Applied Mathematical Sciences, Vol. 8, 2014, no. 8, 405 - 413HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.311643

Stochastic Study for SIR Model

EL BERRAI Imane, BOUYAGHROUMNI Jamal,NAMIR Abdelouahed and EZZBADY Souad

LAMS (Laboratory of analysis, modeling and simulation)Department of mathematics and computer sciences, Faculty of sciences

Ben M’Sik-Casablanca, Hassan II University Mohammedia-Casablanca, Morocco

Copyright c© 2014 EL BERRAI Imane et al. This is an open access article distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribu-

tion, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present work, we consider a mathematical modeling of anSIR epidemic model with constant population size. The way to incor-porate randomness into a mathematical model is to formulate the termsas probabilities at which an event occurs and not, as in deterministicmodels, as rates. In this work we focus only on one kind of discretestochastic process called a Markov Chain. We study what happens inthe long term by making a calculation of probabilities. In addition wepresent the numerical results.

Mathematics Subject Classification: 92B05, 92D30, 60G05, 60J10

Keywords: SIR model, numerical simulation, stochastic process, Markovchain

1 Introduction

In recent years epidemiological modeling of infectious disease transmission hashad an increasing influence on the theory and practice of disease management.Mathematical modeling of the spread of infectious diseases has become partof epidemiology policy decision making in many countries. (See for example,Capasso and Serio [6], Hethcote and Tudor [14], Liu et al. [21][22], Hethcote etal. [15], Hethcote and van den Driessche [16], Derrick and van den Driessche[10], Beretta and Takeuchi [3][4], Beretta et al. [5], Ma et al. [23][24], Ruan

406 EL BERRAI et al.

and Wang [27], Song and Ma [28], Song et al. [29], D’Onofrio et al. [13], Xiaoand Ruan [31]).Other deterministic epidemiology models were then developed in papers byRoss, Ross and Hudson, Martini, and Lotka [1][11][12]. Starting in 1926 Ker-mack and McKendrick published papers on epidemic models and obtained theepidemic threshold result that the density of susceptibles must exceed a crit-ical value in order for an epidemic outbreak to occur [20][25]Mathematicalepidemiology seems to have grown exponentially starting in the middle of the20th century (the first edition in 1957 of Baileys book [1] is an importantlandmark), so that a tremendous variety of models have now been formu-lated, mathematically analyzed, and applied to infectious diseases. Reviews ofthe literature [2][7][8][9][17][18][19][30] show the rapid growth of epidemiologymodeling.The population is divided into three distinct classes: the susceptible S, ”healthyindividuals who can catch the disease”, the infected I, ”those who have thedisease and can transmit it”, and the removed R, ”individuals who have hadthe disease and are now immune to the infection” (or removed from furtherpropagation of the disease by some other means). Schematically, the individ-ual goes through consecutive states S → I → R. Such models are often calledthe SIR models.The remaining parts of this paper are organized as follows: section 2 presentsthe SIR model, In section 3 we study what happens in the long term by makinga calculation of probabilities, the four section to put spotlight on numericalresults. The last section provides concluding remarks.

2 SIR model

The SIR epidemic model is considered by:

S(t) = rcS(t)

(1− S(t)

k

)− αS(t)I(t)

1 + aI(t)+ uS(t)

I(t) =αS(t)I(t)

1 + aI(t)− γI(t)

R(t) = γI(t)− uS(t)

(1)

The model has a susceptible group designated by S, an infected group I anda recovered group R with permanent immunity, rc is the intrinsic growth rateof susceptible, k is the carrying capacity of the susceptible in the absence ofinfective, α is the maximum values of per capita reduction rate of S du to I, ais half saturation constants, u is the vaccination coverage of susceptible, γ isthe natural recover rate from infection.

Stochastic study for SIR model 407

This model is an appropriate one to use under the following assumptions:1) The population is fixed.2) The only way a person can leave the susceptible group is to become in-fected. The only way a person can leave the infected group is to recover fromthe disease. Once a person has recovered, the person received immunity.3) Age, sex, social status, and race do not affect the probability of being in-fected.

3 stochastic Study

An important public health problem is to predict whether an epidemic mayor may not move in a population. Conventional models in epidemiology usingdifferential equations that involve three types of individuals, classified accord-ing to their status. We consider:

s(t) =S(t)

N

i(t) =I(t)

N

r(t) =R(t)

N

In this model s(t) denotes the fraction of the population that can catch thedisease at time t, i(t) denotes the fraction of the population that has thedisease and can spread it to the susceptibles and r(t) denotes the fraction ofthe population that has recovered from the disease and cannot catch it again.This model is appropriate for the spread of a flu epidemic since once a personhas had a particular strain of flu, their immune system prevents them fromcatching that strain again. Since flu spreads fairly quickly, we can assumethat time is measured in days. While most people recover from the flu fairlyeasily, there is a low mortality rate. Those who do not survive are included inr(t). We assume everyone in the population is either susceptible, infected, orrecovered, that is:

s(t) + i(t) + r(t) = 1

We replace in the system, we obtain:s(t) = rcs(t)

(1− s(t)N

k

)− αs(t)Ni(t)

1 + aNi(t)+ us(t)

i(t) =αs(t)Ni(t)

1 + aNi(t)− γi(t)

r(t) = γi(t)− us(t)

(2)

408 EL BERRAI et al.

Let X(n): ”the state of the individual at the time tn ”

At the moment t0 we have:

P{X(0) = R} = Pr(0)

P{X(0) = I} = Pi(0)

P{X(0) = S} = Ps(0)

With:

Pr(0) + Ps(0) + Pi(0) = 1

Following the typical Markov chain understanding, let the vector

P (0) = [Ps(0), Pi(0), Pr(0)]

represent the initial proportions of susceptibles, infecteds, and recovereds.

Likewise at the moment t1, we ask P the following vector:

P (1) = [Ps(1), Pi(1), Pr(1)]

With:

Pr(1) + Ps(1) + Pi(1) = 1

We’re looking for state of the individual at time t1 according to its state attime t0.

Then the one-step forward proportion vector, P (1), is given by:

Stochastic study for SIR model 409

P (1) = P (0)A

We model the dynamics of susceptibles, infecteds, and recovereds by buildinga transition matrix as follows:

A =

1−m+ u m −u0 1− n n0 0 1

where m represents the probability of susceptibles becoming infecteds, and nrepresents the probability of infecteds becoming recovereds. From the Marko-vian point-of-view, the entry A1,1 is the probability of susceptibles who staysusceptibles (at the next step), A1,2 the proportion of susceptibles who becomeinfecteds (at the next step), and, naturally, A1,3 the proportion of susceptibleswho, at the next step, become recovereds. Then A2,1 is the proportion ofinfecteds who become susceptibles (zero), A2,2 is the proportion of infectedsthat stay infected in the next time period, and A2,3 is the proportion of infect-eds recovering. Finally, A3,1 and A3,2 represent the proportion of recoveredsthat become susceptible or infected (zero) respectively, and A3,3 represents theproportion of recovereds who stay recovered. In other words, the transitionmatrix represents a birth process of susceptibles into infecteds and infectedsinto recovereds.

the SIR differential equations suggests that how susceptibles change intime is proportional to both the susceptibles and the infecteds at the time;this furthermore affects the rate of change of the infecteds. Taking a clue, wecan manipulate the ”transition matrix” to incorporate this term. Thus:

B =

1−mpi + u mpi −u0 1− n n0 0 1

So we obtain:

P (1) = P (0)B

We found long-term:BP = PP (I −B) = 0

(3)

So we obtain: Ps =

n

mPr = 1− n

m− us

mPi =

us

m

(4)

410 EL BERRAI et al.

4 Numerical simulation

For the numerical simulations we use the following data.

Table 1: Value of the parameters

Parameter s0 i0 a α γ u NValue 0.5 0.3 2.3 1.49 0.611 0.1 100

Using the below formulas:

m =αsNi

1 + aNin = γi

(5)

we obtain: Ps = 0.56Pr = 0.15Pi = 0.29

(6)

5 Conclusion

The SIR Model is used in the modeling of infectious diseases by computingthe amount of people in a closed population that are susceptible, infected, orrecovered at a given period of time.The model is also used by researchers and health officials to explain the in-crease and decrease in people needing medical care for a certain disease duringan epidemic. The SIR model is the basis for other similar models. The SImodel, also known as the SIS model, is the model where once a person is nolonger infectious, this person becomes susceptible once again. The commoncold can be modeled with the SI model. There is also the SEIR model, wherepeople are categorized as susceptible, exposed, infected, or recovered. The SIRmodel can be adjusted to include variation due to seasonal changes as seen byBauch and Earn.[26]

References

[1] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, 2nd ed.,Hafner, New York, 1975.

[2] N. Becker, The use of epidemic models, Biometrics, 35 (1978), pp. 295305.

Stochastic study for SIR model 411

[3] E. Beretta, Y. Takeuchi, Global stability of a SIR epidemic model withtime delay, J. Math. Biol., 33, pp. 250-260, 1995.

[4] E. Beretta, Y. Takeuchi, Convergence results in SIR epidemic model withvarying population sizes, Nonlinear Anal., 28, pp. 1909-1921, 1997.

[5] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability ofa SIR epidemic model with distributed time delay, Nonlinear Anal., 47,pp. 4107-4115, 2001.

[6] V. Capasso, G. Serio, A generalization of the Kermack-McKendrick de-terministic epidemic model, Math. Biosci., 42, pp. 43-61, 1978.

[7] C. Castillo-Chavez, ed., Mathematical and Statistical Approaches toAIDS Epidemiology, Lecture Notes in Biomath. 83, Springer-Verlag,Berlin, 1989.

[8] K. Dietz, Density dependence in parasite transmission dynamics, Parasit.Today, 4 (1988), pp. 9197.

[9] K. Dietz and D. Schenzle, Mathematical models for infectious diseasestatistics, in A Celebration of Statistics, A. C. Atkinson and S. E. Fein-berg, eds., Springer-Verlag, New York, 1985, pp. 167204.

[10] W.R. Derrick, P. van den Driessche, A disease transmission model innonconstant population, J. Math. Biol., 31, pp. 495-512, 1993.

[11] K. Dietz, Epidemics and rumours: A survey, J. Roy. Statist. Soc. Ser. A,130 (1967), pp. 505 528.

[12] K. Dietz, The first epidemic model: A historical note on P. D. Enko,Austral. J. Statist., 30A (1988), pp. 56-65.

[13] A. D’Onofrio, P. Manfredi, E. Salinelli, Bifurcation thresholds in an SIRmodel with information-dependent vaccination, Mathematical Modellingof Natural Phenomena, 2, pp. 23-38, 2007.

[14] H.W. Hethcote, D.W. Tudor, Integral equation models for endemic infec-tious diseases, J. Math. Biol., 9, pp. 37-47, 1980.

[15] H.W. Hethcote, M.A. Lewis, P. van den Driessche, An epidemiologicalmodel with delay and a nonlinear incidence rate, J. Math. Biol., 27, pp.49-64, 1989.

[16] H.W. Hethcote, P. van den Driessche, Some epidemiological model withnonlinear incidence, J. Math. Biol., 29, pp. 271-287, 1991.

412 EL BERRAI et al.

[17] H. W. Hethcote, A thousand and one epidemic models, in Frontiers inTheoretical Biology, S. A. Levin, ed., Lecture Notes in Biomath. 100,Springer-Verlag, Berlin, 1994, pp. 504515.

[18] H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, inApplied Mathematical Ecology, L. Gross, T. G. Hallam, and S. A. Levin,eds., Springer-Verlag, Berlin, 1989, pp. 193211.

[19] H. W. Hethcote, H. W. Stech, and P. van den Driessche, Periodicity andstability in epidemic models: A survey, in Differential Equations and Ap-plications in Ecology, Epidemics and Population Problems, S. N. Busen-berg and K. L. Cooke, eds., Academic Press, New York, 1981, pp. 6582.

[20] W. O. Kermack and A. G. McKendrick, Contributions to the mathemati-cal theory of epidemics, part 1, Proc. Roy. Soc. London Ser. A, 115 (1927),pp. 700721.

[21] W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behaviour of epidemi-ological models with nonlinear incidence rates, J. Math. Biol., 25, pp.359-380, 1987.

[22] W. M. Liu, S. A. Levin, Y. Iwasa,Influence of nonlinear incidence ratesupon the behaviour of SIRS epidemiological models, J. Math. Biol., 23,pp. 187-204, 1986.

[23] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic modelwith time-delay, Appl. Math. Lett., 17, pp. 1141-1145, 2004.

[24] W.Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemicmodel with distributed time delays, Tohoku Math. J., 54, pp. 581-591,2002.

[25] A. G. McKendrick, Applications of mathematics to medical problems,Proc. Edinburgh Math. Soc., 44 (1926), pp. 98130.

[26] Rhodes, John A., and Elizabeth S. Allman. Mathematical Models in Bi-ology : An Introduction. New York: Cambridge UP, 2003. 280-301.

[27] S. Ruan, W. Wang, Dynamical behaviour of an epidemic model withnonlinear incidence rate, J. Differ. Equations, 188, pp. 135-163, 2003.

[28] M. Song, W. Ma, Asymptotic properties of a revised SIR epidemic modelwith density dependent birth rate and tie delay, Dynamic of Continuous,Discrete and Impulsive Systems, 13, pp. 199-208, 2006.

Stochastic study for SIR model 413

[29] M. Song, W. Ma, Y. Takeuchi, Permanence of a delayed SIR epidemicmodel with density dependent birth rate, J. Comput. Appl. Math., 201,pp. 389-394, 2007.

[30] K. Wickwire, Mathematical models for the control of pests and infectiousdiseases: A survey, Theoret. Population Biol., 11 (1977), pp. 182238.

[31] D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotoneincidence rate, Math. Biosci., 208, pp. 419-429, 2007.

Received: November 17, 2013