c 2016 Society for Industrial and Applied Mathematics SIAM J APPLIED DYNAMICAL SYSTEMS Vol 15, No 2, pp 1062–1084 Classification of Asymptotic Behavior in a Stochastic SIR Model∗ Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N T Dieu† , D H Nguyen‡ , N H Du§ , and G Yin‡ Abstract Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides sufficient conditions that are very close to the necessary ones for the permanence In addition, this paper develops ergodicity of the underlying system It is proved that the transition probabilities converge in total variation norm to the invariant measure Our result gives a precise characterization of the support of the invariant measure Rates of convergence are also ascertained It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree Key words SIR model, extinction, permanence, stationary distribution, ergodicity AMS subject classifications 34C12, 60H10, 92D25 DOI 10.1137/15M1043315 Introduction Since epidemic models were first introduced by Kermack and McKendrick in [15, 16], the study of mathematical models has flourished Much attention has been devoted to analyzing, predicting the spread of, and designing controls of infectious diseases in host populations; see [1, 2, 4, 6, 9, 18, 19, 15, 16, 24, 26, 29] and the references therein One of the classic epidemic models is the SIR (susceptible-infected-removed) model which is suitable for modeling some diseases with permanent immunity such as rubella, whooping cough, measles, and smallpox In the SIR model, a homogeneous host population is subdivided into the following three epidemiologically distinct types of individuals: • (S) The susceptible class: those individuals who are capable of contracting the disease and becoming infected • (I) The infected class: those individuals who are capable of transmitting the disease to others • (R) The removed class: infected individuals who are deceased, or have recovered and are either permanently immune or isolated ∗ Received by the editors October 9, 2015; accepted for publication (in revised form) by L Billings March 7, 2016; published electronically May 26, 2016 http://www.siam.org/journals/siads/15-2/M104331.html † Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam (dieunguyen2008@gmail com) The research of this author was supported in part by the Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training under grant B2015-27-15 ‡ Department of Mathematics, Wayne State University, Detroit, MI 48202 (dangnh.maths@gmail.com, gyin@math.wayne.edu) The research of these authors was supported in part by National Science Foundation grant DMS-1207667 § Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (dunh@vnu.edu.vn) The research of this author was supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 101.03-2014.58 1062 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSIFICATION IN A STOCHASTIC SIR MODEL 1063 If we denote by S(t), I(t), and R(t) the number of individuals at time t in classes (S), (I), and (R), respectively, the spread of infection can be formulated by the following deterministic system of differential equations:   dS(t) = α − βS(t)I(t) − µS(t))dt, (1.1) 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 disease-free death rate, ρ is the excess per capita death rate of the infective class, β is the effective per capita contact rate, and γ is the per capita recovery rate of the infected individuals For the above deterministic model (1.1), if λd = βα µ − (µ + ρ + γ) ≤ 0, then the population tends to the α disease-free equilibrium ( µ , 0, 0); if λd > 0, the population approaches an endemic equilibrium Thus, using the critical threshold value λd , the asymptotic behavior of the system has been completely classified In [29], similar results were given for a general epidemic model with reaction-diffusion in terms of basic reproduction numbers It is well recognized that random effect is often not avoidable, and a population is always subject to random disturbances Thus, it is important to investigate stochastic epidemic models A resurgent effort has been devoted to finding the corresponding classification by means of threshold levels Jiang et al [14] investigated the asymptotic behavior of global positive solution for the nondegenerate stochastic SIR model   dS(t) = α − βS(t)I(t) − µS(t))dt + σ1 S(t)dB1 (t), (1.2) dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σ2 I(t)dB2 (t),   dR(t) = (γI(t) − µR(t))dt + σ3 R(t)dB3 (t), where B1 (t), B2 (t), and B3 (t) are mutually independent Brownian motions, and σ1 , σ2 , and σ3 are the intensities of the white noise The model is more difficult to deal with compared with the deterministic counterpart Moreover, in reality, the classes (S), (I), and (R) are usually subject to the same random factors such as temperature, humidity, pollution, and other extrinsic influences As a result, it is more plausible to assume that the random noises perturbing the three classes are from the same source If we assume that the Brownian motions B1 (t), B2 (t), and B3 (t) are the same, we obtain the model   dS(t) = α − βS(t)I(t) − µS(t))dt + σ1 S(t)dB(t), (1.3) dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σ2 I(t)dB(t),   dR(t) = (γI(t) − µR(t))dt + σ3 R(t)dB(t), which has been considered in [20] An important question is whether the transition to either a disease-free state or the disease state will become permanent In [20], the authors attempted to answer this question for (1.3) for the cases σ1 > and σ2 > By using Lyapunov-type functions, they provided some sufficient conditions for extinction or permanence, as well as ergodicity, for the solution of system (1.3) Using these same methods, the authors of [13, 30] studied extinction and permanence in some different stochastic SIR models Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1064 N T DIEU, D H NGUYEN, N H DU, AND G YIN In contrast to (1.2), (1.3) is more difficult to deal with due to the degeneracy of the diffusion Moreover, although one may assume the existence of an appropriate Lyapunov function, it is fairly difficult to find an effective Lyapunov function in practice Because of these difficulties, there has been no decisive classification for stochastic SIR models that is similar to the deterministic case Our main goal in this paper is to provide such a classification We shall derive a sufficient and almost necessary condition for permanence (as well as ergodicity) and extinction of the disease for the stochastic SIR model (1.3) by using a value λ, which is similar to λd in the deterministic model Note that such results are obtained for a stochastic susceptible-infective-susceptible (SIS) model in [10] However, the model studied there can be reduced to a one-dimensional equation that is much easier to investigate The method used in [10] cannot treat the stochastic SIR model (1.3) Estimation for the convergence rate is also not given in [10] Therefore, a more general method needs to be introduced The new method not only can remove most assumptions in [20] but also can treat cases σ1 > and σ2 < 0, which have not been taken into consideration in [20] Note that cases σ1 > and σ2 < indicate that the random factors have opposite effects on healthy and infected individuals For instance, patients with tuberculosis or some other pulmonary disease not endure well in cool and humid weather, while healthy people may be fine in such weather In addition, individuals with a disease usually have weaker resistance to other diseases Our new method is also suitable for dealing with other stochastic variants of (1.1) such as models introduced in [5, 13, 30] The rest of the paper is arranged as follows Section derives a threshold that is used to classify the extinction and permanence of the disease To establish the desired result, by considering the dynamics on the boundary we obtain a threshold λ that enables us to determine the asymptotic behavior of the solution In particular, it is shown that if λ < 0, the disease will decay at an exponential rate In case λ > 0, the solution converges to a stationary distribution in total variation This means that the disease is permanent The rate of convergence is proved to be bounded above by any polynomial decay The ergodicity of the solution process is also proved Finally, section is devoted to some discussion and comparison to existing results in the literature Some numerical examples are provided to illustrate our results Threshold between extinction and permanence Let (Ω, F, {Ft }t≥0 , P) be a complete probability space with the filtration {Ft }t≥0 satisfying the usual condition, i.e., it is increasing and right continuous while F0 contains all P-null sets Let B(t) be an Ft -adapted, Brownian motion Because the dynamics of the recovered class has no effect on the disease transmission dynamics, we consider only the following system: (2.1) dS(t) = [α − βS(t)I(t) − µS(t)]dt + σ1 S(t)dB(t), dI(t) = [βS(t)I(t) − (µ + ρ + γ)I(t)]dt + σ2 I(t)dB(t) Assume that σ1 , σ2 = By the symmetry of Brownian motion, without loss of generality we suppose throughout this paper that σ1 > Using standard arguments, it can be easily shown that for any positive initial value (u, v) ∈ R2,◦ + := {(u , v ) : u > 0, v > 0}, there exists a unique global solution (Su,v (t), Iu,v (t)), t ≥ 0, that remains in R2,◦ + with probability (see, e.g., [14]), where the subscripts u and v denote the dependence on the initial data (u, v) To Copyright © by SIAM Unauthorized reproduction of this article is prohibited CLASSIFICATION IN A STOCHASTIC SIR MODEL 1065 obtain further properties of the solution, we first consider the equation on the boundary, dS(t) = (α − µS(t))dt + σ1 S(t)dB(t) Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.2) Let Su (t) be the solution to (2.2) with initial value u It follows from the comparison theorem [11, Thm 1.1, p 437] that Su,v (t) ≤ Su (t) ∀t ≥ a.s By solving the Fokker–Planck equation, the process Su (t) has a unique stationary distribution with density f ∗ (x) = (2.3) ba −(a+1) −b x ex, Γ(a) x > 0, σ2 , b = 2α , and Γ(·) is the Gamma function By the strong law of where c1 = µ + 21 , a = 2c σ12 σ12 large numbers [27, Thm 3.16, p 46] we deduce that t→∞ t (2.4) ∞ t lim Su (s)ds = xf ∗ (x)dx := α a.s µ To proceed, we define the threshold as follows: (2.5) λ := σ2 αβ − µ+ρ+γ+ µ Remark 2.1 Because λ is key, we explain its definition and use (i) To determine whether or not Iu,v (t) converges to 0, we consider the Lyapunov exponent of Iu,v (t) when Iu,v (t) is small for a sufficiently long time Hence, we look at the following equation derived from Itˆ o’s formula: ln Iu,v (T ) ln v σ2 = + B(T ) + T T T T T βSu,v (t) − µ + ρ + γ + σ22 dt When T is large, the first and second terms on the right-hand side are small Intuitively, if Iu,v (t) is small for t ∈ [0, T ], Su,v (t) is close to Su (t) Using the ergodicity (2.4), we arrive at the following approximation: T T βSu,v (t) − µ + ρ + γ + σ22 dt T σ2 β Su,v (t) − µ + ρ + γ + T 2 αβ σ ≈ − µ + ρ + γ + = λ µ ≈ ln I dt (T ) u,v Thus, is close to λ As a result, if λ < 0, Iu,v (t) is likely to decay exponentially In T contrast, if λ > 0, Iu,v (t) cannot be small for a long time; that is, the disease will survive permanently (ii) It is interesting to note that λ does not depend on σ1 This is because the long-term ∞ average of S(t), which is xf ∗ (x)dx, does not depend on σ1 Based on the above idea, we now consider two cases λ < and λ > separately Then we provide rigorous proofs of the desired results Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1066 N T DIEU, D H NGUYEN, N H DU, AND G YIN Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 2.1 Case λ < Theorem 2.1 If λ < 0, then for any initial value (S(0), I(0)) = (u, v) ∈ R2,◦ + we have ln Iu,v (t) lim supt→∞ ≤ λ a.s., and the distribution of Su,v (t) converges weakly to the unique t invariant probability measure µ∗ with the density f ∗ Proof Let Iv (t) be the solution to dI(t) = I(t) − (µ + ρ + γ) + β Su (t) dt + σ2 I(t)dB(t), I(0) = v, where S(t) is the solution to (2.2) By the comparison theorem, Iu,v (t) ≤ Iv (t) a.s given that S(0) = S(0) = u, I(0) = I(0) = v In view of Itˆo’s formula and the ergodicity of Su (t), lim sup (2.6) t→∞ σ2 B(t) t − µ + ρ + γ + + β Su (τ ) dτ + σ2 ln Iv (t) = lim sup t t t t→∞ αβ σ = − µ + ρ + γ + = λ < a.s µ That is, Iu,v (t) converges almost surely to at an exponential rate For any ε > 0, it follows from (2.6) that there exists t0 > such that P(Ωε ) > − ε, where λt Ωε := Iu,v (t) ≤ exp ∀t ≥ t0 = ln Iu,v (t) ≤ λt ∀t ≥ t0 λt0 Clearly, we can choose t0 satisfying − 2β λ exp{ } < ε Let Su (t), t ≥ t0 , be the solution to (2.2) given that S(t0 ) = S(t0 ) We have from the comparison theorem that P{Su,v (t) ≤ Su (t) ∀t ≥ t0 } = In view of Itˆ o’s formula, for almost all ω ∈ Ωε we have t ≤ ln Su (t) − ln Su,v (t) = α t0 t ≤β t0 − dτ + β Su,v (τ ) t Iu,v (τ )dτ Su (τ ) t0 2β λt0 λt λτ dτ = − exp − exp exp λ 2 < ε As a result, (2.7) P{| ln Su,v (t) − ln Su (t)| > ε} ≤ − P(Ωε ) < ε ∀t ≥ t0 Let υ ∗ be the distribution of a random variable ln X provided that X admits µ∗ as its distribution In lieu of proving that the distribution of S(t) converges weakly to µ∗ , we claim an equivalent statement that the distribution of ln S(t) converges weakly to υ ∗ By the Portmanteau theorem (see [3, Thm 1]), it is sufficient to prove that for any g(·) : R → R satisfying |g(x) − g(y)| ≤ |x − y| and |g(x)| < ∀x, y ∈ R, we have ∞ g(x)υ ∗ (dx) = Eg(ln Su (t)) → g := R g(ln x)µ∗ (dx) Copyright © by SIAM Unauthorized reproduction of this article is prohibited CLASSIFICATION IN A STOCHASTIC SIR MODEL 1067 Since the diffusion given by (2.2) is nondegenerate, it is well known that the distribution of Su (t) weakly converges to µ∗ as t → ∞ (see, e.g., [11]) Thus, Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.8) lim Eg(ln Su (t)) = g t→∞ Note that (2.9) |Eg(ln Su,v (t)) − g| ≤ |Eg(ln Su,v (t)) − Eg(ln Su (t))| + |Eg(ln Su (t)) − g| ≤ εP{| ln Su,v (t) − ln Su (t)| ≤ ε} + 2P{| ln Su,v (t) − ln Su (t)| > ε} + |Eg(ln Su (t)) − g| Applying (2.7) and (2.8) to (2.9) yields lim sup |Eg(ln Su,v (t)) − g| ≤ 3ε t→∞ Since ε is taken arbitrarily, we obtain the desired conclusion The proof is complete 2.2 Case λ > Let P (t, (u, v), ·) be the transition probability of (Su,v (t), Iu,v (t)) Since the diffusion is degenerate, to obtain properties of P (t, (u, v), ·) we check its hypoellipticity First, we rewrite (2.1) in Stratonovich’s form, dS(t) = [α − c1 S(t) − βS(t)I(t)]dt + σ1 S(t) ◦ dB(t), dI(t) = [−c2 I(t) + βS(t)I(t)]dt + σ2 I(t) ◦ dB(t), (2.10) where c1 = µ + σ12 , c2 =µ+ρ+γ+ A(x, y) = σ22 Put α − c1 x − βxy −c2 y + βxy and B(x, y) = σ1 x σ2 y To proceed, we first recall the notion of Lie bracket If Φ(x, y) = (Φ1 , Φ2 ) and Ψ(x, y) = (Ψ1 , Ψ2 ) are vector fields on R2 , then the Lie bracket [Φ, Ψ] is a vector field given by [Φ, Ψ]j (x, y) = Φ1 ∂Ψj ∂Φj ∂Ψj ∂Φj (x, y) − Ψ1 (x, y) + Φ2 (x, y) − Ψ2 (x, y) ∂x ∂x ∂y ∂y Denote by L(x, y) the Lie algebra generated by A(x, y), B(x, y) and by L0 (x, y) the ideal in L(x, y) generated by B We have the following lemma Lemma 2.1 For σ1 > 0, σ2 = 0, the Hă ormander condition holds for the diffusion (2.10) To be more precise, we have dim L0 (x, y) = at every (x, y) ∈ R2,◦ + or, equivalently, the set 2,◦ of vectors B, [A, B], [A, [A, B]], [B, [A, B]], spans R at every (x, y) ∈ R+ As a result, the transition probability P (t, (u, v), ·) has smooth density p(t, u, v, u , v ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1068 N T DIEU, D H NGUYEN, N H DU, AND G YIN Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Proof This lemma has been proved in [20] for the case σ2 > Assume that r = − σσ21 > It is easy to obtain C := B(x, y) = σ1 x , −ry D :=[A, C](x, y) = α − rβxy , −βxy E :=[C, D](x, y) = −α + r2 βxy , −βxy F :=[C, E](x, y) = α − r3 βxy −βxy Observe that det(D, F ) = only if r2 = or r = (since r > 0) When r = 1, solving det(D, E) = obtains βxy = α which implies det(C, D) = x −y −α = 2,◦ As a result, B, D, E, F span R2 ∀ (x, y) ∈ R+ The lemma is proved In order to describe the support of the invariant measure π ∗ (if it exists) and to prove the ergodicity of (2.1), we need to investigate the following control system on R2,◦ + : (2.11) u˙ φ (t) = σ1 uφ (t)φ(t) + α − βuφ (t)vφ (t) − c1 uφ (t), v˙ φ (t) = σ2 vφ (t)φ(t) + βuφ (t)vφ (t) − c2 vφ (t), where φ is taken from the set of piecewise continuous real-valued functions defined on R Let (uφ (t, u, v), vφ (t, u, v)) be the solution to (2.11) with control φ and initial value (u, v) Denote 2,◦ by O+ (u, v) the reachable set from (u, v) ∈ R+ , that is, the set of (u , v ) ∈ R2,◦ + such that there exist a t ≥ and a control φ(·) satisfying uφ (t, u, v) = u , vφ (t, u, v) = v We now recall some concepts introduced in [17] Let X be a subset of R2,◦ + satisfying the property that for + any w1 , w2 ∈ X, w2 ∈ O (w1 ) Then there is a unique maximal set Y ⊃ X such that this property still holds for Y Such a Y is called a control set A control set W is said to be invariant if O+ (w) ⊂ W ∀ w ∈ W r Putting r := −σ σ1 and zφ (t) = uφ (t)vφ (t), we have an equivalent system (2.12) u˙ φ (t) = σ1 φ(t)uφ (t) + g(uφ (t), zφ (t)), z˙φ (t) = h(uφ (t), zφ (t)), where g(u, z) = −c1 u + α − βzu1−r and h(u, z) = u−r z − (c1 r + c2 )ur + βu1+r + αrur−1 − βrz Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSIFICATION IN A STOCHASTIC SIR MODEL 1069 Lemma 2.2 For the control system (2.12), the following claims hold For any u0 , u1 , z0 > 0, and ε > 0, there exist a control φ and T > such that uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ε For any < z0 < z1 , there are a u0 > 0, a control φ, and T > such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ ≤ t ≤ T Let d∗ = inf u>0 {−(c1 r + c2 )ur + βu1+r + αrur−1 } (a) If d∗ ≤ 0, then for any z0 > z1 there are u0 > 0, a control φ, and T > such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ ≤ t ≤ T d∗ (b) Suppose that d∗ > and z0 > c∗ := βr If c∗ < z1 < z0 , there are u0 > and a control φ and T > such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ ≤ t ≤ T However, there are no control φ and T > such that zφ (T, u0 , z0 ) < c∗ for any u0 > Proof Suppose u0 < u1 and ε > are as in claim Let ρ1 = sup{|g(u, z)|, |h(u, z)| : u0 ≤ u ≤ u1 , |z − z0 | ≤ ε} We choose φ(t) ≡ ρ2 with ( σ1 ρρ21 u0 − 1)ε ≥ u1 − u0 It is easy to check that with this control, there is ≤ T ≤ ρε1 such that uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ε If u0 > u1 , we can construct φ(t) similarly Then claim is proved By choosing u0 to be sufficiently large, there is a ρ3 > such that h(u0 , z) > ρ3 ∀z0 ≤ z ≤ z1 This property, combined with (2.12), implies the existence of a feedback control φ and T > satisfying that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ ≤ t ≤ T We now prove claim If r < 0, then lim − (c1 r + c2 )ur + βu1+r + αrur−1 = −∞, u→0 and if r > 1, then lim − (c1 r + c2 )ur + βu1+r + αrur−1 = u→0 As a result, d∗ ≤ if r ∈ / (0, 1], which implies that for any z0 > z1 we choose u0 such that supz∈[z1 ,z0 ] h(u0 , z) < 0, which implies that there are a feedback control φ and T > satisfying zφ (T, u0 , z0 ) = z1 and uφ (t, u0 , z0 ) = u0 ∀ ≤ t ≤ T = d∗ If d∗ ≤ 0, + αrur−1 If r ∈ (0, 1], there exists u0 such that −(c1 r + c2 )ur0 + βu1+r 0 then for any z0 > z1 > we have supz∈[z1 ,z0 ] h(u0 , z) ≤ u−r supz∈[z1 ,z0 ] {−βrz } < 0, which implies the desired claim Consider the remaining case when r ∈ (0, 1] and d∗ > First, assume c∗ < z1 < z0 Let + αrur−1 = d∗ = βrc∗ Hence, u0 satisfy −(c1 r + c2 )ur0 + βu1+r 0 sup {h(u0 , z)} = u−r z∈[z1 ,z0 ] = sup z∈[z1 ,z0 ] ∗ βru−r z1 (c z − (c1 r + c2 )ur0 + βu1+r + αrur−1 − βrz 0 − z1 ) < Thus, there are a feedback control φ and T > satisfying zφ (T, u0 , z0 ) = z1 and uφ (t, u0 , z0 ) = u0 ∀0 ≤ t ≤ T The final assertion follows from the fact that h(u, c∗ ) ≥ ∀ u > Note that our main goal is not only to prove the permanence of the disease when λ > but also to estimate the convergence rate of the transition probability to an invariant probability measure In order to this, we construct a function V : R2,◦ + → [1, ∞) satisfying EV Su,v (t∗ ), Iu,v (t∗ ) ≤ V (u, v) − κ1 V γ (u, v) + κ2 1{(u,v)∈K} Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1070 N T DIEU, D H NGUYEN, N H DU, AND G YIN Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php for some petite set K and some γ ∈ (0, 1), κ1 , κ2 > 0, t∗ > 1, and then apply [12, Thm 3.6] Recall that a set K is said to be petite with respect to the Markov chain (Su,v (nt∗ ), Iu,v (nt∗ )), 2,◦ n ∈ N, if there exist a measure ψ with ψ(R+ ) > and a probability distribution ν(·) concentrated on N such that ∞ P (nt∗ , u, v, Q)ν(n) ≥ ψ(Q) K(u, v, Q) := 2,◦ ∀(u, v) ∈ K, Q ∈ B(R+ ); n=1 we refer the reader to [22, pp 106 and 121–124] for further details on petite sets We also have to prove that the skeleton Markov chain Su,v (nt∗ ), Iu,v (nt∗ ) , n ∈ N, is irreducible and aperiodic We refer the reader to [22] and [25] for definitions and properties of irreducibility, aperiodicity, and petite sets The estimation of the convergence rate is divided into some lemmas and propositions Because the proofs are quite technical, we explain briefly the main ideas and steps to obtain the convergence rate In the literature (see [7, 12, 22, 23, 28] and references therein), the key to examining convergence rate of ergodic Markov processes (or chains) is to estimate how fast the processes (or chains) enter a petite set (usually a compact set) and practical criteria are often given in terms of Lyapunov functions Basically, our proofs adopt that idea First, we estimate how fast the solution (Su,v (t), Iu,v (t)) enters a compact subset K := [κ−1 , κ]×[κ−1 , κ] for suitable κ > To estimate how fast the solution (Su,v (t), Iu,v (t)) starting −1 or v > κ} enters K, we use Lemma 2.3 in which in D1 := {(u , v ) ∈ R2,◦ + : u > κ or u < κ a Lyapunov function U (u, v) is utilized We estimate the moment (Su,v (t), Iu,v (t)) hits K for −1 (u, v) ∈ D2 := {(u , v ) ∈ R2,◦ + : v < κ } in Propositions 2.1 and 2.2 in which the function [ln v]− (which is − ln v if v < 1) is considered based on the idea in Remark 2.1 Lemmas 2.4 and 2.5 are auxiliary results needed for Propositions 2.1 and 2.2 Finally, we show that the Markov chain Su,v (nt∗ ), Iu,v (nt∗ ) , n ∈ N, is irreducible and aperiodic and that every compact set is petite in Lemma 2.6 with help of Lemmas 2.1 and 2.2 ∗ Lemma 2.3 For any < p∗ < min{ 2µ , 2(µ+ρ+γ) }, let U (u, v) = (u + v)1+p + u− σ12 σ22 exist positive constants K1 , K2 such that (2.13) eK1 t E(U (Su,v (t), Iu,v (t))) ≤ U (u, v) + There K2 (eK1 t − 1) K1 ∗ Proof Consider the Lyapunov function U (u, v) = (u + v)1+p + u− lating the differential operator LU (u, v) associated with (2.1), we have (2.14) p∗ p∗ By directly calcu- ∗ LU (u, v) = (1 + p∗ )(u + v)p (α − µu − (µ + ρ + γ)v) (1 + p∗ )p∗ ∗ + (u + v)p −1 (σ1 u + σ2 v)2 p∗ − p∗ −1 p∗ (2 + p∗ ) − p∗ − u (α − βuv − µu) + σ1 u 2 p∗ ∗ ∗ = (1 + p∗ )α(u + v)p − (1 + p∗ )(u + v)p −1 µ − σ12 u2 p∗ 2 + µ + ρ + γ − σ2 v + (2µ + ρ + γ − p∗ σ1 σ2 )uv p∗ p∗ α − 2+p∗ βp∗ − p∗ p∗ (2 + p∗ )σ12 − u + u v+ + µ u− 2 Copyright © by SIAM Unauthorized reproduction of this article is prohibited CLASSIFICATION IN A STOCHASTIC SIR MODEL 1071 By Young’s inequality, we have u− Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.15) p∗ v≤ ∗ ∗ 4+3p∗ 4+3p∗ 3p∗ − 4+3p − 4+3p 6 u v + ≤ u + (u + v) + 3p∗ + 3p∗ ∗ , 2(µ+ρ+γ) } that K1 := min{µ − p2 σ12 , µ + ρ + It follows from the assumption < p∗ < min{ 2µ σ2 σ2 p∗ σ +p∗ σ ∗ 2 γ− p2 σ22 } > and that p∗ σ1 σ2 < ≤ µ+(µ+ρ+γ), which implies 2µ+ρ+γ−p∗ σ1 σ2 > Applying the two inequalities above and (2.15) to (2.14), we obtain ∗ LU (u, v) + K1 U (u, v) ≤ κ1 (u + v)p − κ2 (u + v)p ∗ +1 + κ3 (u + v) 3p∗ +1 + −κ4 u−1 + κ3 u− + κ5 u− where κ1 = (1 + p∗ )α, κ2 = p∗ K1 , κ3 = easy to derive from this estimate that K2 = sup βp∗ , κ4 = p∗ α , and κ5 = p∗ (2+p∗ )σ12 p∗ , + µ + K1 It is {LU (u, v) + K1 U (u, v)} < ∞ (u,v)∈R2,◦ + As a result, LU (u, v) ≤ K2 − K1 U (u, v) ∀(u, v) ∈ R2,◦ + (2.16) For n ∈ N, define the stopping time ηn = inf{t ≥ : U (Su,v (t), Iu,v (t)) ≥ n} Then Itˆo’s formula and (2.16) yield that E(eK1 (t∧ηn ) U (Su,v (t ∧ ηn ), Iu,v (t ∧ ηn ))) t∧ηn ≤ U (u, v) + E eK1 τ LU (Su,v (τ ), Iu,v (τ )) + K1 U (Su,v (τ ), Iu,v (τ )) dτ K2 (eK1 t − 1) ≤ U (u, v) + K1 By letting n → ∞, we obtain from Fatou’s lemma that (2.17) EeK1 t (U (Su,v (t), Iu,v (t))) ≤ U (u, v) + K2 (eK1 t − 1) K1 The lemma is proved Lemma 2.4 There are positive constants K3 , K4 such that, for any u ≥ 0, v > 0, t ≥ 1, and A ∈ F, (2.18) E [ln Iu,v (t)]2− 1A ≤ [ln v]2− P(A) + K3 P(A)t[ln v]− + K4 t2 where [ln x]− = max{0, − ln x} Copyright © by SIAM Unauthorized reproduction of this article is prohibited P(A), 1072 N T DIEU, D H NGUYEN, N H DU, AND G YIN Proof We have t − ln Iu,v (t) = − ln Iu,v (0) − β Su,v (τ )dτ + c2 t − σ2 B(t) Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ≤ − ln v + c2 t + |σ2 B(t)|, where c2 = µ + ρ + γ + σ22 Therefore, [ln Iu,v (t)]− ≤ [ln v]− + c2 t + |σ2 B(t)| This implies that [ln Iu,v (t)]2− 1A ≤ [ln v]2− 1A + c22 t2 + σ22 B (t) 1A + 2c2 t[ln v]− 1A + 2|σ2 B(t)|1A [ln v]− + 2c1 t|σ2 B(t)|1A By using the Hă older inequality, we obtain for t that EB (t)P(A) ≤ E|B(t)|1A ≤ tP(A) ≤ t P(A) and EB (t)1A ≤ EB (t)P(A) ≤ 6t2 P(A) ≤ 3t P(A) Taking expectation on √ both sides and using the two above estimates as well as the fact that P(A) < P(A) and t ≤ t for t ≥ 1, we have E[ln Iu,v (t)]2− 1A ≤ [ln v]2− P(A) + K3 t P(A)[ln v]− + K4 t2 P(A) for some positive constants K3 , K4 We now choose ε ∈ (0, 1) satisfying (2.19) − √ 3λ (1 − ε) + K3 ε < −λ and − √ 3λ λ (1 − ε) + 2K3 ε < − βH − 2c2 2σ22 < ε Choose H so large that (2.20) βH − 2c2 ≥ + λ; exp − and exp − λ(βH − c2 ) ε < 2 4σ2 Lemma 2.5 For ε and H chosen as above, there are δ ∈ (0, 1) and T ∗ > such that (2.21) P ln v + 3λt ≤ ln Iu,v (t) < ∀ t ∈ [T ∗ , 2T ∗ ] ≥1−ε ∀ u ∈ [0, H], v ∈ (0, δ] Copyright © by SIAM Unauthorized reproduction of this article is prohibited CLASSIFICATION IN A STOCHASTIC SIR MODEL 1073 Proof Let θ ∈ (0, 1) such that βα 11λ − c2 ≥ µ + βθ 12 Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.22) Let S˜u (t) be the solution with initial value u to ˜ = [α − (βθ + µ)S(t)]dt ˜ ˜ dS(t) + σ1 S(t)dB(t) (2.23) Similar to (2.4), t→∞ t t lim P S˜u (τ )dτ = α µ + βθ = ∀ u ∈ [0, ∞) In view of the strong law of large numbers for martingales, P{limt→∞ there exists T ∗ > such that (2.24) σ1 B(t) −λ ≥ ∀t ≥ T∗ t 12 P and P t t S˜0 (τ )dτ ≥ ≥1− α λ − ∀t ≥ T∗ µ + βθ 12β B(t) t = 0} = Hence, ε ε ≥1− By the uniqueness of solutions to (2.23), P S˜0 (t) ≤ S˜u (t) ∀t ≥ = ∀ u ≥ Hence, (2.25) P t t S˜u (τ )dτ ≥ α λ − ∀t ≥ T∗ µ + βθ 12β ε ≥1− We deduce from Itˆ o’s formula that ESu,v (2T ∗ ∧ η˜n ) ≤ 2αT ∗ + u, where ηn := inf{t > : Su,v (t) > n} As a result, there exists nH = nH (ε) > such that P{ηnH < 2T ∗ } < ˜ ∗ , ε) > such that standard arguments, there exists C˜ = C(T sup {|σ2 B(t)|} > C˜ P t∈[0,2T ∗ ] ε ∀u ∈ [0, H] By ε < ˜ then for any t ∈ [0, 2T ∗ ] and By Itˆo’s formula, if ηnH > 2T ∗ and supt∈[0,2T ∗ ] {|σ2 B(t)|} < C, v ≤ δ := θ exp(−βnH T ∗ − C˜ − 1), we have t ln Iu,v (t) ≤ ln v + β ˜ < ln θ Su,v (t)dt + σ2 B(t) ≤ ln v + (βnH T ∗ + C) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1074 N T DIEU, D H NGUYEN, N H DU, AND G YIN As a result, Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.26) P {ζu,v ≤ 2T ∗ } ≤ ε ∀v ≤ δ, u ∈ [0, H], where ζu,v = inf{t ≥ : Iu,v (t) ≥ θ} Observe also that P Su,v (t) ≥ S˜u (t) ∀ t ≤ ζu,v (2.27) = 1, which we have from the comparison theorem From (2.22), (2.24), (2.25), (2.26), and (2.27) we can show that with probability greater than 1−ε, ∀ u ∈ [0, H], v ∈ (0, δ), and t ∈ [T ∗ , 2T ∗ ], t Su,v (τ )dτ − c2 t + σ2 B(t) > ln θ ≥ ln(Iu,v (t)) = ln v + β ≥ ln v + βαt λt λt 3λ − − c2 t − ≥ ln v + t (µ + βθ) 12 12 The proof is complete Proposition 2.1 Assume λ > Let δ, H, and T ∗ be as in Lemma 2.5 There exists a positive constant K5 such that E[ln Iu,v (t)]2− ≤ [ln v]2− − λt[ln v]− + K5 t2 (2.28) for any v ∈ (0, ∞), ≤ u ≤ H, t ∈ [T ∗ , 2T ∗ ] Proof First, consider v ∈ (0, δ], ≤ u ≤ H We have P(Ωu,v ) ≥ − ε, where Ωu,v = ln v + 3λt ≤ ln Iu,v (t) < ∀ t ∈ [T ∗ , 2T ∗ ] In Ωu,v we have − ln v − 3λt ≥ − ln Iu,v (t) > Hence, ≤ [ln Iu,v (t)]− ≤ [ln v]− − 3λt ∀ t ∈ [T ∗ , 2T ∗ ] As a result, [ln Iu,v (t)]2− ≤ [ln v]2− − 3λt 9λ2 t2 [ln v]− + 16 ∀ t ∈ [T ∗ , 2T ∗ ] This implies that (2.29) E 1Ωu,v [ln Iu,v (t)]2− ≤ P(Ωu,v )[ln v]2− − 3λt 9λ2 t2 P(Ωu,v )[ln v]− + P(Ωu,v ) 16 In Ωcu,v = Ω\Ωu,v , we have from Lemma 2.4 that (2.30) E 1Ωcu,v [ln Iu,v (t)]2− ≤ P(Ωcu,v )[ln v]2− + K3 t P(Ωcu,v )[ln v]− + K4 t2 Copyright © by SIAM Unauthorized reproduction of this article is prohibited P(Ωcu,v ) CLASSIFICATION IN A STOCHASTIC SIR MODEL 1075 Adding (2.29) and (2.30), we obtain Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.31) E[ln Iu,v (t)]2− ≤ [ln v]2− + − √ 3λ (1 − ε) + K3 ε t[ln v]− + 9λ2 + K4 t2 16 In view of (2.19) we deduce E[ln Iu,v (t)]2− ≤ [ln v]2− − λt[ln v]− + 9λ2 + K4 t2 16 Now, for v ∈ [δ, ∞) and ≤ u ≤ H we have from Lemma 2.4 that E[ln Iu,v (t)]2− ≤ [ln v]2− + K3 t[ln v]− + K4 t2 ≤ | ln δ|2 + K3 t| ln δ| + K4 t2 Letting K5 be sufficiently large such that K5 > K5 t2 ∀t ∈ [T ∗ , 2T ∗ ], we obtain the desired result 9λ2 16 + K4 and | ln δ|2 + K3 t| ln δ| + K4 t2 ≤ Proposition 2.2 Assume λ > There exists K6 > such that E[ln Iu,v (2T ∗ )]2− ≤ [ln v]2− − λ [ln v]− + K6 T ∗ 2 for v ∈ (0, ∞), u > H ∗ Proof First, consider v ≤ exp{− λT2 } Define the stopping time ξu,v = T ∗ ∧ inf{t > : Su,v (t) ≤ H} Let Ω1 = − σ2 B(2T ∗ ) − (βH − 2c2 )T ∗ ≤1 and Ω2 = − σ2 B(t) − (βH − c2 )t ≤ λ ∀ t ∈ [0, 2T ∗ ] By the exponential martingale inequality [21, Thm 7.4, p 44] and (2.20), (2.32) P(Ω1 ) ≥ − exp − βH − 2c2 2σ22 ≥1− ε and P(Ω2 ) ≥ − exp − λ(βH − c2 ) 4σ22 ε ≥1− Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1076 N T DIEU, D H NGUYEN, N H DU, AND G YIN Define Ω3 = Ω1 ∩ {ξu,v = T ∗ }, Ω4 = {− ln Iu,v (ξu,v ) ≤ − ln v + λ8 } ∩ {ξu,v < T ∗ }, and Ω5 = Ω\(Ω3 ∪ Ω4 ) If ω ∈ Ω3 , we have 2T ∗ ∗ Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php − ln Iu,v (2T ) = − ln v − (βSu,v (τ ) − c2 )dτ − σ2 B(2T ∗ ) T∗ ≤ − ln v − 2T ∗ (βSu,v (τ ) − c2 )dτ + T∗ c2 dτ − σ2 B(2T ∗ ) ≤ − ln v − T ∗ (βH − c2 ) + T ∗ c2 − σ2 B(2T ∗ ) ≤ − ln v − T ∗ (βH − 2c2 ) − σ2 B(2T ∗ ) T ∗ (βH − 2c2 ) ≤ − ln v − + (in view of (2.32)) λT ∗ ≤ − ln v − (by (2.20)) ∗ If v ≤ exp{− λT2 }, then − ln v − λT ∗ ≥ Therefore, λT ∗ + [ln v]− and then taking expectation on both sides yield [ln Iu,v (2T ∗ )]− ≤ − Squaring and then multiplying by 1Ω3 E [ln Iu,v (2T ∗ )]2− 1Ω3 ≤ [ln v]2− P(Ω3 ) − λT ∗ [ln v]− P(Ω3 ) + (2.33) λ2 T ∗ If ω ∈ Ω2 , ξu,v − ln Iu,v (ξu,v ) = − ln v − [βSu,v (τ ) − c2 ]dτ − σ2 B(ξu,v ) ≤ − ln v − (βH − c2 )ξu,v − σ2 B(ξu,v ) ≤ − ln v + λ As a result, Ω2 ∩ {ξu,v < T ∗ } ⊂ Ω4 Hence, P(Ω5 ) = P(Ω5 ∩ {ξu,v < T ∗ }) + P{Ω5 ∩ {ξu,v = T ∗ }) ≤ P(Ωc1 ) + P(Ωc2 ) ≤ ε Let t < T ∗ , u > 0, and v satisfy − ln v ≤ − ln v + λ8 In view of Proposition 2.1 and the strong Markov property, we can estimate the conditional expectation E [ln Iu,v (2T ∗ )]2− ξu,v = t, Iu,v (ξu,v ) = v , Su,v (ξu,v ) = u ≤ [ln v ]2− − λ(2T ∗ − t)[ln v ]− + K5 (2T ∗ − t)2 ≤ [ln v ]2− − λT ∗ [ln v ]− + 4K5 T ∗ ≤ − ln v + λ − λT ∗ (− ln v) + 4K5 T ∗ since − ln v + λ λ2 (− ln v) + 4K5 T ∗ + 64 ∗ 3λT λ [ln v]− + 4K5 T ∗ + ≤ [ln v]2− − 64 ≤ − ln v − λT ∗ − Copyright © by SIAM Unauthorized reproduction of this article is prohibited λ >0 CLASSIFICATION IN A STOCHASTIC SIR MODEL 1077 As a result, Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.34) E 1Ω4 [ln Iu,v (2T ∗ )]2− ≤ [ln v]2− P(Ω4 ) − 3λT ∗ λ2 [ln v]− P(Ω4 ) + 4K5 T ∗ + 64 In view of Lemma 2.4, (2.35) E 1Ω5 [ln Iu,v (2T ∗ )]2− ≤ [ln v]2− P(Ω5 ) + K3 P(Ω5 )2T ∗ [ln v]− + 4K4 T ∗ Adding (2.33), (2.34), and (2.35), we have E [ln Iu,v (2T ∗ )]2− ≤ [ln v]2− − T ∗ ≤ [ln v]2− − (2.36) √ 3λ (1 − ε) + 2K3 ε [ln v]− + K7 T ∗ λT ∗ [ln v]− + K7 T ∗ 2 ∗ for some K7 > We note that if v ≥ exp{− λT2 }, then − ln v ≤ from Lemma 2.4 that E [ln Iu,v (2T ∗ )]2− ≤ λT ∗ Therefore, it follows λ λ2 + 2K3 + 4K4 T ∗ 2 Letting K6 = max{K7 , λ4 + K3 λ2 + 4K4 }, we have E [ln Iu,v (2T ∗ )]2− ≤ [ln v]2− − λT ∗ [ln v]− + K6 T ∗ 2 ∀u ≥ H, v ∈ (0, ∞) The proof is complete Lemma 2.6 Every compact subset K is petite for the Markov chain (S(2nT ∗ ), I(2nT ∗ )) (n ∈ N) The irreducibility and aperiodicity of (Su,v (2nT ∗ ), Iu,v (2nT ∗ )) (n ∈ N) is a by-product (see [25, 22]) Proof Note that we can always choose φ∗ ∈ R such that (c1 − σ1 φ∗ ) > 0, (c2 − σ2 φ∗ ) > 0, and αβ −(c1 −σ1 φ∗ )(c2 −σ2 φ∗ ) > Hence, with the constant control φ∗ we can show that the solution to (2.11) with control φ∗ , (uφ∗ (t, u, v), vφ∗ (t, u, v)) converges to a positive equilibrium 2,◦ ∗ (u∗ , v∗ ) ∀ (u, v) ∈ R2,◦ + Let (u , v ) ∈ R+ such that p(2T , u∗ , v∗ , u , v ) > By the ∗ smoothness of p(2T , ·, ·, ·, ·), there exists a neighborhood Wδ = (u∗ −δ, u∗ +δ)×(v∗ −δ, v∗ +δ) that is invariant under (2.11) with control φ∗ and an open set G (u , v ) such that (2.37) p(1, u, v, u , v ) ≥ m > ∀ (u, v) ∈ Wδ , (u , v ) ∈ G Since (uφ∗ (t, u, v), vφ∗ (t, u, v)) converges to a positive equilibrium (u∗ , v∗ ), in view of the support theorem (see [11, Thm 8.1, p 518]), there is nu,v ∈ N such that P (2nu,v T ∗ , u, v, Wδ ) := 2ρu,v > Since (Su,v (t), Iu,v (t)) is a Markov–Feller process, there exists an open set Vu,v (u, v) such that P (nu,v , u , v , Wδ ) ≥ ρu,v ∀(u , v ) ∈ Vu,v Since K is a compact set, there is a finite Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1078 N T DIEU, D H NGUYEN, N H DU, AND G YIN number of Vui ,vi , i = 1, , l, satisfying K ⊂ li=1 Vui ,vi Let ρK = min{ρui ,vi , i = 1, , l} For each (u, v) ∈ K, there exists nui ,vi such that P (nui ,vi , u, v, Wδ ) ≥ ρK Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2.38) From (2.37), (2.38), ∀ (u, v) ∈ K there exists nui ,vi such that p((2nui ,vi + 2)T ∗ , u, v, u , v ) ≥ ρK m (2.39) ∀ (u , v ) ∈ G Define the kernel K(u, v, Q) := l l P ((2nui ,vi + 2)T ∗ , u, v, Q) ∀Q ∈ B(R2,◦ + ) i=1 We derive from (2.39) that K(u, v, Q) ≥ ρK m µ(G ∩ Q) l (2.40) 2,◦ ∀Q ∈ B(R+ ), where µ(·) is the Lebesgue measure on R2,◦ + Equation (2.40) means that every compact set 2,◦ K ⊂ R+ is petite for the Markov chain (Su,v (2T ∗ n), Iu,v (2T ∗ n)) π∗ Theorem 2.2 Let λ > 0, d∗ as in Lemma 2.2 There exists an invariant probability measure such that ∗ lim tq P (t, (u, v), ·) − π ∗ (·) = (2.41) t→∞ ∀(u, v) ∈ R2,◦ + , where · is the total variation norm and q ∗ is any positive number The support of π ∗ is 2,◦ ∗ r ∗ ∗ R2◦ + if d ≤ and is {(u, v) ∈ R+ : u v ≥ d } if d > Moreover, for any initial value 2,◦ ∗ (u, v) ∈ R+ and a π -integrable function f we have (2.42) P T →∞ T T lim f Su,v (t), Iu,v (t) dt = R2,◦ + f (u , v )π ∗ (du , dv ) = Proof By virtue of Lemma 2.3, there are h1 > 0, H1 > satisfying (2.43) EU Su,v (2T ∗ ), Iu,v (2T ∗ ) ≤ (1 − h1 )U (u, v) + H1 2,◦ ∀(u, v) ∈ R+ Let V (u, v) = U (u, v) + [ln v]2− In view of Propositions 2.1 and 2.2 and (2.43), there is a compact set K ⊂ R2,◦ + , h2 > 0, H2 > satisfying (2.44) EV (Su,v (2T ∗ ), Iu,v (2T ∗ )) ≤ V (u, v) − h2 V (u, v) + H2 1{(u,v)∈K} Applying (2.44) and Lemma 2.6 to [12, Thm 3.6] we obtain that (2.45) n P (2nT ∗ , (u, v), ·) − π ∗ → as n → ∞ Copyright © by SIAM Unauthorized reproduction of this article is prohibited ∀(u, v) ∈ R2,◦ + Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSIFICATION IN A STOCHASTIC SIR MODEL 1079 for some invariant probability measure π ∗ of the Markov chain (S(2nT ∗ ), I(2nT ∗ )) It is shown in the proof of [12, Thm 3.6] that (2.44) implies EτK < ∞, where τK = inf{n ∈ N : (Su,v (2nT ∗ ), Iu,v (2nT ∗ )) ∈ K} In view of [17, Thm 4.1], the Markov process (Su,v (t), Iu,v (t)) has an invariant probability measure ν ∗ As a result, ν ∗ is also an invariant probability measure of the Markov chain (Su,v (2nT ∗ ), Iu,v (2nT ∗ )) In light of (2.45), we must have ν ∗ = π ∗ , that is, π ∗ is an invariant measure of the Markov process (Su,v (t), Iu,v (t)) In the proofs, we use the function [ln v]2− for the sake of simplicity In fact, we can treat [ln v]1+q for any small q ∈ (0, 1) in the same manner We can show that there is hq , Hq > 0, − and a compact set Kq satisfying (2.46) EVq (Su,v (2T ∗ ), Iu,v (2T ∗ )) ≤ Vq (u, v) − hq [Vq (u, v)] 1+q + Hq 1{(u,v)∈Kp } ∀(u, v) ∈ R2,◦ + , where Vq (u, v) = U (u, v) + [ln v]1+q − Then applying [12, Thm 3.6], we can obtain n1/q P (2nT ∗ , (u, v), ·) − π ∗ → as n → ∞ Since P (t, (u, v), ·) − π ∗ is decreasing in t, we easily deduce ∗ tq P (t, (u, v), ·) − π ∗ → as t → ∞, where q ∗ = 1/q ∈ (1, ∞) On the one hand, in view of Lemma 2.2, the invariant control set of (2.11), denoted by C, r ∗ ∗ is R2,◦ if d∗ ≤ and {(u, v) ∈ R2,◦ + : u v ≥ d } if d > By [17, Lem 4.1], C is exactly the support of the unique invariant measure π ∗ The strong law of large numbers can be obtained by using [23, Thm 8.1] or [17] Discussion and numerical examples 3.1 Discussion We have shown that the extinction and permanence of the disease in a stochastic SIR model can be determined by the sign of a threshold value λ Only the critical case λ = is not studied in this paper To illustrate the significance of our results, let us compare them with those in [20] Theorem 3.1 (see [20, Thm 3.1]) Assume that σ1 > 0, σ2 > Let (S(t), I(t)) be a solution of system (2.10) If µ > σ12 , µ + ρ + γ > σ22 , R0 > 1, and δ < µ2 (µ + ρ + γ)2 ∗ ∗2 S , I , µ − σ12 µ + ρ + γ − σ22 then there exists a stationary distribution π ∗ for the Markov process (S(t), I(t)), which is the limit in total variation of transition probability P (t, (u, v), ·) Here, δ= S∗ = µσ12 ∗ (µ + ρ + γ)σ22 ∗ µ + ρ + γ ∗ I σ2 , S + I + 2β µ − σ12 µ + ρ + γ − σ22 µ+ρ+γ , β I∗ = α µ − , µ+ρ+γ β R0 = βα à(à + + ) Copyright â by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1080 N T DIEU, D H NGUYEN, N H DU, AND G YIN By straightforward calculations, or by the arguments in section of [8], we can show that the conditions of [20] are much more restrictive than the condition λ > Moreover, it should be noted that Theorem 2.1 is the same as Lemma 3.5 in [20] In contrast to [20], we provide a rigorous proof of Theorem 2.1 here Moreover, the conclusions in Theorems 2.1 and 2.2 still hold for the nondegenerate model (1.2) As a result we have the following theorem for model (1.2) Theorem 3.2 Let (Su,v (t), Iu,v (t)) be the solution to (1.2) with initial value (S(0), I(0)) = 2,◦ (u, v) ∈ R+ Define λ as (2.5) If λ < 0, then limt→∞ Iu,v (t) = a.s., and the distribution of Su,v (t) converges weakly to µ∗ , which has the density (2.3) If λ > 0, the solution 2,◦ process (Su,v (t), Iu,v (t)) has a unique invariant probability measure ϕ∗ whose support is R+ ∗ Moreover, the transition probability P (t, (u, v), ·) of (Su,v (t), Iu,v (t)) converges to ϕ (·) in total variation The rate of convergence is bounded above by any polynomial rate Moreover, for any ϕ∗ -integrable function f , we have P t→∞ t t lim f Su,v (τ ), Iu,v (τ ) dτ = R2,◦ + f (u , v )ϕ∗ (du , dv ) =1 2,◦ ∀(u, v) ∈ R+ Remark 3.1 We articulate the similarities and differences between the nondegenerate diffusion model given in (1.2) and the degenerate diffusion model in (2.1) To obtain Theorem 3.2 for (1.2), we not need Lemmas 2.1, 2.2, and 2.6 because of its nondegeneracy Unlike the invariant probability measure π ∗ of (2.1), the support of ϕ∗ (·) is always the whole space R2,◦ + On the other hand, although the generators of (1.2) (after removing R(t)) and (2.1) are slightly different, the result of Lemma 2.3 still holds for (1.2) with a slight modification To estimate [ln Iu,v (t)]− , we mainly estimate the difference Su (t) − Su,v (t) and the value of ln Iu,v (t) The estimates are almost the same for both (1.2) and (2.1) since for (2.1), there is only one Brownian motion, while for (1.2), each of Su (t) − Su,v (t) and the value of ln Iu,v (t) is driven directly by one Brownian motion Hence, Lemmas 2.4 and 2.5 and Propositions 2.1 and 2.2 still hold for (1.2) However, for the critical case when λ = 0, degeneracy or nondegeneracy would probably make a difference in studying the asymptotic behavior of the two equations This is an interesting open question to consider in the future It should be emphasized that our techniques can also be used to improve results in [5, 13, 30] 3.2 Example Let us finish this paper by providing some numerical examples Example 3.1 Consider (2.1) with parameters α = 20, β = 4, µ = 1, ρ = 10, γ = 1, σ1 = 1, and σ2 = −1 Direct calculation shows that λ = 67.5 > 0, d∗ = 7.75 > 0, and c∗ = 1.9375 By virtue of Theorem 2.2, (2.1) has a unique invariant probability measure π ∗ whose support is {(u, v) : u ≥ 1.9375 v } Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold A sample path of solution to (2.1) is illustrated by Figure 1, while the phase portrait in Figure demonstrates that the support ∗ of π ∗ lies above and includes the curve u = cv = 1.9375 as well as the empirical density of π ∗ v In nondegenerate case (1.2), with this same set of parameters the empirical density of π ∗ is illustrated by Figure Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSIFICATION IN A STOCHASTIC SIR MODEL 1081 Figure Trajectories of Su,v (t), Iu,v (t) in Example 3.1 Figure Phase portrait of (2.1); the boundary s = of π in Example 3.1 ∗ 1.9375 i of the support of π ∗ and the empirical density 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.5 1.5 2.5 3.5 4 Figure The empirical density of ϕ∗ in Example 3.1 for the nondegenerate equation (1.2) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1082 N T DIEU, D H NGUYEN, N H DU, AND G YIN Figure Trajectories of Su,v (t), Iu,v (t) in Example 3.2 Figure Phase portrait of (2.1) and the empirical density of π ∗ in Example 3.2 Example 3.2 Consider (2.1) with parameters α = 7, β = 3, µ = 1, ρ = 1, γ = 2, σ1 = 1, and σ2 = For these parameters, the conditions in Theorem 3.1 are not satisfied We obtain λ = 16.5 > 0, d∗ = −∞ As a result of Theorem 2.2, (2.1) has a unique invariant probability measure π ∗ whose support is R2,◦ + Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold A sample path of solution to (2.1) is depicted in Figure 4, while the phase portrait in Figure demonstrates the support of π ∗ and the empirical density of π ∗ Example 3.3 Consider (2.1) with parameters α = 5, β = 5, µ = 4, ρ = 1, γ = 1, σ1 = 2, and σ2 = −1 It can be shown that λ = −1.75 < As a result of Theorem 2.1, Iu,v (t) → a.s as t → ∞ This claim is supported by Figure That is, the population will eventually be disease free The distribution of Su,v (t) converges to f ∗ (x) as t → ∞ The graphs of f ∗ (x) and empirical density of Su,v (t) at t = 50 are illustrated in Figure Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1083 Figure Trajectories of Su,v (t), Iu,v (t) in Example 3.3 y=f*(x) y=f (x) 0.9 S 0.8 0.7 0.6 y Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CLASSIFICATION IN A STOCHASTIC SIR MODEL 0.5 0.4 0.3 0.2 0.1 0 0.5 1.5 2.5 x 3.5 4.5 Figure The graph of the stationary density f ∗ (in blue) and the graph of the empirical density of S(50) (in red) in Example 3.3 Acknowledgments We gratefully thank the reviewers and the editor for constructive comments and detailed suggestions, which led to much improvement in the presentation of the paper N T Dieu, D H Nguyen, and G Yin would also like to thank the Vietnam Institute for Advance Study in Mathematics (VIASM) for providing a fruitful research environment and extending support and hospitality during their visit REFERENCES [1] M E Alexander, C Bowman, S M Moghadas, R Summers, A B Gumel, and B M Sahai, A vaccination model for transmission dynamics of influenza, SIAM J Appl Dyn Syst., (2004), pp 503–524, http://dx.doi.org/10.1137/030600370 [2] F Ball and D Sirl, An SIR epidemic model on a population with random network and household structure, and several types of individuals, Adv Appl Probab., 44 (2012), pp 63–86 [3] M Barczy and G Pap, Portmanteau theorem for unbounded measures, Statist Probab Lett., 76 (2006), pp 1831–1835 [4] F Brauer and C C Chavez, Mathematical Models in Population Biology and Epidemiology, SpringerVerlag, New York, 2012 [5] Y Cai, Y Kang, M Banerjee, and W Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J Differential Equations, 259 (2015), pp 7463–7502 [6] V Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, Berlin, 1993 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 05/27/16 to 129.93.16.3 Redistribution subject to SIAM license or copyright; 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