LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC

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1 LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic processes converge to the trajectories of a two dimensional dynamical system. If the reproductive number, R, is greater than one then the limiting system has an interior fixed point so that the epidemic processes started with a nontrivial proportion of initially infected are endemic. We show global stability of the interior fixed point of the limiting dynamical system. 1. Introduction The SIRS epidemic models were introduced in 1933 by Kermack and McKendrick 5) to describe endemic infections. In any time interval t, t + t) a susceptible individual is equally likely to come into contact with any of the currently infected individuals and the average number of such contacts is proportional to a fixed fraction of currently infected individuals, θ 1 I t /N. Susceptible individuals are equally susceptible to the disease and the probability of infection upon a contact with an infected individual is θ 2. In the same time interval, any infected individual may recover and become temporarily immune at rate ρ and an immune individual may lose his immunity and become susceptible at rate 1. That is, the unit of time is taken to be equal to the average immune period. The population is assumed closed so that there is no inflow or outflow of individuals. Let N IN be the population size parameter. For t, let S t = the number of susceptible individuals at time t, I t = the number of infected individuals at time t, R t = the number of recovered immune) individuals at time t, and let s t = S t /N, i t = I t /N and r t = R t /N. Since S t + I t + R t N, the pair S t, I t ) or s t, i t ) completely describes the state of the system at any time t. We say that a transition occurs whenever an individual changes state. Under the assumptions, the process S t, I t ) is a pure jump Markov process Key words and phrases. Stochastic epidemic model, SIRS epidemic processes, Law of large numbers, Endemic infection, Threshold phenomena in epidemic processes. 1

2 2 R. G. DOLGOARSHINNYKH S θ 1 I ρ R Figure 1. SIRS model. The letters S, I, and R refer to the possible states an individual can assume in turn susceptible, infected and recovered immune). The individual transition rates are shown above the arrows. with the following instantaneous transition rates q{s, I) S 1, I + 1)} = θsi/n, q{s, I) S, I 1)} = ρi, q{s, I) S + 1, I)} = R, where θ = θ 1 θ 2. It will often be more convenient to work with the scaled process γt N = γ t = s t, i t ). This is a continuous time Markov Chain with state space K N = {s, i) : Ns, Ni) Z 2 +, s + i 1} and transition rates 1) q{s, i) s N 1, i + N 1 )} = Nθsi, q{s, i) s, i N 1 )} = Nρi, q{s, i) s + N 1, i)} = Nr. The transition rules 1) imply that for small h > Es t+h F t ) = s t + r t h θi t s t h + oh), Ei t+h F t ) = i t + θi t s t h ρi t h + oh), where F t contains all events that happen before or at time t. Letting h suggests that the random SIRS paths at least over short periods of time) may be approximated in mean by the solutions of the system of ordinary differential equations 2) ds t dt = r t θi t s t, di t dt = θi ts t ρi t.

3 i LLN FOR THE SIRS EPIDEMIC 3 1 Mean field: theta=3,rho= s Figure 2. Mean field. Trajectories of 2) for θ > ρ with started at initial points with i >. We will show in sec. 3 that the random SIRS paths, γ N, tend to solutions of system 2) as the populations size N tends to infinity. This is a first order approximation to the random paths, or the strong law of large numbers for the problem. Here and later we will often refer to the random SIRS paths by γ N or γ and to the solutions of the mean path ODE by γ. Trajectories of the dynamical system 2) lie in K = {s, i) : s, i, s + i 1}. The system has fixed points, that is points such that the rates of change in s and i at these points are all equal to. Solving for such points, we get two possibilities; either s = ρ θ s and i = θ ρ θ1 + ρ) = 1 ρ/θ 1 + ρ i or s = 1 and i =. When ρ < θ the fixed point γ = s, i ) is in the interior of the triangle K; when ρ θ the only fixed point of the dynamical system in K is s = 1, i =. We will show in sec. 5 that the solutions of 2) started at initial points with i > converge to γ when ρ < θ and to 1, ) if ρ θ as t tends to infinity. Once we establish that the random paths γ N converge to solutions γ of 2) we can conclude that if the rate of infection θ is less than or equal to the rate of recovery ρ, the infection tends to become rapidly extinct by following the mean path to the point where s = 1 and i =. On the other hand, if ρ is less than θ then with a significant fraction of the population initially infected we expect the epidemic to follow the mean path to the fixed point in the interior of K and become endemic; that is, the fraction of infected individuals in the population remains significant for a long time. In the case when ρ < θ, the state of the system turns out to be well approximated by the endemic level, γ, for a long time.

4 i i i 4 R. G. DOLGOARSHINNYKH N=1 N= population fractions s time N=4 N= population fractions s time N=25 N= population fractions s time Figure 3. Initial infection spread. The simulations are for population sizes N = 1, 4, 25 and for parameters θ = 2, ρ = 1. For this choice of parameters the fixed point is γ = s, i ) = 1/2, 1/4). All three simulations were started at s, i ) = 1/4, 1/2) N. The left plots show the dynamics of the infection spread in terms of fractions, s and i, of susceptibles and infecteds for the three population sizes over time period of length 5. As N increases the random paths are seen to follow the limiting deterministic paths more closely. In the plots on the right side values of s and i are plotted over time. We see that the fluctuations around the fixed point γ decrease with N.

5 LLN FOR THE SIRS EPIDEMIC 5 The existence of a threshold at the level ρ = θ, that is, qualitatively different behavior of the model for ρ < θ and ρ θ, is not unexpected. The reproduction number, R, is defined as the average) number of secondary cases that are produced by one infected individual in wholly susceptible population. It has been established that large epidemics can occur when R > 1, and the infection quickly becomes extinct when R < 1 see e.g. 6)). For the SIRS model with transitions given by 1) the reproduction number R = θ 1/ρ = θ/ρ. Therefore, the epidemics die out quickly when R < 1 or θ < ρ and but not when R > 1, that is when θ > ρ. The case R = 1 for the SIRS model also leads to extinction of the infection after a short period of time. 2. SIRS Processes as Time Changed Poisson Processes Before we proceed with the proof of our main theorem we construct a version of the random SIRS processes γ N by applying random time changes to a collection of Poisson processes see e.g. 3) chap 4, 4. In particular, let Y 1, Y 2, Y 3 be independent rate one Poisson processes and let y k ) = N 1 Y k N ) for k = 1, 2, 3 and N IN. If the initial population fractions γ = s, i ) are nonrandom then there is a version of γ satisfying ) ) s t = s y 1 θs u i u du + y 3 r u du 3) ) i t = i + y 1 θs u i u du y 2 In what follows it will be useful to define ) ρi u du. 4) β 1 γ) = θsi, β 2 γ) = ρi, β 3 γ) = r = 1 s i. Note that for any t > and k = 1, 2, 3 β k γ u ) du θ + ρ + 1)t < and therefore the Y k Nt) s are well defined for all t [, ). For random initial conditions we can construct the processes γ N satisfying 3) by first generating initial conditions independently of Y 1, Y 2, Y Proof of the Law of Large Numbers Fix T >. Let γ = γ N be a sequence of SIRS processes, indexed by the population size parameter N, satisfying 3) for t T. Theorem 1. If lim N γ = γ and γ is a solution of 2) on [, T ] with an initial condition γ then lim sup γ γ = a.s. N t T

6 6 R. G. DOLGOARSHINNYKH Proof. Since γ satisfies 3) for t T, s t s t s s + ỹ1 β 1 t γ u ) du) + ỹ3 β 3 γ u ) du) + β 1 γ u ) du β 1 t γ u ) du + β 3 γ u ) du β 3 γ u ) du where ỹ k s) = y k s) s are centered and scaled Poisson processes and β k s. are defined by 4). Let ɛ k t) = ỹ k βk γ u ) du) From the strong law of large numbers for a Poisson process, we have for v < and k = 1, 2, 3 lim sup ỹ k u) = a.s. N u v Since β k γ u ) θ + ρ + 1) it follows that lim sup ɛ k t) = N t T for k = 1, 2, 3. Because the functions β k are Lipschitz continuous as functions of γ, there exists a constant M > such that s t s t s s + ɛ 1 t) + ɛ 3 t) + a.s. M γ u γ u du. The same argument leads to a similar bound on i t ī t and combining the two we get ) γ t γ t 2 γ γ + ɛ 1 t) + ɛ 2 t) + ɛ 3 t) + M γ u γ u du ) 2 γ γ + sup k sup ɛ k t) + t T M γ u γ u du Hence by Gronwall s inequality stated below for proof see e.g. 4), chap 3) γ t γ t 2 γ γ + sup k and the conclusion of the theorem follows. Gronwall s Inequality. sup ɛ k t) ) e 2MT t T Let f be an integrable function on [, T ]. If M > and then ft) ɛ + M fs) ds, for t T, ft) ɛe Mt, for t T..

7 LLN FOR THE SIRS EPIDEMIC 7 4. Construction of Lyapunov Function Let fx) be a continuous function of x defined on an open set containing. Let x : [, ) R d solve ẋ t = fx t ). A function V x) defined in a neighborhood of x = is a Lyapunov function if it has continuous partial derivatives, V x) and the trajectory derivative of V satisfies V x t ) = grad V ) fx t ), where the dot denotes scalar multiplication and grad V = V/ x 1,..., V/ x d ). Constructing a Lyapunov function allows us to show global asymptotic stability of trajectories of 2) started at initial points such that i > ; that is, if γ is a solution of 2) such that i > then lim t γ t = γ. To construct a Lyapunov function for the mean path vector field we need to find a function that is positive on the domain and has a negative derivative along the directions of the vector field. We will find it convenient to switch to i, r) coordinates. In these coordinates our vector field becomes di dt = θ1 i t r t )i t ρi t dr dt = ρi t r t. Let x t = i t i, y t = r t r. Showing that, ) is globally asymptotically stable in the x, y) coordinates is equivalent to showing that γ is globally asymptotically stable in the old coordinates. In x, y) coordinates, dx dt = θ 1 x t + i ) y t + r ) ) x t + i ) ρx t + i ) dy dt = ρx t + i ) y t + r ) and simplifying we get dx dt = θx2 t θx t y t θ ρ 1 + ρ x t θ ρ 1 + ρ y t 5) dy dt = ρx t y t. Following the prescription that can be found in e.g. 2) we seek a Lyapunov function of the form vx, y) = F x) + Gy). Then dvx t, y t ) dt = df dx θx 2 t θx t y t θ ρ 1 + ρ x t θ ρ ) 1 + ρ y t + dg dy ρx t y t ).

8 8 R. G. DOLGOARSHINNYKH We further restrict our search to functions v such that the cross terms disappear, that is omitting index t to simplify expressions) df dx θxy θ ρ ) 1 + ρ y + dg ρx ; dy implying that df dx y df dx This suggests the possibility θx + θ ρ 1 + ρ θx + θ ρ 1 + ρ df dx = which when integrated gives and 6) F x) = x i log ) df dx x + i x x x + i, = dg dy ρx ) 1 ρx = dg dy 1 y = dg dy s y. dg dy = y s, 1 + x ), Gy) = y2 i 2s vx, y) = x i log 1 + x ) + y2. i 2s It is not hard to see that vx, y) on the domain of interest, namely on {x, y) x > i, y r, x + y s } and dv dt = df dx θx 2 θ ρ ) 1 + ρ x + dg dy y) = x θxx + i ) y 2 /s = θ x 2 + y 2 /ρ). x + i 5. Stability of Interior Fixed Points In this section we examine stability of the interior fixed point, γ, for the limiting deterministic dynamical system when ρ < θ. We show that for any initial condition with i > and hence x = i i > i lim γ t = γ. t Recall that x t = i t i and y t = r t r and it is enough to show that the system 5) is globally asymptotically stable. In this section, we continue to work in x, y) coordinates.

9 LLN FOR THE SIRS EPIDEMIC 9 Lemma 1. Let x t, y t ) be the solution of 5) started at x, y ), where x > i. There exists x > i such that x t > x for all t. Proof. Since vx, ) + as x i there exists an x < x such that vx, ) > vx, y ) and x > i. Suppose that the solution of 2) started at x, y ) intersects level x = x at some point x, y ). Let v be the Lyapunov function defined by 6). Since v is nonincreasing along the trajectories of 5), but by construction vx, y ) vx, y ), vx, y ) vx, ) > vx, y ) and we arrive at a contradiction. Therefore x t > x for all t. Lemma 2. Any solution of 5) started at a point x, y ) such that x > i is asymptotically stable. Proof. The linearization of 5) near the point,) is dx dt = θ ρ 1 + ρ x θ ρ 1 + ρ y := A x, y) dy dt = ρx y and A has eigenvalues λ 1,2 = θ 1 + ρ ± 1 ) + θ ρ T ρθ ρ), 1 + ρ with negative real parts, since θ > ρ >. Therefore, system 5) is Lyapunov stable, that is for any ε > there exists a δ > such that a solution started in a δ-neighborhood, U δ, of the stable point never leaves the ε-neighborhood of the stable point, U ε, see e.g. 1) chap. 3. Let ε > and δ be as above and x be as in Lemma 1. Suppose the trajectory started at x, y ) never enters U δ. Then the trajectory will lie in K U δ, where K = {x, y) x > x, y r, x + y s }. But dv dt = θ x2 + y 2 /ρ) < C in K U δ for some constant C >. So that vx t, y t ) = vx, y ) + v t dt vx, y ) Ct. If t, v will eventually have to become negative and we arrive at a contradiction. Therefore the trajectory falls into U δ eventually and hence x t, y t ) U ɛ for all t after that happens. Since ɛ was arbitrary the trajectory tends to the stable point as t goes to infinity.

10 1 R. G. DOLGOARSHINNYKH References [1] V. Arnold 1992) Ordinary Differential Equations. Springer, New York [2] Barabashin 197) Introduction to the Theory of Stability. Wolters- Noordhoff, Netherlands [3] S. Ethier and T. Kurtz 1985) Markov Processes. Characterization and Convergence. Wiley & Sons, NY [4] P. Hartman 1964) Ordinary Differential Equations. Wiley & Sons, NY [5] W. Kermack and A. McKendrick1933) Contributions to the mathematical theory of epidemics, III - Further studies of the problem of endemicity. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics [6] D. Mollison, ed. 1995) Epidemic Models: Their Structure and Relation to Data. Cambridge, Cambridge