Classification of the asymptotic behavior of a stochastic SIR model

In this paper, we investigate the asymptotic behavior of a stochastic SIR epidemic system. We give sufficient conditions for the permanence and ergodicity of the solution to the system. The conditions obtained in fact are very close to the necessary conditions. We also characterize the support of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to the invariant measures. Our results can be considered as an significant improvement of the result given by Lin, Y. et al. 18.

Classification of the asymptotic behavior of a stochastic SIR model N. T. Dieu ∗ N.H. Dang,† N.H. Du,‡ July 30, 2015 Abstract In this paper, we investigate the asymptotic behavior of a stochastic SIR epidemic system. We give sufficient conditions for the permanence and ergodicity of the solution to the system. The conditions obtained in fact are very close to the necessary conditions. We also characterize the support of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to the invariant measures. Our results can be considered as an significant improvement of the result given by Lin, Y. et al. [18]. Keywords. SIR model; Extinction; Permanence; Stationary Distribution; Ergodicity. Subject Classification. 34C12, 60H10, 92D25. 1 Introduction Since epidemic models were first introduced by Kermack and McKendrick [13, 14], mathematical models have been used for the purpose of analyzing, predicting the spread and the control of infectious diseases in host populations (see [1, 3, 4, 13, 14, 17, 19, 22]). One of classic epidemic models is the SIR (Susceptible-Infected-Removed) model which is suitable for some diseases with permanent immunity such as rubella, whooping cough, measles, ∗ Department of Mathematics,Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam, dieunguyen2008@gmail.com. This research was supported in part by the Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training. No. B2015-27-15. Author would like also to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for supporting and providing a fruitful research environment and hospitality. † Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, dangnh.maths@gmail.com. This research was supported in part by the National Science Foundation under grant DMS-1207667 and was finished when the author was in the Institute for Advance Study in Mathematics (VIASM). ‡ Corresponding author: Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam, dunh@vnu.edu.vn. 1 smallpox, etc. In the SIR model, a homogeneous host population is subdevided into three epidemiologically distinct types of individuals. • (S), The susceptible class, i.e., the class of those individuals who are capable of contracting the disease and becoming infective, • (I), the infective class, i.e., the class of those individuals who are capable of transmitting the disease to others, • (R), the removed class, i.e., the class of infected individuals who are dead, or have recovered and are permanently immune, or are isolated. If we denote by S(t), I(t), R(t) the number of individuals in classes (S),(Z),(R) respectively at time t, the spread of infection is formulated by the following system of differential equations:   dS(t) = α − βS(t)I(t) − µS)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 and ρ is the excess per capita death rate of infective class, β is the effective per capita contact rate, γ is per capita recovery rate of the infective individuals. On the other hand, it is well-recognized that the population is always subject to random factors and we obviously desire to learn how randomness effects our models. It is therefore of significant importance to investigate stochastic epidemic models. Jiang et al. [8] investigated the asymptotic behavior of global positive solution for the non-degenerate stochastic SIR model   dS(t) = α − βS(t)I(t) − µS)dt + σ1 S(t)dB1 (t) dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σ2 I(t)dB3 (t)   dR(t) = (γI(t) − µR(t))dt + σ3 R(t)dB3 (t), (1.2) where B1 (t), B2 (t) and B3 (t) are mutually independent Brownian motions, σ1 , σ2 , σ3 are the intensities of the white noises. This model has been extended to multi-group ones in [11, 25, 26]. However, in reality, the classes (S), (I), (R) are usually subject to the same random factors such as temperature, humidity, polution and other extrinsic influences. As a result, it is more plausible to assume that the random noise perturbing the three classes is correlated. If we assume that the Brownian motions B1 (t), B2 (t) and B3 (t) are the same, 2 we obtain the following model which has been considered in [18]   dS(t) = α − βS(t)I(t) − µS)dt + σ1 S(t)dB(t) 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). (1.3) When studying epidemic models, it is naturally important to know whether the models tend to a disease free state or the disease will survive permanently. For the deterministic model (1.1), the asymptotic behavior has been classified completely as follows. If λd = α βα − (µ + ρ + γ) ≤ 0 then the population tends to the disease-free equilibrium ( , 0, 0) µ µ while the population approaches an endemic equilibrium in case λd > 0. In [18], the authors attempted to answer the aforesaid question for the model (1.3) in case σ1 > 0, σ2 > 0. By using Lyapunov-type functions, they provided some sufficient conditions for extinction or permanence as well as ergordicity for the solution of system (1.3). Using the same methods, the extinction and permanence in some different stochastic SIR models have been studied in [10, 12, 24, 27]. In practice, it is, however, very difficult to find an effective Lyapunov function in practice so their conditions are restrictive and not close to a necessary condition. In other words, there has been no classification for stochastic SIR models that is similar to the deterministic case. 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 (1.3) in using a value λ, which is similar to λd in the deterministic model. The new method introduced in this paper can remove most assumptions in [18] as well as can consider the case σ1 > 0, σ2 < 0 which has not been taken into consideration in [18]. It is also suitable to deal with other stochastic variants of (1.1) such as models introduced in [10, 12, 24, 27], etc. The rest of the paper is arranged as follows. Section 2 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 in an exponential rate. In case λ > 0, the solution converges to a stationary distribution in total variation. It means that the disease is permanent. The ergodicity of the solution process is also proved. Finally, Section 3 is devoted to some discussion and comparison to existing results in literature. Some numerical examples and figures are also provided to illustrate our results. 3 2 Threshold Between Extinction and Permanence Let (Ω, F, {Ft }t≥0 , P) be a 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 motions. Because the dynamics of class of recover has no effect on the disease transmission dynamics, we only consider the following system: 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). (2.1) Assume that σ1 , σ2 = 0. By the symmetry of Brownian motion, without loss of generality, we suppose throughout this paper that σ1 > 0. Using standard arguments, it can be easily shown that for any positive initial value (S(0), I(0)) = (u, v) ∈ R2,◦ + := {(x, y) : x > 0, y > 0}, there exists uniquely a global solution (S(t), I(t)), t ≥ 0 that remains in R2,◦ + with probability 1 (see e.g. [8]). To 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). (2.2) It follows from the comparison theorem ([7, Theorem 1.1, p. 352]) that S(t) ≤ S(t) ∀t ≥ 0 a.s. provided that S(0) = S(0) > 0 and I(0) > 0. By solving the Fokker-Planck equation, it is shown that the process S(t) has a unique stationary distribution with density given by f ∗ (x) = where c1 = µ + σ12 ,a 2 ∗ = 2c1 ,b σ12 = 2α σ12 ba −(a+1) −b x ex, Γ(a) (2.3) and Γ(·) is the Gamma function. Since g1 (x) = x−1 , and g2 (x) = x, are f -integrable (i.e., integrable with respect to the function f ∗ ), by the strong law of large number we deduce that, 1 t→∞ t ∞ t S −1 (s)ds = lim −∞ 0 1 t→∞ t x−1 f ∗ (x)dx := ∞ t lim xf ∗ (x)dx := S(s)ds = −∞ 0 c1 a.s. α α a.s. µ (2.4) (2.5) Otherwise, 1 1 ln S(t) = t t t − c1 + αS −1 (s) ds + α 0 B1 (t) . t (2.6) Consequently, lim sup t→∞ 1 1 ln S(t) ≤ lim sup ln S(t) = 0 a.s., t t→∞ t 4 (2.7) and 1 lim sup t→∞ t t S(τ )dτ ≤ 0 α a.s. µ (2.8) To proceed, we estimate E[S(t)]−p . It follows from Itˆo’s formula that S −1 (t) satisfies a stochastic logistic equation dS −1 (t) = S −1 (t) (µ + σ12 ) − αS −1 (t) dt − σ1 S −1 (t)dB(t). It is well known (see e.g. [9, Lemma 2.3, p. 591]) that if S(0) = u ∈ (0, ∞) then lim sup ES −p (t) ≤ Hp < ∞ ∀ p > 1. (2.9) t→∞ We now define the threshold λ := σ2 αβ − µ+ρ+γ+ 2 . µ 2 (2.10) ln I(t) ≤ t t→∞ λ a.s. and the distribution of S(t) converges weakly to the unique invariant probability mea2,◦ Theorem 2.1. If λ < 0, then for any initial value (S(0), I(0)) ∈ R+ we have lim sup sure µ∗ with the density f ∗ . Proof. Let I(t) be the solution to the equation dI(t) = I(t) − (µ + ρ + γ) + β S(t) dt + σ1 I(t)dB(t), where S(t) is the solution to (2.2). By comparison theorem, I(t) ≤ I(t) a.s. given that S(0) = S(0), I(0) = I(0). In view of Itˆo’s formula and the ergodicity of S(t), lim sup t→∞ 1 1 t σ2 B(t) ln I(t) = lim sup − (µ + ρ + γ + 2 ) + β S(τ ) dτ + σ1 t t 0 2 t t→∞ 2 αβ σ − µ + ρ + γ + 2 = λ < 0 a.s. = µ 2 (2.11) That is, I(t) converges almost surely to 0 at an exponential rate. For any ε > 0, it follows from (2.11) that there exists t0 > 0 such that P(Ωε ) > 1 − ε where Ωε := I(t) ≤ exp λt 2 ∀t ≥ t0 = ln I(t) ≤ λt ∀t ≥ t0 . 2 2β λt0 exp < ε. Let S(t), t ≥ t0 be the solution λ 2 to (2.2) given that S(t0 ) = S(t0 ). We have from the comparison theorem that P{S(t) ≤ Clearly, we can choose t0 satisfying − 5 S(t) ∀t ≥ t0 } = 1. In view of Itˆo’s formula, for almost ω ∈ Ωε we have t 0 ≤ ln S(t) − ln S(t) =α t0 t ≤β t0 1 − 1 dτ + β S(τ ) t I(τ )dτ. S(τ ) t0 λτ 2β λt0 λt exp dτ = − exp − exp 2 λ 2 2 < ε. As a result, P{| ln S(t) − ln S(t))| > ε} ≤ 1 − P(Ωε ) < ε ∀t ≥ t0 . (2.12) Let ν ∗ 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 will claim an equivalent statement that the distribution of ln S(t) converges weakly to ν ∗ . By the Portmanteau theorem (see [2, Theorem 1, p. 1832]), it is sufficient to prove that for any g(·) : R → R satisfying |g(x) − g(y)| ≤ |x − y| and |g(x)| < 1 ∀x, y ∈ R, we have Eg(ln S(t)) → g := R ∞ 0 g(x)ν ∗ (dx) = g(ln x)µ∗ (dx). Since the diffusion (2.2) is non-degenerate, it is well know that the distribution of S(t) weakly converges to µ∗ as t → ∞ (see e.g. [6]). Thus lim Eg(ln S(t)) = g. t→∞ (2.13) On the one hand, |Eg(ln S(t)) − g| ≤ |Eg(ln S(t)) − Eg(ln S(t))| + |Eg(ln S(t)) − g| ≤ εP{| ln S(t) − ln S(t)| ≤ ε} + 2P{| ln S(t) − ln S(t)| > ε} + |Eg(ln S(t)) − g|. (2.14) Applying (2.12) and (2.13) to (2.14) yields lim sup |Eg(ln S(t)) − g| ≤ 3ε. t→∞ Since ε is taken arbitrarily, we obtain the desired conclusion. The proof is complete. Lemma 2.1. For any 0 < p < min{ 2µ , 2(µ+ρ+γ) }, there exists a K < ∞ such that σ2 σ2 1 2 lim sup E[S(t) + I(t)]1+p ≤ K, (2.15) t→∞ for any (S(0), I(0)) = (u, v) ∈ R2+ . In particular, when v = 0 we have lim sup E[S(t)]1+p ≤ K ∀ S(0) = u ∈ (0, ∞). t→∞ 6 (2.16) Proof. Consider the Lyapunov function V (s, i) = (s + i)1+p . By directly calculating the differential operator LV (s, i) associated with equation (2.1), we have (1 + p)p (s + i)p−1 (σ1 s + σ2 i)2 2 p p (µ − σ12 )s2 + (µ + ρ + γ − σ22 )i2 ) 2 2 LV (s, i) = (1 + p)(s + i)p (α − µs − (µ + ρ + γ)i) + = (1 + p)α(s + i)p − (1 + p)(s + i)p−1 + (2µ + ρ + γ − pσ1 σ2 )si . It follows from 2µ + ρ + γ > pσ12 2 + pσ22 2 ≥ pσ1 σ2 that 2µ + ρ + γ − pσ1 σ2 > 0. By choosing a number K2 satisfying 0 < K2 < min{µ − p2 σ12 , µ + ρ + γ − p2 σ22 } we see that K1 = sup {LV (s, i) + K2 V (s, i)} < ∞. s,i∈R+ We obtain LV (s, i) ≤ K1 − K2 V (s, i) ∀(s, i) ∈ R2+ . For n ∈ N, define the stopping time θn = inf{t ≥ 0 : S(t) + I(t) ≥ n}. In view of Itˆo’s formula, we yield that eK2 (t∧θn ) V (S(t ∧ θn ), I(t ∧ θn )) t∧θn [LV (S(τ ), I(τ )) + K2 V (S(τ ), I(τ ))]eK2 τ dτ =V (u, v) + 0 t∧θn eK2 τ [(2 + p)(S(τ ) + I(τ ))1+p (σ1 µS(τ ) + σ2 I(τ ))]dB(τ ). + 0 t∧θn K1 eK2 τ dτ ≤V (u, v) + 0 t∧θn eK2 τ [(2 + p)(S(τ ) + I(τ ))1+p (σ1 µS(τ ) + σ2 I(τ ))]dB(τ ). + 0 Taking expectation to both side, we have E(eK2 (t∧θn ) V (S(t ∧ θn ), I(t ∧ θn ))) ≤ V (u, v) + K1 (eK2 (t∧θn ) − 1) . K2 By letting n → ∞ we obtain from Fatou’s Lemma that EeK2 t (V (S(t), I(t))) ≤ V (u, v) + 7 K1 (eK2 t − 1) . K2 Dividing both sides by eK2 t and letting t → ∞, we have lim sup E(V (S(t), I(t))) ≤ t→∞ K1 := K. K2 Lemma is proved. Theorem 2.2. If λ > 0, the process (S(t), I(t)) has an invariant probability measure concentrated on R2,◦ + . Proof. Let (S(0), I(0)) = (u, v) ∈ R2,◦ + . Since 0 ∨ [ln S(t)] ≤ S(t), it follows from (2.15) that E[ln S(t)] exists (although it may be −∞) and lim sup t→∞ E[ln S(t)] ES(t) ≤ lim sup = 0. t t (2.17) E[ln I(t)] EI(t) ≤ lim sup = 0. t t (2.18) Similarly, lim sup t→∞ From (2.7) we have lim sup t→∞ 1 t t α − c1 − βI(τ ) dτ ≤ 0. S(τ ) E 0 This implies that lim sup t→∞ 1 t t E 0 α α α +E − c1 − βE[I(τ )] dτ ≤ 0. − S(τ ) S(τ ) S(τ ) (2.19) In view of (2.4) and (2.9), it follows from the uniform integrability of S −1 (t) that 1 t→∞ t t lim α E S(τ ) 0 − c1 dτ = 0. (2.20) Similarly, it follows from (2.5) and (2.16) that 1 t→∞ t t lim E S(τ )]dτ = 0 α . µ (2.21) From (2.19) and (2.20), we obtain 1 lim sup t→∞ t t E 0 α α − − βE[I(τ )] dτ ≤ 0. S(τ ) S(τ ) Let m > 0 to be chosen later, not depend on the initial value. We aim to show that 1 lim inf t→∞ t t E[I(τ )]dτ ≥ m. 0 8 (2.22) If (2.22) does not hold, there exists a sequence {tn }n≥0 with tn → ∞ as n → ∞ such that tn 1 n→∞ tn lim Hence 1 lim sup tn n→∞ E[I(τ )]dτ < m. (2.23) 0 tn E 0 α α − dτ < βm. S(τ ) S(τ ) Therefore, for any κ > 0, 1 lim sup n→∞ tn 1 = lim sup n→∞ tn 1 ≤ lim sup n→∞ tn tn E(S(τ ) − S(τ ))dτ 0 tn E (S(τ ) − S(τ ))1{S(τ )≤κ} dτ + lim sup n→∞ 0 tn E 0 tn 1 tn E (S(τ ) − S(τ ))1{S(τ )≥κ} dτ 0 1 1 1 S(τ )S(τ )1{S(τ )≤κ} dτ +lim sup − S(τ ) S(τ ) n→∞ tn < κ2 βm + 1 1 lim sup p κ n→∞ tn tn E S(τ ) tn E S(τ )1{S(τ )≥κ} dτ 0 1+p dτ < 0 κ2 βm K + p. α κ We can choose κ sufficiently large and then choose m sufficiently small such that and κ2 βm α ≤ λ . 4β K κp ≤ λ 4β Therefore, lim sup n→∞ tn 1 tn E(S(τ ) − S(τ ))dτ ≤ 0 λ , 2β which in the combination with (2.10) and (2.21) implies that lim inf n→∞ B(t) E ln I(tn ) 1 tn σ2 = lim inf E − (µ + ρ + γ + 2 ) + βS(τ ) dτ + σ1 n→∞ tn tn 0 2 t tn 2 1 σ = lim E − (µ + ρ + γ + 2 ) + β S(τ ) dτ n→∞ tn 0 2 tn 1 − β lim sup E S(τ ) − S(τ ) dτ tn 0 n→∞ λ 1 tn λ ≥λ − β lim sup E S(τ ) − S(τ ) dτ ≥ λ − β × := > 0. tn 0 2β 2 n→∞ This contradicts (2.18). As a result, there is a positive constant m > 0 satisfying (2.22). For 0 < 1 t < m < H < ∞, H¨older’s inequality yields that t E 1{I(τ )≥ } I(τ ) dτ ≤ E 0 1 t t 1{I(τ )≥ } dτ p 1+p 0 ≤ 1 t 1 t t 1+p [I(τ )] 0 p 1+p t E 1{I(τ )≥ 0 9 } dτ 1 1+p dτ 1 t 1 1+p t 1+p E[I(τ )] 0 dτ . Hence, lim inf t→∞ 1 t lim inf t E 1{I(τ )≥ } dτ ≥ t→∞ 0 t 0 1 t E 1{I(τ )≥ } I(τ ) dτ lim sup 1t ≥K 1 lim inf t→∞ t − p1 1+p p t→∞ t t 0 E[I(τ )]1+p dτ 1+p p E[I(τ )]dτ − 1 p 1 ≥ K − p (m − ) 1+p p > 0. (2.24) 0 We also have from (2.15) that the following inequality holds with probability 1. lim sup t→∞ 1 t t E1{S(τ )+I(τ )≥H} dτ ≤ 0 1 H 1+p lim sup t→∞ 1 t t E(S(τ ) + I(τ ))1+p dτ ≤ 0 It follows from (2.24) and (2.25) that we can choose H sufficiently large and K . (2.25) H 1+p sufficiently small such that 1 lim inf t→∞ t t E1{(S(τ ),I(τ ))∈D} dτ ≥ 0 (m − ) K 1 p 1+p p − K > 0, H 1+p (2.26) where D = {(s, i) : i ≥ , s+i ≤ H}. By virtue of the invariance of M = {(s, i) : s ≥ 0, i > 0} under equation (2.1), we can consider the Markov process (S(t), I(t)) on the state space M. It is easy to show that (S(t), I(t)) has the Feller property. Thus, in view of inequality (2.26) and the compactness of D in M, we implies that there is an invariant probability measure π ∗ on M (see [23] or [20]). Since I(t) → 0 provided that S(0) = 0, limt→∞ P (t, (0, I(0)), K) = 0 for all compact set K ⊂ M. Thus, we must have π ∗ ({(0, i) : i > 0}) = 0, equivalently 2,◦ ∗ π ∗ (R2,◦ + ) = 1. Furthermore, by the invariance of R+ , we derive that π is an invariant 2,◦ probability measure of (S(t), I(t)) on R+ . To obtain properties of π ∗ , we first rewrite equation (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). where c1 = µ + σ12 , c2 2 =µ+ρ+γ+ σ22 . 2 (2.27) Denote by (S s,i (t), I s,i (t)) the solution to (2.1) with initial value (s, i) and let P (t, (s, i), ·) be its transition probabilities. Put A(x, y) = 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) , j = 1, 2. ∂x ∂x ∂y ∂y 10 Denote by L(x, y) the Lie algebra generated by A(x, y), B(x, y) and L0 (x, y) the ideal in L(x, y) generated by B. We have the following lemma. Lemma 2.2. For σ1 > 0, σ2 = 0, the H¨ormader condition holds for the diffusion (2.27). To 2,◦ be more precise, we have dimL0 (x, y) = 2 at every (x, y) ∈ R+ or equivalently, the set of vectors B, [A, B], [A, [A, B]], [B, [A, B]], . . . spans R2 at every (x, y) ∈ R2,◦ + . Proof. This lemma has been proved in [18] for the case σ2 > 0. Assume that r = − σ2 > 0. σ1 It is easy to obtain C := 1 B(x, y) = σ1 x , −ry D :=[A, C](x, y) = α − rβxy , −βxy E :=[C, D](x, y) = −α + r2 βxy , −βxyn F :=[C, E](x, y) = α − r3 βxy . −βxy Since det(D, F ) = 0 only if r2 = 1 or r = 1 (since r > 0). When r = 1, solving det(D, E) = 0 obtains βxy = α which implies det(C, D) = x 0 −y −α = 0. 2,◦ As a result, B, D, E, F span R2 for all (x, y) ∈ R+ . The lemma is proved. In order to describe the support of the invariant measure π ∗ and to prove the ergodicity of (2.1), we need to investigate the following control system on R2,◦ 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), (2.28) 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 equation (2.28) with control φ and initial value 2 (u, v). Denote by O1+ (u, v) the reachable set from (u, v) ∈ R2,◦ + , that is the set of (u , v ) ∈ R such that there exists a t ≥ 0 and a control φ(·) satisfying uφ (t, u, v) = u , vφ (t, u, v) = v . We now recall some concepts introduced in [16]. Let X be a subset of R2 satisfying the property that for any w1 , w2 ∈ X, we have w2 ∈ O1+ (w1 ). Then there is a unique maximal set Y ⊃ X such that this property still holds for Y . Such Y is called a control set. A control set W is said to be invariant if O1+ (w) ⊂ W for all w ∈ W . 11 Putting r := −σ2 σ1 and zφ (t) = urφ (t)vφ (t), we have an equivalent system u˙ φ (t) = σ1 φ(t)uφ (t) + g(uφ (t), zφ (t)), z˙φ (t) = h(uφ (t), zφ (t)), (2.29) where g(u, z) = −c1 eu + α − βzu1−r , and h(u, z) = u−r z − (c1 r + c2 )ur + βu1+r + αrur−1 − βrz . 2,◦ Denote by O2+ (u, z) the set of (u , z ) ∈ R+ such that there is t > 0 and a control φ(·) such that uφ (t, u, z) = u , zφ (t, u, v) = z . Lemma 2.3. For the control system (2.28), the following claims hold 1. For any u0 , u1 , z0 > 0 and ρ > 0, there exists a control φ and T > 0 such that uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ρ. 2. For any 0 < z0 < z1 , there is a u0 > 0, a control φ, and T > 0 such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤ t ≤ T . 3. Let d∗ = inf {−(c1 r + c2 )ur + βu1+r + αrur−1 }. u>0 (a) If d∗ ≤ 0 then for any z0 > z1 , there is u0 > 0, a control φ, and T > 0 such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤ t ≤ T . d∗ . If c∗ < z1 < z0 , there is u0 > 0 and a (b) Suppose that d∗ > 0 and z0 > c∗ := βr control φ and T > 0 such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤ t ≤ T . However, there is no control φ and T > 0 such that zφ (T, u0 , z0 ) < c∗ . Proof. Suppose that u0 < u1 and let ρ1 = sup{|g(u, z)|, |h(u, z)| : u0 ≤ u ≤ u1 , |z − z0 | ≤ ρ}. σ1 ρ2 u0 We choose φ(t) ≡ ρ2 with −1 ρ ≥ u1 −u0 . It is easy to check that with this control, ρ1 there is 0 ≤ T ≤ ρρ1 such that uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ρ. If u0 > u1 , we can construct φ(t) similarly. Then the item 1 is proved. Now, by choosing u0 to be sufficiently large, there is a ρ3 > 0 such that h(u0 , z) > ρ3 ∀z0 ≤ z ≤ z1 . This property, combining with (2.29), implies the existence of a feedback control φ and T > 0 satisfying that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 , ∀ 0 ≤ t ≤ T . 12 We now prove item 3. If r < 0 then lim − (c1 r + c2 )ur + βu1+r + αrur−1 ] = −∞ u→0 r and lim − (c1 r + c2 )u + βu u→0 1+r r−1 + αru ] = 0 if r > 1. As a result, d∗ ≤ 0 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 is a feedback control φ and T > 0 satisfying zφ (T, u0 , z0 ) = z1 and uφ (t, u0 , z0 ) = u0 ∀0 ≤ t ≤ T . + αrur−1 = d∗ . If d∗ ≤ 0, then If r ∈ (0, 1] there exists u0 such that −(c1 r + c2 )ur0 + βu1+r 0 0 2 for any z0 > z1 > 0 we have supz∈[z1 ,z0 ] h(u0 , z) ≤ u−r 0 supz∈[z1 ,z0 ] {−βrz } < 0 which implies the desired claim. Consider the remaining case when r ∈ (0, 1] and d∗ > 0. First, assume c∗ < z1 < z0 . Let u0 satisfy −(c1 r + c2 )ur0 + βu1+r + αru0 er−1 = d∗ = βrc∗ . Hence 0 sup {h(u0 , z)} =u−r sup 0 z∈[z1 ,z0 ] z − (c1 r + c2 )ur0 + βu1+r + αrur−1 − βrz 0 0 z∈[z1 ,z0 ] ∗ = − βru−r 0 z1 (c − z1 ) < 0. Thus, there is a feedback control φ and T > 0 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∗ ) ≥ 0 for all u ∈ R. Proposition 2.1. The control system (2.28) has only one invariant control set C. If d∗ ≤ 0, C = R2,◦ . If d∗ > 0, C = {(u, v) : ur v ≥ c∗ }. Proof. If d∗ ≤ 0, it follows from items 1, 2, and item 3a of Lemma 2.3 that for any (u1 , z1 ), (u2 , z2 ) ∈ R2 , (u2 , z2 ) ∈ O2+ (u1 , z1 ). Hence, for any (u1 , v1 ), (u2 , v2 ) ∈ R2 , we have (u2 , v2 ) ∈ O1+ (u1 , v1 ). This implies that R2 is an unique invariant control set. If d∗ > 0, items 1, 2 and 3b of Lemma 2.3 imply that O2+ (u, z) ⊃ {(u , z ) : z ≥ c∗ } for all (u, v) ∈ R2 and ∗ O2+ (u, z) = {(u , z ) : z ≥ c∗ } for all u > 0, z ≥ c∗ . As a result, {(u, z) ∈ R2,◦ + : z ≥ c } is r ∗ a unique invariant control set for (2.29). In conclusion, C := {(u, z) ∈ R2,◦ + : u v ≥ c } is a unique invariant control set for (2.28). Note that if λ > 0, there is an invariant probability measure π ∗ of the process (S(t), I(t)). Since there is only one invariant control set C, it follows from Lemma 2.2 that π ∗ is the unique invariant probability measure with support C. Moreover, for any initial value (S(0), I(0)) = (u, v) ∈ C and a π ∗ -integrable function f we have P 1 T →∞ T T lim f (u , v )π ∗ (du , dv ) = 1. f S(t), I(t) dt = R2,◦ + 0 13 (2.30) These results are proved in [16]. Moreover, it follows from [6, Proposition 5.1] lim P (t, (u, v), ·) − π ∗ (·) = 0 ∀(u, v) ∈ C, t→∞ (2.31) where P (t, (u, v), ·) is the transition probability of the Markov process (S(t), I(t)) and · is the total variation norm. 2,◦ We aim to prove that (2.30) and (2.31) hold for all (u, v) ∈ R+ . We need only consider ∗ the case d∗ > 0 since C = R2,◦ + in case d ≤ 0. Proposition 2.2. Suppose that d∗ > 0. Then, for any initial value (S(0), I(0)) = (u, v) ∈ 2,◦ R+ , we have τCu,v < ∞ almost surely with τCu,v = inf{t > 0 : (S(t), I(t)) ∈ C}. Proof. Note that d∗ > 0 only if r ∈ (0, 1]. We have τCu,v = inf{t : βrS r (t)I(t) ≥ d∗ }. Let Z(t) = ln[I(t)S r (t)]. By virtue of Itˆo’s formula, we obtain dZ(t) = [−c3 + βS(t) + αrS −1 (t) − rβI(t)]dt = S −r (t)[−c3 S r (t) + βS 1+r (t) + αrS −1+r (t) − βrS r (t)I(t)]dt, where c3 = c1 r + c2 > 0. Suppose that P{τCu,v < ∞} < 1, then there exists even Ω∗ such that P(Ω∗ ) > 0 and τCu,v = ∞ in Ω∗ . If ω ∈ Ω∗ , then d∗ − βrS r (t)I(t) ≥ 0 ∀t ≥ 0. Thus dZ(t) ≥ 0 ∀t ≥ 0. (2.32) On the other hand, lim −c3 xr + βx1+r + αrx−1+r = +∞ if r ∈ (0, 1), x→0 and lim −c3 xr + βx1+r + αrx−1+r = αr > d∗ if r = 1. x→0 Hence, for any r ∈ (0, 1], there exists m > 0, δ ∗ > 0 such that inf {−c3 xr + βx1+r + αrx−1+r } > d∗ + δ ∗ . 0≤x≤m It follows from (2.32) and (2.33) that for almost all ω ∈ Ω∗ dZ(t) ≥ δ ∗ S −r (t)1{S(t)≤m} dt ≥ δ ∗ S −r (t)1{S(t)≤m} dt, 14 (2.33) where S(t) is the solution to (2.2) with S(0) = eu . By the ergodicity of S(t) we have m t 1 t→∞ t s−r f ∗ (s)ds := m > 0, a.s. [S(τ )]−r 1{S(τ )≤m} dτ = lim 0 0 It implies that t [S(τ )]−r 1{S(τ )≤m} dτ = +∞, a.s. lim t→∞ 0 ∗ Therefore, for almost ω ∈ Ω , t [S(τ )]−r 1{S(τ )≤m} dτ = +∞, a.s. lim Z(t) ≥ lim t→∞ t→∞ 0 which contradict the assumption that for all t ≥ 0, eZ(t) < d∗ βr in Ω∗ . Thus, P{τCu,v < ∞} = 1. The proof is complete. Since we have already shown that for any initial value (S(0), I(0)) = (u, v) ∈ R2,◦ , (S(t), I(t)) eventually enters C. So we obtain the following theorem. Theorem 2.3. Suppose σ1 , σ1 = 0, λ > 0. Then, (2.27) has a unique invariant probability measure π ∗ with support C. For any π ∗ -integrable function f , and initial value (S(0), I(0)) = (u, v) ∈ R2,◦ + , we have 1 P lim T →∞ T T f (u , v )π ∗ (du , dv ) = 1. f S(t), I(t) dt = R2,◦ + 0 Moreover, the transition probability P (t, (u, v), ·) converges to π ∗ (·) in total variation. Proof. The assertions can be proved using (2.30), (2.31), and Propositions 2.2. 3 Discussion and Numerical Examples 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 = 0 is not studied in this paper. To illustrate the significance of our results, let us compare our results with those in [18]. Theorem 3.1. [18, Theorem 3.1] Assume that σ1 > 0, σ2 > 0. Let (ξ(t), η(t)) be a solution of system (2.27). If µ > σ12 , µ + ρ + γ > σ22 , R0 > 1 and δ < min µ2 (µ + ρ + γ)2 ∗ 2 ∗2 S , I , µ − σ12 µ + ρ + γ − σ22 15 then there exists a stationary distribution π ∗ for the Markov process (ξ(t), η(t)) which is the limit in total variation of transition probability P (t, (u, v), ·). Here δ= S∗ = µσ12 ∗ 2 (µ + ρ + γ)σ22 ∗ 2 (µ + ρ + γ ∗ 2 S + I + I σ2 , µ − σ12 µ + ρ + γ − σ22 2β µ+ρ+γ ∗ α µ βα , I = − ; R0 = . β µ+ρ+γ β µ(µ + ρ + γ) By straightforward calculations or by arguments in Section 4 of [5] we can show that their conditions are much more restrictive than the condition λ > 0. Moreover, it should be noted that Theorem 2.1 is the same as Lemma 3.5 in [18]. However, the weak convergence of S(t) to µ∗ was not be proved carefully in that paper. For this reason, we re-stated Theorem 2.1 and provided a rigorous proof in this paper. Moreover, the estimates in Theorems 2.1 and 2.2 still hold for the non-degenerate model (1.2). Note that for a non-degenerate diffusion, the existence of an invariant probability measure implies the ergodicity of the diffusion as well as the convergence in total variation of the transition probability to the invariant measure (see [6, 15]). As a result we have the following theorem for the model (1.2). Theorem 3.2. Let (S(t), I(t)) be the solution to (1.2) with initial value (S(0), I(0)) ∈ R2,◦ + . Define λ as (2.10). If λ < 0, then lim I(t) = 0 a.s. and the distribution of S(t) converges weakly to µ∗ , t→∞ which has the density (2.3). If λ > 0, the solution process (S(t), I(t)) has a unique invariant probability measure ϕ∗ whose support is R2,◦ + . Moreover, the transition probability P (t, (u, v), ·) of (S(t), I(t)) converges to ϕ∗ (·) in total variation and for any ϕ∗ -integrable function f , we have 1 t→∞ t t f S(u), I(u) du = P lim 0 R2,◦ + 2,◦ f (u , v )ϕ∗ (du , dv ) = 1 ∀(u, v) ∈ R+ . It should be emphasized that our techniques can be also used to improve results in [10, 12, 24, 27]. 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, σ2 = −1. Direct calculation shows that λ = 67.5 > 0, d∗ = 7.75 > 0, c∗ = 1.9375. As a result of Theorem 2.2 that (2.1) has a unique invariant probability measure π ∗ whose support 16 is {(S, I) : S ≥ 1.9375 }. I 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 Figures 1, while the phase portrait in Figure 2 demonstrates that the support of π ∗ lies above and includes the curve S = c∗ I = 1.9375 I as well as the empirical density of π ∗ . Figure 1: Trajectories of S(t), I(t) in Example 3.1. Figure 2: Phase portait of (2.1); the boundary I = empirical density of π ∗ in Example 3.1. 1.9375 S of the support of π ∗ and the Example 3.2. Consider (2.1) with parameters α = 50, β = 5, µ = 3, ρ = 2.5, γ = 4.5, σ1 = 4.3 σ2 = 0.5. For these parameters, the conditions in Theorem 3.1 are not satisfied. We obtain λ = 73.2083 > 0, d∗ = −∞. As a result of Theorem 2.2 that (2.1) has a unique 2,◦ invariant probability measure π ∗ whose support is R+ . Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold. A 17 sample path of solution to (2.1) is illustrated by Figures 3, while the phase portrait in Figure 4 demonstrates that the support of π ∗ and the empirical density of π ∗ . Figure 3: Trajectories of S(t), I(t) in Example 3.2. Figure 4: Phase portrait of (2.1) and the support of π ∗ in Example 3.2. 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, σ2 = −1. By calculation, λ = −1.75 < 0 As a result of Theorem 2.1 that I(t) → 0 a.s. as t → ∞. This claim is justified in Figures 5. This means, population will eventually have no disease. References [1] R.M. Anderson, R.M. May, Population biology of infectious diseases, Part I, Nature, 280(1979), 361-367. 18 Figure 5: Trajectories of S(t), I(t) in Example 3.3. [2] M. Barczy, G. Pap, Portmanteau theorem for unbounded measures, Statist. Probab. Lett., 76(2006), 1831-1835. [3] F. Brauer, C. C. Chavez, Mathematical models in population biology and epidemiology, Springer-Verlag New York, 2012. [4] V. Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, Berlin, 1993. [5] N.H. Du, D.H. Nguyen, G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, to appear in J. Appl. Probab. [6] K. Ichihara, H. 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Shi, Ningzhong Global stability of two-group SIR model with random perturbation, J. Math. Anal. Appl., 360(2009), no. 1, 235-244. [26] X. Zhong, F. Deng, Extinction and Persistent of a Stochastic Multi-group SIR Epidemic Model, Journal of Control Science and Engineering, 1(2013) 13-22. [27] Y. Zhou, W. Zhang, S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 244(1)(2014), 118-131. 21 [...]... non-degenerate model (1.2) Note that for a non-degenerate diffusion, the existence of an invariant probability measure implies the ergodicity of the diffusion as well as the convergence in total variation of the transition probability to the invariant measure (see [6, 15]) As a result we have the following theorem for the model (1.2) Theorem 3.2 Let (S(t), I(t)) be the solution to (1.2) with initial value (S(0),... Anal Appl., 340(2008), 588-597 [10] C Ji, D Jiang, Threshold behaviour of a stochastic SIR model, Appl Math Model. , 38(2014), no 21-22, 5067-5079 [11] C.Y Ji, D.Q Jiang, Q.S Yang, N.Z Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48(2012), 121-131 19 [12] C.Y Ji, D.Q Jiang, N.Z Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stochastic. .. Springer-Verlag, Berlin, 1993 [5] N.H Du, D.H Nguyen, G Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, to appear in J Appl Probab [6] K Ichihara, H Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z Wahrsch Verw Gebiete, 30(1974), 235-254 Corrections in 39(1977), 81-84 [7] N Ikeda, S Watanabe, Stochastic. .. differential equations and diffusion processes, second edition, North-Holland Publishing Co., Amsterdam, (1989) [8] D.Q Jiang, J.J Yu, C.Y Ji, N.Z Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math Comput Modell., 54(2011), 221-232 [9] D Jiang, N Shi, X Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J Math Anal... 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 Figures 1, while the phase portrait in Figure 2 demonstrates that the support of π ∗ lies above and includes the curve S = c∗ I = 1.9375 I as well as the empirical density of π ∗ Figure 1: Trajectories of S(t), I(t) in Example 3.1... Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann Probab., 15(1987), no 2, 690-707 [17] A Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull Math Biol., 68(2006), 615-626 [18] Y Lin, D Jiang, P Xia, Long-time behavior of a stochastic SIR model, Appl Math Comput., 236(2014), 1-9 [19] R.M May, Stability... 2: Phase portait of (2.1); the boundary I = empirical density of π ∗ in Example 3.1 1.9375 S of the support of π ∗ and the Example 3.2 Consider (2.1) with parameters α = 50, β = 5, µ = 3, ρ = 2.5, γ = 4.5, σ1 = 4.3 σ2 = 0.5 For these parameters, the conditions in Theorem 3.1 are not satisfied We obtain λ = 73.2083 > 0, d∗ = −∞ As a result of Theorem 2.2 that (2.1) has a unique 2,◦ invariant probability... probability measure π ∗ whose support is R+ Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold A 17 sample path of solution to (2.1) is illustrated by Figures 3, while the phase portrait in Figure 4 demonstrates that the support of π ∗ and the empirical density of π ∗ Figure 3: Trajectories of S(t), I(t) in Example 3.2 Figure 4: Phase... Stochastic Anal Appl., 30(2012), 755-773 [13] W O Kermack, A G McKendrick, Contributions to the mathematical theory of epidemics (part I), Proc R Soc Lond Ser A, 115(1927), 700-721 [14] W O Kermack, A G McKendrick, Contributions to the mathematical theory of epidemics, part II, Proc Roy Sot Ser A, 138(1932), 55-83 [15] R.Z Khas’minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin... Population biology of infectious diseases, Part I, Nature, 280(1979), 361-367 18 Figure 5: Trajectories of S(t), I(t) in Example 3.3 [2] M Barczy, G Pap, Portmanteau theorem for unbounded measures, Statist Probab Lett., 76(2006), 1831-1835 [3] F Brauer, C C Chavez, Mathematical models in population biology and epidemiology, Springer-Verlag New York, 2012 [4] V Capasso, Mathematical Structures of Epidemic ... the per capita birth rate of the population, µ is the per capita disease-free death rate and ρ is the excess per capita death rate of infective class, β is the effective per capita contact rate,... permanence and ergodicity of certain stochastic predator-prey models, to appear in J Appl Probab [6] K Ichihara, H Kunita, A classification of the second order degenerate elliptic operators and... individuals who are capable of contracting the disease and becoming infective, • (I), the infective class, i.e., the class of those individuals who are capable of transmitting the disease to others,