Stationary distribution and extinction of a stochastic

Available online at www.sciencedirect.com

Journal of the Franklin Institute 355 (2018) 8891–8914 www.elsevier.com/locate/jfranklin

Stationary distribution and extinction of a stochastic dengue epidemic model Qun Liu a, Daqing Jiang a,b,c,∗, Tasawar Hayat b,d, Ahmed Alsaedi b a School

of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin 130024, PR China b Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia c College of Science, China University of Petroleum, Qingdao, Shandong 266580, PR China d Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan Received 17 July 2017; received in revised form 5 September 2018; accepted 4 October 2018 Available online 11 October 2018

Abstract In this paper, we investigate the dynamical behavior of a stochastic dengue epidemic model. First of all, by constructing a suitable stochastic Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the diseases. The existence of stationary distribution implies stochastic weak stability. © 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Dengue is a viral disease transmitted by the bite of an Aedes mosquito infected with one of the four dengue virus serotypes (DEN-1, DEN-2, DEN-3 and DEN-4) [1,2]. Dengue can affect almost all age groups (infant to adult), and symptoms appear 3–14 days after the infected mosquito bite [2]. In recent decades, the burden of dengue increases rapidly and in ∗ Corresponding author at: School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin Province 130024, PR China. E-mail address: [email protected] (D. Jiang).

https://doi.org/10.1016/j.jfranklin.2018.10.003 0016-0032/© 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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Q. Liu et al. / Journal of the Franklin Institute 355 (2018) 8891–8914

2012, World Health Organization (WHO) reported that there may be 50–100 million dengue cases occur worldwide each year. It is estimated that approximately 3.6 billion people are living in dengue affected parts of the world [3]. In most epidemic models, the bilinear incidence is always used which is given by βSI, where β denotes the probability of transmission per contact and S and I are the susceptible and infected populations, respectively. However, there are ample reasons for using nonlinear incidence such as saturating and nearly bilinear. For example, Levin and his coworkers [4,5] have adopted a nonlinear incidence form like βSq Ip with p, q > 0, which depends on different infective diseases and environments. To prevent the unboundedness of contact rate, Capasso and Serio [6] used a saturated incidence of the form βSI /(1 + αI ), α > 0. Recently, in order to understand and control dengue infection, a large number of mathematical models have been proposed and analyzed [7–11]. For instance, Newton and Reiter [7] developed the first SEIR model for dengue in the form of a system of Ordinary Differential Equations (ODEs), in which the mosquito population was not modeled. In [8,9], Focks et al. began describing mosquitoes populations by using a Dynamic Table Models, where later the human population and the disease were introduced [10]. Cai et al. [11] formulated a dengue epidemic model with saturation and bilinear incidence. They investigated the global stability of the disease-free equilibrium and the endemic equilibrium. Their model can be described by the following system of differential equations: ⎧ bβ1 SH IV ⎪ ⎪ SH = μK − − μSH , ⎪ ⎪ 1 + αIV ⎪ ⎪ ⎪ ⎨I = bβ1 SH IV − (μ + γ + γ )I , 0 1 H H 1 + αIV ⎪ ⎪ RH = (γ0 + γ1 )IH − μRH , ⎪ ⎪ ⎪ ⎪ S = A − bβ2 IH SV − mSV , ⎪ ⎩ V IV = bβ2 IH SV − mIV ,

(1.1)

where SH = SH (t ), IH = IH (t ), RH = RH (t ) denote the number of susceptible, infective and recovered (or treated) hosts at time t, respectively, NH (t ) = SH (t ) + IH (t ) + RH (t ) denotes the total number, SV = SV (t ), IV = IV (t ) denote the number of susceptible and infective at time t in the vector population, respectively, NV (t ) = SV (t ) + IV (t ) is the total population sizes of the vector. The parameter μK is the recruitment rate of the human population, μ, γ 0 , γ 1 are the natural death rate, recovered rate and get treated rate of the human population, respectively. The parameter A is the recruitment rate of the mosquito population, mNV (t) denotes the total deaths in the mosquito population and so m is the per capita mortality rate of mosquitoes, b denotes the number of bites vector per unit time, β 1 is the transmission probability from vector to human and β 2 is the transmission probability from human to vector. All parameter values in system (1.1) are assumed to be positive constants. For convenience in mathematics, let γ = γ0 + γ1 . Because the variable RH does not appear in the first two equations, thus system (1.1) becomes ⎧ bβ1 SH IV ⎪ ⎪ ⎪SH = μK − 1 + αI − μSH , ⎪ ⎪ V ⎨ bβ1 SH IV IH = − (μ + γ )IH , 1 + αIV ⎪ ⎪ ⎪ ⎪SV = A − bβ2 IH SV − mSV , ⎪ ⎩ IV = bβ2 IH SV − mIV .

(1.2)


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1 β2 KA In system (1.2), the basic reproduction number R0 = bm2β(μ+ is the threshold which deγ) termines whether the epidemic occurs or not. System (1.2) always has a disease-free equilibrium E0 = (K, 0, 0, 0) and if R0 ≤ 1 it is globally asymptotically stable in the positively invariant set , where = {(SH , IH , SV , IV ) ∈ R4+ : 0 ≤ SH + IH ≤ K, 0 ≤ SV + IV = mA }. If R0 > 1, then E0 is unstable and there is a locally asymptotically stable endemic equilibm2 (μ+γ )(1+αI ∗ ) m2 IV∗ rium E ∗ = (SH∗ , IH∗ , SV∗ , IV∗ ) in the interior of , where SH∗ = b2 β1 β2 (A−mI ∗V) , IH∗ = bβ2 (A−mI ∗), 2

V

V

γ )(R0 −1) SV∗ = mA − IV∗ , IV∗ = (μmα+μm(μ+ . bβ1 m)(μ+γ )+b2 β1 β2 μK On the other hand, it is well known that epidemic models are always affected by the environmental noise which is an important component in an ecosystem since it can provide an additional degree of realism in comparison to their deterministic counterparts [12]. Therefore, many authors have developed stochastic models for epidemic populations [13–20]. Britton and Traoré [13] considered a stochastic model describing the spread of a vector borne disease in a community where individuals (hosts and vectors) die and new individuals (hosts and vectors) are born. They obtained some important dynamical properties of the vector-borne epidemic model. In [14], Otero and Solari studied a spatially explicit stochastic dengue model which includes the evolution of the mosquitoes population [15,16] but in which humans are regarded as a spatially fixed population, i.e., the spatial spreading of the disease is possible due to mosquito dispersal. Barmak et al. [17] explored the effect of human mobility on the spatio-temporal dynamics of Dengue with a stochastic model that takes into account the epidemiological dynamics of the infected mosquitoes and humans, with different mobility patterns of the human population. Inspired by the above mentioned works, in this paper, by taking into account the effect of randomly fluctuating environment, we assume that the parameters involved in the model (1.2) fluctuate around some average value due to the continuous fluctuation in the environment. Following this approach, we study a stochastic dengue epidemic model in which we assume that the environmental noise is proportional to the variables SH , IH , SV and IV . Then corresponding to system (1.2), we obtain the following stochastic model bβ1 SH IV dSH = μK − − μSH dt + σ1 SH dB1 (t ), 1 + αIV bβ1 SH IV dIH = − (μ + γ )IH dt + σ2 IH dB2 (t ), 1 + αIV dSV = [A − bβ2 IH SV − mSV ]dt + σ3 SV dB3 (t ),

dIV = [bβ2 IH SV − mIV ]dt + σ4 IV dB4 (t ),

(1.3)

where B˙ i (t ) are the white noise, namely, Bi (t) are mutually independent standard Brownian motions, σi2 > 0 denote the intensities of the white noise, i = 1, 2, 3, 4. The paper is organized as follows: In Section 2, we show that there is a unique global positive solution of system (1.3) with any positive initial value. In Section 3, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to system (1.3). In Section 4, we obtain sufficient conditions for extinction of the diseases. Finally, some concluding remarks and future directions are presented to end this paper. Throughout this paper, unless otherwise specified, let (, 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) and we also let Bi (t) be mutually inde-

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pendent standard Brownian motions defined on the complete probability space, i = 1, 2, 3, 4. Define Rd+ = {x = (x1 , . . . , xd ) ∈ Rd : xi > 0, 1 ≤ i ≤ d}. Here we present some basic theory in stochastic differential equations (SDEs) which is introduced in [21]. In general, consider the d-dimensional stochastic differential equation dx(t ) = f (x(t ), t )dt + g(x(t ), t )dB(t ) for t ≥ t0 ,

(1.4)

with the initial value x(0) = x0 ∈ Rd . B(t) denotes a d-dimensional standard Brownian motion defined on the complete probability space (, 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. (1.4) is defined by [21] L=

d d ∂ ∂ 1 T ∂2 + fi (x, t ) + [g (x, t )g(x, t )]i j . ∂t ∂xi 2 i, j=1 ∂ xi ∂ x j i=1

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 ) + t race[gT (x, t )Vxx (x, t )g(x, t )], 2 where Vt =

∂V ∂t

∂V ∂V , Vx = ( ∂x , . . . , ∂x ), Vxx = ( ∂∂xi ∂Vx j )d×d . According to Itô’s formula [21], if 1 d 2

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 Since SH (t), IH (t), SV (t) and IV (t) in system (1.3) denote the number of individuals at time t, they should be nonnegative. So for further study, we should first give some condition under which system (1.3) has a unique global positive solution. To this end, we establish the following theorem. Theorem 2.1. For any initial value (SH (0), IH (0), SV (0), IV (0)) ∈ R4+ , there exists a unique solution (SH (t), IH (t), SV (t), IV (t)) of system (1.3) on t ≥ 0 and the solution will remain in R4+ with probability one, namely, (SH (t ), IH (t ), SV (t ), IV (t )) ∈ R4+ for all t ≥ 0 almost surely (a.s.). Proof. Since the coefficients of system (1.3) satisfy the local Lipschitz condition, then for any initial value (SH (0), IH (0), SV (0), IV (0)) ∈ R4+ , there is a unique local solution (SH (t ), IH (t ), SV (t ), IV (t )) ∈ R4+ on t ∈ [0, τ e ) a.s., where τ e denotes the explosion time [21]. To prove this solution is global, we only need to prove that τe = ∞ a.s. To this end, let k0 ≥ 1 be sufficiently large such that SH (0), IH (0), SV (0) and IV (0) all lie within the interval [ k10 , k0 ]. For each integer k ≥ k0 , define the stopping time as follows [21] τk = inf t ∈ [0, τe ) : min{SH (t ), IH (t ), SV (t ), IV (t )}


≤

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1 or max{SH (t ), IH (t ), SV (t ), IV (t )} ≥ k , k

where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). It is easy to see that τ k is increasing as k → ∞. Set τ∞ = limk→∞ τk , whence τ ∞ ≤ τ e a.s. If we can verify that τ∞ = ∞ a.s., then τe = ∞ and (SH (t ), IH (t ), SV (t ), IV (t )) ∈ R4+ a.s. for all t ≥ 0. In other words, to complete the proof all we need to prove is that τ∞ = ∞ a.s. If this assertion is not true, then there exists a pair of constants T > 0 and ∈ (0, 1) such that P{τ∞ ≤ T } > . Thus there is an integer k1 ≥ k0 such that P{τk ≤ T } ≥ for all k ≥ k1 .

(2.1)

Define a C2 -function V : R4+ → R+ ∪ {0} as follows SH + (IH − 1 − ln IH ) V (SH , IH , SV , IV ) = SH − a1 − a1 ln a1 SV + SV − a2 − a2 ln + (IV − 1 − ln IV ), a2 where a1 , a2 are positive constants to be determined later. The nonnegativity of this function can be seen from u − 1 − ln u ≥ 0 for any u > 0. Let k ≥ k0 and T > 0 be arbitrary. Making use of Itô’s formula to V yields dV (SH , IH , SV , IV ) = LV (SH , IH , SV , IV )dt + σ1 (SH − a1 )dB1 (t ) + σ2 (IH − 1)dB2 (t ) +σ3 (SV − a2 )dB3 (t ) + σ4 (IV − 1)dB4 (t ), where LV : R4+ → R is defined by LV (SH , IH , SV , IV ) bβ1 SH IV a1 σ12 σ2 a1 bβ1 SH IV 1 μK − = 1− − μSH + + 1− − (μ + γ )IH + 2 SH 1 + αIV 2 IH 1 + αIV 2 2 2 a2 σ3 σ a2 1 [A − bβ2 IH SV − mSV ] + [bβ2 IH SV − mIV ] + 4 + 1− + 1− SV 2 IV 2 2 2 2 2 a1 σ1 σ a2 σ3 σ = μK + A + μ + γ + m + a1 μ + a2 m + + 2 + + 4 − μ(SH + IH ) − γ IH 2 2 2 2 a1 bβ1 IV a1 μK bβ1 SH IV a2 A bβ2 IH SV − + − − mSV − mIV − + a2 bβ2 IH − SH 1 + αIV IH (1 + αIV ) SV IV a1 σ12 σ22 a2 σ32 σ42 ≤ μK + A + μ + γ + m + a1 μ + a2 m + + + + 2 2 2 2 +(a1 bβ1 − m)IV + (a2 bβ2 − γ )IH . Choose a1 =

m , bβ1

a2 =

γ bβ2

such that a1 bβ1 − m = 0 and a2 bβ2 − γ = 0. Hence we obtain

LV (SH , IH , SV , IV ) ≤ μK + A + μ + γ + m + a1 μ + a2 m +

a1 σ12 σ2 a2 σ32 σ2 + 2 + + 4 := K˜ , 2 2 2 2

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where K˜ is a positive constant. Therefore, we have dV (SH , IH , SV , IV ) ≤ K˜ dt + σ1 (SH − a1 )dB1 (t ) + σ2 (IH − 1)dB2 (t ) + σ3 (SV − a2 )dB3 (t ) +σ4 (IV − 1)dB4 (t ).

(2.2)

Integrating Eq. (2.2) from 0 to τk ∧ T = min{τk , T } and then taking the expectation on both sides, we have EV (SH (τk ∧ T ), IH (τk ∧ T ), SV (τk ∧ T ), IV (τk ∧ T )) ≤ V (SH (0), IH (0), SV (0), IV (0)) + K˜ E(τk ∧ T ). Therefore EV (SH (τk ∧ T ), IH (τk ∧ T ), SV (τk ∧ T ), IV (τk ∧ T )) ≤ V (SH (0), IH (0), SV (0), IV (0)) + K˜ T . (2.3) Set k = {τk ≤ T } for k ≥ k1 and by Eq. (2.1) we have P(k ) ≥ . Note that for every ω ∈ k , there is SH (τ k , ω) or IH (τ k , ω) or SV (τ k , ω) or IV (τ k , ω) equals either k or 1k . Hence V(SH (τ k , ω), IH (τ k , ω), SV (τ k , ω), IV (τ k , ω)) is no less than either k k k − a1 − a1 ln ∧ (k − 1 − ln k) ∧ k − a2 − a2 ln a1 a2 or 1 1 1 1 1 1 − a1 − a1 ln = − a1 + a1 ln (ka1 ) ∧ − 1 − ln = − 1 + ln k k ka1 k k k k 1 1 1 ∧ − a2 − a2 ln = − a2 + a2 ln (ka2 ) . k ka2 k Consequently, we have V (SH (τk , ω), IH (τk , ω), SV (τk , ω), IV (τk , ω)) 1 k k ∧ (k − 1 − ln k) ∧ k − a2 − a2 ln ∧ ≥ k − a1 − a1 ln − a1 + a1 ln (ka1 ) a1 a2 k 1 1 ∧ − 1 + ln k ∧ − a2 + a2 ln (ka2 ) . k k In view of Eq. (2.3), we have V (SH (0), IH (0), SV (0), IV (0)) + K˜ T ≥ E[Ik (ω) (SH (τk , ω), IH (τk , ω), SV (τk , ω), IV (τk , ω))] 1 k k ∧ (k − 1 − ln k) ∧ k − a2 − a2 ln ∧ ≥ k − a1 − a1 ln − a1 + a1 ln (ka1 ) a1 a2 k 1 1 ∧ − 1 + ln k ∧ − a2 + a2 ln (ka2 ) , k k where Ik denotes the indicator function of k . Letting k → ∞, we have ∞ > V (SH (0), IH (0), SV (0), IV (0)) + K˜ T = ∞, which leads to the contradiction and so we must have τ∞ = ∞. This completes the proof.


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3. Existence of ergodic stationary distribution of system (1.3) In this section, we will establish sufficient conditions for the existence of a unique ergodic stationary distribution. First of all, we present a lemma which will be used later. Let X(t) be a homogeneous Markov process in Ed (Ed denotes a d-dimensional Euclidean space) and be described by the following stochastic differential equation dX (t ) = b(X )dt +

k

gr (X )dBr (t ).

r=1

The diffusion matrix is defined as follows A˜ (x) = (ai j (x)), ai j (x) =

k

gir (x)grj (x).

r=1

Lemma 3.1 ([22]). The Markov process X(t) has a unique ergodic stationary distribution μ( · ) if there exists a bounded domain D ⊂ Ed with

regular boundary and (B.1): there is a positive number M such that di, j=1 ai j (x)ξi ξ j ≥ M|ξ |2 , x ∈ D, ξ ∈ Rd . (B.2): there exists a nonnegative C2 -function V such that LV is negative for any Ed ࢨD. Then

1 T Px lim f (X (t ))dt = f (x)μ(dx) = 1 T →∞ T 0 Ed for all x ∈ Ed , where f( · ) is a function integrable with respect to the measure μ. Define a parameter as follows b2 β1 β2 μK A R0S = . 2 σ1 σ22 σ32 σ42 μ+ μ+γ + m+ m+ 2 2 2 2 Theorem 3.1. Assume that R0S > 1, then for any initial value (SH (0), IH (0), SV (0), IV (0)) ∈ R4+ , system (1.3) admits a unique stationary distribution μ( · ) and it has the ergodic property. Proof. In order to prove Theorem 3.1, we only need to validate conditions (B.1) and (B.2) in Lemma 3.1. Now we prove the condition (B.1). The diffusion matrix of system (1.3) is given by ⎛ 2 2 ⎞ σ1 SH 0 0 0 ⎜ 0 σ22 IH2 0 0 ⎟ ⎟. A˜ = ⎜ ⎝ 0 0 σ32 SV2 0 ⎠ 0 0 0 σ42 IV2 Choose M˜ = min (SH ,IH ,SV ,IV )∈D¯ k ⊂R4+ {σ12 SH2 , σ22 IH2 , σ32 SV2 , σ42 IV2 }, we get ⎛ 4 i, j=1

ai j (SH , IH , SV , IV )ξi ξ j = σ1 SH ξ1

σ2 IH ξ2

σ3 SV ξ3

⎞ σ1 SH ξ1 ⎜ σ2 IH ξ2 ⎟ ⎟ σ4 IV ξ4 ⎜ ⎝ σ3 SV ξ3 ⎠ σ4 IV ξ4

= (σ1 SH )2 ξ12 + (σ2 IH )2 ξ22 + (σ3 SV )2 ξ32 + (σ4 IV )2 ξ42

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≥ M˜ ξ 2 for any (SH , IH , SV , IV ) ∈ D¯ k , ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ R4 , where D¯ k = [ 1k , k] × [ 1k , k] × [ 1k , k] × [ 1k , k] and k > 1 is a sufficiently large integer, then the condition (B.1) in Lemma 3.1 holds. : R4+ → R Now we will prove the condition (B.2). To this end, we define a C2 -function V as follows c4 α c2 bβ2 c4 α V (SH , IH , SV , IV ) = M − ln IH − c1 ln SH − c2 ln SV − c3 ln IV + IV − IV + IH m m μ+γ 1 (SH + IH + SV + IV )θ+1 , − ln SH − ln IH − ln SV + θ +1 where ci (i = 1, 2, 3, 4) are positive constants to be chosen later, θ > 0 is a constant satisfying (μ ∧ m) > θ2 (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) and M > 0 is a sufficiently large number satisfying the following condition −Mλ + C ≤ −2,

(3.1)

where σ2 λ = μ + γ + 2 (R0S − 1) > 0, 2 b2 β1 β2 μK A R0S = , σ12 σ22 σ32 σ42 μ+ μ+γ + m+ m+ 2 2 2 2

1 θ 2 2 2 2 (μK + A )(SH + IH + SV + IV ) − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) B= sup 2 2 (SH ,IH ,SV ,IV )∈R4+

×(SH + IH + SV + IV )θ+1 and

θ

1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) (SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) C= sup 4 2 (SH ,IH ,SV ,IV )∈R4+

σ2 σ2 σ2 +bβ1 IV + bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 . 2 2 2 (SH , IH , SV , IV ) is not only continuous, but also tends to ∞ as (SH , IH , Moreover, note that V SV , IV ) approaches the boundary of R4+ and as (SH , IH , SV , IV ) → ∞, where · denotes the Euclidean norm of a point in R4+ . So it must be lower bounded and achieve this lower bound at a point (SH0 , IH0 , SV0 , IV0 ) in the interior of R4+ . Then, we define a C2 -function V : R4+ → R+ ∪ {0} as c4 α c2 bβ2 c4 α IH V (SH , IH , SV , IV ) = M − ln IH − c1 ln SH − c2 ln SV − c3 ln IV + IV − IV + m m μ+γ 1 (SH0 , IH0 , SV0 , IV0 ) (SH +IH +SV + IV )θ+1 −V − ln SH −ln IH − ln SV + θ +1


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:= MV1 (SH , IH , SV , IV ) + V2 (SH ) + V3 (IH ) + V4 (SV ) + V5 (SH , IH , SV , IV ), where (SH , IH , SV , IV ) ∈ ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k) and k > 1 is a sufficiently large 2 bβ2 integer, V1 (SH , IH , SV , IV ) = − ln IH − c1 ln SH − c2 ln SV − c3 ln IV + cm4 α IV − cm4 α IV + cμ+ I , γ H 1 V2 (SH ) = − ln SH , V3 (IH ) = − ln IH , V4 (SV ) = − ln SV , V5 (SH , IH , SV , IV ) = θ+1 (SH + IH + (SH0 , IH0 , SV0 , IV0 ), SV + IV )θ+1 − V b2 β1 β2 μK A c1 = , σ2 σ2 σ2 μ+ 1 2 m+ 3 m+ 4 2 2 2 b2 β1 β2 μK A c2 = σ12 σ32 2 μ+ m+ 2 m+ 2

σ42 2

,

b2 β1 β2 μK A c3 = , σ2 σ2 σ2 μ+ 1 m+ 3 m+ 4 2 2 2 2 b2 β1 β2 μK A c4 = . σ12 σ32 σ42 μ+ m+ m+ 2 2 2

(3.2)

(3.3)

(3.4)

(3.5)

Applying Itô’s formula to V1 (SH , IH , SV , IV ), we have c1 μK c2 A σ22 c1 bβ1 IV σ12 bβ1 SH IV − − LV1 = − + μ+γ + + + c1 μ + (1 + αIV )IH 2 SH 1 + αIV 2 SV 2 2 c3 bβ2 IH SV σ σ c4 αbβ2 +c2 bβ2 IH + c2 m + 3 − + c3 m + 4 + IH SV −c4 (1 + αIV ) 2 IV 2 m c2 b2 β1 β2 SH IV c4 αbβ2 +c4 − IH SV + c4 αIV + − c2 bβ2 IH m (μ + γ )(1 + αIV ) c1 bβ1 IV bβ1 SH IV c1 μK c2 A c3 bβ2 IH SV = − − − − − c4 (1 + αIV ) + (1 + αIV )IH SH SV IV 1 + αIV σ22 σ12 σ32 σ42 + c1 μ + + c2 m + + c3 m + + c4 + c4 αIV + μ+γ + 2 2 2 2 c2 b2 β1 β2 SH IV + (μ + γ )(1 + αIV ) σ2 σ2 σ2 ≤ −5 5 c1 c2 c3 c4 b2 β1 β2 μK A + μ + γ + 2 + c1 μ + 1 + c2 m + 3 2 2 2 σ42 c1 bβ1 IV c2 b2 β1 β2 SH IV + c4 + +c3 m + + + c4 αIV 2 1 + αIV (μ + γ )(1 + αIV ) σ22 c1 bβ1 IV b2 β1 β2 μK A μ + + = − γ + + 2 1 + αIV σ12 σ32 σ42 μ+ m+ m+ 2 2 2

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c2 b2 β1 β2 SH IV + + c4 αIV (μ + γ )(1 + αIV ) σ2 c1 bβ1 IV c2 b2 β1 β2 SH IV = − μ + γ + 2 (R0S − 1) + + + c4 αIV 2 1 + αIV (μ + γ )(1 + αIV ) c1 bβ1 IV c2 b2 β1 β2 SH IV := −λ + + + c4 αIV , 1 + αIV (μ + γ )(1 + αIV )

(3.6)

where in the third equality, we have used Eqs. (3.2), (3.3), (3.4), (3.5) and σ2 λ = μ + γ + 2 (R0S − 1) > 0. 2 Similarly bβ1 IV σ2 μK + +μ+ 1 SH 1 + αIV 2 σ12 μK ≤− + bβ1 IV + μ + , SH 2

LV2 = −

(3.7)

LV3 = −

σ2 bβ1 SH IV +μ+γ + 2 , (1 + αIV )IH 2

(3.8)

LV4 = −

σ2 A + bβ2 IH + m + 3 SV 2

(3.9)

and LV5 = (SH + IH + SV + IV )θ [μK + A − μSH − (μ + γ )IH − m(SV + IV ] θ + (SH + IH + SV + IV )θ−1 × (σ12 SH2 + σ22 IH2 + σ32 SV2 + σ42 IV2 ) 2 ≤ (SH + IH + SV + IV )θ [μK + A − (μ ∧ m)(SH + IH + SV + IV )] θ + (SH + IH + SV + IV )θ+1 × (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 θ = (μK + A )(SH + IH + SV + IV )θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 ×(SH + IH + SV + IV )θ+1 θ 2 1 2 2 2 (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) (SH + IH + SV + IV )θ+1 ≤ B− 2 2 θ 2 1 2 2 2 (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) (SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ), ≤ B− 2 2 where

(3.10)

1 θ 2 2 2 2 (μK + A )(SH + IH + SV + IV ) − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) B= sup 2 2 (SH ,IH ,SV ,IV )∈R4+

×(SH + IH + SV + IV )θ+1 . θ


8901

Therefore, in view of Eqs. (3.6), (3.7), (3.8), (3.9) and (3.10), we obtain M c1 bβ1 IV M c2 b2 β1 β2 SH IV μK bβ1 SH IV A LV ≤ −M λ + + + Mc4 αIV − − − 1 + αIV (μ + γ )(1 + αIV ) SH (1 + αIV )IH SV θ 2 1 − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) (SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV 2 2 σ2 σ2 σ2 +bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 . 2 2 2 Define a bounded closed set as follows

1 1 1 1 , D = (SH , IH , SV , IV ) ∈ R4+ : ≤ SH ≤ , 3 ≤ IH ≤ 3 , ≤ SV ≤ , ≤ IV ≤ where 0 < < 1 is a sufficiently small number. In the set R4+ \ D , we can choose sufficiently small such that the following conditions hold
3 , D5 = (SH , IH , SV , IV ) ∈ R4+ : SH >

1 1 , D8 = (SH , IH , SV , IV ) ∈ R4+ : IV > . D7 = (SH , IH , SV , IV ) ∈ R4+ : SV > Clearly, R4+ \ D = D c = D1 ∪ D2 ∪ D3 ∪ D4 ∪ D5 ∪ D6 ∪ D7 ∪ D8 . Next, we will prove that LV (SH , IH , SV , IV ) ≤ −1 for any (SH , IH , SV , IV ) ∈ D c , which is equivalent to proving it on the above eight domains, respectively. θ+IVθ +1 θ θ+1 Case 1. If (SH , IH , SV , IV ) ∈ D1 , due to SH IV < IV ≤ θ+1 = θ+1 + θ+1 IV , we have Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV μK + + + Mc4 αIV SH 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 2

LV ≤ −

×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m +

σ12 σ2 σ2 + 2 + 3 2 2 2

Mc1 bβ1 IV Mc2 b2 β1 β2 θ μK + + + Mc4 αIV SH 1 + αIV (μ + γ )(θ + 1)(1 + αIV ) Mc2 b2 β1 β2 1 θ (μ ∧ m) − + − (μ + γ )(θ + 1) 4 2 1 θ 2 2 2 2 2 θ+1 2 2 2 (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) ×(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) IV − 4 2

≤−

(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) σ12 σ2 σ2 + 2 + 3 2 2 2 Mc1 bβ1 IV Mc2 b2 β1 β2 θ μK ≤− + + + Mc4 αIV SH 1 + αIV (μ + γ )(θ + 1)(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 4 2 +bβ1 IV + bβ2 IH + B + 2μ + γ + m +

×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m + μK +D SH μK +D ≤− ≤ −1,

σ12 σ2 σ2 + 2 + 3 2 2 2

≤−

which follows from Eqs. (3.11) and (3.12) and Mc1 bβ1 IV Mc2 b2 β1 β2 θ D= sup + + Mc4 αIV 1 + αIV (μ + γ )(θ + 1)(1 + αIV ) (SH ,IH ,SV ,IV )∈R4+

(3.20)


−

1 θ (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 4 2

×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m +

8903

σ12 σ2 σ2 + 2 + 3 . 2 2 2 (3.21)

Hence LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D1 . Case 2. If (SH , IH , SV , IV ) ∈ D2 , we obtain Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV bβ1 SH IV + + + Mc4 αIV (1 + αIV )IH 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 2

LV ≤ −

×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m + bβ1 SH IV +E (1 + αIV )IH bβ1 ≤− +E (1 + α ) ≤ −1,

σ12 σ2 σ2 + 2 + 3 2 2 2

≤−

(3.22)

which follows from Eq. (3.13) and Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV E = sup + + Mc4 αIV 1 + αIV (μ + γ )(1 + αIV ) (SH ,IH ,SV ,IV )∈R4+ 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) 2 2 ×(SHθ+1

+

IHθ+1

+

SVθ+1

+

IVθ+1 )

σ12 σ22 σ32 . + bβ1 IV + bβ2 IH + B + 2μ + γ + m + + + 2 2 2 (3.23)

Therefore LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D2 . Case 3. If (SH , IH , SV , IV ) ∈ D3 , we have Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV A + + + Mc4 αIV SV 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 2

LV ≤ −

×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m + ≤−

A +F SV

σ12 σ2 σ2 + 2 + 3 2 2 2

8904


A +F ≤ −1, ≤−

(3.24)

which follows from Eq. (3.14) and Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV F = sup + + Mc4 αIV 1 + αIV (μ + γ )(1 + αIV ) (SH ,IH ,SV ,IV )∈R4+ 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) 2 2 ×(SHθ+1

+

IHθ+1

+

SVθ+1

+

IVθ+1 )

σ12 σ22 σ32 . + bβ1 IV + bβ2 IH + B + 2μ + γ + m + + + 2 2 2 (3.25)

Hence LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D3 . Case 4. If (SH , IH , SV , IV ) ∈ D4 , due to SH IV < SH ≤

θ+SHθ +1 θ+1

=

θ θ+1

+

S θ+1 , θ+1 H

we get

Mc2 b2 β1 β2 SH IV + M c4 αIV LV ≤ −Mλ + Mc1 bβ1 IV + μ+γ 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) 2 2 σ2 σ2 σ2 ×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 2 2 2 Mc2 b2 β1 β2 Mc2 b2 β1 β2 θ ≤ −Mλ + Mc1 bβ1 IV + + Mc4 αIV + (μ + γ )(θ + 1) (μ + γ )(θ + 1) 1 θ − (μ ∧ m) − 4 2 1 θ 2 2 2 2 2 θ+1 2 2 2 (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) ×(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) SH − 4 2 (SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) +bβ1 IV + bβ2 IH + B + 2μ + γ + m +

σ12 σ2 σ2 + 2 + 3 2 2 2

Mc2 b2 β1 β2 θ ≤ −Mλ + Mc1 bβ1 IV + + Mc4 αIV (μ + γ )(θ + 1) θ 1 − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 4 2 ×(SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m + Mc2 b2 β1 β2 θ + Mc4 αIV + C (μ + γ )(θ + 1) Mc2 b2 β1 β2 θ ≤ −Mλ + Mc1 bβ1 + + Mc4 α + C (μ + γ )(θ + 1) ≤ −Mλ + Mc1 bβ1 IV +

σ12 σ2 σ2 + 2 + 3 2 2 2


≤ −1,

8905

(3.26)

which follows from Eq. (3.15) and 1 θ − (μ ∧ m) − C= sup 4 2 (SH ,IH ,SV ,IV )∈R4+ (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) (SHθ+1 + IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH

σ2 σ2 σ2 +B + 2μ + γ + m + 1 + 2 + 3 . 2 2 2 Consequently LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D4 . Case 5. If (SH , IH , SV , IV ) ∈ D5 , we derive 1 θ LV ≤ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) SHθ+1 4 2 Mc1 bβ1 IV 1 θ 2 − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) SHθ+1 + 4 2 1 + αIV Mc2 b2 β1 β2 SH IV 1 θ 2 + + Mc4 αIV − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) (μ + γ )(1 + αIV ) 2 2 (IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV σ2 σ2 σ2 +bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 2 2 2 1 θ 2 ≤ − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) SHθ+1 + G 4 2 1 1 θ 2 2 2 2 ≤ − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) θ+1 + G 4 2 ≤ −1,

(3.27)

which follows from Eq. (3.16) and 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) G= sup 4 2 (SH ,IH ,SV ,IV )∈R4+ Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV SHθ+1 + + + Mc4 αIV 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) 2 2 (IHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ

σ2 σ2 σ2 +γ + m + 1 + 2 + 3 . 2 2 2 Thus LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D5 .

(3.28)

8906


Case 6. If (SH , IH , SV , IV ) ∈ D6 , we have 1 θ LV ≤ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) IHθ+1 4 2 Mc1 bβ1 IV 1 θ 2 − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) IHθ+1 + 4 2 1 + αIV 2 Mc2 b β1 β2 SH IV 1 θ (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) + + Mc4 αIV − (μ + γ )(1 + αIV ) 2 2 (SHθ+1 + SVθ+1 + IVθ+1 ) + bβ1 IV σ2 σ2 σ2 +bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 2 2 2 1 θ 2 ≤ − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) IHθ+1 + H 4 2 1 1 θ 2 2 2 2 ≤ − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) 3θ+3 + H 4 2 ≤ −1,

(3.29)

which follows from Eq. (3.17) and 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) H= sup 4 2 (SH ,IH ,SV ,IV )∈R4+ Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV + + Mc4 αIV IHθ+1 + 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) (SHθ+1 + SVθ+1 + IVθ+1 ) 2 2 +bβ1 IV + bβ2 IH + B + 2μ

σ2 σ2 σ2 +γ + m + 1 + 2 + 3 . 2 2 2 So LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D6 . Case 7. If (SH , IH , SV , IV ) ∈ D7 , we get 1 θ LV ≤ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) SVθ+1 4 2 Mc1 bβ1 IV 1 θ 2 − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) SVθ+1 + 4 2 1 + αIV 2 Mc2 b β1 β2 SH IV 1 θ (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) + + Mc4 αIV − (μ + γ )(1 + αIV ) 2 2 (SHθ+1 + IHθ+1 + IVθ+1 ) + bβ1 IV σ2 σ2 σ2 +bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 2 2 2 1 θ 2 ≤ − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) SVθ+1 + J 4 2

(3.30)


1 1 θ (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) θ+1 + J 4 2 ≤ −1,

8907

≤−

which follows from Eq. (3.18) and Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV J= sup + + Mc4 αIV 1 + αIV (μ + γ )(1 + αIV ) (SH ,IH ,SV ,IV )∈R4+ 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) 2 2 ×(SHθ+1 + IHθ+1 + IVθ+1 ) + bβ1 IV + bβ2 IH + B + 2μ + γ + m +

(3.31)

σ12 σ2 σ2 + 2 + 3 . (3.32) 2 2 2

Therefore LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D7 . Case 8. If (SH , IH , SV , IV ) ∈ D8 , we obtain 1 θ LV ≤ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) IVθ+1 4 2 Mc1 bβ1 IV 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) IVθ+1 + 4 2 1 + αIV 2 Mc2 b β1 β2 SH IV 1 θ (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) + + Mc4 αIV − (μ + γ )(1 + αIV ) 2 2 (SHθ+1 + IHθ+1 + SVθ+1 ) + bβ1 IV σ2 σ2 σ2 +bβ2 IH + B + 2μ + γ + m + 1 + 2 + 3 2 2 2 1 θ 2 ≤ − (μ ∧ m) − (σ1 ∨ σ22 ∨ σ32 ∨ σ42 ) IVθ+1 + Q 4 2 1 1 θ 2 2 2 2 ≤ − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) θ+1 + Q 4 2 ≤ −1,

(3.33)

which follows from Eq. (3.19) and 1 θ 2 2 2 2 − (μ ∧ m) − (σ1 ∨ σ2 ∨ σ3 ∨ σ4 ) IVθ+1 Q= sup 4 2 (SH ,IH ,SV ,IV )∈R4+ Mc1 bβ1 IV Mc2 b2 β1 β2 SH IV + + Mc4 αIV 1 + αIV (μ + γ )(1 + αIV ) 1 θ − (μ ∧ m) − (σ12 ∨ σ22 ∨ σ32 ∨ σ42 ) (SHθ+1 + IHθ+1 + SVθ+1 ) 2 2 +bβ1 IV + bβ2 IH + B + 2μ

σ12 σ22 σ32 . +γ + m + + + 2 2 2 +

(3.34)

8908


Consequently LV ≤ −1 for any (SH , IH , SV , IV ) ∈ D8 . Clearly, from Eqs. (3.20), (3.22), (3.24), (3.26), (3.27), (3.29), (3.31) and (3.33), we can obtain that for a sufficiently small , LV (SH , IH , SV , IV ) ≤ −1 for all (SH , IH , SV , IV ) ∈ R4+ \ D . Therefore, the condition (B.2) in Lemma 3.1 is satisfied. In view of Lemma 3.1, we obtain that system (1.3) is ergodic and admits a unique stationary distribution. This completes the proof. 4. Extinction of the diseases In this section, we will establish sufficient conditions for extinction of the diseases. We establish the following theorem. Theorem 4.1. Let (SH (t), IH (t), SV (t), IV (t)) be the solution of system (1.3) with any initial σ2 σ2 value (SH (0), IH (0), SV (0), IV (0)) ∈ R4+ . If μ > 21 and m > 23 , then for almost ω ∈ , we have √ bβ2 A 1 R0 ≤ ν a.s., lim sup ln I (t ) + I (t ) H V m2 (μ + γ ) m t→∞ t √ √ 2 β1 β2 AKσ1 where ν = min{μ + γ , m}( R0 −1)1{√R0 ≤1} + max{μ + γ , m}( R0 −1)1{√R0 >1} + b m(μ+ γ) R0 1 2 + σ (μ + γ )( ( R0 (2μ−σ ) 2 − (2(σ2−2 + σ4−2 ))−1 . Especially, if ν < 0, then the dis2 ) 3 2m−σ32 1) eases IH and IV go to extinction with probability one, i.e., 1

1

lim IH (t ) = 0 and lim IV (t ) = 0 a.s.

t→∞

t→∞

Furthermore, the distributions of SH (t) and SV (t) converge weakly to the measures which have the densities π1 (x) = Q1 σ1−2 x

−2− 2μ2 − 2μK 2 σ

1

σ2

e

σ x 1

, π2 (y) = Q2 σ3−2 y

2μ +1 σ12

−2− 2m2 − σ

3

e

2A σ2y 3

, x, y ∈ (0, ∞ ), σ2

2m 2 +1

1 where Q1 = [σ1−2 ( 2μK ) ( 2μ + 1)]−1 and Q2 = [σ3−2 ( 2A3 ) σ3 ( 2m + 1)]−1 are two conσ12 σ32 ∞ ∞ stants satisfying 0 π1 (x)dx = 1 and 0 π2 (y)dy = 1, respectively.

Proof. Since for any initial value (SH (0), IH (0), SV (0), IV (0)) ∈ R4+ , the solution of system (1.3) is positive, we have d SH ≤ [μK − μSH ]d t + σ1 SH d B1 (t ) and dSV ≤ [A − mSV ]dt + σ3 SV dB3 (t ). Consider the following auxiliary logistic equations with random perturbations d X = [μK − μX ]d t + σ1 X d B1 (t )

(4.1)

and d Y = [A − mY ]dt + σ3Y dB3 (t ).

(4.2)


8909

It is easy to check that Eq. (4.1) has a stationary solution X˜ (t ) which has the density (see [23]) π1 (x) = Q1 σ1−2 x

−2− 2μ2 − 2μK 2 σ

1

σ x 1

e

, x ∈ (0, ∞ ),

∞ ( 2μ + 1)]−1 is a constant satisfying 0 π1 (x)dx = 1. Moreσ12 over, we can also check that Eq. (4.2) has a stationary solution Y˜ (t ) which has the density (see [23]) 2μ 2 +1

σ2

1 where Q1 = [σ1−2 ( 2μK ) σ1

π2 (y) = Q2 σ3−2 y

−2− 2m2 − σ

3

e

2A σ2y 3

, y ∈ (0, ∞ ),

2m ∞ σ 2 2 +1 where Q2 = [σ3−2 ( 2A3 ) σ3 ( 2m + 1)]−1 is a constant satisfying 0 π2 (y)dy = 1. Let X(t) be σ32 the solution of SDE Eq. (4.1) with the initial value X (0) = SH (0) > 0, then using the comparison theorem of 1-dimensional stochastic differential equation [24], we have SH (t) ≤ X(t) for any t ≥ 0 a.s. Similarly, let Y(t) be the solution of SDE Eq. (4.2) with the initial value Y (0) = SV (0) > 0, then applying the comparison theorem of 1-dimensional stochastic differential equation, we get SV (t) ≤ Y(t) for any t ≥ 0 a.s. On the other hand, we have ∞ I1 := x π1 (x )dx 0 ∞ 2μ − 2 −1 − 2μK 2 = Q1 σ1−2 x σ1 e σ1 x dx 0 ∞ 2μK − σ2μ2 −1 σ2μ2 −1 −t 2μK −2 1 = Q1 σ1 t 1 e dt σ12 σ12 0 2 2μ σ1 2μ σ12 = Q1 σ1−2 2μK σ12

2μ 2μK ( σ12 ) = σ12 ( 2μ + 1) σ2 1

2μK σ12 = σ12 2μ = K and

∞

x 2 π1 (x)dx ∞ 2μ 2μK − 2 − 2 −2 = Q1 σ1 x σ1 e σ1 x dx 0 ∞ 2μK − σ2μ2 ( σ2μ2 −1)−1 −t 2μK −2 1 t 1 = Q1 σ1 e dt σ12 σ12 0 2μ 2μK 1− σ 2 2μ 1 = Q1 σ1−2 −1 2 σ1 σ12

I2 :=

0

8910


=

2μK σ12

( 2μ − 1) σ2 2

1

( 2μ + 1) σ12 2 2μK 2 σ1 2μ = − 1 −1 2μ σ12 σ12 =

2μK 2 . 2μ − σ12

Therefore, we have ∞ 2 (x − K ) π1 (x)dx = 0

∞

(x 2 − 2K x + K 2 )π1 (x)dx

0

= I2 − 2K I1 + K 2 2μK 2 = − 2K 2 + K 2 2μ − σ12 =

σ12 K 2 . 2μ − σ12

Similarly, we have ∞ σ 2 A2 A 2 y− π2 (y )dy = 2 3 . m m (2m − σ32 ) 0 Moreover, by Theorem 1.4 of [25], p.27, we get that there exists a left eigenvector of M0 = ⎛ ⎞ bβ1 K 0 √ ⎜ ⎟ bβ A √ ⎝ bβ A μ + γ ⎠ corresponding to R0 , which is denoted as (ω1 , ω2 ) = ( m22 , R0 ), i.e., 2 0 m2 √ R0 (ω1 , ω2 ) = (ω1 , ω2 )M0 . Define a C2 -function V : R2+ → R+ by V (IH , IV ) = α1 IH + α2 IV , where α1 =

ω1 μ+γ

, α2 =

d (ln V ) = L(ln V )d t +

ω2 . m

Making use of Itô’s formula to ln V leads to

1 (α1 σ2 IH d B2 (t ) + α2 σ4 IV dB4 (t )), V

(4.3)

where

α1 bβ1 SH IV α2 α2 σ 2 I 2 α2 σ 2 I 2 L(ln V ) = − (μ + γ )IH + [bβ2 IH SV − mIV ] − 1 22 H − 2 42 V . V 1 + αIV V 2V 2V Moreover, we obtain 1 1 1 2 1 V 2 = α1 σ2 IH + α2 σ4 IV ≤ α12 σ22 IH2 + α22 σ42 IV2 + σ2 σ4 σ22 σ42 and

bβ1 SH IV 1 α1 − (μ + γ )IH + α2 [bβ2 IH SV − mIV ] V 1 + αIV

(4.4)


α1 bβ1 IV α2 bβ2 IH A SV − (SH − K ) + V (1 + αIV ) V m

bβ1 K IV bβ2 A 1 α1 + − (μ + γ )IH + α2 IH − mIV V 1 + αIV m α1 bβ1 IV α2 bβ2 IH A SV − ≤ (SH − K ) + V V m

bβ2 A 1 α1 [bβ1 K IV − (μ + γ )IH ] + α2 + IH − mIV V m α1 bβ1 IV α2 bβ2 IH A Y− ≤ (X − K ) + V V m

ω1 1 ω2 bβ2 A + IH − mIV [bβ1 K IV − (μ + γ )IH ] + V μ+γ m m α1 bβ1 α2 bβ2 A 1 Y − + (ω1 , ω2 )(M0 (IH , IV )T − (IH , IV )T ) ≤ |X − K | + α2 α1 m V α1 bβ1 α2 bβ2 A 1 √ Y − + ( R0 − 1)(ω1 IH + ω2 IV ) = |X − K | + α2 α1 m V α1 bβ1 α2 bβ2 A 1 √ Y − + ( R0 − 1)[(μ + γ )α1 IH + mα2 IV ] = |X − K | + α2 α1 m V √ √ ≤ min{μ + γ , m}( R0 − 1)1{ R0 ≤1} + max{μ + γ , m} √ α1 bβ1 α2 bβ2 A Y . ( R0 − 1)1{√R0 >1} + |X − K | + − α α m

8911

=

2

(4.5)

1

In view of Eqs. (4.4) and (4.5), we get √ L(ln V ) ≤ min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m} √ α1 bβ1 ( R0 − 1)1{√R0 >1} + |X − K | α2 α2 bβ2 A Y − − (2(σ2−2 + σ4−2 ))−1 . + α1 m According to Eq. (4.3), we get √ d (ln V ) ≤ min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m} √ α1 bβ1 ( R0 − 1)1{√R0 >1} + |X − K | α2 α2 bβ2 α1 σ2 IH α2 σ4 IV A −2 −2 −1 Y − (2(σ dt + + − + σ )) d B2 (t ) + dB4 (t ). (4.6) 2 4 α1 m V V Integrating Eq. (4.6) from 0 to t and then dividing by t on both sides, we have √ √ ln V (t ) ln V (0) ≤ + min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m}( R0 − 1)1{√R0 >1} t t α1 bβ1 t α2 bβ2 t A Y (s) − ds − (2(σ2−2 + σ4−2 ))−1 + |X (s) − K |ds + α2t 0 α1t 0 m

8912


1 t α1 σ2 IH (s) dB2 (s) t 0 V (s) 1 t α2 σ4 IV (s) + dB4 (s) t 0 V (s) √ √ ln V (0) = + min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m}( R0 − 1)1{√R0 >1} t α1 bβ1 t α2 bβ2 t A Y ds − (2(σ2−2 + σ4−2 ))−1 + |X (s) − K |ds + (s) − α2t 0 α1t 0 m M1 (t ) M2 (t ) + + , (4.7) t t t t IH (s) IV (s) where M1 (t ) = 0 α1 σV2(s) dB2 (s), M2 (t ) = 0 α2 σV4(s) dB4 (s) are local martingales t whose quadratic variations are M1 , M1 t = σ22 0 ( α1VIH(s)(s) )2 ds ≤ σ22t and M2 , M2 t = t σ42 0 ( αV2 IV(s)(s) )2 ds ≤ σ42t. Using the strong law of large numbers for local martingale [21] yields +

lim

t→∞

Mi (t ) = 0 a.s., i = 1, 2. t

(4.8)

∞ ∞ Since X(t) and Y(t) are ergodic and 0 x π1 (x )dx < ∞, 0 yπ2 (y )dy < ∞, we obtain ∞ ∞ 1 t 1 2 lim |X (s) − K |ds = |x − K |π1 (x)dx ≤ (x − K ) π1 (x )dx 2 (4.9) t→∞ t 0 0 0 and 1 t→∞ t lim

∞ t ∞ A 2 1 Y (s) − A ds = y − A π2 (y)dy ≤ 2. y − π (y ) dy 2 m m m 0 0 0

(4.10)

Taking the superior limit on both sides of Eq. (4.7) and combining with Eqs. (4.8), (4.9) and (4.10), we have lim sup t→∞

√ √ ln V (t ) ≤ min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m}( R0 − 1)1{√R0 >1} t K 2 σ12 A2 σ32 α1 bβ1 α2 bβ2 1 1 2 + 2 − (2(σ −2 + σ −2 ))−1 + 2 4 2 2 2 α2 α1 2μ − σ1 m (2m − σ3 ) √ √ = min{μ + γ , m}( R0 − 1)1{√R0 ≤1} + max{μ + γ , m}( R0 − 1)1{√R0 >1} 1 R0 b2 β1 β2 AK σ1 1 1 2 2 + + σ3 (μ + γ ) 2 2 m(μ + γ ) R0 (2μ − σ1 ) 2m − σ3 −(2(σ2−2 + σ4−2 ))−1 := ν a.s.,

which is the required assertion. In addition, if ν < 0, we can easily conclude that lim sup t→∞

ln IH (t ) ln IV (t ) < 0 and lim sup < 0 a.s., t t t→∞

which implies that limt→∞ IH (t ) = 0 and limt→∞ IV (t ) = 0 a.s. That is to say, the diseases IH and IV die out with probability one. This completes the proof.


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5. Concluding remarks and future directions In this paper, we study the dynamical behavior of a stochastic dengue epidemic model. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to model Eq. (1.3). Then we obtain sufficient conditions for extinction of the diseases. The existence of stationary distribution implies stochastic weak stability. Some interesting topics deserve further consideration. On the one hand, one may propose some more realistic but complex models, such as considering the effects of impulsive perturbations on system (1.3). On the other hand, our model is autonomous, it is interesting to investigate the nonautonomous case and study other interesting dynamical properties, such as the existence of positive periodic solutions. Moreover, in our model (1.2), we only introduce the white noise into it, one can also introduce the colored noise into model (1.2) and study the existence of an ergodic stationary distribution of the positive solutions to the considered model. We leave these problems for our future work. Acknowledgments This work was supported by the National Natural Science Foundation of P.R. China (No.11871473), Natural Science Foundation of Guangxi Province (No. 2016GXNSFBA380006), the Fundamental Research Funds for the Central Universities (No.15CX08011A), KY2016YB370 and 2016CSOBDP0001. References [1] World health organization. dengue and severe dengue, 2014, http:// www.searo.who.int/ thailand/ factsheets/ fs0008/ en/. [2] World health organization. health topics (dengue), 2013, http:// www.searo.who.int/ topics/ dengue/ en/ . [3] J.E.M. Pessanha, Risk assessment and risk maps using a simple dengue fever model, Dengue Bull. 36 (2012) 73–86. [4] W. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23 (1986) 187–204. [5] W. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987) 359–380. [6] V. Capasso, G. Serio, A generalization of the Kermack–Mckendrick deterministic epidemic model, Math. Biosci. 42 (1978) 43–61. [7] E.A. Newton, P. Reiter, A model of the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on dengue epidemics, Am. J. Trop. Med. Hyg. 47 (1992) 709–720. [8] D.A. Focks, D.G. Haile, E. Daniels, G.A. Mount, Dynamic life table model for aedes aegypti (diptera: Culicidae): analysis of the literature and model development, J. Med. Entomol. 30 (1993a) 1003–1017. [9] D.A. Focks, D.G. Haile, E. Daniels, G.A. Mount, J. Med. 30 (1993b) 1018–1028. [10] D.A. Focks, E. Daniels, D.G. Haile, J.E. Keesling, A simulation model of the epidemiology of urban dengue fever: literature analysis, model development, preliminary validation, and samples of simulation results, Am. J. Trop. Med. Hyg. 53 (1995) 489–506. [11] L. Cai, S. Guo, X. Li, M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos Solitons Fractals 42 (2009) 2297–2304. [12] Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A 392 (2013) 4916–4927. [13] T. Britton, A. Traoré, A stochastic vector-borne epidemic model: quasi-stationarity and extinction, Math. Biosci. 289 (2017) 89–95.

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