The threshold of a stochastic SIS epidemic model with imperfect vaccination

Accepted Manuscript The threshold of a stochastic SIS epidemic model with imperfect vaccination Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi

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S0378-4754(17)30232-X http://dx.doi.org/10.1016/j.matcom.2017.06.004 MATCOM 4472

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Mathematics and Computers in Simulation

Received date : 27 June 2016 Revised date : 19 June 2017 Accepted date : 19 June 2017 Please cite this article as: Q. Liu, D. Jiang, N. Shi, T. Hayat, A. Alsaedi, The threshold of a stochastic SIS epidemic model with imperfect vaccination, Math. Comput. Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.06.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The threshold of a stochastic SIS epidemic model with imperfect vaccination Qun Liua,b , Daqing Jianga,c,d1 , Ningzhong Shia , Tasawar Hayatc,e , Ahmed Alsaedic a

School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin 130024, P.R.

China b

School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, P.R. China c Nonlinear

Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia d College of Science, China University of Petroleum (East China), Qingdao 266580, Shandong Province, P.R. China e Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

Abstract In this paper, we analyze the threshold RvS of a stochastic SIS epidemic model with partially protective vaccination of efficacy e ∈ [0, 1]. Firstly, we show that there exists a unique global positive solution of the stochastic system. Then RvS > 1 is verified to be sufficient for persistence in the mean of the system. Furthermore, three conditions for the disease to die out are given, which improve the previously-known results on extinction of the disease. We also obtain that large noise will exponentially suppress the disease from persisting regardless of the value of the basic reproduction number RvS . Keywords: Stochastic SIS epidemic model; Imperfect vaccination; Threshold; 1

Corresponding author at School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin 130024, P.R. China. E-mail address: [email protected] (D. Jiang). Preprint submitted to Mathematics and Computers in Simulation

June 19, 2017

Persistence in the mean; Extinction. 1. Introduction Recently, due to the negative effect of infectious diseases on the population growth, it has been important to understand the dynamical behavior of such diseases and to predict what may happens [21]. Thus mathematical modeling has become an important tool in investigating a diverse range of such diseases to obtain a better understanding of transmission mechanisms (see e.g. [26, 2, 28, 7, 15]). Controlling infectious diseases has been an increasingly complex issue and vaccination has been a commonly used method for eliminating diseases such as measles, polio, diphtheria, tetanus, tuberculosis, etc. And routine vaccination is now provided in all developing countries against all these diseases (see e.g. [6, 14, 19, 22]). It is well known that the SIS epidemic model is one of the most important models in epidemiology and mathematical ecology. Most models for the transmission of infectious diseases descend from the classical SIS epidemic model of Kermack and McKendrick established in 1927, see [9]. In recent years, many authors have analyzed the SIS epidemic model allowing vaccination, i.e., the SIVS model and much research has been done on SIVS models (see e.g. [1, 5, 10, 29, 30, 11]). However, it has been thought that the immune system will create antibody against disease because vaccination doses are taken during this process. But it may be not in a fully protective level, that is to say, the efficacy of the vaccine is less than one. Motivated by this fact, Safan and Rihan [21] considered the following SIS epidemic model with imperfect vaccination  dS   = (1 − p)µ + αI − (µ + ψ)S − βSI,    dt dI (1.1) = βSI + (1 − e)βV I − (µ + α)I,  dt     dV = pµ + ψS − µV − (1 − e)βV I, dt

where S denotes the fractions of susceptible individuals, I denotes the fractions of infected individuals, V denotes the density of vaccinees who have begun the vaccination process. Individuals are assumed to be born susceptible with rate µ where a proportion p of them obtains vaccinated immediately after birth. Susceptible individuals can either die with rate µ, vaccinated 2

with rate ψ or obtain infected with force of infection βI where β is the successful contact rate between infected and susceptible individuals. Infected individuals can either die with rate µ or be removed with rate α. Vaccinated individuals can either die with rate µ or obtain infected with force of infection (1 − e)βI where e measures the efficacy of the vaccine-induced protection against infection. If e = 1, then the vaccine is perfectly effective in preventing infection, while e = 0 means that the vaccine has no effect. All parameter values in system (1.1) are assumed to be nonnegative and µ > 0. In system (1.1), the basic reproduction number Rv = β[(1−p)µ+(1−e)(pµ+ψ)] (µ+ψ)(α+µ) is the threshold which determines whether the epidemic occurs or not. If ¯ 0, V¯ ) = Rv ≤ 1, system (1.1) has only the infection-free equilibrium E¯0 = (S, (1−p)µ ( µ+ψ , 0, pµ+ψ ) and it is globally asymptotically stable in the invariant Ω, µ+ψ where Ω = {(S, I, V ) : S ≥ 0, I ≥ 0, V ≥ 0, S + I + V ≤ 1}. This means that the disease will disappear and the entire population will become susceptible. If Rv > 1, then E¯0 is unstable and there is a globally asymptotically stable endemic equilibrium E¯1 = (S¯1 , I¯1 , V¯1 ) which shows that the disease will persist in a population, where S¯1 > 0, I¯1 > 0, V¯1 > 0 and satisfy the following equations   (1 − p)µ + αI¯1 − (µ + ψ)S¯1 − β S¯1 I¯1 = 0, β S¯1 I¯1 + (1 − e)β V¯1 I¯1 − (µ + α)I¯1 = 0,  pµ + ψ S¯1 − µV¯1 − (1 − e)β V¯1 I¯1 = 0.

However, for human infectious diseases, the nature of epidemic growth and spread is inherently variability owing to the unpredictability of individuals contacts (see e.g. [25, 3]). Hence it is important to reveal how the environmental noise affects the epidemic model. Stochastic differential equation models could be a more appropriate way of modeling infectious diseases and many stochastic models for epidemic populations have been developed (see e.g. [29, 30, 11, 13, 20, 12, 23, 24, 27]). Especially, the stochastic vaccination epidemic models have been extensively investigated under the assumption that the vaccinated individuals could not be infected (see e.g. [29, 30, 11, 12]) or the vaccinated individuals still could be infected (see e.g. [23, 24, 27]) due to the partial effectiveness of the vaccine. For instance, under the assumption of the vaccine, Zhao et al. [29] established sufficient conditions for extinction and persistence of the epidemic in the stochastic model, and Lin et al. [11] considered the asymptotic stability of the stochastic SISV epidemic model and the existence of a stationary distribution. By considering the partial effectiveness of the vaccine, Tornatore et al. [23, 24] 3

formulated and investigated another version of stochastic epidemic models with vaccination and analyzed the stability of the disease-free equilibrium. In this paper, we assume that stochastic perturbations in the environment will mainly affect the parameter β, as in Tornatore et al. [23], so that β → ˙ β + σ B(t), where B(t) is a standard Brownian motion with the intensity 2 σ > 0. By this way, the stochastic version corresponding to system (1.1) can be expressed as follows   dS = [(1 − p)µ + αI − (µ + ψ)S − βSI]dt − σSIdB(t), dI = [βSI + (1 − e)βV I − (µ + α)I]dt + σSIdB(t) + (1 − e)σV IdB(t),  dV = [pµ + ψS − µV − (1 − e)βV I]dt − (1 − e)σV IdB(t). (1.2) For system (1.2), the main difference is that the vaccination number considered in this paper is also perturbed by the white noise. The difference is the main difficulty to be conquered in obtaining the threshold of system (1.2), which has never been considered in previously literatures. For system (1.2), there are two interesting questions yet to be answered: Question A. Under what conditions, will the disease be persistent? Question B. Does there is a threshold under appropriate condition, which is above one or below one can completely determine the persistence and extinction of the disease? With these two questions in mind, in this paper, we will concentrate our attention on system (1.2) to (i): obtain explicit conditions which can guarantee the disease to be persistent; (ii): further develop the conditions for the infectious compartment to be extinct. Then by comparison, we get a threshold whose value above one or below one can guarantee the persistence and extinction of the disease under mild extra condition. Remark 1.1. It is easy to see that d(S + I + V ) = [µ − µ(S + I + V )]dt. Then S(t) + I(t) + V (t) = 1 + e−µt (S(0) + I(0) + V (0) − 1)  1, if S(0) + I(0) + V (0) ≤ 1, ≤ S(0) + I(0) + V (0), if S(0) + I(0) + V (0) > 1 := K. Thus the region Γ = {(S, I, V ) : S > 0, I > 0, V > 0, S + I + V ≤ 1} 4

is a positively invariant set of system (1.2). Hence from now on, we always assume that the initial value (S(0), I(0), V (0)) ∈ Γ. R For convenience, in what follows, we introduce the notation hx(t)i = 1 t x(s)ds if x is an integrable function on t ≥ 0. Define a parameter t 0 1 h β[(1 − p)µ + (1 − e)(pµ + ψ)] σ 2  (1 − p)µ + (1 − e)(pµ + ψ) 2 i . RvS = − µ+α µ+ψ 2 µ+ψ

The rest of this paper is organized as follows. In Section 2, we prove the existence and uniqueness of the global positive solution of system (1.2) by the way mentioned in [4, 18]. In Section 3, we verify that the disease will persist if RvS > 1. In Section 4, we show that the disease goes to extinction exponentially under three conditions. We also obtain that large noise will exponentially suppress the disease from persisting regardless of the value of the basic reproduction number RvS . Finally, some conclusion is given. Throughout this paper, unless otherwise specified, we let (Ω, F , {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). Moreover, we let B(t) be a Brownian motion defined on the complete probability space. We also let Rn+ = {x ∈ Rn : xi > 0 f or all 1 ≤ i ≤ n}. Here we present some basic theory in stochastic differential equations which is introduced in [16]. In general, consider the d-dimensional stochastic differential equation dx(t) = f (x(t), t)dt + g(x(t), t)dB(t) f or t ≥ t0 ,

(1.3)

with the initial value x(t0 ) = x0 ∈ Rd . B(t) denotes an n-dimensional standard Brownian motion defined on the complete probability space (Ω, F , {Ft }t≥0 , P). Denote by C 2,1 (Rd × [t0 , ∞]; R+ ) the family of all nonnegative functions V (x, t) defined on Rd × [t0 , ∞] such that they are continuously twice differentiable in x and once in t. The differential operator L of Eq. (1.3) is defined by [16] d d X ∂ 1X T ∂2 ∂ L= + [g (x, t)g(x, t)]ij . + fi (x, t) ∂t i=1 ∂xi 2 i,j=1 ∂xi ∂xj

If L acts on a function V ∈ C 2,1 (Rd × [t0 , ∞]; R+ ), then

1 LV (x, t) = Vt (x, t) + Vx (x, t)f (x, t) + trace[g T (x, t)Vxx (x, t)g(x, t)], 2 5

2

∂V ∂V ∂V V where Vt = ∂V , Vx = ( ∂x , , . . . , ∂x ), Vxx = ( ∂x∂i ∂x )d×d . In view of Itˆo’s ∂t 1 ∂x2 j d formula, if x(t) ∈ Rd , then

dV (x(t), t) = LV (x(t), t)dt + Vx (x(t), t)g(x(t), t)dB(t). 2. Existence and uniqueness of the global positive solution In this section, we shall prove that the solution of system (1.2) is global and positive with any initial value (S(0), I(0), V (0)) ∈ Γ, which is the first concerning thing in studying the dynamical behavior of infection models. As we know, in order for a stochastic differential equation to have a unique global (i.e., no explosion at any finite time) solution, the coefficients of the system are generally required to satisfy the linear growth condition and local Lipschitz condition (see [16]). However, the coefficients of system (1.2) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (1.2) may explode at a finite time. In this section, inspired by the method in [18], we show that there is a unique global positive solution of system (1.2). We establish the following theorem. Theorem 2.1. For any initial value (S(0), I(0), V (0)) ∈ Γ, there is a unique positive solution (S(t), I(t), V (t)) of system (1.2) on t ≥ 0 and the solution will remain in Γ with probability one, that is to say, (S(t), I(t), V (t)) ∈ Γ for all t ≥ 0 almost surely (a.s.). Proof. Our proof is motivated by the works of Mao, Marion and Renshaw [18]. Since the coefficients of system (1.2) satisfy the local Lipschitz condition, for any initial value (S(0), I(0), V (0)) ∈ Γ, there is a unique local solution (S(t), I(t), V (t)) on t ∈ [0, τe ), where τe is the explosion time [16]. To show this solution is global, we only need to prove that τe = ∞ a.s. To this end, let k0 ≥ 1 be sufficiently large such that S(0), I(0) and V (0) all lie within the interval [ k1 , k]. For each integer k ≥ k0 , define the stopping time through

o n 1 τk = inf t ∈ [0, τe ) : min{S(t), I(t), V (t)} ≤ or max{S(t), I(t), V (t)} ≥ k , k

where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). Clearly, τk is increasing as k → ∞. Set τ∞ = limk→∞ τk , whence τ∞ ≤ τk a.s. If τ∞ = ∞ a.s. is true, then τe = ∞ a.s. and (S(t), I(t), V (t)) ∈ Γ a.s. for all t ≥ 0. That is to say, to complete the proof, we need to prove τ∞ = ∞ a.s. If this statement is false, then there exists a 6

pair of constants T > 0 and ǫ ∈ (0, 1) such that P{τ∞ ≤ T } > ǫ. Therefore, there is an integer k1 ≥ k0 such that P{τk ≤ T } ≥ ǫ f or all k ≥ k1 .

(2.1)

Define a nonnegative C 2 -function V1 : Γ → R+ as follows 

S V1 (S, I, V ) = S − a − a ln + (I − 1 − ln I) + (V − 1 − ln V ). a

The nonnegativity of this function can be seen from

u − 1 − ln u ≥ 0 f or ∀u > 0. Let k ≥ k0 and T > 0 be arbitrary. An application of Itˆo’s formula [16], we obtain dV1 (S, I, V ) = LV1 (S, I, V )dt + σ[aI − S − (1 − e)V + (1 − e)I]dB(t), where LV1 : Γ → R+ is defined by LV1 (S, I, V ) = (1 − p)µ + αI − (µ + ψ)S − βSI −

aµ(1 − p) I − aα + (µ + ψ)a S S

1 +aβI + aσ 2 I 2 + βSI + (1 − e)βV I − (µ + α)I − βS − (1 − e)βV 2 1 1 +µ + α + σ 2 S 2 + (1 − e)2 σ 2 V 2 + pµ + ψS − µV − (1 − e)βV I 2 2 pµ ψS 1 − − + µ + (1 − e)βI + (1 − e)2 σ 2 I 2 V V 2 I aµ(1 − p) − aα ≤ −(µ + β)S + (aβ − µ)I − [µ + (1 − e)β]V − S S 1 2 1 2 pµ ψS − + µ + a(µ + ψ) + (1 − e)β + 2µ + α + aσ + σ − V V 2 2 1 2 1 2 2 2 + σ (1 − e) + σ (1 − e) 2 2 ≤ −(µ + β)S + (aβ − µ)I − [µ + (1 − e)β]V + µ + a(µ + ψ) + β 1 3 +2µ + α + aσ 2 + σ 2 . (2.2) 2 2 7

Choose a = to

µ β

such that aβ − µ = 0, then substituting this into (2.2) leads

3 1 LV1 (S, I, V ) ≤ µ + a(µ + ψ) + β + 2µ + α + aσ 2 + σ 2 := K, 2 2 where K is a positive constant. Then dV1 (S, I, V ) ≤ Kdt + σ[aI − S − (1 − e)V + (1 − e)I]dB(t).

(2.3)

Integrating (2.3) from 0 to τk ∧ T = min{τk , T } and then taking the expectations on both sides, we get EV1 (S(τk ∧ T ), I(τk ∧ T ), V (τk ∧ T )) ≤ V1 (S(0), I(0), V (0)) + KE(τk ∧ T ). Hence EV1 (S(τk ∧ T ), I(τk ∧ T ), V (τk ∧ T )) ≤ V1 (S(0), I(0), V (0)) + KT.

(2.4)

Set Ωk = {τk ≤ T } for k ≥ k1 and by (2.1), one can see that P(Ωk ) ≥ ǫ. Note that for every ω ∈ Ωk , there is S(τk , ω) or I(τk , ω) or V (τk , ω) equals either k or k1 . Therefore V1 (S(τk , ω), I(τk , ω), V (τk , ω)) is no less than either k − 1 − ln k or

1 1 1 − 1 − ln = − 1 + ln k. k k k

Consequently V1 (S(τk , ω), I(τk , ω), V (τk , ω)) ≥ (k − 1 − ln k) ∧ In view of (2.4), we have

1 k

 − 1 + ln k .

V1 (S(0), I(0), V (0)) + KT ≥ E[1Ωk (ω) V1 (S(τk , ω), I(τk , ω), V (τk , ω))]  1 − 1 + ln k , ≥ ǫ(k − 1 − ln k) ∧ k

where 1Ωk (ω) denotes the indicator function of Ωk . Letting k → ∞, then ∞ > V1 (S(0), I(0), V (0)) + KT = ∞,

which arises the contradiction, so we must have τ∞ = ∞ a.s. This means that the solution (S(t), I(t), V (t)) will not explode in a finite time a.s. This completes the proof. 8

3. Persistence in the mean In the dynamical behavior of the epidemic models, we are interested in two things, one is when the disease goes to extinction, the other is when the disease prevails. In this section, we shall investigate the persistence of the disease. Theorem 3.1. Let (S(t), I(t), V (t)) be the solution of system (1.2) with any initial value (S(0), I(0), V (0)) ∈ Γ. If RvS > 1 holds, then the disease will be persistent in the mean a.s., i.e., h  e(α + β)  3σ 2  e2 (α2 + β 2 ) i−1 S lim inf hI(t)i ≥ β (1−e)+ + (1−e)2 + (Rv −1)(µ+α) > 0. t→∞ µ+ψ 2 (µ + ψ)2 Furthermore

and

e(α + β)  (1 − p)µ α(µ + α) h  β (1 − e) + lim inf hS(t)i ≥ + t→∞ µ+ψ+β µ+ψ+β µ+ψ 2 2 2 2 i−1 3σ e (α + β ) + (1 − e)2 + (RvS − 1) a.s. 2 2 (µ + ψ)

n (1 − p)µ ψ α(µ + α) pµ + + lim inf hV (t)i ≥ t→∞ µ + β(1 − e) µ + β(1 − e) µ + ψ + β µ + ψ + β h  e(α + β)  3σ 2  e2 (α2 + β 2 ) i−1 × β (1 − e) + + (1 − e)2 + µ+ψ 2 (µ + ψ)2 o ×(RvS − 1) a.s. To verify Theorem 3.1, we first present some lemmas which will be used later. Lemma 3.1 [17]. Let M(t), t ≥ 0 be a continuous local martingale with M(0) = 0. Let θ > 1 and υk and γk be two sequences of positive numbers with υk → ∞ as k → ∞. Then for almost all ω ∈ Ω there exists a random integer k0 = k0 (ω) such that for all k ≥ k0 , θ 1 ln k, 0 ≤ t ≤ υk , M(t) ≤ γk hM, Mi(t) + 2 γk

where hM, Mi(t) is the quadratic variation of M(t). Lemma 3.2. Let g(t) be a continuous and bounded function on [0, ∞), then Z 1 t lim sup √ g(s)dB(s) ≤ θ a.s., t ln t 0 t→∞ 9

and for any constant ξ > 0, 1 lim sup √ t ln t t→∞

Proof. Choose γk = large k, we obtain

−ξk e√ k

Z

t

−ξ(t−s)

g(s)e

0

dB(s) ≤ θ a.s.

and υk = k in Lemma 3.1, then for sufficiently

Z k ˜2 √ √ √ L 1 ξs k + θ k ln k, sup g(s)e dB(s) ≤ √ g 2 (s)ds + θ k ln k ≤ 2 0≤s≤t 2 k 0

˜ = supt≥0 {g(t)}, which implies that where L 1 lim sup √ t ln t t→∞

If we choose γk =

−ξk e√ k

Z

g(s)dB(s) ≤ θ.

t

0

and υk = k in Lemma 3.1, one can get that

Z t g(s)e−ξ(t−s) dB(s) ≤ e−ξt sup

Z t g(s)eξs dB(s) 0≤s≤t 0   1 e−ξk Z k √ −ξt 2 2ξs ξk √ k ln k ≤ e g (s)e ds + θe 2 k 0 √ e supt≥0 {g 2 (t)} √ k + θ k ln k a.s. ≤ 4ξ

0

Then we have 1 lim sup √ t ln t t→∞

Z

t

−ξ(t−s)

g(s)e

0

dB(s) ≤ θ a.s.

This completes the proof. Lemma 3.3. Let (S(t), I(t), V (t)) be the solution of system (1.2) with the initial value (S(0), I(0), V (0)) ∈ Γ, then S(t) + (1 − e)V (t) =

(1 − p)µ + (1 − e)(pµ + ψ) − H(t, I(t)) + ϕ(t) µ+ψ Z t −eσ S(s)I(s)e−(µ+ψ)(t−s) dB(s), 0

10

where H(t, I(t)) = (1−e)I(t)−eα

Z

t

−(µ+ψ)(t−s)

I(s)e

ds+eβ

0

Z

t

S(s)I(s)e−(µ+ψ)(t−s) ds

0

and  (1 − p)µ  ϕ(t) = (1 − e)[S(0) + I(0) + V (0) − 1]e−µt − e − S(0) e−(µ+ψ)t . µ+ψ Proof. It follows from system (1.2) that S(t) + I(t) + V (t) = 1 + e−µt [S(0) + I(0) + V (0) − 1] and Z t  (1 − p)µ  (1 − p)µ −(µ+ψ)t − − S(0) e −β S(t) = S(s)I(s)e−(µ+ψ)(t−s) ds µ+ψ µ+ψ 0 Z t Z t +α I(s)e−(µ+ψ)(t−s) ds − σ S(s)I(s)e−(µ+ψ)(t−s) dB(s). (3.1) 0

0

Then we obtain S(t) + (1 − e)V (t) = (1 − e)[S(t) + V (t)] + eS(t) = (1 − e){1 + e−µt [S(0) + I(0) + V (0) − 1] − I(t)}  (1 − p)µ  (1 − p)µ   +e − − S(0) e−(µ+ψ)t µ+ψ µ+ψ Z t  Z t −(µ+ψ)(t−s) −e β S(s)I(s)e ds − α I(s)e−(µ+ψ)(t−s) ds 0 Z t0  +σ S(s)I(s)e−(µ+ψ)(t−s) dB(s) 0

(1 − p)µ + (1 − e)(pµ + ψ) − H(t, I(t)) + ϕ(t) = µ+ψ Z t −eσ S(s)I(s)e−(µ+ψ)(t−s) dB(s), 0

where H(t, I(t)) = (1−e)I(t)−eα

Z

t

−(µ+ψ)(t−s)

I(s)e

0

ds+eβ

Z

0

11

t

S(s)I(s)e−(µ+ψ)(t−s) ds

and

 (1 − p)µ  ϕ(t) = (1 − e)[S(0) + I(0) + V (0) − 1]e−µt − e − S(0) e−(µ+ψ)t . µ+ψ This completes the proof. Lemma 3.4 [8]. Let f ∈ C[[0, ∞) × Ω, R]. If there exist positive constants λ0 , λ such that Z t ln f (t) ≥ λt − λ0 f (s)ds + F (t) a.s. 0

for all t ≥ 0, where F ∈ C[[0, ∞) × Ω, R] and limt→∞ Z 1 t λ lim inf f (s)ds ≥ a.s. t→∞ t 0 λ0

F (t) t

= 0 a.s., then

Proof of Theorem 3.1. Applying Itˆo’s formula to system (1.2) and then taking an integration lead to E M(t) σ2 D 1 I(t) 2 (S(t)+(1−e)V (t)) + ln = βhS(t)+(1−e)V (t)i−(µ+α)− , t I(0) 2 t (3.2) Rt where M(t) = 0 (S(s) + (1 − e)V (s))dB(s) is a local martingale. According to Lemma 3.2, one can see that M(t) lim = 0 a.s. (3.3) t→∞ t In view of Lemma 3.3, one can get that (1 − p)µ + (1 − e)(pµ + ψ) hS(t) + (1 − e)V (t)i = − hH(t, I(t))i + hϕ(t)i µ+ψ Z Z eσ t v − S(s)I(s)e−(µ+ψ)(v−s) dB(s)dv t 0 0 and  (1 − p)µ + (1 − e)(pµ + ψ) 2 + hH 2 (t, I(t))i h(S(t) + (1 − e)V (t))2 i ≤ µ+ψ 2[(1 − p)µ + (1 − e)(pµ + ψ)] +hϕ2 (t)i + hϕ(t)i µ+ψ 2[(1 − p)µ + (1 − e)(pµ + ψ)] K(t) eσ − µ+ψ t i2 2 2 Z thZ v e σ + S(s)I(s)e−(µ+ψ)(v−s) dB(s) dv. t 0 0 12

It follows from (3.2) that 1 I(t) β[(1 − p)µ + (1 − e)(pµ + ψ)] σ 2  (1 − p)µ + (1 − e)(pµ + ψ) 2 ln ≥ − (µ + α) − t I(0) µ+ψ 2 µ+ψ 2 K(t) σ − hH 2 (t, I(t))i − βhH(t, I(t))i + βhϕ(t)i − βeσ 2 t 2h σ 2[(1 − p)µ + (1 − e)(pµ + ψ)] − hϕ2 (t)i + hϕ(t)i 2 µ+ψ 2[(1 − p)µ + (1 − e)(pµ + ψ)] K(t) i eσ − µ+ψ t i2 2 2 2 Z thZ v σ eσ −(µ+ψ)(v−s) − S(s)I(s)e dB(s) dv, (3.4) 2 t 0 0

where

hH(t, I(t))i Z t Z t D E −(µ+ψ)(t−s) −(µ+ψ)(t−s) = (1 − e)I(t) − eα I(s)e ds + eβ S(s)I(s)e ds 0 0 h eβ i eα hI(t)i + ≤ (1 − e) + µ+ψ µ+ψ h e(α + β) i = (1 − e) + hI(t)i, (3.5) µ+ψ

Z t D 2 2 2 2 2 hH (t, I(t))i ≤ 3 (1 − e) I (t) + e α I(s)e−(µ+ψ)(t−s) ds 0 Z t 2 E +e2 β 2 S(s)I(s)e−(µ+ψ)(t−s) ds 2

0

e2 β 2 i e2 α2 hI(t)i + (µ + ψ)2 (µ + ψ)2 h e2 (α2 + β 2 ) i 2 = 3 (1 − e) + hI(t)i (µ + ψ)2

h ≤ 3 (1 − e)2 +

and

Z tZ

v

(3.6)

S(s)I(s)e−(µ+ψ)(v−s) dB(s)dv 0 0 Z Z t  1  t S(s)I(s)dB(s) − S(s)I(s)e−(µ+ψ)(t−s) dB(s) . = µ+ψ 0 0

K(t) =

13

Note that (S(t), I(t), V (t)) ∈ Γ and in view of Lemma 3.2, we have lim

t→∞

K(t) = 0 a.s. t

(3.7)

Now we compute that Z t hZ v i2 S(s)I(s)e−(µ+ψ)(v−s) dB(s) dv 0 Z t0 hZ v i2 −2(µ+ψ)v = e S(s)I(s)e(µ+ψ)s dB(s) dv 0 0 Z t hZ v i 1 S(v)I(v) S(s)I(s)e−(µ+ψ)(v−s) dB(s) dB(v) = µ+ψ 0 0 Z t  2 1 −(µ+ψ)(t−s) S(s)I(s)e dB(s) − 2(µ + ψ) 0 Z t hZ v i 1 ≤ S(v)I(v) S(s)I(s)e−(µ+ψ)(v−s) dB(s) dB(v). µ+ψ 0 0

(3.8)

By (3.1), one can obtain that Z t σ S(s)I(s)e−(µ+ψ)(t−s) dB(s)

0 Z t  (1 − p)µ  (1 − p)µ + + S(0) + β S(s)I(s)e−(µ+ψ)(t−s) ds ≤ µ+ψ µ+ψ 0 Z t 2(1 − p)µ + α + β + 2. +α I(s)e−(µ+ψ)(t−s) ds + S(t) ≤ µ+ψ 0

An application of Lemma 3.2, we have Z hZ v i 1 t −(µ+ψ)(v−s) S(v)I(v) S(s)I(s)e dB(s) dB(v) = 0 a.s. lim sup t 0 t→∞ 0 This, together with (3.8), implies that Z Z i2 1 th v lim S(s)I(s)e−(µ+ψ)(v−s) dB(s) dv = 0 a.s. t→∞ t 0 0

(3.9)

Obviously lim hϕ(t)i = 0 and lim hϕ2 (t)i = 0.

t→∞

t→∞

14

(3.10)

Substituting (3.5), (3.6), (3.7), (3.9) and (3.10) into (3.4) and then applying Lemma 3.4 result in h  e(α + β)  3σ 2  e2 (α2 + β 2 ) i−1 S lim inf hI(t)i ≥ (µ+α) β (1−e)+ + (1−e)2 + (Rv −1) > 0. t→∞ µ+ψ 2 (µ + ψ)2

Next, we shall show that system (1.2) is persistent in the mean, i.e., we shall show that S and V are persistent. It follows from system (1.2) that Z S(t) − S(0) σ t S(s)I(s)dB(s) + t t 0 = (1 − p)µ + αhI(t)i − (µ + ψ)hS(t)i − βhS(t)I(t)i ≥ (1 − p)µ + αhI(t)i − (µ + ψ + β)hS(t)i, Z V (t) − V (0) σ(1 − e) t + V (s)I(s)dB(s) t t 0 = pµ + ψhS(t)i − β(1 − e)hV (t)I(t)i − µhV (t)i ≥ pµ + ψhS(t)i − [µ + β(1 − e)]hV (t)i. Making use of Lemmas 3.2 and 3.4, we have (1 − p)µ α + lim inf hI(t)i µ + ψ + β µ + ψ + β t→∞ e(α + β)  α(µ + α) h  (1 − p)µ β (1 − e) + + ≥ µ+ψ+β µ+ψ+β µ+ψ 2 2 2 2 i−1 3σ e (α + β ) + (1 − e)2 + (RvS − 1) a.s. (3.11) 2 (µ + ψ)2

lim inf hS(t)i ≥ t→∞

and

ψ pµ + lim inf hS(t)i µ + β(1 − e) µ + β(1 − e) t→∞ n (1 − p)µ pµ ψ α(µ + α) ≥ + + µ + β(1 − e) µ + β(1 − e) µ + ψ + β µ + ψ + β h  e2 (α2 + β 2 ) i−1 e(α + β)  3σ 2  + (1 − e)2 + × β (1 − e) + µ+ψ 2 (µ + ψ)2 o ×(RvS − 1) a.s.

lim inf hV (t)i ≥ t→∞

This completes the proof.

15

4. Extinction of the disease In this section, we shall discuss the extinction of the disease describing by system (1.2). Theorem 4.1. Let (S(t), I(t), V (t)) be the solution of system (1.2) with the initial value (S(0), I(0), V (0)) ∈ Γ. If one of the following three assumptions holds (A)

β2 < µ + α; 2σ 2

β[(1 − p)µ + (1 − e)(pµ + ψ)] σ 2 n (1 − p)µ α(µ + α) − (µ + α) − + µ+ψ 2 µ+ψ+β µ+ψ+β o2 h  e2 (α2 + β 2 ) i−1 S e(α + β)  3σ 2  + (1 − e)2 + (R − 1) < 0; × β (1 − e) + v µ+ψ 2 (µ + ψ)2

(B)

(1 − p)µ + (1 − e)(pµ + ψ) ≤ β and RvS < 1, µ+ψ then the disease will die out exponentially a.s., i.e., (C) σ 2

lim sup t→∞

lim sup t→∞

β2 1 I(t) ln ≤ 2 − (µ + α) < 0 a.s. if (A) holds; t I(0) 2σ

(4.1)

1 I(t) β[(1 − p)µ + (1 − e)(pµ + ψ)] ln ≤ − (µ + α) t I(0) µ+ψ e(α + β)  α(µ + α) h  σ 2 n (1 − p)µ β (1 − e) + + − 2 µ+ψ+β µ+ψ+β µ+ψ o 2 2 2 2 i−1 2 3σ e (α + β ) + (1 − e)2 + (RvS − 1) < 0 a.s. 2 (µ + ψ)2 if (B) holds; (4.2)

lim sup t→∞

1 I(t) ln ≤ (µ + α)(RvS − 1) < 0 a.s. if (C) holds. t I(0)

(4.3)

Proof. If (A) holds, then according to (3.2), one can see that 1 I(t) σ2 M(t) ln ≤ βhS(t) + (1 − e)V (t)i − (µ + α) − h(S(t) + (1 − e)V (t))i2 + t I(0) 2 t 16

β i2 M(t) σ2 h β2 h(S(t) + (1 − e)V (t))i − − (µ + α) − + 2σ 2 2 σ2 t β2 M(t) ≤ − (µ + α) + . 2σ 2 t

≤

In view of (3.3), we obtain lim sup t→∞

β2 1 I(t) ln ≤ 2 − (µ + α) < 0 a.s., t I(0) 2σ

which shows that limt→∞ I(t) = 0 a.s. If (B) holds, then in view of (3.2), we obtain 1 I(t) σ2 M(t) ln = βhS(t) + (1 − e)V (t)i − (µ + α) − h(S(t) + (1 − e)V (t))2 i + t I(0) 2 t 2 σ M(t) β[(1 − p)µ + (1 − e)(pµ + ψ)] − (µ + α) − hS(t)i2 + + βhϕ(t)i ≤ µ+ψ 2 t K(t) −βeσ . t An application of (3.3), (3.7), (3.10) and (3.11), we get lim sup t→∞

β[(1 − p)µ + (1 − e)(pµ + ψ)] 1 I(t) ln ≤ − (µ + α) t I(0) µ+ψ σ 2 n (1 − p)µ e(α + β)  α(µ + α) h  − β (1 − e) + + 2 µ+ψ+β µ+ψ+β µ+ψ o 2 2 2 i−1 2 2 e (α + β ) 3σ (1 − e)2 + (RvS − 1) < 0, + 2 2 (µ + ψ)

which implies that limt→∞ I(t) = 0 a.s. If (C) holds, then by (3.2), one can see that 1 I(t) σ2 M(t) ln ≤ βhS(t) + (1 − e)V (t)i − (µ + α) − hS(t) + (1 − e)V (t)i2 + t I(0) 2 t 2 β[(1 − p)µ + (1 − e)(pµ + ψ)] σ (1 − p)µ + (1 − e)(pµ + ψ) 2 ≤ − (µ + α) − µ+ψ 2 µ+ψ   (1 − p)µ + (1 − e)(pµ + ψ) − β hH(t, I(t))i + Ψ(t), + σ2 µ+ψ 17

where Ψ(t) = 2

 (1 − p)µ + (1 − e)(pµ + ψ)

µ+ψ  K(t) 2 + hϕ(t)i − eσ . t

− hH(t, I(t))i + β



K(t)  hϕ(t)i − eσ t

Note that (3.7) and (3.10) show that limt→∞ Ψ(t) = 0 a.s., one can derive that 1 I(t) ≤ (µ + α)(RvS − 1) < 0 a.s., lim sup ln I(0) t→∞ t which indicates that limt→∞ I(t) = 0 a.s. In summary, if one of the above three assumptions holds, the disease will go to extinction exponentially with probability one. This completes the proof. Remark 4.1. Theorem 4.1 shows that the disease will die out with probability one. More precisely, • (4.1) and (4.2) show that large noise will exponentially suppress the disease from prevailing. • From (4.3), when the noise is sufficiently small, RvS < 1 is sufficient for guaranteeing the infectious compartment to die out. This, together with Theorem 4.1, yields the conclusion that RvS can be considered as the threshold of system (1.2), whose value is above one or below one completely determines the persistence and extinction of the disease in case that the noise is sufficiently small. 5. Conclusion This paper deals with the threshold dynamics of a stochastic SIS epidemic model with imperfect vaccination. First of all, we discuss the existence and uniqueness of the global positive solution of system (1.2). Then if the basic reproduction number RvS > 1, we establish sufficient conditions for persistence in the mean of system (1.2). In addition, we give three conditions for the disease to die out. We also obtain that large noise will exponentially suppress the disease from prevailing regardless of the value of RvS . Some interesting topics deserve further investigations. On the one hand, one may propose some more realistic but complex models, such as considering the effects of impulsive perturbations on system (1.2). On the other hand, our model is autonomous, it is also interesting to investigate the nonautonomous case and consider other important dynamical properties of system 18

(1.2), such as the existence of nontrivial periodic solutions, stochastic ultimate boundedness, pathwise estimation and so on. We leave these cases as our future work. Acknowledgments This work was supported by NSFC of China Grant No.11371085, the Fundamental Research Funds for the Central Universities (No.15CX08011A), 2016GXNSFBA380006 and KY2016YB370. References [1] J. Arino, C.C. McCluskey, P. wan den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math. 64 (2003) 260-276. [2] J.R. Artalejo, M.J. Lopez-Herrero, Stochastic epidemic models: New behavioral indicators of the disease spreading, Appl. Math. Model. 38 (2014) 4371-4387. [3] Y. Cai, Y. Kang, M. Banerjee, W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equ. 259 (2015) 7463-7502. [4] N. Dalal, D. Greenhalgh, X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl. 341 (2008) 1084-1101. [5] A. d’Onofrio, Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times, Appl. Math. Comput. 151 (2004) 181-187. [6] H.C. Frederick, A susceptible-infected epidemic model with voluntary vaccinations, J. Math. Biol. 53 (2006) 253-272. [7] C. Ji, D. Jiang, Q. Ynag, N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica 48 (2012) 121-131. [8] C. Ji, D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38 (2014) 5067-5079. [9] W. Kermack, A. McKendrick, Contributions to the mathematical theory of epidemics (Part I), Proc. Soc. Lond. Ser. A 115 (1927) 700-721. 19

[10] J. Li, Z. Ma, Qualitative analyses of SIS epidemic model with vaccination and varying total population size, Math. Comput. Model. 35 (2002) 1235-1243. [11] Y. Lin, D. Jiang, S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A 394 (2014) 187-197. [12] D. Liu, B. Wang, A novel time delayed HIV/AIDS model with vaccination and antiretroviral therapy and its stability analysis, Appl. Math. Model. 37 (2013) 4608-4625. [13] Q. Liu, Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A 428 (2015) 140153. [14] X. Liu, Y. Takeuchi, S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol. 253 (2008) 1-11. [15] Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real World Appl. 14 (2013) 1286-1299. [16] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [17] X. Mao, Almost sure asymptotic bounds for a class of stochastic differential equations, Stoch. Stoch. Reports 41 (1992) 57-69. [18] X. Mao, G. Marion, E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stoch. Process. Appl. 97 (2002) 95-110. [19] I.A. Moneim, D. Greenhalgh, Threshold and stability results for an sirs epidemic model with a general periodic vaccination strategy, J. Biol. Syst. 13 (2005) 131-150. [20] E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991. [21] M. Safan, F.A. Rihan, Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation, Math. Comput. Simul. 96 (2014) 195-206. 20

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21

Highlights (for review)

Highlights ●A stochastic SIS epidemic model with imperfect vaccination is investigated. ●When R_{v}^{S}>1, we establish sufficient conditions for persistence in the mean of the system. ●Three conditions for the disease to die out are given, which improve the previous results on extinction of the disease. ●Large noise will exponentially suppress the disease from persisting regardless of the value of R_{v}^{S}.