The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence

Accepted Manuscript The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence Buyu Wen, Zhidong Teng, Zhiming Li

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S0378-4371(18)30600-9 https://doi.org/10.1016/j.physa.2018.05.056 PHYSA 19596

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Physica A

Received date : 14 December 2017 Revised date : 23 March 2018 Please cite this article as: B. Wen, Z. Teng, Z. Li, The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence, Physica A (2018), https://doi.org/10.1016/j.physa.2018.05.056 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.

The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence Buyu Wen, Zhidong Teng∗, Zhiming Li College of Mathematics and Systems Science, Xinjiang University Urumqi 830046, People’s Republic of China

Abstract:In this paper, a stochastic periodic SIVS epidemic model with nonlinear incidence and vaccination is investigated. The threshold conditions on the existence of stochastic positive periodic solutions and the extinction of disease with probability one are established by constructing the new stochastic Lyapunov functions and using the new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic model. The numerical simulations are given to illustrate the main theoretical results and present some new interesting conjectures. Key words: Stochastic SIVS epidemic model; nonlinear incidence; vaccination; stochastic periodic solution; extinction; Lyapunov function.

1

Introduction Mathematical models describing realistic epidemiological systems and disease controls

for a long time have played an important role. It has been confirmed that vaccination is an important strategy for the control and elimination of infectious diseases. In recent years, many scholars have investigated various types of epidemic models with vaccination (See, for example [1–7]). However, in the real world, epidemic models are always affected by the environmental white noise which is an important component in an ecosystem (See, for example [8–11]). Consequently, it is essential to reveal that how the environmental white noise disturbs the epidemic models. Stochastic differential equation models play an important role in many kinds of branches of applied sciences including disease dynamics, as they can predict the future dynamics of their deterministic counterpart accurately. Therefore, lots of scholars ∗

Corresponding author. Tel.: +86 0991 8585505. E-mail address: zhidong [email protected] (Z. Teng)

1

have studied the stochastic epidemic models (See, for example [12–26]). Particularly, in [16], Zhao and Jiang investigated the following stochastic SIVS epidemic model with vaccination   dS(t) =[(1 − q)A − βS(t)I(t) − (µ + p)S(t) + γI(t) + εV (t)]dt      + σ1 S(t)dB1 (t),  dI(t) =[βS(t)I(t) − (µ + γ + α)I(t)]dt + σ2 I(t)dB2 (t),      dV (t) =[qA + pS(t) − (µ + ε)V (t)]dt + σ3 V (t)dB3 (t).

(1.1)

σ22 ˜ 0 = βS0 − with S0 = A[(1−q)µ+ε] , and further They gave threshold value R µ+γ+α 2(µ+γ+α) µ(µ+p+ε) ˜ proved that when R0 < 1, then the disease I in model (1.1) is extinct with probability ˜ 0 > 1, then model (1.1) is permanent in the mean with probability one. one, and when R

On the other hand, due to the seasonal variation, individual life cycle, hunting, food supplies, mating habits, harvesting and so on, the parameters will not remain constant, but exhibit a more or less periodicity. In order to derive sustained oscillatory, many authors have investigated the effect of seasonal variation and randomness (See, for example [27–33]). In [27], the authors studied the following stochastic θ-periodic SIVS epidemic model with vaccination    dS(t) =[(1 − q(t))A(t) − β(t)S(t)I(t) − (µ(t) + p(t))S(t) + γ(t)I(t)     + ε(t)V (t)]dt + σ1 (t)S(t)dB1 (t),

 dI(t) =[β(t)S(t)I(t) − (µ(t) + γ(t) + α(t))I(t)]dt + σ2 (t)I(t)dB2 (t),      dV (t) =[q(t)A(t) + p(t)S(t) − (µ(t) + ε(t))V (t)]dt + σ3 (t)V (t)dB3 (t).

They proposed threshold value hR0 iθ = c2 (t)q(t)A(t) − (µ(t) + γ(t) + α(t) +

1 θ

1 2 σ (t)) 2 2

Rθ 0

(1.2)

R0 (t)dt with R0 (t) = c1 (t)(1 − q(t))A(t) +

(See, Theorem 3.1 in [27]), and further proved

that when hR0 iθ > 0, then the disease I in model (1.2) is permanent in the mean with probability one, and model (1.2) has at least one stochastic positive θ-periodic solution. ˜ 0 and hR0 iθ , we see that they are completely difComparing two threshold values R ferent. Particularly, there exist two parameters c1 (t) and c2 (t) in hR0 iθ . Therefore, our first question is whether we can propose a new threshold value for model (1.2) which is ˜ 0 and does not include c1 (t) and c2 (t)? similar to R

As well as we know, in modelling the dynamics of epidemic systems the incidence rate is an important substance. In many practicality the nonlinear incidence is frequently used for achieving more exact results. We see that some deterministic and stochastic epidemic models with nonlinear incidence have been extensively studied (see, for example [13, 18–20, 23–26, 34–36]). Particularly, in [34–36] the autonomous epidemic models with

2

nonlinear incidence and vaccination are discussed. The dynamical behaviors, such as the stability of equilibria, extinction and persistence of the disease and the bifurcation phenomena, are established. However, we see that stochastic epidemic models with nonlinear incidence and vaccination are studied barely. Therefore, our second question is whether we also can establish a series similar results on the extinction, permanence and the existence of stochastic positive periodic solution for the periodic stochastic SIVS epidemic model with nonlinear incidence and vaccination? Therefore, in this paper we consider the following periodic stochastic SIVS epidemic model with nonlinear incidence and vaccination   dS(t) =[(1 − q(t))A(t) − β(t)f (S(t))g(I(t)) − (µ(t) + p(t))S(t) + γ(t)I(t)      + ε(t)V (t)]dt + σ1 (t)S(t)dB1 (t),  dI(t) =[β(t)f (S(t))g(I(t)) − (µ(t) + γ(t) + α(t))I(t)]dt + σ2 (t)I(t)dB2 (t),      dV (t) =[q(t)A(t) + p(t)S(t) − (µ(t) + ε(t))V (t)]dt + σ3 (t)V (t)dB3 (t).

(1.3)

˜ 0 . By using We will propose a threshold value for model (1.3) which is similar to R the Lyapunov function method, the Khasminskii’s boundary periodic Markov processes, theory of stochastic process, and introducing new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic model, we will establish a series of new threshold criteria on the existence of a stochastic positive periodic solution and the extinction of disease with probability one. This paper is organized as follows. In Section 2, we will give some useful lemmas and fundamental assumptions for nonlinear incidence functions. In Section 3, we will get the existence of stochastic positive periodic solution. In Section 4, the results on the extinction of disease with probability one are stated and proved. In Section 5, the numerical examples are presented. Finally, in Section 6, a conclusion is given.

2

Preliminaries

n Denote R+ = {x = (x1 , x2 , · · · , xn ) : xi > 0, i = 1, 2, · · · , n} and R+0 = [0, ∞). Let Rt f (t) be an integrable function on R+0 , we define hf it = 1t 0 f (s)ds for any t > 0. If f (t)

also is bounded on R+0 , we denote f u = supt≥0 f (t) and f l = inf t≥0 f (t).

Throughout this paper, we assume that model (1.3) is defined on a complete probability space (Ω, {Ft }t≥0 , P ) with a filtration {Ft }t≥0 satisfying the usual conditions. In model (1.3), S(t), I(t) and V (t) represent the numbers of susceptible, infectious and immune, respectively; A(t) is the recruitment of new numbers into the population;

3

q(t) is a fraction of vaccinated for new number; β(t) represents the disease transmission coefficient; µ(t) stands for the natural death rate of the population; γ(t) is the recovery of infectious; p(t) represents the proportional coefficient of vaccinated for the susceptible; ε(t) denotes the rate of losing their immunity for vaccinated individuals; α(t) denotes the disease-caused death rate of infectious; Bi (t) (i = 1, 2, 3) are independent standard Brownian motions with Bi (0) = 0, and σi denotes the intensity of Bi (t). For model (1.3), we introduce the following assumptions. (H1 ) All parameters A(t), q(t), p(t), β(t), µ(t), γ(t), α(t), ε(t), and σi (t) (i = 1, 2, 3) are nonnegative periodic continuous functions with common period ω > 0 defined on R+0 , 0 ≤ p(t) < 1, µl > 0, β l > 0 and Al > 0.

(H2 ) 2µl − ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 ) > 0. (H3 ) Functions f (S) and g(I) are nonnegative and continuously differentiable for S ≥ 0

and I ≥ 0, respectively.

g(I) I

is nonincreasing for I > 0. f (0) = g(0) = 0, and g 0 (0) > 0.

(H4 ) f 0 (S) is nondecreasing for S ≥ 0.

(H5 ) f 0 (S) is nonincreasing for S ≥ 0. 0 1 (H6 ) f 0 (S) ≥ 0 for all S ≥ 0, limS→∞ f (S) = 0 and maxS>0 { ff(S)S } < ∞. (S) (H7 ) supS>0 { f (S) } < ∞. S 0

(0) (H8 ) maxI>0 { gg(I) − I1 )} < ∞.

Remark 2.1. When f (S) =

Sm 1+ω1 S

with constants ω1 ≥ 0 and m ≥ 2, then (H3 ), (H4 )

and (H6 ) clearly are satisfied. When f (S) = ln(1 + ω1 S) with constant ω1 > 0, then (H3 ),

(H5 ) and (H6 ) are satisfied. When f (S) = 1+ωS 1 S with constant ω1 ≥ 0, then (H3 ), (H5 ) and (H7 ) are also satisfied. When g(I) = 1+ωI 2 I with constant ω2 ≥ 0, then (H8 ) clearly is satisfied.

Firstly, we introduce the definition of stochastic process and a criterion for the existence of stochastic periodic solution of stochastic differential equations which can be found in [37]. Definition 2.2. (See [37]) A stochastic process ξ(t) = ξ(t, ω) (−∞ < t < +∞) is said to be periodic with period θ if for every finite sequence of number t1 , t2 , · · · , tn the joint distribution of random variables ξ(t1 + h), · · · , ξ(tn + h) is independent of h, where h = kθ and k = ±1, ±2, · · · .

Consider the following stochastic differential equation dx(t) = b(t, x(t))dt +

k X r=1

4

σr (t, x(t))dBr (t),

(2.1)

where t ∈ R+ , x(t) = (x1 (t), x2 (t), · · · , xl (t))T , b(t, x) = (b1 (t, x), b2 (t, x), · · · , bl (t, x))T

and σr (t, x) = (σr1 (t, x), σr2 (t, x), · · · , σrl (t, x))T (r = 1, 2, · · · , k) are the l-dimensional vec-

tor function defined on R+ × Rl and B1 (t), B2 (t), · · · , Bk (t) are the standard Brownian motions. Lemma 2.3. (See [37]) Suppose that functions b(t, x) and σr (t, x) (r = 1, 2, . . . , k) in equation (2.1) are θ-periodic in t and satisfy that for any compact set D ⊂ Rl , |b(t, x) − b(t, y)| + |b(t, x)| +

k X

r=1 k X r=1

|σr (t, x) − σr (t, y)| ≤ B|x − y|, |σr (t, x)| ≤ B(1 + |x|),

for all (t, x) ∈ R+ × D, where B is a constant. Suppose further that there exist a compact

set U ⊂ Rl and a function V (t, x) ∈ C 1,2 (R+ × Rl , R+0 ) which is θ-periodic in t such that the following conditions are satisfied

inf V (t, x) → ∞ as H → ∞,

|x|>H

LV (t, x) =

l l X ∂V ∂V 1X ∂ 2V + bi (t, x) + ≤ −1, aij (t, x) ∂t ∂x 2 ∂x ∂x i i j i=1 i,j=1

(2.2)

(2.3)

for all (t, x) ∈ R+ × (Rl \ U ), where aij =

k X

σri (t, x)σrj (t, x),

i, j = 1, 2, . . . , l.

r=1

Then there exists a solution of equation (2.1) which is a stochastic θ-periodic process. Next, we introduce some useful basic lemmas for model (1.3) which will be used in the proof of the main results of this paper. On the existence of unique global positive solution with any positive initial condition for model (1.3), we have the following result. Lemma 2.4. Assume that (H1 ), (H3 ), and (H6 ) or (H7 ) hold. Then for any initial 3 value (S(0), I(0), V (0)) ∈ R+ solution (S(t), I(t), V (t)) of model (1.3) uniquely exists and 3 remains in R+ for any t ≥ 0 with probability one.

This lemma can be proved by using the following Lyapunov functions, V (S, I, V ) =

Z

a

S

(1 −

f (a) )dS + (I − 1 − ln I) + (V − 1 − ln V ) f (S) 5

l

l

where the constant a is chosen such that f (a) = βαu g+µ 0 (0) if assumption (H6 ) is satisfied, and S V (S, I, V ) = (S − a − a ln ) + (I − 1 − ln I) + (V − 1 − ln V ) a where the constant a = We here omit it.

αl +µl β u M ∗ g 0 (0)

with M ∗ = supS>0 { f (S) } if assumption (H7 ) is satisfied. S

Lemma 2.5. Assume that (H1 ) − (H3 ), and (H6 ) or (H7 ) hold. Let (S(t), I(t), V (t)) be 3 , then the solution of model (1.3) with initial value (S(0), I(0), V (0)) ∈ R+

1 (S(t) + I(t) + V (t)) = 0 a.s. t→∞ t lim

Lemma 2.6. Assume that (H1 ) − (H3 ), and (H6 ) or (H7 ) hold. Let (S(t), I(t), V (t))

3 , then for any be the solution of model (1.3) with initial value (S(0), I(0), V (0)) ∈ R+ continuous positive ω-periodic function c(t) defined for t ≥ 0,

1 lim t→∞ t

Z

t

0

Z 1 t c(s)σ1 (s)S(s)dB1 (s) = 0, lim c(s)σ2 (s)I(s)dB2 (s) = 0, t→∞ t 0 Z t 1 lim c(s)σ3 (s)V (s)dB3 (s) = 0 a.s. t→∞ t 0

Lemma 2.5 and Lemma 2.6 can be proved by using the similar methods which are given in [16]. We hence omit them. Lemma 2.7. The following equation (

S00 (t) = (1 − q(t))A(t) − (µ(t) + p(t))S0 (t) + ε(t)V0 (t), V00 (t) = q(t)A(t) + p(t)S0 (t) − (µ(t) + ε(t))V0 (t).

(2.4)

has a unique positive ω -periodic solution (S0 (t), V0 (t)). Proof. Equation (2.4) is equivalent to the following integral equation  R t+ω R t (µ(τ )+p(τ ))dτ  es [(1 − q(s))A(s) + ε(s)V0 (s)]ds  t  Rω ,  S0 (t) = 1 − e− 0 (µ(τ )+p(τ ))dτ R R t+ω t (µ(τ )+ε(τ ))dτ  es (q(s)A(s) + p(s)S0 (s))ds    V0 (t) = t R . − 0ω (µ(τ )+ε(τ ))dτ 1−e

(2.5)

We only need to show that equation (2.5) has a unique positive periodic solution (S0 (t), V0 (t)). Define two sets of functions as follows 2 X ={b ∈ C(R+0 ; R+ ), b(t) = b(t + ω), b(t) = (S0 (t), V0 (t))},

D ={b ∈ X, 0 < m1 ≤ S0 (t) ≤ M1 , 0 < m2 ≤ V0 (t) ≤ M2 }. 6

We also define a map: A = (A1 , A2 ) : D → X, where for any b(t) = (S0 (t), V0 (t)) ∈ D, (A1 b)(t) = (A2 b)(t) =

R t+ω t

Rt

e

s (µ(τ )+p(τ ))dτ

[(1 − q(s))A(s) + ε(s)V0 (s)]ds Rω

1 − e− 0 (µ(τ )+p(τ ))dτ R t+ω R t (µ(τ )+ε(τ ))dτ (q(s)A(s) + p(s)S0 (s))ds es t 1 − e−

Rω 0

(µ(τ )+ε(τ ))dτ

,

.

We will find positive constants m1 , M1 , m2 and M2 such that A(D) ⊂ D. Firstly, (A1 b)(t) ≥ ≥

R t+ω t

e

Rt

s (µ(τ )+p(τ ))dτ

−

Rω

[(1 − q(s))A(s)]ds

1 − e 0 (µ(τ )+p(τ ))dτ R t+ω R t (1 − q)A l t e s (µ(τ )+p(τ ))dτ [µ(τ ) + p(τ )]ds R [ ] − 0ω (µ(τ )+p(τ ))dτ (µ + p)

=[

(1 − q)A l ] := m1 . (µ + p)

Similarly, we have (A2 b)(t) ≥ [

(2.6)

1−e

qA l ] := m2 . (µ + ε)

(2.7)

On the other hand, we have (A1 b)(t) ≤

R t+ω t

e

Rt

s (µ(τ )+p(τ ))dτ

[(1 − q(s))A(s) + ε(s)M2 ]ds

1 − e−

Rω 0

(µ(τ )+p(τ ))dτ

R t+ω R t (µ(τ )+p(τ ))dτ es [µ(τ ) + p(τ )]ds ε (1 − q)A u Rω ] + M2 [ ]u ) t ≤([ − (µ(τ )+p(τ ))dτ (µ + p) (µ + p) 1−e 0 (1 − q)A u ε =[ ] + M2 [ ]u := M1 . (µ + p) (µ + p)

(2.8)

Similarly, we also have (A1 b)(t) ≤ [

qA u ε ] + M1 [ ]u := M2 . (µ + ε) (µ + ε)

(2.9)

From (2.8) and (2.9), we can get M1 =

qA u ε [ (1−q)A ]u + [ (µ+p) ]u [ µ+ε ] µ+p

1−

ε ε [ (µ+ε) ]u [ (µ+p) ]u

,

M2 = [

qA u ε ] + M1 [ ]u . µ+ε (µ + ε)

Finally, from (2.6)-(2.9) it follows that A(D) ⊂ D. On the other hand, it is easy to see that map A is completely continuous. In view of Leray-Schsuder’s fixed point theorem, map A has a fixed point (S0 (t), V0 (t)) in D. This shows that equation (2.4) has a positive periodic solution (S0 (t), V0 (t)).

7

Next, we will prove the uniqueness of periodic solution (S0 (t), V0 (t)). Let (S¯0 (t), V¯0 (t)) be another positive periodic solution of equation (2.4), and let also z1 (t) = S0 (t) − S¯0 (t) and z2 (t) = V0 (t) − V¯0 (t). We have z10 (t) = −(µ(t) + p(t))z1 (t) + ε(t)z2 (t), z20 (t) = p(t)z1 (t) − (µ(t) + ε(t))z2 (t).

(2.10)

Define function V (t) = |z1 (t)| + |z2 (t)|. Calculating the upper right derivative of V (t) along the solution of (2.10) leads to

D+ V (t) ≤sign(z1 (t))[−(µ(t) + p(t))z1 (t) + ε(t)z2 (t)] + sign(z2 (t))[p(t)z1 (t) − (µ(t) + ε(t))z2 (t)] ≤ − µ(t)(|z1 (t)| + |z2 (t)|) = −µ(t)V (t). It follows that limt→∞ V (t) = 0. Hence, we can obtain z1 (t) = z2 (t) = 0. Thus, equation (2.4) has unique positive periodic solution (S0 (t), V0 (t)). This completes the proof. Lemma 2.8. The following equation (

c01 (t) = (µ(t) + p(t))c1 (t) − β(t)f 0 (S0 (t))g 0 (0) − p(t)c2 (t), c02 (t) = (µ(t) + ε(t))c2 (t) − ε(t)c1 (t).

(2.11)

has a unique positive ω-periodic solution (c1 (t), c2 (t)). Proof. Equation (2.11) is equivalent to the following integral system  R t+ω R t (µ(τ )+p(τ ))dτ  es (β(s)f 0 (S0 (s))g 0 (0) + p(s)c2 (s))ds  t  Rω ,  c1 (t) = 1 − e− 0 (µ(τ )+p(τ ))dτ R R t+ω t (µ(τ )+ε(τ ))dτ  es ε(s)c1 (s)ds    c2 (t) = t R . − 0ω (µ(τ )+ε(τ ))dτ 1−e

(2.12)

We only need to show that system (2.12) has a unique positive periodic solution (c1 (t), c2 (t)). A similar argument as in the proof of Lemma 2.7 imlies that system (2.12) has a positive periodic solution (c1 (t), c2 (t)). Next, we will prove the uniqueness of the periodic solution. Let (¯ c1 (t), c¯2 (t))T be another positive periodic solution of equation (2.11). Take z1 (t) = c1 (t) − c¯1 (t) and z2 (t) = c2 (t) − c¯2 (t). We have

z10 (t) = −(µ(t) + p(t))z1 (t) + p(t)z2 (t), z20 (t) = ε(t)z1 (t) − (µ(t) + ε(t))z2 (t). 0

Consequently, z10 (t) − z2 (t) = (µ(t) + ε(t) + p(t))(z2 (t) − z1 (t)). Hence, z2 (t) − z1 (t) = 0.

Thus, z10 (t) = µ(t)z1 (t) and z20 (t) = µ(t)z2 (t). From this, we can obtain z1 (t) = 0 and 8

z2 (t) = 0. Hence, c¯1 (t) = c1 (t) and c¯2 (t) = c2 (t). This completes the proof. Lemma 2.9. Assume that (H1 ), (H3 ), and (H6 ) or (H7 ) hold. Let (S(t), I(t), V (t)) be a 3 solution of model (1.3) with initial value (S(0), I(0), V (0)) ∈ R+ , then we have

Z

t

β(s)f 0 (S0 (s))g 0 (0)(S(s) − S0 (s))ds 0 Z t =Φ(t) − (β(t)f 0 (S0 (s))g 0 (0) + c1 (s)α(s) + c2 (s)p(s) − c1 (s)p(s))I(t)ds Z t 0 Z t + c1 (s)σ1 (s)S(s)dB1 (s) + c1 (s)σ2 (s)I(s)dB2 (s) 0 0 Z t + c2 (s)σ3 (s)V (s)dB3 (s), 0

where Φ(t) =c1 (t)S0 (t) + c2 (t)V0 (t) − c1 (t)(S(t) + I(t)) − c2 (t)V (t) − c1 (0)S0 (0) − c2 (0)V0 (0) + c1 (0)(S(0) + I(0)) + c2 (0)V (0), and (S0 (t), V0 (t)) is given in Lemma 2.7, (c1 (t), c2 (t)) is given in Lemma 2.8. Proof. An application of Itˆ o0 s formula gives d[c1 (t)(S + I) + c2 (t)V ] =[c1 (t)(1 − q(t))A(t) + c2 (t)q(t)A(t) − β(t)f 0 (S0 (s))g 0 (0)S

− (β(t)f 0 (S0 (s))g 0 (0) + c1 (t)α(t) + c2 (t)p(t) − c1 (t)p(t))I]dt

+ c1 (t)σ1 (t)SdB1 (t) + c1 (t)σ2 (t)IdB2 (t) + c2 (t)σ3 (t)V dB3 (t) and d[c1 (t)S0 (t) + c2 (t)V0 (t)] =c1 (t)(1 − q(t))A(t) + c2 (t)q(t)A(t) − β(t)f 0 (S0 (t))g 0 (0)S0 (t). Therefore, we have d[c1 (t)S0 (t) + c2 (t)V0 (t) − c1 (t)(S + I) − c2 (t)V ]

=[β(t)f 0 (S0 (t))g 0 (0)(S − S0 (t))

+ (β(t)f 0 (S0 (t))g 0 (0) + c1 (t)α(t) + c2 (t)p(t) − c1 (t)p(t))I]dt − c1 (t)σ1 (t)SdB1 (t) − c1 (t)σ2 (t)IdB2 (t) − c2 (t)σ3 (t)V dB3 (t).

9

(2.13)

Integrating from 0 to t leads to c1 (t)S0 (t) + c2 (t)V0 (t) − c1 (t)(S(t) + I(t)) − c2 (t)V (t) − c1 (0)S0 (0) − c2 (0)V0 (0) + c1 (0)(S(0) + I(0)) + c2 (0)V (0) Z t β(s)f 0 (S0 (s))g 0 (0)(S − S0 (s))ds = 0 Z t + (β(s)f 0 (S0 (s))g 0 (0) + c1 (s)α(s) + c2 (s)p(s) − c1 (s)p(s))Ids 0 Z t Z t Z t − c1 (s)σ1 (s)SdB1 (s) − c1 (s)σ2 (s)IdB2 (s) − c2 (s)σ3 (s)V dB3 (s). 0

0

0

From this, we can obtain the conclusion of the lemma. This completes the proof.

3

Existence of stochastic ω-periodic solution Define the function as follows 0 σ22 (t) ˜ 0 (t) = β(t)f (S0 (t))g (0) − R . µ(t) + γ(t) + α(t) 2(µ(t) + γ(t) + α(t))

We have the following result. ˜ 0 iω > 1, Theorem 3.1. Suppose that (H1 ) − (H4 ), (H6 ) or (H7 ), and (H8 ) hold. If hR then model (1.3) has at least one positive stochastic ω-periodic solution. ˜ 0 iθ > 1 is equivalent to hR0 iθ > 0, where Proof. It is clear that hR 1 R0 (t) = β(t)f (S0 (t))g 0 (0) − (µ(t) + γ(t) + α(t) + σ22 (t)). 2

(3.1)

Take two constants θ > 0 and M > 0 such that θ µl − ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 ) > 0, 2

−M hR0 iω + f1u + f3u ≤ −2,

(3.2)

where functions f1 (x) and f3 (x) will be determined later. Define the auxiliary function V (t, x) in the following form V (t, S, I, V ) =

1 (S + I + V )θ+1 − θ+1

Z

1

S

1 ds − ln V + M [− f (S)

− c1 (t)(S − S0 (t) + I) − c2 (t)(V − V0 (t)) − w(t)]

10

Z

1

I

g 0 (0) dI g(I)

if assumption (H6 ) is satisfied, and 1 (S + I + V )θ+1 − ln S − ln V + M [− V (t, S, I, V ) = θ+1

Z

1

I

g 0 (0) dI g(I)

− c1 (t)(S − S0 (t) + I) − c2 (t)(V − V0 (t)) − w(t)] if assumption (H7 ) is satisfied, where S0 (t) and V0 (t) are given in Lemma 2.7, c1 (t) and c1 (t) are given in Lemma 2.8, and w(t) is a function defined on R+0 satisfying w0 (t) = hR0 iω − R0 (t) and w(0) = 0. It is easy to verify that w(t) is a ω-periodic function on R+0 .

Therefore, U (t, S, I, V ) is ω-periodic in t. We here only give the proof under assumption (H6 ). The proof under assumption (H7 ) is similar. By (H6 ), without loss of generality, we can assume U (S, I, V ) ≥ P (S) +

Q(I) + R(V ) for any S ≥ 1, I > 0 and V > 0, where

1 S θ+1 − M3 S − M cu1 S + M cl1 S0l , θ+1 1 θ+1 Q(I) = I − M ln I − M M2 I − M cu1 I − M wu , θ+1 1 R(V ) = V θ+1 − ln V − M cu2 V + M cl2 V0l θ+1 P (S) =

with M3 = supS≥1

1 . f (S)

Thus, we can easily obtain that U (S, I, V ) satisfies

inf{U (t, S, I, V ) : t ≥ 0, max{S, I, V } ≥ H} → ∞ as H → ∞. On the other hand, by (H3 ), since limS→0+ and limV →0+ ln V = −∞, we further have

RS

1 ds 1 f (S)

= −∞, limI→0+

RI

1 dI 1 g(I)

= −∞

inf{U (t, S, I, V ) : t ≥ 0, min{S, I, V } ≤ h} → ∞ as h → 0+ . Therefore, U (t, S, I, V ) satisfies condition (2.2) in Lemma 2.3. Next, We prove that condition (2.3) in Lemma 2.3 is satisfied. We will find a closed 3 set U ∈ R+ such that 3 LV (t, S, I, V ) ≤ −1 for all t ≥ 0, (S, I, V ) ∈ R+ \U.

Denote Z S 1 1 θ+1 V1 = (S + I + V ) , V2 = − ds, V3 = − ln V, θ+1 1 f (S) Z I 0 g (0) V4 = − dI − c1 (t)(S − S0 (t) + I) − c2 (t)(V − V0 (t)) − w(t). 1 g(I) 11

We have LV = LV1 + LV2 + LV3 + M LV4 . Using Itˆ o0 s formula, we have LV1 =(S + I + V )θ [A(t) − (µ(t) + α(t))I − µ(t)S − µ(t)V ] 1 + θ(S + I + V )θ−1 (σ12 (t)S 2 + σ22 (t)I 2 + σ32 (t)V 2 ) 2 ≤(S + I + V )θ Au − µl (S + I + V )θ+1 1 + θ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 )(S + I + V )θ+1 2 ≤C − B(S θ+1 + I θ+1 + V θ+1 ), where B = 21 [µl − 12 θ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 )] > 0 by (3.2), and 1 1 C = sup {(S + I + V )θ Au − [µl − θ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 )](S + I + V )θ+1 }. 2 2 S,I,V ≥0 1 S I (1 − q(t))A(t) + (µ(t) + p(t)) − γ(t) f (S) f (S) f (S) 0 V 1 f (S) 2 − ε(t) + β(t)g(I) + σ (t)S 2 f (S) 2 f 2 (S) 1 S 1 S 1 (1 − q u )Al + (µu + pu ) + M4 (σ1u )2 , ≤β u g 0 (0)I − f (S) f (S) 2 f (S)

LV2 = −

0

where M4 = maxS>0 { ff(S)S } by (H6 ). (S) 1 1 1 q(t)A(t) − p(t)S + µ(t) + ε(t) + σ32 (t) V V 2 1 1 l l ≤ − q A + µu + εu + (σ3u )2 . V 2

LV3 = −

and LV4 = − (S − S0 (t) + I)[(µ(t) + p(t))c1 (t) − β(t)f 0 (S0 (t))g 0 (0) − p(t)c2 (t)] − (V − V0 (t))[(µ(t) + ε(t))c2 (t) − ε(t)c1 (t)] − hR0 iω + R0 (t) − c1 (t)[−(µ(t) + p(t))(S − S0 (t)) + ε(t)(V − V0 (t)) − (µ(t) + α(t))I] − β(t)g 0 (0)[f (S0 (t)) + f (S) − f (S0 (t))] + (µ(t) + γ(t) + α(t))

g 0 (0)I − 1] − c2 (t)[p(t)(S − S0 (t)) g(I) 1 1 g 0 (0)g 0 (I) 2 − (µ(t) + ε(t))(V − V0 (t))] + σ22 (t) + σ22 (t)[ I − 1]. 2 2 g 2 (I) + (µ(t) + γ(t) + α(t))[

By (H4 ) and the mean value theorem, we have for any S ≥ 0 f (S) − f (S0 (t)) = f 0 (ξ1 (t))(S − S0 (t)) ≥ f 0 (S0 (t))(S − S0 (t)), 12

(3.3)

where ξ1 (t) is situated between S0 (t) and S. g 0 (I)I−g(I) I2

)0 = From (H3 ) and (H8 ), since ( g(I) I

≤ 0, we have for any I > 0 g 0 (I)

I g 0 (0) 1 I ≤ 1, g 0 (0) −1=( − )I ≤ M2 I, g(I) g(I) g(I) I

(3.4)

0

(0) where M2 = supI>0 { gg(I) − I1 }.

By (3.3), (3.4) and the expression of R0 (t), we finally can obtain LV4 ≤ − hR0 iω + [p(t)c2 (t) + c1 (t)α(t) + β(t)f 0 (S0 (t))g 0 (0)]I 1 g 0 (0)I + [µ(t) + γ(t) + α(t) + σ22 (t)]( − 1) 2 g(I)

1 ≤ − hR0 iω + [pu cu2 + αu cu1 + β u f 0 (S0u )g 0 (0) + (µu + γ u + αu + (σ2u )2 )M2 ]I. 2 Accordingly, we can get that LV ≤C − B(S θ+1 + I θ+1 + V θ+1 ) + β u g 0 (0)I S 1 S 1 − (1 − q u )Al + (µu + pu ) + M3 (σ1u )2 f (S) f (S) 2 f (S) 1 1 l l − q A + µu + εu + (σ3u )2 + M [−hR0 iω V 2 1 + [pu cu2 + αu cu1 + β u f 0 (S0u )g 0 (0) + (µu + γ u + αu + (σ2u )2 )M2 ]I] 2 =f1 (S) + f2 (I) + f3 (V ), where f1 (S) = − BS θ+1 −

(3.5)

1 1 [(1 − q u )Al − (µu + pu )S − M4 S(σ1u )2 ], f (S) 2

f2 (I) = − BI θ+1 + β u g 0 (0)I + M [−hR0 iω + (pu cu2 + αu cu1 1 + β u f 0 (S0 (t))g 0 (0) + (µu + γ u + αu + (σ2u )2 )M2 )I], 2 1 1 f3 (V ) = − BV θ+1 + C − q l Al + µu + εu + (σ3u )2 . V 2 We can obtain that, since limS→∞

1 f (S)

= 0 by (H6 ), then f1 (S) + f2u + f3u → −∞ as

S → +∞. Furthermore, we also can obtain that

f1 (S) + f2u + f3u → −∞ as S → 0+ ,

f3 (V ) + f1u + f2u → −∞ as V → 0+ or V → +∞, f2 (I) + f1u + f3u → −∞ as I → +∞,

and f2 (I) + f1u + f3u → −M hR0 iθ + f1u + f3u ≤ −2 as I → 0+ . 13

Take ε0 > 0 sufficiently small and let 3 U = {(S, I, V ) ∈ R+ : ε0 ≤ S ≤

1 1 1 , ε0 ≤ I ≤ , ε0 ≤ V ≤ }. ε0 ε0 ε0

Then from (3.5) we have LV ≤ −1,

3 \U. (S, I, V ) ∈ R+

Therefore, by Lemma 2.3 we finally get that model (1.3) has at least one positive stochastic ω-periodic solution. This completes the proof. Remark 3.2. For f (S) =

Sm 1+ω1 S

and g(I) =

I 1+ω2 I

with constants m ≥ 2, ω1 ≥ 0 and

ω2 ≥ 0, by Remark 2.1, then (H3 ), (H4 ), (H6 ) and (H8 ) hold, and we have S m (t)

˜ 0 (t) = R

β(t) 1+ω01 S0 (t) µ(t) + γ(t) + α(t)

−

σ22 (t) . 2(µ(t) + γ(t) + α(t))

˜ 0 iω > 1, then model (1.3) has a positive stochastic Therefore, from Theorem 3.1, if hR ω-periodic solution. Remark 3.3. Take f (S) = S and g(I) = σ22 (t) ˜ 0 (t) = β(t)S0 (t) − We have R µ(t)+γ(t)+α(t)

I , 1+ω2 I

2(µ(t)+γ(t)+α(t))

then all assumptions (H3 ) − (H8 ) hold. ˜ 0 iω > 1 , from Theorem 3.1 we have that if hR

then model (1.3) has a positive stochastic ω-periodic solution.

When model (1.3) degenerates to the autonomic case, we have   dS(t) =[(1 − q)A − βf (S(t))g(I(t)) − (µ + p)S(t) + γI(t)      + εV (t)]dt + σ1 S(t)dB1 (t),

 dI(t) =[βf (S(t))g(I(t)) − (µ + γ + α)I(t)]dt + σ2 I(t)dB2 (t),      dV (t) =[qA + pS(t) − (µ + ε)V (t)]dt + σ3 V (t)dB3 (t),

(3.6)

where A, q, β, µ, p, γ, ε, α and σi (i = 1, 2, 3) are nonnegative constants, and 0 ≤ q ≤ 1, A > 0, µ > 0 and β > 0. We further have that equation (2.4) in Lemma 2.7 becomes to (

S00 (t) = (1 − q)A − (µ + p)S0 (t) + εV0 (t), V00 (t) = qA + pS0 (t) − (µ + ε)V0 (t),

(3.7)

which exists a unique positive equilibrium (S0 , V0 ), where S0 = A[(1−q)µ+ε] and V0 = µ(µ+p+ε) A[µq+p] ˜ 0 iθ becomes to the following form for model . We also see that threshold value hR µ(µ+p+ε)

(3.6)

0 σ22 ˜ 0 = βf (S0 )g (0) − R . µ+γ+α 2(µ + γ + α)

14

Particularly, when f (S) = S and g(I) = I, then model (3.6) degenerates to model (1.1) σ22 ˜ 0 = βS0 − and threshold value is R . µ+γ+α 2(µ+γ+α)

Based on the existence of unique ergodic stationary distribution for stochastic differential equations (See [37,38]), as a consequence of Theorems 3.1 we can obtain the following result for model (3.6). Corollary 3.4. Assume that (H3 ), (H4 ), (H6 ) or (H7 ) and (H8 ) hold, and 2µ > (σ12 ∨ ˜ 0 > 1, then model (3.6) has a unique stationary distribution and it has the σ 2 ∨ σ 2 ). If R 2

3

ergodic property.

Remark 3.5. From Theorem 3.1 and Corollary 3.4 we easily see that the existence of unique ergodic stationary distribution is obtained for model (1.3) and the corresponding results on the existence of positive stochastic ω-periodic solutions established in [27] are extended and improved.

4

Extinction of disease

˜ 0 iω < 1 Theorem 4.1. Suppose that (H1 ) − (H3 ), (H5 ), and (H6 ) or (H7 ) hold. If hR

and α(t) ≥ p(t) for all t ≥ 0, then for any solution (S(t), I(t), V (t)) of model (1.3) with 3 , limt→∞ I(t) = 0 a.s. exponentially, limt→∞ hSit = initial value (S(0), I(0), V (0)) ∈ R+ hS0 iω a.s. and limt→∞ hV it = hV0 iω a.s., where (S0 (t), V0 (t)) is given in Lemma 2.7.

˜ 0 iθ < 1 is equivalent to hR0 iθ < 0. Using Itˆ Proof. It is clear that hR o0 s formula to ln I(t), since

g(I) I

≤ g 0 (0) for all I > 0 by (H3 ), we can obtain

1 d ln I(t) ≤ [β(t)f (S(t))g 0 (0) − (µ(t) + γ(t) + α(t)) − σ22 (t)]dt + σ2 (t)dB2 (t). 2

(4.1)

By (H5 ) and the mean value theorem, we have f (S(t)) − f (S0 (t)) = f 0 (ξ2 (t))(S(t) − S0 (t)) ≤ f 0 (S0 (t))(S(t) − S0 (t)),

(4.2)

where ξ2 (t) is situated between S0 (t) and S(t). From (4.1) and (4.2) we can obtain Z

t 1 ln I(t) ≤ ln I(0) + [β(s)f (S0 (s))g 0 (0) − (µ(s) + γ(s) + α(s)) − σ22 (s)]ds 2 0 Z t Z t − β(s)f 0 (S0 (s))(S(s) − S0 (s))g 0 (0)ds + σ2 (s)dB2 (s) 0

0

15

(4.3)

Substituting (2.13) given in Lemma 2.9 into (4.3) we further have ln I(t) ln I(0) 1 ≤ + t t t

Z

t

1 R0 (s)ds − t

Z

t

(β(t)f 0 (S0 (s))g 0 (0) + c1 (s)α(s) Z 1 t c1 (s)σ1 (s)S(s)dB1 (s) + c2 (s)p(s) − c1 (s)p(s))I(s)ds + t 0 Z Z 1 t 1 t + c1 (s)σ2 (s)I(s)dB2 (s) + c2 (s)σ3 (s)V (s)dB3 (s) t 0 t 0 Z 1 t Φ(t) + σ2 (s)dB2 (s) + t 0 t φ(t) Φ(t) ≤hR0 it + + , t t

where

Z

0

0

Z

t

t

φ(t) = ln I(0) + c1 (s)σ1 (s)S(s)dB1 (s) + c1 (s)σ2 (s)I(s)dB2 (s) 0 0 Z t Z t + c2 (s)σ3 (s)V (s)dB3 (s) + σ2 (s)dB2 (s), 0

0

and Φ(t) is given in Lemma 2.9. By Lemma 2.5, Lemma 2.6 and the strong law of large numbers, we obtain that limt→∞ φ(t) = 0 a.s. and limt→∞ Φ(t) = 0 a.s. Therefore, we t t finally have lim sup t→∞

ln I(t) ≤ lim suphR0 it = hR0 iω < 0 a.s. t t→∞

This shows that limt→∞ I(t) = 0 a.s. exponentially. Let u(t) = S(t) − S0 (t) and v(t) = V (t) − V0 (t). Define the auxiliary function W (t) = |u(t)| + |v(t)|. Using Itˆ o0 s formula to W (t), we have dW (t) =sign(u(t))(dS(t) − dS0 (t)) + sign(v(t))(dV (t) − dV0 (t)) =sign(u(t))(−β(t)f (S(t))g(I(t)) − (µ(t) + p(t))(S(t) − S0 (t)) + ε(t)(V (t) − V0 (t)) + γ(t)I(t) + σ1 (t)S(t)dB1 (t)) + sign(v(t))(p(t)(S(t) − S0 (t)) − (µ(t) + ε(t))(V (t) − V0 (t)) + σ3 (t)V (t)dB3 (t)) ≤ − µ(t)W (t) + β(t)f (S(t))g(I(t)) + γ(t)I(t) + σ1 (t)sign(u(t))S(t)dB1 (t) + σ2 (t)sign(v(t))V (t)dB3 (t).

16

Therefore, Z t Z W (t) − W (0) 1 t l1 ≤−µ W (s)ds + β(s)f (S(s))g(I(s))ds t t 0 t 0 Z Z 1 t 1 t γ(s)I(s)ds + σ1 (s)sign(u(s))S(s)dB1 (s) + t 0 t 0 Z 1 t + σ3 (s)sign(v(s))V (s)dB3 (s). t 0

(4.4)

(0) From Lemma 2.5 we have limt→∞ W (t)−W = 0 a.s.. From limt→∞ I(t) = 0 a.s. expot R 1 t nentially, it follows that limt→∞ t 0 γ(s)I(s)ds = 0 a.s.. From (H1 ), (H3 ) and (4.2), we

obtain

1 t

Z

t

β(s)f (S(s))g(I(s))ds Z t 0 u1 ≤g (0)β [f (S0 (s))I(s) + f 0 (S0 (s))S(s)I(s) − f 0 (S0 (s))S0 (s)I(s)]ds t 0 Z t 0 u ∗1 [2I(s) + S(s)I(s)]ds, ≤g (0)β M t 0 0

(4.5)

where M ∗ = sup0≤t≤ω {f (S0 (t))+|f 0 (S0 (t))|(1+S0 (t))} < ∞. Let N (t) = S(t)+I(t)+V (t), then we easily obtain

dN (t) ≤ Au − µl N (t) + σ1 (t)S(t)dB1 (t) + σ2 (t)I(t)dB2 (t) + σ3 (t)V (t)dB3 (t), Rt u from Lemmas 2.5 and 2.6, we further have lim supt→∞ 1t 0 N (s)ds ≤ Aµl a.s.. For any Rt u ε > 0 there is a T > 0 such that I(t) < ε a.s. and 1t 0 S(s)ds ≤ Aµl + ε a.s. for all t > T , then we have

Z Z 1 T 1 t S(s)I(s)ds ≤ S(s)I(s)ds + ε S(s)ds t 0 t T 0 Z 1 T Au ≤ S(s)I(s)ds + ( l + ε)ε. t 0 µ R t Thus, by the arbitrariness of ε, we further have limt→∞ 1t 0 S(s)I(s)ds = 0 a.s.. There1 t

Z

t

fore, from (4.5) we finally can obtain 1 lim t→∞ t

Z

t

β(s)f (S(s))g(I(s))ds = 0 a.s.

0

Let ω(t) = S(t) + I(t) + V (t). By (H2 ), choosing a constant θ > 2 such that µl − θ−1 ((σ1u )2 ∨ (σ2u )2 ∨ (σ3u )2 ) > 0. Then, using the similar argument given in [16] we can 2

17

¯ > 0 such that the expectation obtain that there is a constant M ¯ E[(1 + ω(t))θ ] ≤ M

for all t ≥ 0.

(4.6)

Rt Let X(t) = 0 σ1 (s)sign(u(s))S(s)dB1 (s). Using the Burkholder-Davis-Gundy inequality (See [38]), we can obtain that there is a constant Cθ > 0 such that for all t ≥ 0 θ

E[ sup |X(s)| ] ≤ Cθ E[ 0≤s≤t

Z

t

θ

S 2 (s)ds] 2 .

0

From (4.6) we further have ¯ t θ2 E[ sup |X(s)|θ ] ≤ Cθ M 0≤s≤t

for all t ≥ 0. From this, a similar argument as in [16], we can finally obtain that 1 lim t→∞ t

Z

t

σ1 (s)sign(u(s))S(s)dB1 (s) = 0 a.s.

0

Similarly, we also have 1 lim t→∞ t

Z

t

σ3 (s)sign(v(s))V (s)dB3 (s) = 0 a.s.

0

From (4.4), it follows that limt→∞

1 t

Rt 0

W (s)ds = 0 a.s.. This shows that limt→∞ hSit =

hS0 iω a.s. and limt→∞ hV it = hV0 iω a.s.. This completes the proof. Remark 4.2. When f (S) = ln(1 + ω1 S) or f (S) =

S , 1+ω1 S

and g(I) =

I , 1+ω2 I

by Remark

2.1, then (H3 ), (H5 ), and (H6 ) or (H7 ) are satisfied. We also have σ22 (t) ˜ 0 (t) = β(t) ln(1 + ω1 S0 (t)) − R µ(t) + γ(t) + α(t) 2(µ(t) + γ(t) + α(t)) or ˜ 0 (t) = R

β(t) 1+ωS01(t) S0 (t) µ(t) + γ(t) + α(t)

−

σ22 (t) . 2(µ(t) + γ(t) + α(t))

˜ 0 iω < 1, then the disease I in model (1.3) is extinct Therefore, from Theorem 4.1, if hR exponentially with probability one.

For model (3.6), directly from Theorem 4.1, we can obtain the following results. Corollary 4.3. Assume that (H3 ), (H6 ) or (H7 ) hold, and µ > 21 (σ12 ∨ σ22 ∨ σ32 ). If (H5 ) ˜ 0 < 1, then for any solution (S(t), I(t), V (t)) of model (3.6) with initial holds, α ≥ p and R 18

3 value (S(0), I(0), V (0)) ∈ R+ , limt→∞ I(t) = 0 a.s. exponentially, limt→∞ hSit = S0 a.s.

and limt→∞ hV it = V0 a.s..

Remark 4.4. From Theorems 3.1 and 4.1, and Corollaries 3.4 and 4.3, we easily see that the corresponding results which are obtained in [16, 27] are extended and improved.

5

Numerical examples In this section, we will give the numerical examples to illustrate the theoretical results

which are established in this paper, and further discuss whether assumptions (H4 ) − (H8 ) in Theorems 3.1 and 4.1 can be weaken. The numerical methods are proposed in [39]. I S , g(I) = 1+I and parameters A = Example 5.1. Take in model (1.3) f (S) = 1+S

5 + 0.6 sin t, q = 1.5 + 0.05 sin t,  = 2 + 0.8 sin t, α = 3, p = 0.7 + 0.2 cos t, β = 5 + 4 sin t, µ = 1.2 + 0.1 cos t, γ = 1 + 0.6 cos t, σ1 = 0.1 + 0.01 sin t, σ2 = 0.5 + 0.1 sin t and σ3 = 0.05 + 0.05 sin t. It is easy to see that (H1 ) − (H3 ), (H5 ) and (H7 ) are satisfied and α(t) ≥ p(t) for all

t ≥ 0. The numerical simulation of positive ω-periodic solution (S0 (t), V0 (t)) of equation

(2.4) which is given in Lemma 2.7 is illustrated in Figure 1. From (3.1), by numerical ˜ 0 iω = 0.5744 < 1. Thus, all the conditions in Theorem 4.1 are computing we obtain hR satisfied. The conclusions given in Theorem 4.1 are numerically simulated in Figure 2. 2.2

3.1

1.6

2.8 V0(t)

3

2.9

S0(t)

2

1.8

1.4

2.7

1.2

2.6

1

2.5

0.8

0

1000

2000

3000 4000 Time T

5000

6000

7000

0

1000

2000

3000 4000 Time T

5000

6000

7000

Figure 1: The numerical simulation of solution (S0 (t), V0 (t)) of equation (2.4) with initial value (S0 (0), V0 (0)) = (1.0960, 2.6448). It shows that (S0 (t), V0 (t)) converges at a positive periodic solution of equation (2.4).

Example 5.2. Take in model (1.3) f (S) =

S2 , 1+S

g(I) =

I 1+I

and parameters A = 7 +

0.8 sin t, q = 1.2+0.1 cos t,  = 3.5+0.2 cos t, α = 1.5+0.5 sin t, p = 0.9, β = 14+1.1 sin t, µ = 1.6 + 0.6 sin t, γ = 0.6 + 0.5 sin t, σ1 = 0.05 + 0.03 sin t, σ2 = 0.1 + 0.1 sin t and σ3 = 0.02 + 0.01 sin t. It is clear that (H1 ) − (H3 ) are satisfied. By computing, we know that (H4 ), (H6 ), (H7 )

and (H8 ) also are satisfied, and g 0 (0) = 1. The numerical simulation of positive ω-periodic solution (S0 (t), V0 (t)) of equation (2.4) is illustrated in Figure 3. From (3.1), by numerical 19

0.1

2.6

0.09

2.4

0.08

2.2

0.07

2

0.06 I(t)

S(t)

2.8

1.8

0.05

1.6

0.04

1.4

0.03

1.2

0.02

1

0.01

0.8

0

1000

2000

3000 4000 Time T

5000

6000

0

7000

0

1000

2000

3000 4000 Time T

5000

6000

7000

3.2 3.1 3 2.9

V(t)

2.8 2.7 2.6 2.5 2.4 2.3

0

1000

2000

3000 4000 Time T

5000

6000

7000

Figure 2: Numerical simulations of solution (S(t), I(t), V (t)) with initial value (S(0), I(0), V (0)) = (2, 0.1, 2.5). It shows that limt→∞ I(t) = 0 a.s.

˜ 0 iω = 7.7072 > 1. Thus, all the conditions in Theorem 3.1 are computing we obtain hR

satisfied. The conclusions established in Theorem 3.1 are numerically simulated in Figure 4. 2.35

3.4

2.3

3.2

2.25

3

2.2

2.8

2.15 V0(t)

0

S (t)

3.6

2.6 2.4

2.1 2.05

2.2

2

2

1.95

1.8 1.6

1.9 0

500

1000

1500 2000 Time T

2500

3000

1.85

3500

0

500

1000

1500 2000 Time T

2500

3000

3500

Figure 3: The numerical simulation of positive periodic solution (S0 (t), V0 (t)) of equation (2.4).

Example 5.3. Take in model (1.3) f (S) =

S2 , 1+S

g(I) =

I 1+I

and parameters A =

5 + 0.4 sin t, q = 1.5 + 0.01 sin t,  = 2 + 0.8 sin t, α = 3 + 0.1 cos t, p = 0.7 + 0.2 cos t, β = 5 + 4 sin t, µ = 1.2 + 0.1 cos t, γ = 1 + 0.6 cos t, σ1 = 0.1 + 0.01 sin t, σ2 = 0.5 + 0.1 sin t and σ3 = 0.05 + 0.05 sin t. It is easy to see that (H1 ) − (H3 ) and (H6 ) are satisfied and α(t) ≥ p(t) for all t ≥ 0.

But, we see that (H5 ) does not hold. Therefore, Theorem 4.1 is not applicable. The numerical simulation of positive ω-periodic solution (S0 (t), V0 (t)) of equation (2.4) ˜ 0 iω = 0.9903 < is illustrated in Figure 5. From (3.1), by numerical computing we obtain hR 20

1.1

1.8 1.6

1.05 1.4 1.2 I(t)

S(t)

1

0.95

1 0.8 0.6

0.9 0.4 0.85

0

1000

2000

3000 4000 Time T

5000

6000

0.2

7000

1.92

0

1000

2000

3000 4000 Time T

5000

6000

7000

1.8

1.9

1.6

1.88 1.4 1.86 1.2

I(t)

V(t)

1.84 1.82 1.8

1 0.8

1.78 0.6 1.76 0.4

1.74 1.72

0

1000

2000

3000 4000 Time T

5000

6000

0.2 0.85

7000

0.9

0.95

1

1.05

1.1

S(t)

Figure 4: Numerical simulations of solution (S(t), I(t), V (t)) with initial value (S(0), I(0), V (0)) = (0.9, 1.3, 1.9). It is shown that I(t) is permanent in the mean with probability one and model (1.3) has a stochastic positive periodic solution.

1. The numerical simulations given in Figure 6 show that I(t) still is extinct with probability one. This seem to show that (H5 ) can be weaken in Theorem 4.1. 3.2

2.2

3.1

2

3 1.8 2.9

0

S (t)

V0(t)

1.6

2.8 2.7

1.4

2.6 1.2 2.5 1

0.8

2.4

0

1000

2000

3000 4000 Time T

5000

6000

2.3

7000

0

1000

2000

3000 4000 Time T

5000

6000

7000

Figure 5: The numerical simulation of positive periodic solution (S0 (t), V0 (t)) of equation (2.4).

Example 5.4. Take in model (1.3) f (S) =

S 1+S 2

and g(I) =

I , 1+I 2

and the parameters

A = 0.9+0.1 sin t, q = 0.8+0.2 cos t,  = 0.5+0.1 cos t, α = 0.3+0.3 sin t, p = 0.1+0.1 cos t, β = 12, µ = 0.1 + 0.1 sin t, γ = 0.1 + 0.05 sin t, σ1 = 0.07 + 0.03 sin t, σ2 = 0.05 + 0.03 sin t and σ3 = 0.05 + 0.01 sin t. It is easy to see that (H1 ) − (H3 ) are satisfied. By computing, we have f 0 (S) =

1 − ω1 S 2 , (1 + ω1 S 2 )2

f 00 (S) =

21

2ω12 S 3 − 6ω1 S , (1 + ω1 S 2 )3

3

0.35

0.3 2.5 0.25 2 I(t)

S(t)

0.2

0.15

1.5

0.1 1 0.05

0.5

0

1000

2000

3000 4000 Time T

5000

6000

0

7000

0

1000

2000

3000 4000 Time T

5000

6000

7000

3.2 3.1 3 2.9

V(t)

2.8 2.7 2.6 2.5 2.4 2.3 2.2

0

1000

2000

3000 4000 Time T

5000

6000

7000

Figure 6: Numerical simulations of solution (S(t), I(t), V (t)) with initial value (S(0), I(0), V (0)) = (1, 0.2, 2.5). It is clear to see that I(t) is extinct with probability one.

sup S>0

f (S) = 1 < ∞, S

lim sup S→∞

1 = ∞, f (S)

max{ S>0

f 0 (S)S } < ∞, f (S)

this shows that (H7 ) holds, and (H4 ) and (H6 ) are not satisfied. In addition, we have 0

(0) g 0 (0) = 1 and maxI>0 { gg(I) − I1 } = ∞. Hence, (H8 ) is also not satisfied. Therefore, Theorem 3.1 is not applicable.

The numerical simulation of positive ω-periodic solution (S0 (t), V0 (t)) for equation ˜ 0 iω = (2.4) is illustrated in Figure 7. From (3.1), by the numerical computing we obtain hR

7.5355 > 1. The numerical simulations given in Figure 8 show that I(t) still is permanent in the mean with probability one, and model (1.3) has a stochastic positive periodic solution. This seem to show that (H4 ), (H6 ) and (H8 ) can be weaken in Theorem 3.1. 7.4

3

7.2 7 6.8

2.5

V0(t)

0

S (t)

6.6 6.4 6.2 2

6 5.8 5.6 5.4

0

1000

2000

3000

4000

1.5

5000

Time T

0

1000

2000

3000

4000

5000

Time T

Figure 7: The numerical simulation of positive periodic solution (S0 (t), V0 (t)) of equation (2.4).

22

0.32

3.5

0.3 3

0.28 0.26

2.5 I(t)

S(t)

0.24 0.22 2 0.2 0.18

1.5

0.16

0

1000

2000

3000 4000 Time T

5000

6000

1

7000

1.45

0

1000

2000

3000 4000 Time T

5000

6000

7000

3.5

1.4 3

1.35 1.3

2.5 V(t)

I(t)

1.25 1.2

2 1.15 1.1

1.5

1.05 1

0

1000

2000

3000 4000 Time T

5000

6000

1

7000

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

S(t)

Figure 8: Numerical simulations of solution (S(t), I(t), V (t)) with initial value (S(0), I(0), V (0)) = (0.2, 2, 1.2). It is shown that I(t) is permanent in the mean and model (1.3) has a stochastic positive periodic solution.

6

Conclusion In this paper, we have investigated a stochastic SIS epidemic model with periodic coef-

ficients, nonlinear incidence and vaccination. We introduced the assumptions (H3 ) − (H8 )

for the nonlinear incidence functions f (S) and g(I) and put forward the new threshold σ22 (t) ˜ 0 iω with R ˜ 0 (t) = β(t)f (S0 (t))g0 (0) − value hR . µ(t)+γ(t)+α(t) 2(µ(t)+γ(t)+α(t)) ˜ 0 iω , we have established Based on the assumptions (H1 ) − (H8 ) and threshold value hR the criteria on the existence of stochastic positive periodic solutions and the extinction of disease I with probability one. These conclusions have been presented in Theorems 3.1 and 4.1. From these results we easily see that the corresponding results obtained in [16, 27] are extended and improved. Furthermore, the conclusions also are illustrated in Examples 5.1 and 5.2 by the numerical simulations. However, from the numerical simulations given in Examples 5.3 and 5.4, we see that when assumption (H5 ) does not satisfy, then the conclusions of Theorem 4.1 also can be obtained, and when assumptions (H4 ), (H6 ) and (H8 ) are not satisfied, then the conclusions of Theorem 3.1 still can hold. Therefore, it is easily seen that assumptions (H4 ) − (H8 ) can be weaken. Here, we propose a conjecture, only under the assumptions ˜ 0 iω < 1 then the disease I in model (1.3) is extinct with probability (H1 ) − (H3 ), when hR ˜ 0 iω > 1 then model (1.3) has a stochastic positive periodic solution. one, and when hR

23

Acknowledgements This research is supported by the Opening Project of Key Laboratory of Xinjiang of China (Grant Nos. 2016D03022).

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26

Highlights (for review) 1. A periodic stochastic SIVS epidemic model with nonlinear incidence and vaccination is proposed. 2. A new threshold value for the model is introduced. 3. The threshold conditions for the existence of stochastic positive periodic solutions are obtained. 4. The threshold conditions for the extinction of disease are established.