion ofsus eptible RBC by extra ellular parasites (n∗k∗), rate of loss of s hizonts due to burstingof infe ted RBCs (k11), rate of loss of infe ted RBCs due to bursting of infe ted RBCs(kb), proportion of merozoites (γ), proportion of RBCs (α), growth rate of e�e tor ells(ωe), natural death of infe ted midgut ells (µimc), natural death of s hizonts inside RBCs(µcr), growth rate of s hizonts (kcl), natural death of sporozoites (µp), natural death ofsporozoites (n3), rate of bursting of infe ted midgut ells (k12), rate of bursting of infe tedliver ells (kl), that have been in�uential in the predi tions of the models outputs. Were ommend that therapy with highly e� a ious drugs would be an e�e tive ontrol measurein eradi ating malaria at all stages but we think that only a malaria va ine ould reliablyprote t against all stages malaria infe tion. We have also shown that hroni infe tions an transform manageable malaria into a more a tive disease. The re ommendation fromour study is that; in malaria endemi areas, individuals with malaria or showing malariasymptoms should be tested for hroni infe tions and those who test positive for any hroni infe tion should be treated for both malaria and the hroni infe tion. We also re ommenda hange in manner treatment is administered. Spe i� ally, malaria treatment should beadministered at health entres6.2 Future workThe models analysed in this work ould be extended as follows:1. Sin e the model did not in orporate the e�e t of ytokines (how do ell ommuni ate),we hope to extend the model by onsidering the e�e t of ytokines.2. Sin e there is relationship between malaria and HIV/AIDS, we hope to in lude themodel of o-infe tion of HIV and Malaria.


Appendix AParameters values and initial variablesused in simulationsIn this Appendix we will show some of the parameters used to draw the �gures on ourmodel.TABLE. A.1. The table that shows the initial variables that used in FIG. (4.3,4.4)Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values(DFE) 5 ∗ 109 0 0 50 0 200Values(EEP) 5 ∗ 109 0 0 107 0 200TABLE. A.2. The table with parameters values used in FIG. (4.3)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.2) 0.01 2 ∗ 10(−9) 0.2 0.0001 0.01 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 20Parameters kpi n1 m ktp reValues 0.08745 24 10(−8) 0.0009 2000

88


Appendix A: Parameters values and initial variables used in simulations 89TABLE. A.3. The table with parameters values used in FIG. (4.4)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.2) 0.01 2 ∗ 10(−9.46) 0.2 0.001 0.008 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 15Parameters kpi n1 m ktp reValues 0.08745 12 10(−8) 0.0009 800TABLE. A.4. The table that shows the initial variables that used in FIG. (4.5,4.6)Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values(DFE) 1011 0 0 10(10) 0 200TABLE. A.5. The table with parameters values used in FIG. (4.5,4.6)Parameters Sr µr k α γ µrl kbValues 2 ∗ 10(9) 0.01 2 ∗ 10(−9) 0.1 0.04 0.01 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 10Parameters kpi m ktp reValues 0.08745 10(−8) 0.01 4000TABLE. A.6. The table that shows the initial variables that used in FIG. (4.7)Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values(DFE) 5 ∗ 109 0 0 5 ∗ 10(7) 0 200 1 1 1TABLE. A.7. The table with parameters values used in FIG. (4.7)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.2) 0.01 2 ∗ 10(−9.46) 0.2 0.0001 0.008 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 20Parameters kpi m ktp re n1Values 0.08745 10(−8) 0.008745 880 12TABLE. A.8. The table that shows the initial variables that used in FIG. (4.8,4.9, 4.10)Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values(4.8) 5 ∗ 109 0 0 5 ∗ 10(7) 0 200Values(4.9) 5 ∗ 109 0 0 5 ∗ 10(8) 10 200


Appendix A: Parameters values and initial variables used in simulations 90TABLE. A.9. The table with parameters values used in FIG. (4.8,4.9,4.10)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.2) 0.01 2 ∗ 10(−9.46) 0.2 0.008 0.008 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 20Parameters kpi m ktp re n1Values 0.08745 10(−8) 0.008745 880 12TABLE. A.10. The table that shows the initial variables that used in FIG.(4.11,4.12,4.13,4.14) Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values 5 ∗ 109 0 0 50 0 200TABLE. A.11. The table with parameters values used in FIG. (4.11,4.12, 4.13,4.14)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.2) 0.01 2 ∗ 10(−9) 0.2 0.0001 0.01 0.4Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.01 10(−8) 0.0208 0.04 20Parameters kpi m ktp re n1Values 0.08745 10(−8) 0.008745 2000 24TABLE. A.12. The table that shows the initial values used in FIG. (4.15,4.16)Variables R(0) Rl(0) Ra(0) Pi(0) Pe(0) E(0)Values 5 ∗ 109 0 0 107 0 200TABLE. A.13. The table with parameters and values used in FIG. (4.15,4.16)Parameters Sr µr k α γ µrl kbValues 2.5 ∗ 10(7.3) 0.01 2 ∗ 10(−9.25) 0.1 0.001 0.008 0.5Parameters µra N k11 n∗k∗ µpe ω SpeValues 0.014 32 0.02 10(−8) 0.0208 0.05 20Parameters kpi n1 m ktp reValues 0.08745 12 10(−8) 0.0009 880TABLE. A.14. The table that shows the initial values used in FIG. (5.2,5.3,5.4)Variables Sl(0) Il(0) P (0) Cl(0) R(0) Ri(0) MrValues 1011 0 0 1 5 ∗ 109 0 1Variables T (0) Cr(0) Smc(0) Imc(0) G(0) E(0)Values 1 1 106 0 1 200


Appendix A: Parameters values and initial variables used in simulations 91TABLE. A.15. The table with parameters and values used in FIG. (5.2,5.3,5.4)Parameters πil βl µsl kl N m µilValues 3 ∗ 10(7.5) 4 ∗ 10(−10) 0.004 0.2 32 10(−8) 0.01Parameters αp k12 n3 µimc µp kcl ktpValues 0.2 0.2 30 0.01 0.01 0.02 0.25Parameters Sr k µr kb µri µil krValues 2.5 ∗ 10(7.2) 2 ∗ 10(−9.25) 0.01 0.4 0.2 0.01 0.001Parameters γ ω µt k11 µcr πmc βmcValues 0.4 10(−4) 0.01 0.02 0.2 2.4 ∗ 104 10(−7)Parameters µg ωe re βp n1 µmr k13Values 0.4 0.04 2.1 ∗ 105 4 ∗ 10(−10) 24 0.0208 0.6Parameters µcl k7 µmc πpValues 0.01 10(−8) 0.025 0.08

TABLE. A.16. The table that shows the initial values used in FIG. (5.5,5.6,5.7)Variables Sl(0) Il(0) P (0) Cl(0) R(0) Ri(0) MrValues 1010.5 0 0 1 5 ∗ 109 0 1Variables T (0) Cr(0) Smc(0) Imc(0) G(0) E(0)Values 1 1 105 0 1 200

TABLE. A.17. The table with parameters and values used in FIG.(5.8,5.9, 5.10,5.5,5.6,5.7,5.11,5.12, 5.13)Parameters πil βl µsl kl N m µilValues 3 ∗ 10(7.7) 4 ∗ 10(−10) 0.01 0.2 32 10(−8) 0.01Parameters αp k12 n3 µimc µp kcl ktpValues 0.5 0.3 5 0.2 0.001 0.02 0.0001Parameters Sr k µr kb µri µil krValues 2.5 ∗ 10(7.3) 2 ∗ 10(−8) 0.01 0.4 0.2 0.01 0.01Parameters γ ω µt k11 µcr πmc βmcValues 0.4 10(−4) 0.01 0.02 0.2 2.4 ∗ 104 10(−7)Parameters µg ωe re βp n1 µmr k13Values 0.4 0.04 2.1 ∗ 105 4 ∗ 10(−9) 12 0.01 0.6Parameters µcl k7 µmc πpValues 0.01 10(−8) 10(−10) 0.08


Bibliography[1℄ R. M. Anderson, R. M. May, and S. Gupta. Non-linear phenomena in host-parasiteintera tions. Parasitology, 99:S59 � S79, 1989.[2℄ R. Antia, A. Yates, and J. C. de Roode. The dynami s of a ute malaria infe tions.I. E�e t of the parasite's red blood ell preferen e. Pro eedings of the Royal So ietyB, 275:1449 � 1458, 2008.[3℄ K. Artavanis-Tsakonas and E. M. Riley. Innate immune response to malaria: Rapidindu tion of IFN-γ from human NK ells by live plasmodium fal iparum-infe tederythro ytes. The Journal of Immunology, 169:2956 � 2963, 2002.[4℄ K. Artavanis-Tsakonas, J. E. Tongren, and E. M. Riley. The war between the malariaparasite and the immune system: immunity, immunoregulation and immunopathol-ogy. Clini al and Experimental Immunology, 133:145 � 152, 2003.[5℄ D. J. Austin, N. J. White, and R. M. Anderson. The dynami s of drug a tion on thewithin-host population growth of infe tious agents: Melding pharma okineti s withpathogen population dynami s. Journal of Theoreti al Biology, 194:313 � 339, 1998.[6℄ P. B. Bloland. Drug resistan e in malaria. World Health Organization (WHO), 2001.[7℄ S. M. Blower. Sensitivity and un ertainty analysis of omplex models of diseasetransmission: An HIV model, as an example. International Statisti al Review, 62:229� 243, 1994.[8℄ L. J. Bru e-Chwatt. Alphonse laveran's dis overy 100 years ago and today's global�ght against malaria. Journal of the Royal So iety of Medi ine, 74:531 � 536, 1981.[9℄ L. Cao, L. Krymskaya, V. Tran, S. Mi, M. C. Jensen, S. Blan hard, and M. Kalos.Development and appli ation of a multiplexable �ow ytometry-based assay to quan-tify ell-mediated ytolysis. Journal of the International So iety for Advan ement ofCytometry, 77A:534 � 545, 2010.[10℄ CDC. Parasite Image Library. http://dpd. d .gov/dpdx. Last a essed 27 September2011. 92


Bibliography 93[11℄ CDC. Malaria, Biology. http://www. d .gov/malaria/about/biology/index.html,2010 (review). Last a essed: 1 Mar h 2011.[12℄ N. Chitnis, J. M. Hyman, and J. M. Cushing. Determining important parametersin the spread of malaria through the sensitivity analysis of a mathemati al model.Bulletin of Mathemati al Biology, 2008.[13℄ C. Chiyaka, W. Garira, and S. Dube. Modelling immune response and drug therapyin human malaria infe tion. Computational and Mathemati al Methods in Medi ine,9:143 � 163, 2008.[14℄ K. Chotivani h, R. Udomsangpet h, J. A. Simpson, P. Newton, S. Pukrittayakamee,S. Looareesuwan, and N. J. White. Parasite multipli ation potential and the severityof fal iparum malaria. Journal of Infe tious Diseases, 181:1206 � 1209, 2000.[15℄ S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, and A. S. Perelson. Modeling the me h-anisms of a ute hepatitis B virus infe tion. Journal of Theoreti al Biology, 247:23 �35, 2007.[16℄ K. L. Cooke and P. van den Driess he. Analysis of an SEIRS epidemi model withtwo delays. Journal of Mathemati al Biology, 35:240 � 260, 1996.[17℄ A. Coppi, R. Natarajan, G. Pradel, B. L. Bennett, E. R. James, M. A. Roggero,G. Corradin, C. Persson, R. Tewari, and P. Sinnis. The malaria ir umsporozoiteprotein has fun tional domains, ea h with distin t roles as sporozoites journey frommosquito to mammalian host. The Journal of the Experimental Medi ine, pages 1 �16, 2011.[18℄ A. F. Cowman and B. S. Crabb. Invasion of red blood ells by malaria parasites.Cell, 124:755 � 766, 2006.[19℄ B. C. de Sousa and C. Cunha. Development of mathemati al models for the analysisof hepatitis delta virus viral dynami s. PLoS ONE, 5:e12512, 2010.[20℄ O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz. On the de�nition and the omputation of the basi reprodu tion ratio R0 in models for infe tious diseases inheterogeneous populations. Journal of Mathemati al Biology, 28:365 � 382, 1990.[21℄ K. Dietz, G. Raddatz, and L. Molineaux. Mathemati al model of the �rst wave ofplasmodium falsiparam asexual parasitemia in non- immune and va inated individ-uals. The Ameri an Journal of Tropi al Medi ine and Hygiene, 75:46 � 55, 2006.[22℄ H. Elsaesser, K. Sauer, and D. G. Brooks. IL-21 Is required to ontrol hroni viralinfe tion. S ien e, 324:1569 � 1572, 2009.[23℄ C. R. Engwerda and M. F. Good. Intera tions between malaria parasites and thehost immune system. Current Opinion in Immunology, 17:381 � 387, 2005.


Bibliography 94[24℄ L. Esteva, A. B. Gumel, and C. V. de Leon. Qualitative study of transmissiondynami s of drug-resistan e malaria. Mathemati al and Computer Modelling, 50:611� 630, 2009.[25℄ A. Friedman, J. Turner, and B. Szomolay. A model on the in�uen e of age onimmunity to infe tion with my oba terium tuber ulosis. Experimental Gerontology,43:275 � 285, 2008.[26℄ H. Fujioka and M. Aikawa. Stu ture and life y le. Chemi al Immunology, 80:1 � 26,2002.[27℄ S. Gandon, M. J. Ma kinnon, S. Nee, and A. F. Read. Imperfe t va ines and theevolution of pathogen virulen e. Nature, 414:751 � 756, 2001.[28℄ M. B. Gravenor, A. R. M Lean, and D. Kwiatkowski. The regulation of malariaparasitaemia: parameter estimates for a population model. Parasitology, 110:115 �122, 1995.[29℄ D. T. Haydon, L. Matthews, R. Timms, and N. Colegrave. Top-down or bottom-upregulation of intra-host blood-stage malaria: do malaria parasites most resemble thedynami s of prey or predator? Pro eedings of the Royal So iety B, 270:289 � 298,2003.[30℄ B. Hellriegel. Modelling the immune response to malaria with e ologi al on epts:short-term behaviour against long-term equilibrium. Pro eedings of the Royal So ietyB, 250:249 � 256, 1992.[31℄ C. Hetzel and R. M. Anderson. The within-host ellular dynami s of bloodstagemalaria theoreti al and experimental studies. Parasitology, 113:25 � 38, 1996.[32℄ A. Hoare, D. G. Regan, and D. P. Wilson. Sampling and sensitivity analysis tools(SaSAT) for omputational modelling. Theoreti al Biology and Medi al Modelling,5:4, 2008.[33℄ M. R. Hollingdale and U. Krzy h. Immune responses to liver-stage parasites: Impli- ations for va ine development. Chemi al Immunology, 80:97 � 124, 2002.[34℄ M. B. Hoshen, R. Heinri h, W. D. Stein, and H. Ginsburg. Mathemati al modellingof the within-host dynami s of plasmodium fal iparum. Parasitology, 121:227 � 235,2000.[35℄ A. Iggidr, J. Kamgang, G. Sallet, and J. Tewa. Global analysis of new malariaintrahost models with a ompetitive ex lusion prin iple. SIAM Journal of AppliedMathemati s, 67:260 � 278, 2006.


Bibliography 95[36℄ IUPUI. ECE680 Modern automati ontrol Routh's stability riterion.http://www.engr.iupui.edu/ skoskie/ECE680/Routh.pdf, 2007. Last a essed 17September 2011.[37℄ B. Jayabalasingham, N. Bano, and I. Coppens. Metamorphosis of the malaria parasitein the liver is asso iated with organelle learan e. Cell Resear h, 20:1043 � 1059, 2010.[38℄ H. J. Kaiser. HIV/Malaria o-Infe tion nearly doubles HIV viral load, in reases han e of transmission, study says. http://www.thebody. om/ ontent/art9124.html,2005. Last a essed: 1 Mar h 2011.[39℄ V. Kalia, S. Sarkar, and R. Ahmed. CD8 T- ell memory di�erentiation during a uteand hroni viral infe tions. Landes Bios ien e and Springer S ien e + BusinessMedia, 2010.[40℄ D. E. Kirs hner. Dynami s of o-infe tion with m. tuber ulosis and HIV-1. Theoret-i al Population Biology, 55:94 � 109, 1999.[41℄ A. J. Knell, editor. Malaria. Oxford University Press, 1991.[42℄ J. W. Koehler, M. Bolton, A. Rollins, K. Snook, E. de Haro, E. Henson, L. Rogers,L. N. Martin, D. J. Krogstad, M. A. James, J. Ri e, B. Davison, R. S. Veazey,R. Prabhu, A. M. Amedee, R. F. Garry, and F. B. Cogswell. Altered immune re-sponses in Rhesus ma aques o-Infe ted with SIV and plasmodium ynomolgi: Ananimal model for oin ident AIDS and relapsing malaria. PLoS ONE, 4:e7139, 2009.[43℄ C. M. Kribs-Zaleta and J. X. Velas o-Hernandez. A simple va ination model withmultiple endemi states. Mathemati al Bios ien es, 164:183 � 201, 2000.[44℄ G. Li and Z. Jin. Global stability of a SEIR epidemi model with infe tious for e inlatent, infe ted and immune period. Chaos, Solitons and Fra tals, 25:1177 � 1184,2005.[45℄ F. J. Lopez-Antunano and G. S hmunis, editors. Diagnosis of Malaria. Number 512.S ienti� Publi ation: Pan Ameri an Health Organization, USA, 1990.[46℄ MalariaSite. Control of malaria. http://www.malariasite. om/malaria/ControlOfMalaria.htm. Last a essed: 17 September 2011.[47℄ MalariaSite. Life y le. http://www.malariasite. om/malaria/LifeCy le.htm. Lasta essed: 30 September 2011.[48℄ G. L. Mandell, J. E. Bennett, and R. Dolin. Prin iples and Pra ti e of Infe tiousDiseases, 7th Edition. Chur hhill Livingstone, 2009.[49℄ D. P. Mason, F. E. M Kenzie, and W. H. Bossert. The blood-stage dynami s ofmixed plasmodium malariae-plasmodium fal iparum infe tions. Journal of Theoret-i al Biology, 198:549 � 566, 1999.


Bibliography 96[50℄ A. Matteelli, F. Castelli, and S. Caligaris. Life Cy le of Malaria Parasites, hapter 2,pages 17 � 23. University of Bres ia (Italy), 1997.[51℄ F. E. M Kenzie. Why model malaria? Parasitology Today, 16:511 � 516, 2000.[52℄ F. E. M Kenzie and W. H. Bossert. The dynami s of plasmodium fal iparum blood-stage infe tion. Journal of Theoreti al Biology, 188:127 � 140, 1997.[53℄ F. E. M Kenzie and W. H. Bossert. An integrated model of plasmodium fal iparumdynami s. Journal of Theoreti al Biology, 232:411 � 426, 2005.[54℄ F. E. M Kenzie, D. L. Smith, W. P. O'Meara, and E. M. Riley. Strain theory ofmalaria: The �rst 50 years. Advan es in Parasitology, 66:1 � 46, 2008.[55℄ P. G. M Queen and F. E. M Kenzie. Age-stru tured red blood ell sus eptibility andthe dynami s of malaria infe tions. Pro eedings of the National A ademy of S ien es,101:9161 � 9166, 2004.[56℄ P. G. M Queen and F. E. M Kenzie. Competition for red blood ells an enhan eplasmodium vivax parasitemia in mixed-spe ies malaria infe tion. The Ameri anJournal of Tropi al Medi ine and Hygiene, 75:112 � 125, 2006.[57℄ P. G. M Queen and F. E. M Kenzie. Host ontrol of malaria infe tions: Con-straints on immune and erythropoeiti response kineti s. PLoS Computation Biology,4:e1000149, 2008.[58℄ N. Mideo, T. Day, and A. F. Read. Modelling malaria pathogenesis. Cellular Mi ro-biology, 10:1947 � 1955, 2008.[59℄ S. A. Mikolaj zak, V. Ja obs-Lorena, D. C. Ma Kellar, N. Camargo, and S. H. I.Kappe. L-FABP is a riti al host fa tor for su essful malaria liver stage development.International Journal for Parasitology, 37:483 � 489, 2007.[60℄ L. H. Miller, D. I. Baru h, K. Marsh, and O. K. Doumbo. The pathogeni basis ofmalaria. Nature, 415:643 � 649, 2002.[61℄ O. R. Millington, C. D. Lorenzo, R. S. Phillips, P. Garside, and J. M. Brewer. Sup-pression of adaptive immunity to heterologous antigens during plasmodium infe tionthrough hemozoin-indu ed failure of dendriti ell fun tion. Journal of Biology, 5:5,2006.[62℄ L. A. Mohammed. Stability of linear ontrol systems: Routh's Criterion.www.uote hnology.edu.iq/dep-produ tion/media/pdf/FCLe 8.pdf. Last a essed 21September 2011.[63℄ R. H. Morrow and W. J. Moss. The Epidemiology and Control of Malaria, hapter 26,pages 1087 � 1138. Jones and Bartlett Publishers, 2007.


Bibliography 97[64℄ MVI. PATH malaria va ine initiative (MVI). http://www.malariava ine.org/. Lasta essed: 17 September 2011.[65℄ A. Niger and A. B. Gumel. Immune response and imperfe t va ine in malariadynami s. Mathemati al Population Studies, 18:54 � 86, 2011.[66℄ F. Nuwaha. The hallenge of hloroquine-resistant malaria in sub-Saharan Afri a.Health Poli y and Planning, 16:1 � 12, 2001.[67℄ F. Nyabadza. Modelling the role of prophylaxis in malaria prevention. InternationalJournal of Biologi al and Medi al S ien es, 1:18 � 23, 2007.[68℄ B. E. Petersen, W. C. Bowen, K. D. Patrene, W. M. Mars, A. K. Sullivan, N. Murase,S. S. Boggs, J. S. Greenberger, and J. P. Go�. Bone marrow as a potential sour e ofhepati oval ells. S ien e, 284:1168 � 1170, 1999.[69℄ M. Pruden io, A. Rodriguez, and M. M. Mota. The silent path to thousands ofmerozoites: the plasmodium liver stage. Nature Reviews Mi robiology, 4:849 � 856,2006.[70℄ RollBa kMalaria. Key malaria fa ts. http://www.rollba kmalaria.org/keyfa ts.html.Last a essed: 17 September 2011.[71℄ P. J. Rosenthal, editor. Anti-malarial Chemotherapy: Me hanisms of a tion, Resis-tan e, and Dire tions in Drug Dis overy. Humana Press In ., 2001.[72℄ D. Sa ks. BAC talk about ell type-spe i� regulation of human IL−10. Pro eedingsof the National A ademy of S ien es, 106:16895 � 16896, 2009.[73℄ K. J. Saliba and K. Kirk. pH Regulation in the intra ellular malaria parasite plas-modium fal iparum. The Journal of Biologi al Chemistry, 274:33213 � 33219, 1999.[74℄ A. Saltelli, S. Tarantola, F. Campolongo, and M. Ratto. Sensitivity Analysis in Pra -ti e: A Guide to Assessing S ienti� Models. John Wiley and Sons, Ltd, England,2004.[75℄ M. A. San hez and S. M. Blower. Un ertainty and sensitivity analysis of the basi reprodu tive rate tuber ulosis as an example. Ameri an Journal of Epidemiology,145:1127 � 1137, 1997.[76℄ C. Shi�. Integrated approa h to malaria ontrol. Clini al Mi robiology Reviews,15:278 � 293, 2002.[77℄ R. Shonkwiler and S. J. Aneke. Some mathemati al models for malaria. S hool ofMathemati s, 1999. Georgia Institute of Te hnology, Atlanta GA 30332.[78℄ L. Simpson. Life y le and transmission. http://dna.kdna.u la.edu/parasite ourse−old/malaria�les /sub hapters/life%20 y le.htm. Last a essed 30 September 2011.


Bibliography 98[79℄ T. Smith, G. F. Killeen, N. Maire, A. Ross, L. Molineaux, F. T. G. Hutton,J. Utzinger, K. Dietz, and M. Tanner. Mathemati al modelling of the impa t ofmalaria va ines on the lini al epidemiology and natural history of plasmodium fal- iparum malaria: Overview. Ameri an Journal of Tropi al Medi ine and Hygiene,75:1 � 10, 2006.[80℄ R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint, and S. I. Hay. The globaldistribution of lini al episodes of plasmodium fal iparum malaria. Nature, 434:214� 217, 2005.[81℄ D. A. Stevens. Combination immunotherapy and antifungal hemotherapy. Clini alInfe tious Diseases, 26:1266 � 1269, 1998.[82℄ M. M. Stevenson and E. M. Riley. Innate immunity to malaria. Nature ReviewsImmunology, 4:169 � 180, 2004.[83℄ A. Sturm, R. Amino, C. van de Sand, T. Regen, S. Retzla�, A. Rennenberg,A. Krueger, J. Pollok, R. Menard, and V. T. Heussler. Manipulation of host hepa-to ytes by the malaria parasites for delivery into liver sinosoids. S ien e, 313:1287 �1290, 2006.[84℄ Y. Su, S. Ruan, and J. Wei. Periodi ity and syn hronization in blood-stage malariainfe tion. Journal of Mathemati al Biology, 63:557 � 574, 2011.[85℄ A. Suhrbier, L. A. Winger, E. Castellano, and R. E. Sinden. Survival and anti-geni pro�le of irradiated malarial sporozoites in infe ted liver ells. Infe tion andImmunity, 58:2834 � 2839, 1990.[86℄ K. R. Tan, S. Mali, and P. M. Arguin. Malaria risk information and prophylaxis, by ountry. http://wwwn . d .gov/travel/yellowbook/2010/ hapter−2/malaria.htm.Last a essed: 29 June 2011.[87℄ M. I. Teboh-Ewungkem and T. Yuster. A within-ve tor mathemati al model ofplasmodium fal iparum and impli ation of in omplete fertilization on optimal game-to yte sex ratio. Journal of Theoreti al Biology, 264:273 � 286, 2010.[88℄ K. A. Trott, J. Y. Chau, S. Lu khart, and K. Abel. Enhan ed severity of malariainfe tion in simian immunode� ien y virus (SIV) infe ted rhesus ma aques. TheJournal of Immunology, 182:129.15, 2009.[89℄ J. Tumwiine, S. Lu khaus, J. Y. T. Mugisha, and L. S. Luboobi. An age-stru turedmathemati al model for the within host dynami s of malaria and the immune system.Journal of Mathemati al Modelling and Algorithms, 7:79 � 97, 2008.[90℄ UNICEF. Promoting rational use of drugs and orre t ase management in ba-si healthy servi es. UNICEF programme division in ooperation with the WorldHealthy Organization, 2000. Last a essed: 15 August 2011.


Bibliography 99[91℄ UNICEF. Malaria and hildren: progress in intervention overage. UNICEF andThe roll ba k Malaria partnership, 2007. Last a essed: 1 Mar h 2011.[92℄ USAID. Angola malaria indi ator survey 2006− 2007, 2007.[93℄ P. van den Driess he and J. Watmough. Reprodu tion numbers and sub-thresholdendemi equilibria for ompartmental models of disease transmission. Mathemati alBios ien es, 180:29 � 48, 2002.[94℄ E. J. Wherry and R. Ahmed. Memory CD8 T- ell di�erentiation during viral infe -tion. Journal of Virology, 78:5535 � 5545, 2004.[95℄ WHO. Guidelines for the treatment of malaria. World Health Organization.[96℄ WHO. Malaria. http://www.who.int/media entre/fa tsheets/fs094/en/, 2010. Lasta essed: 1 Mar h 2011.[97℄ E. N. Wiah, I. K. Dontwi, and I. A. Adetunde. Using mathemati al model to depi tthe immune response to hepatitis B virus infe tion. Journal of Mathemati s Resear h,3:157 � 167, 2011.[98℄ Wikipedia. Ronald Ross. http://en.wikipedia.org/wiki/RonaldRoss. Last a essed:17 September 2011.[99℄ Wikipedia. Sensitivity analysis. http://en.wikipedia.org/wiki/Sensitivityanalysis.Last a essed 23 September 2011.[100℄ Wikipedia. Adaptive immune system. http://en.wikipedia.org/wiki/Adaptive-immune-system, 2010. Last a essed: 26 January 2010.[101℄ Wikipedia. Red blood ell. http://en.wikipedia.org/wiki/Red-blood- ells, 2010. Lasta essed: 17 June 2010.[102℄ A. Yeka, K. Banek, N. Bakyaita, S. G. Staedke, M. R. Kamya, A. Talisuna,F. Kironde, S. L. Nsobya, A. Kilian, M. Slater, A. Reingold, P. J. Rosenthal,F. Wabwire-Mangen, and G. Dorsey. Artemisinin versus nonartemisinin ombina-tion therapy for un ompli ated malaria: Randomized lini al trials from four sites inUganda. PLOS Medi ine, 2:0654 � 0662, 2005.[103℄ S. Yeung, W. Pongtavornpinyo, I. M. Hastings, A. J. Mills, and N. J. White. An-timalarial drug resistan e, artemisinin-based ombination therapy, and the ontri-bution of modelling to elu idating poli y hoi es. Ameri an Journal of Tropi alMedi ine and Hygiene, 71:179� 186, 2004.[104℄ D. Zhang, P. Shankar, Z. Xu, B. Harnis h, G. Chen, C. Lange, S. J. Lee, H. Valdez,M. M. Lederman, and J. Lieberman. Most antiviral CD8 T ells during hroni viralinfe tion do not express high levels of perforin and are not dire tly ytotoxi . Blood,101:226 � 235, 2003.


Documents

Stellenbosch University ://core.ac.uk/download/pdf/37401067.pdf · 2020. 1. 7. · Lungu y ersit (Univ of ana) Botsw Co-Promoter: Prof. John e v Hargro y ersit (Univ of bh) Stellenosc