Long-time behavior of an SIR model with perturbed disease transmission coefficient

In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used 
to classify the extinction and permanence of the disease. Precisely, $\lambda 0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].


1.
Introduction. There have been many kinds of infectious diseases, which have permanent immunity upon recovered, like morbilli, HBV, rubella, whooping cough, smallpox, etc. Such epidemic spreads are often described by SIR (Susceptible-Infective-Recovered) epidemic models. Under the assumption that population develop in a certain environment, the densities S(t), I(t), R(t) of individuals in susceptible, infective, recovered classes, respectively at time t, satisfy the following differential equations:      dS(t) = α − βS(t)I(t) − µS(t))dt dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt dR(t) = (γI(t) − µR(t))dt, where α is the per capita birth rate of the population, µ is the per capita diseasefree death rate and ρ is the excess per capita death rate of infective class, β is the effective per capita contact rate, γ is the per capita recovery rate of the infective individuals. However, it is well-known that this assumption is in general not true since the evolution of a population is always subject to unpredictable factors. The problem to learn how randomness affects to the long term behavior of population in stochastic models is interesting. If the disease transmission coefficient β in the equation (1) is subject to the environmental white noise, then it becomes β → β + white noise. Thus, under the language of stochastic calculus, the epidemic model (1) becomes      dS(t) = α − βS(t)I(t) − µS(t))dt − σS(t)I(t)dB(t) dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σS(t)I(t)dB(t) dR(t) = (γI(t) − µR(t))dt, (2) where B(t) is a Brownian motion.
This model has been investigated in [6,7,11,18] and extended in [1,8,16,20,19,21]. When studying epidemic models, it is naturally important to know whether the population will result in a disease free state or the disease will remain permanently.
For the deterministic model (1), the asymptotic behavior has been classified completely by the value λ d = βα µ − (µ + ρ + γ). Precisely, if λ d ≤ 0 then the population will result in the disease-free equilibrium ( α µ , 0, 0) while the population approach an endemic equilibrium in case λ d > 0.
For the stochastic case, in [7,11,18,21], the authors attempted to answer the afore-mentioned question for the model (2). In particular, in [11], by using Lyapunov-type functions, they provided some sufficient conditions for the exponential extinction of the disease as well as sufficient conditions for the existence of a stationary distribution to the system (2) and described the support of the invariant density. Unfortunately, their conditions are restrictive and not close to any necessary one. With these results it is unable to classify completely stochastic SIR models similar to the deterministic case. Furthermore, there is a gap in [11]. Specially, when Hörmander's condition was verified and when a control system was investigated, they used drift and diffusion coefficients of Itô stochastic differential equation instead of those of Stratonovich one.
Our main goal in this paper is to provide a sufficient and almost necessary condition for permanence (as well as ergodicity) and extinction of the disease in the stochastic SIR model (2) in using a value λ to be similar to λ d in the deterministic model. Note that such kind of results are obtained for a stochastic SIS model in [3]. However, the model studied there can be reduced to one-dimensional equation that is much easier to investigate.
The rest of the paper is arranged as follows. Section 2 derives a threshold λ that is used to classify the model. Clearly, it is shown that if λ < 0, the disease is eradicated as a disease-free equilibrium ( α µ , 0, 0), which is exponentially asymptotically stable. This situation is the eradication of the disease among the population. Meanwhile, in case λ > 0, the solution converges to a stationary distribution in total variation, i.e., the disease is permanent. The ergodicity of the solution process is also examined. Finally, Section 3 is reserved for some discussion and comparison to existing results in the literature. Some numerical examples and figures are also provided to illustrate our results.
2. Sufficient and almost necessary conditions for permanence. Let (Ω, F, P) be a complete probability space and B(t) be an one-dimensional Brownian motion. Because the dynamics of recover class has no effect on the disease transmission dynamics, we need only considering the following system It is shown in [7] that R 2 We first rewrite the equation (3) in Stratonovich's form: (4) Denote by (S s,i (t), I s,i (t)) the solution with initial value (s, i) to (4). In case there is no confusion, we simply write (S(t), I(t)) for (S s,i (t), I s,i (t)). Let P (t, s, i, · ) be its transition probability and let .
By direct calculation we see that where [·, ·] is the Lie bracket of the vector fields. Therefore, det(A, [A, B])(s, i) = σ 2 s 2 i 2 (γ + ρ) = 0 ∀ (s, i) ∈ R 2,• + . Thus, the Lie algebra L(s, i) generated by vector fields A(s, i) and B(s, i) and the ideal L 0 (s, i) in L(s, i) generated by B(s, i) are non degenerate. In particular, the Hörmander condition (see [14,15]) holds for the diffusion equation (4). As a result, the transition probability P (t, s, i, ·) of (S s,i (t), I s,i (t)) has density p(t, s, i, u, v) which is smooth in (s, i, u, v) ∈ R 4,• + . In order to study the control set and the support of diffusion process (4), we analyze the following control system.
where φ is taken from the set of piecewise continuous real valued functions defined on R + . Let (u φ (t, u, v), v φ (t, u, v)) be the solution to the equation (5) with control φ and initial value (u, v). Denote by O + 1 (u, v) the reachable set from (u, v) ∈ R 2,• + , that is the set of (u , v ) ∈ R 2 such that there exist a t ≥ 0 and a control φ(·) We now recall some concepts introduced in [10]. Let X be a subset of R 2 having the property: w 2 ∈ O + 1 (w 1 ) for any w 1 , w 2 ∈ X. Then, there is a unique maximal set Y ⊃ X such that this property still holds for Y . Such Y is called a control set. where ) be the solution to the equation (6) with control φ and initial value (z 0 , v 0 ).
It is easy to see that for all 0 for all control φ and t > 0. We have the following claims.
if v 0 is sufficiently small. As a result, there are a control φ, and a T > 0 Claim 4. For any z 0 > α µ and 0 < v 0 < z 0 , there are a control φ, and a T > 0 Therefore, we cannot find any control φ and T > 0 satisfying From these five claims, we conclude that the control system (6) has a unique invariant control set C = (z, v) : z ∈ α µ+ρ+γ , α µ , 0 < v < z . This means the control system (5) has only one invariant control set, namely S = (u, v) ∈ R 2,• + :

LONG-TIME BEHAVIOR OF A STOCHASTIC SIR MODEL 3433
We are now in position to provide a condition for the existence of a unique invariant probability measure for the process (S(t), I(t)) and investigate some properties of the invariant probability measure.
Firstly, define the threshold By adding side by side in system (3), we have Using the comparison theorem yields Theorem 2.1. Let (S(t), I(t)) be the solution to equation (3). If λ > 0, the process (S(t), I(t)) has an unique invariant probability measure whose support is S.
Proof. We derive from (3) that From (9) and the strong law of large numbers for local martingales we have Thus, Otherwise, by Itô's formula, we obtain ln I(t) = ln I(0) By (9) and (12), we conclude that Multiplying (10) with − β µ and adding it into (13) we obtain Let be a positive number, lim inf for sufficiently small. Also, using (9) yields lim t→∞ E1 {S(t)+I(t)> α µ } = 0, which implies that lim sup We show that (S(t), I(t)) is a strong Feller Markov process. Indeed, from the righthand side of inequalities (8), we have As a consequence, for any m > 0 the solution (S(t), I(t)), starting in the domain A m = {(s, i) : s + i < α µ + m} remains in A m for all t. Therefore, the solution (S(t), I(t)) of equation (4) with initial conditions in A m forms a Markov process with states space A m . By virtue of boundedness of A m , the smoothness of the coefficients and the Hörmander condition for the diffusion equation (4), it follows that (S(t), I(t)) is a strong Feller Markov process (see [4,10,9] for details).
By the well-known results in [4] and [10] and the Hörmander condition, it follows that for any ϕ * -integrable function f , and (s, i) ∈ S we have P lim and where · is the total variation norm.
We now show that for any initial value (s, i) ∈ R 2,• + , (S s,i (t), I s,i (t)) eventually enters S. Indeed, suppose in the contrary that P( On the other hand from (14) it yields lim t→∞ t 0 I(τ )dτ = ∞ a.s., which implies that lim t→∞ (S(t) + I(t)) = −∞ for all ω ∈ Ω 1 . This is a contradiction.
Summing up we obtain the following result.
Proof. We proceed in the following steps. (i) By using a Lyapunov function, we can show that for any ε > 0, there exists a δ > 0 such that (23) (ii) There is 0 < δ < δ such that for any (s 0 , i 0 ) ∈ R 2,• + , the process (S s0,i0 (t), I s0,i0 (t)) is recurrent relative to S δ = (s, i) ∈ R 2 + : s + i ≤ α µ + δ, s ≥ δ . (iii) There exists a T > 0 such that for any (u, v) ∈Ŝ δ , there exists a control φ such that (u φ (t, u, v), v φ (t, u, v)) ∈ U δ for some t ∈ [0, T ]. (iv) Using Markov property of the solution and the support theorem we show that U δ is absorptive and then we obtain the desired conclusion.
For (s, i) ∈ R 2,• + , set τ s,i δ = inf{t > 0 : (S s,i (t), I s,i (t)) ∈ U δ }. Then, it is easy to see that Define a sequence of stopping times Since (S s0,i0 (t), I s0,i0 (t)) is recurrent relative to S δ , η k is finite for every k. Consider the events To obtain last item (iv), we deduce from the strong Markov property of (S(t), I(t)) and (27) that As a result, In light of the strong Markov property of (S(t), I(t)), (23) and (28) yield Since ε can be taken arbitrarily, we obtain Finally, it follows from Itô's formula that Hence (22) follows straightforward from (30) and (31).

Discussion and numerical examples.
We have classified whenever the disease in a stochastic SIR model is extinct or permanent by the sign of a threshold value λ. Only the critical case λ = 0 is not studied in this paper.
Let us finish this paper by providing some numerical examples.
A sample path of solution to (3) is illustrated by Figures 1, while the phase portrait in Figure 2 demonstrates that the support of π * is a domain surrounded by the lines s + i = 1, 25; i = 0; s = 0 and s + i = 2. The empirical density of π * is shown in this picture as well.