The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV. Jan P. Medlock

Transcription

1 The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV A Thesis Presented to The Faculty of the Division of Graduate Studies by Jan P. Medlock In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics Georgia Institute of Technology June 1999

2 The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV Approved: Ronald Shonkwiler, Chairman Konstantin Mischaikow Xu-Yan Chen Date Approved by Chairman

3 Acknowledgments First, I would like to thank Professor Ron Shonkwiler for countless hours of guidance which made this thesis possible. I would like to thank Mac Hyman for presenting me with this problem and for getting it off the ground. I would like to thank Professors Konstantin Mischaikow and Xu-Yan Chen for their time in serving on my examining committee. I would like to thank all of the students and faculty who listened to me rant about this thesis and made invaluable suggestion. Finally, I would to thank Professor Jim Herod for sparking my interest in mathematical biology. iii

4 Contents Acknowledgments List of Tables List of Figures Summary iii vi viii ix 1 Introduction HIV Background Modeling ODEs and Monte Carlo History of ODE Models in HIV The Importance of Stochastic Migration The Deterministic Model Model Formulation The Disease-Free Equilibrium The Reproductive Number, R The Endemic Equilibrium Parameter Estimation The carrying capacity, S The natural death rate, µ The removal rate, γ iv

5 2.5.4 The partner acquisition rate, r The relative infection rate, β The Stochastic Model Model Formulation The Disease-Free Equilibrium Solving the Stochastic ODE The Fokker-Planck Equation Parameter Estimation the noise coefficient, η Numerical Simulation R 0 < Numerical Simulation R Numerical Simulation R 0 > Numerical Simulation Asymmetric Convergence to the Mean Approximating the Probability Density of S Conclusion 26 Bibliography 27 v

6 List of Tables 3.1 Baseline Parameters for Numerical Simulations vi

7 List of Figures 2.1 A schematic of system (2.1). Here S is the susceptibles, I is the infectives, R is the removeds, µ > 0, a constant, is the death rate, µs 0 > 0, a constant, is the migration term, γ, a constant, is the removal rate from I to R and λ = rβi S+I is the infection rate A schematic of system (3.2). Here S is the susceptibles, I is the infectives, R is the removeds, µ > 0, a constant, is the death rate, µs 0 > 0, a constant, is the migration term, γ, a constant, is the removal rate from I to R, λ = rβi S+I is the infection rate and η is the stochastic term The mean of S versus time for 10,000 runs of the disease-free case. The dashed line is the analytic value. The mean of S is S 0 with a Gaussian distribution The variance of S versus time for 10,000 runs of the disease-free case. The dashed line is the analytic value. The variance of S is (µs 0 η) 2 /2µ and S is Gaussian The mean of S and I versus time for 10,000 runs with an initial infection of 50% for the stochastic SIR model with R 0 < 1. The stochastic system is very close to the deterministic system and the infection dies out as in the deterministic case The mean of S and I versus time for 10,000 runs with an initial infection of 50% for the stochastic SIR model with R 0 1. The stochastic system is very close to the deterministic system and exhibits the same behavior vii

8 3.6 The mean of S and I versus time for 10,000 runs with an initial infection of 1% for the stochastic SIR model with R 0 > 1. The mean values of the susceptibles in the stochastic system are slightly greater than the values in the deterministic system. This is due to a difference in convergence rates of the deterministic system from above and below the endemic equilibrium The numerical stationary distribution of ρ. The solid line is ρ(δ) and the dashed line is ρ( δ) The numerical and analytic stationary distribution of ρ. The dashed line is the analytic value and the solid line is the value from numerical simulation. 25 viii

9 Summary Here a simple deterministic susceptible-infective-removed (SIR) model of HIV transmission in a high-risk population will be presented as a system of ordinary differential equations and analyzed using techniques from dynamical systems. Once the dynamics of this deterministic model are understood, a stochastic migration term will be added to the deterministic model to model variation in the influx of individuals to the susceptible group. This model will be presented as a system of stochastic ordinary differential equations. The Fokker-Planck equation will be used to transform this system to a single deterministic partial differential equation which will then be analyzed. Using this analysis and numerical simulation of the system of stochastic ordinary differential equations it will be shown that the system is insensitive to small fluctuations in migration. It will further be shown that in the case of an infection which persists in the population, the mean number of infectives at equilibrium in the model with stochastic migration is slightly lower than the number of infectives at equilibrium in the model without stochastic migration. In this case, it will be shown that the stochastic migration acts to slightly lessen the infection in the population. ix

10 Chapter 1 Introduction 1.1 HIV Background AIDS (acquired immune deficiency syndrome) and HIV (human immunodeficiency virus), the virus which causes AIDS, have gone to the forefront of epidemiological research in the last 15 years. Many consider HIV to be the most serious world epidemic of this century and perhaps the next. While much is still unknown about HIV, researchers have discovered much about its transmission dynamics in populations. HIV is primarily a sexually transmitted disease it spreads in shared bodily fluid such as blood and semen. Here we will consider the spread of HIV only by sexual contacts. A key characteristic of HIV is its long infection time (anywhere from a few months to years) before the onset of AIDS. In fact, some individuals may carry the disease but never develop AIDS. During this infection time the individual is infective and may infect others. Here it will be assumed that once an individual progresses from this infective stage to AIDS, the individual will no longer be sexually active and cannot infect others. 1.2 Modeling ODEs and Monte Carlo In the study of the transmission of infectious diseases in general and HIV in particular two basic kinds of models are used: discrete probabilistic Monte Carlo-type models and 1

11 continuous systems of ordinary differential equations (ODE) models. The ODE models assume large populations so that parameter values can be considered to take on mean values with no variation. Such models are well suited to the analysis techniques of dynamical systems. The Monte Carlo models make no mean value assumptions and are for that reason more accurate. However these models are difficult to analyze and often large scale models are not practical computationally. Somewhere between these two types of models is a third: systems of stochastic ODEs (SODEs). These models have the advantage of analytic tools similar to those available for ODEs with fewer assumptions on population size and mean values. Here an SODE model will be developed History of ODE Models in HIV A standard model in the study of infectious disease is the susceptible-infective-removed (SIR) model. This is a compartment model with three compartments: individuals susceptible to the infection, individuals who are infectious and can infect others and individuals who are removed from the infection and can no longer infect others or contract the infection themselves. In Anderson et al. [3], the authors discuss one of the first attempts to model the transmission of HIV using the standard SIR model and estimate some epidemiological parameters (these are refined in [1] and [2]). It is this model which will be considered. However, due to the wide estimates for parameters (in particular, transmission probability and the partner acquisition rate), a small sub-population will be considered in which these parameters have much narrower estimates. 2

12 1.3 The Importance of Stochastic Migration The model presented will have stochastic fluctuation in the migration of new susceptibles into the susceptible class. Clearly the influx of people into a population is stochastic. Therefore, it will be shown that the standard SIR model is insensitive to this stochastic migration term. 3

13 Chapter 2 The Deterministic Model 2.1 Model Formulation Consider an SIR model for HIV transmission in a population of individuals who are at high-risk for HIV. This might be a population of homosexual men or prostitutes in a large city or any other population whose behavior makes it high-risk to spread HIV. In this population let S denote the susceptibles, I denote the infectives and R denote the individuals removed from the infective class, i.e. they are no longer infecting susceptibles. Note that S, I, R 0 because they represent numbers of people. Assume a constant migration of individuals into the high-risk population as new susceptibles, that is, into S, µs 0 > 0. Further assume that the number of people removed from each group due to natural causes such as death or leaving the high-risk population (not HIV or AIDS related) is proportional to the number of individuals in the group, µs, µi and µr, where µ > 0 will be called the natural death rate for historical reasons, which is constant. Additionally the number of individuals removed from the infective class into the removed class (by progression from HIV to AIDS) is proportional to the number of individuals in the infective class, γi, where γ > 0 is the removal rate which is a constant. The infection rate, λ, depends on the number of partners per individual per unit time, r > 0, the transmission probability per partner, β > 0, which are both taken to be constants, and the proportion of infected individuals to sexually active individuals, I/(S + I). Note here that the removed individuals are taken to be sexually inactive so that there are no new infections due to the removed class. For 4

14 µs 0 λs S γi I R µs µi µr Figure 2.1: A schematic of system (2.1). Here S is the susceptibles, I is the infectives, R is the removeds, µ > 0, a constant, is the death rate, µs 0 > 0, a constant, is the migration term, γ, a constant, is the removal rate from I to R and λ = rβi is the infection rate. S+I simplicity further assume that the high-risk population is homogeneous, neglect variation in susceptibility and risk behavior, assume the incubation period for the disease is negligible and assume random mixing of the high-risk population. The following system of ODEs describes this SIR model where ds dt =µ(s0 S(t)) λ(t)s(t), di =λ(t)s(t) (µ + γ)i(t), dt dr =γi(t) µr(t), dt λ(t) = (2.1) rβi(t) S(t) + I(t). (2.2) Figure 2.1 illustrates the system (2.1). This system is nonlinear due to the form of λ. Since R does not affect S or I, consider the equivalent system with λ as in (2.2). ds dt =µ(s0 S(t)) λ(t)s(t), di dt =λ(t)s(t) (µ + γ)i(t), (2.3) 5

15 2.2 The Disease-Free Equilibrium Consider the case where there is no infection, i.e. I 0. Then (2.3) reduces to Setting ds dt ds dt = µ(s0 S(t)). (2.4) = 0 here shows an equilibrium of (2.3) at E 0 = (S 0, 0). This is the diseasefree equilibrium. Thus, in the absence of infectives the susceptibles have an equilibrium value of S 0. Now investigating the stability of this equilibrium will derive the so-called reproductive number, R The Reproductive Number, R 0 In order to determine the stability of system (2.3) at E 0, we will linearize the system about the point E 0 by taking the Jacobian. Given a system of ODEs written in vector form, that is, an n-vector X(t) and some n-vector-valued function of n-vectors F so that Then the Jacobian of (2.5) at some point X 0 is defined as F 1 F 1 F X 1 1 X 2 X n J( X F 2 F 2 F 0 ) = X 1 X 2 2 X n F n F X n 2 X n d X dt = F ( X(t)). (2.5) F n X 1 X= X 0. (2.6) Thus the Jacobian of (2.3) is rβi2 µ J(S, I) = rβi 2 (S+I) 2 (S+I) 2 rβs2 (S+I) 2 (µ + γ) + rβs2 (S+I) 2. (2.7) 6

16 Finding the eigenvalues of this Jacobian matrix results in the characteristic equation where λ here is an eigenvalue. (S + I) 2 λ 2 + [(2µ + γ)(s + I) rβ(s I)] (S + I)λ + [ µ(µ + γ)(s + I) 2 rβ ( µs 2 (µ + γ)i 2)] = 0 (2.8) Find the eigenvalues at E 0 by substituting S = S 0 and I = 0 into (2.8). Solve for λ to get the eigenvalues, µ and rβ µ γ. In order for the equilibrium E 0 to be asymptotically stable, both eigenvalues must be negative. Clearly µ < 0. Thus if rβ µ γ < 0 then both eigenvalues of J(E 0 ) are negative and the equilibrium E 0 is asymptotically stable. This implies that a small population of infectives introduced into the system would not cause a persistent infection. That is, if a small number of infectives were added to the population the population would return to the disease-free state after some time. Conversely, if rβ µ γ > 0 then the equilibrium E 0 is unstable and an introduction of infectives will result in a persistent infection, a so-called endemic infection. where Now define the reproductive number, R 0 = rβ µ + γ = rβ τ, (2.9) τ = 1 µ + γ. (2.10) Note that τ is the mean duration of infection because the (µ + γ) term in (2.3) can be thought of as the probability that a person in I is removed from I either by natural causes (the µ term) or progression to R (the γ term). Thus if R 0 > 1 then there is an endemic infection and if R 0 < 1 then there is no endemic infection. R 0 can be thought of as follows: if the system is near the disease-free equilibrium and one infective person is added to the population then R 0 is the number of newly infective persons caused by this one added person. So if the added infective produces more than one new infective then the infection will persist; if the added infective produces less than one new infective the infection will die out. 7

17 2.4 The Endemic Equilibrium Now consider the case where R 0 > 1 so that the system has an endemic infection. Then E 0 is unstable. To find the endemic equilibrium set the right-hand sides of (2.3) to zero. This gives 0 =µ(s 0 S) λs, 0 =λs (µ + γ)i. (2.11) It follows that λs =µ(s 0 S), λs =(µ + γ)i, (2.12) so then S = S 0 I(1 + γ/µ). (2.13) Also from (2.12) λs = rβi S = (µ + γ)i, (2.14) S + I so then I = rβ µ + γ S S = (R 0 1)S. (2.15) Finally substituting (2.15) into (2.13) and solving for S, the endemic equilibrium point is S S 0 = 1 + (R 0 1)(1 + γ/µ), I (R 0 1)S 0 = 1 + (R 0 1)(1 + γ/µ). (2.16) Thus the endemic equilibrium is E e = (S, I ). (2.17) 8

18 To show that E e is asymptotically stable, substitute S = S, I = I into the characteristic equation (2.8). Using (2.15) and (2.9) the characteristic equation is ( ) µγ λ 2 γ3 + (rβ γ)λ + (µ + 2γ) (µ + 2γ)(µ + γ) + rβ(µ + γ) + = 0. (2.18) rβ rβ Since R 0 > 1, by (2.9), rβ γ > µ > 0, so in order for the real part of both eigenvalues to be negative, Factoring this gives µγ γ3 (µ + 2γ) (µ + 2γ)(µ + γ) + rβ(µ + γ) + rβ rβ > 0. (2.19) (rβ γ)(γ + µ)(rβ µ γ) rβ > 0, (2.20) which holds using again the fact that rβ γ > µ > 0. Thus both eigenvalues of the Jacobian at E e for R 0 > 1 have negative real part, so E e is asymptotically stable. Note that the system (2.3) is invariant in the first quadrant, i.e. if an initial point (S 0, I 0 ) is in the first quadrant then its solution remains in the first quadrant. All initial points are in the first quadrant because S and I represent numbers of people and are therefore nonnegative. Note also that this equilibrium E e exists (and is unstable) for R 0 < 1 but is out of the first quadrant and is therefore unimportant when R 0 < 1. The dynamics of (2.3) are now known: if R 0 < 1 then the only equilibrium in the invariant first quadrant is E 0 and this is asymptotically stable; if R 0 > 1 then there are two equilibria in the first quadrant, E 0, unstable, and E e, asymptotically stable. 2.5 Parameter Estimation The carrying capacity, S 0 S 0 is the carrying capacity of population in the absence of infectives. This was chosen to be 1 so that S and I can be thought of as proportions of the population. 9

19 2.5.2 The natural death rate, µ µ has been called here the natural death rate for historical reasons. It should be the removal rate from the high-risk group. Thus µ 1 is the average time spent in the high-risk group. It is assumed that people entering the high-risk group are young adults. The amount of time that an individual stays in the high risk group is not known and certainly varies between populations. Here the µ 1 is taken to be 20 years The removal rate, γ γ is the removal rate from the infective group to the removed group. γ 1 is the average amount of time from which an individual is infective until that individual gets full-blown AIDS. Here this average time is taken to be about 3 years The partner acquisition rate, r r is the partner acquisition rate the mean number of partners per year per individual in the high-risk group. This varies greatly from population to population. Populations in which HIV spreads have reported having hundreds of partners per year (e.g. prostitutes) to just one or two per year [6]. Here r is taken to be 3 as a conservative estimate of the number of partners per year in the high-risk group The relative infection rate, β The relative infection rate, β, in [6] is estimated between and These give a reproductive number between 0.09 and Here the baseline β is chosen to be 0.15 so that R 0 = 1.29 is near 1, where the bifurcation occurs. 10

20 Chapter 3 The Stochastic Model 3.1 Model Formulation Now that the deterministic model is understood, consider the case of a stochastic migration into the susceptibles. That is the case when the migration into the susceptibles has stochastic fluctuations around the deterministic value, µs 0. The new migration term will take the form µs 0 = µs 0 (1 + η dw t dt ) (3.1) where W t is a Wiener process and η > 0 is the noise coefficient which determines the size of the stochastic term. Note that a Wiener process is a type of continuous random walk with mean 0. Thus the SIR model with additive noise in the migration term corresponding to (2.3) written in differential form is ds = ( µ(s 0 S(t)) λ(t)s(t) ) dt + µs 0 η dw t, di = ( λ(t)s(t) (µ + γ)i(t) ) dt, (3.2) dr = ( γi(t) µr(t) ) dt, where λ(t) = rβi(t) S(t) + I(t). (3.3) 11

21 µs 0 (1 + η) λs S γi I R µs µi µr Figure 3.1: A schematic of system (3.2). Here S is the susceptibles, I is the infectives, R is the removeds, µ > 0, a constant, is the death rate, µs 0 > 0, a constant, is the migration term, γ, a constant, is the removal rate from I to R, λ = rβi is the infection rate and η is S+I the stochastic term. Figure 3.1 illustrates the system (3.2). Again the R term is ignored, because it has no effect on the dynamics of S and I, to give the equivalent system ds = ( µ(s 0 S(t)) λ(t)s(t) ) dt + µs 0 η dw t, di = ( λ(t)s(t) (µ + γ)i(t) ) dt, (3.4) with λ as in (3.3). 3.2 The Disease-Free Equilibrium In the absence of infection, i.e. I 0, system (3.4) reduces to ds = µ(s 0 S(t)) dt + µs 0 η dw t. (3.5) Solving the Stochastic ODE To solve this, make the change of variables X(t) = S(t) S 0 so (3.5) becomes dx = µx(t) dt + µs 0 η dw t. (3.6) 12

22 The Fokker-Planck Equation The Fokker-Plank Equation is a transformation from a system of stochastic ODEs in a variable x into a system of deterministic partial differential equations (PDEs) for the probability distribution of the variable x. Given the system of n SODEs written in vector form dx = A( X, t) dt + B( X, t) dw t (3.7) where A is the n-vector of the deterministic terms of the system, B is the n n matrix of the stochastic terms of the sequence and d W t is an n-dimensional Wiener process, then the corresponding PDE is the Fokker-Planck Equation ρ n t = A i ρ + 1 n x i 2 i=1 i=1 n j=1 2 x i x j B i,j ρ (3.8) where x i and A i are the i th components of x and A, respectively, B i,j is the element (i, j) of the matrix B and ρ( x, t x 0, 0) = Pr[ X(t) = x X(0) = x 0 ]. (See [4] for more on the Fokker-Planck Equation). Now applying the Fokker-Planck Equation to (3.6) ρ t = x (µxρ) + (µs0 η) 2 2 ρ 2 x, (3.9) 2 with the initial condition ρ(x, 0 x 0, 0) = δ(x x 0 ), where δ is the Dirac delta. Note here that this transformation ignores the condition S 0. This is acceptable due to the fact that this condition has almost no effect on the dynamics of the system because it is far away from the mean and the variance is relatively small. Take the Fourier transform of (3.9) in x and let φ(s) be the Fourier transform of ρ, i.e. φ(s, t) = + ρ(x, t)e isx dx, (3.10) where i = 1. This φ is known as the characteristic function of ρ. Now the PDE is transformed to φ t + µs φ s + (µs0 η) 2 s 2 φ = 0, (3.11) 2 13

23 with the initial condition φ(s, 0) = e isx 0. Solving this by the method of characteristics results in two ODEs for the characteristic curves, Solving these by separation of variables gives dt = 1, t(0) = 0 dz (3.12) ds dz = µs, s(0) = ξ. t = z, s = ξe µz, (3.13) which is z = t, ξ = se µt. (3.14) This yields the ODE dφ dz + (µs0 η) 2 s 2 Φ = 0, Φ(0) = e iξx 0, (3.15) 2 which, by (3.13), is dφ dz = η) 2 (µs0 ξ 2 e 2µz Φ, Φ(0) = e iξx 0. (3.16) 2 Solving this by separation of variables gives [ (µs 0 η) 2 Φ = exp ξ ( 2 1 e 2µz) ] iξx 0. (3.17) 4µ Now using (3.14), the solution to (3.11) is φ(s, t) = exp [ (µs 0η) 2 s ( 2 1 e 2µt) ] isx 0 e µt. (3.18) 4µ This is the characteristic function of a Gaussian; take the inverse Fourier transform to get 1 ρ(x, t) = exp [ µ(x x ] 0e µt ) 2, (3.19) π (µs 0η) 2 (µs 0 η) 2 µ 14

24 Table 3.1: Baseline Parameters for Numerical Simulations Parameters are as per table unless otherwise specified SIR Parameters Carrying capacity S Natural death rate µ 0.05 yrs 1 Removal rate γ 0.3 yrs 1 Partner acquisition rate r 3.0 partners/year Relative infection rate β 0.15 partners 1 Noise coefficient η 0.4 the solution to (3.9). Thus X is Gaussian with E X(t) = X(0) e µt, Var X(t) = (µs0 η) 2 ( ) 1 e 2µt (3.20) 2µ where E is the mathematical expectation (mean) and Var is the variance. So S is Gaussian with E S(t) = S 0 + S(0) e µt, Var S(t) = (µs0 η) 2 ( ) 1 e 2µt. (3.21) 2µ Letting t + the stationary solution gives E S = S 0, Var S = (µs0 η) 2. (3.22) 2µ Parameter Estimation the noise coefficient, η The noise coefficient, η, describes the amount of fluctuation in the migration term. The parameter η here is chosen so that Var S here is about 5% as a typical fluctuation in a population Numerical Simulation The mean and variance of S(t) in the disease-free case can be estimated by simulating system (3.5) numerically and these values can be compared to the analytic values just derived. 15

25 1.01 Mean of 10,000 runs for S(t) mean of S(t) S(t) time in years Figure 3.2: The mean of S versus time for 10,000 runs of the disease-free case. The dashed line is the analytic value. The mean of S is S 0 with a Gaussian distribution. 4.5 x 10 3 Variance of 10,000 runs for S(t) 4 S(t) variance of S(t) time in years Figure 3.3: The variance of S versus time for 10,000 runs of the disease-free case. The dashed line is the analytic value. The variance of S is (µs 0 η) 2 /2µ and S is Gaussian. 16

26 Mean of 10,000 runs for S(t), I(t) 1 S(t) 0.8 mean of S(t),I(t) I(t) time in years Figure 3.4: The mean of S and I versus time for 10,000 runs with an initial infection of 50% for the stochastic SIR model with R 0 < 1. The stochastic system is very close to the deterministic system and the infection dies out as in the deterministic case. The parameters are as in Table 3.1 with S(0) = S 0 and I(0) = 0. Using a stochastic integration package (SDEInt, a C++ library, by Everett Carter) for 10,000 runs of (3.5) to t = 300 the mean of S was found to be S 0 and the variance of S was found to be (µs 0 η) 2 /2µ which agrees with the solution to the Fokker-Planck Equation (3.9). S is Gaussian as was also found by (3.9). Figures 3.2 and 3.3 show plots of the mean and variance of S versus time and the analytic values. The numerical values follow closely the analytic values. 3.3 R 0 < Numerical Simulation For the parameter values in Table 3.1 and β = 0.1, then R < 1 which gives a stable disease-free equilibrium for the deterministic case. The initial values are S 0 = 0.5 and I 0 = 0.5, where the infectives are 50% of the population. Using the stochastic integration package for 10,000 runs to time 300 the mean value of the system is very close to the 17

27 Mean of 10,000 runs for S(t), I(t) 1 S(t) 0.8 mean of S(t), I(t) I(t) time in years Figure 3.5: The mean of S and I versus time for 10,000 runs with an initial infection of 50% for the stochastic SIR model with R 0 1. The stochastic system is very close to the deterministic system and exhibits the same behavior. deterministic solution and the infection dies out as in the deterministic case. The variance of S is again on the order of (µs 0 η) 2 /2µ while the variance of I is very small. Figure 3.4 shows the mean values of S and I for 10,000 runs. 3.4 R Numerical Simulation For the parameter values in Table 3.1 and β = , then R 0 = Here the deterministic system (2.3) is near a critical value of the parameter R 0 where the system undergoes a bifurcation from a stable disease-free equilibrium with no endemic equilibrium to an unstable disease-free equilibrium with a stable endemic equilibrium. If the stochastic system (3.4) is sensitive to the noise term it would be expected to be sensitive at this critical value. Again the initial values are S 0 = 0.5 and I 0 = 0.5, where the infectives are 50% of the population. Using the stochastic integration package for 10,000 runs to time 300 the 18

28 Mean of 10,000 runs for S(t), I(t) mean of S(t), I(t) S(t) 0.2 I(t) time in years Figure 3.6: The mean of S and I versus time for 10,000 runs with an initial infection of 1% for the stochastic SIR model with R 0 > 1. The mean values of the susceptibles in the stochastic system are slightly greater than the values in the deterministic system. This is due to a difference in convergence rates of the deterministic system from above and below the endemic equilibrium. mean value of the system is very close to the deterministic solution and the variance of S is on the order of (µs 0 η) 2 /2µ while the variance of I is very small. Figure 3.5 shows the mean values of S and I for 10,000 runs. 3.5 R 0 > Numerical Simulation For the parameter values in Table 3.1 and β = 0.15, then R > 1 which gives a stable endemic equilibrium. The initial values are S 0 = 0.99 and I 0 = 0.01, where the infectives are 1% of the population. Using the stochastic integration package for 10,000 runs to time 300 the mean value of the system is close to the deterministic solution. The variances of S and I are on the order of (µs 0 η) 2 /2µ. Figure 3.6 shows the mean values of 19

29 S and I for 10,000 runs. However, in this case the mean value of S, S, is on the order of 10 3 greater (.335 compared to.333) than the deterministic value of the endemic equilibrium. Similarly, the mean value of I is on the order of 10 3 less than the deterministic value of the endemic equilibrium Asymmetric Convergence to the Mean To see why this is the case, let S = S + δ(t) and I = I where S and I are the values of S and I at the endemic equilibrium (2.16) and δ(t) is some small perturbation to S. Since S 0, δ S. Then from the deterministic system (2.3) ds dt = d dt (S + δ(t)) = dδ dt =µ(s 0 S δ(t)) rβi S + δ(t) + I (S + δ(t)) =µ(s 0 S ) rβi S + I S + rβi S + I S rβi µδ(t) S + I + δ(t) (S + δ(t)). (3.23) By (2.11) and (2.15), µ(s 0 S ) rβi S +I S = µ(s 0 S ) λ S = 0 and I = (R 0 1)S so dδ dt = µδ + rβ(r 0 1)S rβ(r 0 1)S R 0 S R 0 S + +δ (S + δ) ( S = µδ rβ(r 0 1)S + δ R 0 S + δ 1 ) R 0 = µδ rβ R 0 (R 0 1) 2 S δ R 0 S + δ Because all the parameters are positive and R 0 > 1, rβ dδ dt = µ δ + R 0 (R 0 1) 2 S δ R 0 S + δ 20. (3.24) (3.25)

30 for δ > R 0 S. (Recall that δ S > R 0 S, so this condition is not restrictive.) Now consider perturbations to S, δ < 0 downward and δ + > 0 upward, with δ = δ + and δ > R 0 S. For δ the denominator R 0 S + δ is smaller than the denominator R 0 S + δ + for δ +. Therefore, since the other terms are all of the same magnitude, dδ dt > dδ δ=δ dt. (3.26) δ=δ+ So the convergence of (2.3) is faster from below the endemic equilibrium than from above. Thus considering the system at equilibrium, an upward perturbation would return more slowly to equilibrium than a downward perturbation. So with the stochastic migration providing perturbations in both directions equally likely, S > S Approximating the Probability Density of S To quantify S S again use S = S +δ(t) and I = I where S and I are the values of S and I at the deterministic endemic equilibrium (2.16) and δ(t) is some small perturbation to S. Now apply this to the stochastic system (3.4) and using the same algebra used to arrive at (3.24), [ ] rβ R dδ = µδ 0 (R 0 1) 2 S δ dt + µs 0 η dw R 0 S t + δ ( ) µδ 2 + µr 0 + rβ R 0 (R 0 1) 2 S δ = dt + µs 0 η dw R 0 S t + δ (3.27) which now includes the stochastic term. Again apply the Fokker-Planck Equation (3.8) which gives ρ t = δ ( ) µδ2 + µr 0 + rβ R 0 (R 0 1) 2 S [ ] δ ρ + 2 (µs 0 η) 2 ρ. (3.28) R 0 S + δ δ 2 2 Now investigate the stationary distribution of ρ by considering ρ t 0. Note that it can be shown that this stationary distribution is stable, i.e. given an initial condition, the 21

31 distribution evolves in time to the stationary distribution. This gives the ODE d 2 ρ dδ + d (T ρ) = 0 (3.29) 2 dδ where T = ( ) 2µδ µr 0 + rβ R 0 (R 0 1) 2 S δ (µs 0 η) 2 (R 0 S + δ) = 2µ (µs 0 η) 2 δ + 2rβ(R 0 1) 2 S R 0 (µs 0 η) 2 2rβ(R 0 1) 2 S 2 (µs 0 η) 2 (R 0 S + δ). (3.30) Note that (3.30) is asymptotically, for large δ, T = (D 1 δ + D 0 )ρ(δ) (3.31) within ɛ, where D 1 and D 0 are positive constants. Integrate equation (3.29) to get or equivalently dρ dδ + T ρ = c 1 (3.32) dρ dδ = c 1 T ρ (3.33) where c 1 is an integration constant. By (3.31), (3.33) is, asymptotically c 1 = dρ dδ + (D 1δ + D 0 )ρ. (3.34) Now to show c 1 = 0 it is sufficient to show that as δ +, (D 1 δ + D 0 )ρ(δ) 0 and dρ dδ 0. Recall that since ρ is a probability density, ρ 0 and ρ = 1. Assume that for some > 0, for δ >, ρ(δ) is not monotonically decreasing. Then dρ dδ sign and repeatedly passes through 0. Where dρ dδ is of varying = 0, ρ(δ) is either a local maximum or a local minimum. Consider a local minimum followed by a local maximum, ρ min = ρ(δ min ) and ρ max = ρ(δ max ) with δ min < δ max and ρ min < ρ max. Since the derivative is 0 at both δ min and δ max, from (3.34), c 1 = (D 1 δ min + D 0 )ρ min and c 1 = (D 1 δ max + D 0 )ρ max. This 22

32 is a contradiction since (D 1 δ min + D 0 )ρ min < (D 1 δ max + D 0 )ρ max. Therefore, for δ >, ρ(δ) is monotonically decreasing. Since ρ 0, ρ 0 and ρ is monotone after some point, dρ dδ 0 by the Mean Value Theorem. Now to show that (D 1δ + D 0 )ρ(δ) 0 it suffices to show that δρ(δ) 0. Assume that δρ(δ) does not go to 0. Then there exists ɛ > 0 and a sequence of δ i, i = 1, 2,... such that δ i + and δ i ρ(δ i ) > ɛ for all i. Thus ρ(δ i ) > ɛ/δ i, for all i. By discarding terms of the sequence if necessary, we may assume without loss of generality that δ i+1 > 2δ i. By the monotonicity of ρ, 1 = > > = ɛ > ɛ + + i=1 + i=1 + i=1 + i=1 = +, ρ(δ) dδ (δ i+1 δi) ρ(δ i+1 ) (δ i+1 δi) ( 1 δ i δ i ɛ δ i+1 which gives a contradiction. Therefore, (D 1 δ + D 0 )ρ(δ) 0. So c 1 = 0. Since c 1 = 0, and then Integrate to get ln ρ(δ) = dρ dt dρ ρ = ) (3.35) = T ρ (3.36) T dδ. (3.37) µ (µs 0 η) 2 δ2 + 2rβ(R 0 1) 2 S R 0 (µs 0 η) 2 δ 2rβ(R 0 1) 2 S 2 (µs 0 η) 2 ln (R 0 S + δ) + c 2 (3.38) 23

33 8 ρ(δ), Probability Density of δ ρ ρ( δ) ρ(δ) δ Figure 3.7: The numerical stationary distribution of ρ. The solid line is ρ(δ) and the dashed line is ρ( δ). so then ρ(δ) = e T dδ = C 2 exp ( µ (µs 0 η) 2 δ2 2rβ(R 0 1) 2 S R 0 (µs 0 η) 2 ) 2rβ(R 0 1) 2 S 2 (3.39) δ (R 0 S (µs + δ) 0 η) 2 where C 2 is e c 2. This is the stationary distribution of δ. The constant C 2 is easily found using the fact that ρ = 1. The stationary distribution is dominated by the e δ2 term which is a Gaussian distribution and the remaining terms skew the stationary distribution slightly from this Gaussian. Figure 3.7 shows the analytic stationary distribution ρ(δ) and ρ( δ). This figure shows that the mean of the analytic stationary distribution is positive as is the numerical stationary distribution because ρ(δ) > ρ( δ). Figure 3.8 shows this stationary distribution compared with the stationary distribution from numerical simulation of system (3.4). Note that the analytic stationary distribution is more spread out than the numerical stationary distribution. In considering perturbations, only perturbations to S were considered and I was considered to be a constant, I. In system (3.4) perturbations in S affect I through the λs term. 24

34 0.04 Distribution of S p S Figure 3.8: The numerical and analytic stationary distribution of ρ. The dashed line is the analytic value and the solid line is the value from numerical simulation. 25

35 Chapter 4 Conclusion For R 0 < 1 and for R 0 > 1 the deterministic system (2.3) is insensitive to a small stochastic migration term. In the case of R 0 > 1 a small stochastic migration term results in a mean value at equilibrium which is very slightly greater than the deterministic equilibrium. This is due to the form of the infection term and can be quantified by using the Fokker-Planck equation to transform the system of stochastic ODEs into a single deterministic PDE. For greater accuracy, perturbations to both S and I should be considered in the stochastic system (3.4) which would result in a deterministic PDE with two spacial variables and the time variable after transformation with the Fokker-Planck Equation. The solution to this PDE gives the distributions of S and I exactly. Here it has been shown that the deterministic SIR model presented is insensitive to small stochastic variation in the migration term. In the case where R 0 > 1, the stochastic variation results in a mean value at equilibrium which is slightly greater than the equilibrium value of the deterministic model. This is due to the form of the infection term and was quantified using the Fokker-Planck equation to transform the system of stochastic ODEs into a single deterministic PDE. Small variation in migration occurs in actual populations naturally. The fact that the deterministic model presented here is insensitive to this variation is important other types of variation may now be considered, for example variation in the number of sexual partners. 26

36 Bibliography [1] R.M. Anderson and R.M. May. Transmission dynamics of HIV infection. Nature, 326(6109): , [2] R.M. Anderson and R.M. May. Epidemiological parameters of HIV transmission. Nature, 333(6173): , [3] R.M. Anderson, G.F. Medley, R.M. May, and A.M. Johnson. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA Journal of Mathematics Applied in Medicine and Biology, 3: , [4] C.W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer-Verlag, New York, [5] H. W. Hethcote. Three Basic Epidemiological Models, pages In Levin et al. [7], [6] J.M. Hyman, J. Li, and E.A. Stanley. The differentiated infectivity and staged progression models for the transmission of HIV. Mathematical Biosciences, 155(2):77 109, [7] S.A. Levin, T.G. Hall, and L.J. Gross, editors. Applied Mathematical Ecology. Springer-Verlag, New York, [8] J.D. Murray. Mathematical Biology. Springer-Verlag, New York,