# Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination.

1. Introduction

As is well known, transmissions of many infectious diseases are inevitably affected by environment white noise, which is an important component in realism, because it can provide some additional degrees of realism compared to their deterministic counterparts. Therefore, in recent years, stochastic differential equation (SDE) has been used widely by many researchers to model the dynamics of spread of infectious disease (see [1-5] and the references cited therein). There are different possible approaches to include effects in the model. Here, we mainly introduce three approaches. The first one is through time Markov chain model to consider environment noise in SIS model (see, e.g.,  and the references cited therein). The second is with parameters perturbation (see [2, 5, 7] and the references cited therein). The last issue to model stochastic epidemic system is to perturb around the positive equilibria of deterministic models (see, e.g., [1, 8, 9] and the references cited therein).

Now, we consider stochastic epidemic models with parameters perturbation. The incidence rate of a disease denotes the number of new cases per unit time, and this plays an important role in the study of mathematical epidemiology. In many epidemic models, the bilinear incidence rate [beta]SI is frequently used (see [2, 5, 7, 8, 10-17]), and the saturated incidence rate [beta]SI/(1 + aI) is also frequently used (see, e.g., [18-22]). Comparing with bilinear incidence rate and saturated incidence rate, Lahrouz and Omari  and Liu and Chen  introduced a nonlinear incidence rate [beta]SI/[phi](I) into stochastic SIRS epidemic models. In , Tang et al. investigated a class of stochastic SIRS epidemic models with nonlinear incidence rate [beta]f(S)g(I):

[mathematical expression not reproducible]. (1)

Lahrouz et al.  studied a deterministic SIRS epidemic model with nonlinear incidence rate [beta]SI/[phi](I) and vaccination. If the transmission of the disease is changed by nonlinear incidence rate [beta]f(S)g(I), and to make the model more realistic, let us suppose that the death rates of the three classes in the population are different, then a more general deterministic SIRS model is described by the following ordinary differential equation:

[mathematical expression not reproducible], (2)

where S(t), I(t), and R(t) denote the numbers of susceptible, infectious, and recovered individuals at time t, respectively. [LAMBDA] denote a constant input of new members into the susceptible per unit time. q is the rate of vaccination for the new members. p is the rate of vaccination for the susceptible individuals. [d.sub.s] is the natural mortality rate or the removal rate of the S. [d.sub.I] is the removal rate of the infectious and usually is the sum of natural mortality rate and disease-induced mortality rate. [d.sub.R] is the removal rate of the recovered individual. [gamma] is the recovery rate of infective individual. [epsilon] is the rate at which the recovered individual loses immunity. [beta] represents the transmission coefficient between compartments S and I, and [beta]f(S)g(I) denotes the incidence rate of the disease. For biological reasons, we usually assume that functions f(S) and g(I) satisfy the following properties:

([H.sub.1]) g(I) is two-order continuously differentiable function; g(I)/I is monotonically nondecreasing with respect to I; g(0) = 0 and g'(0) > 0.

([H.sub.2]) f(S) is two-order continuously differentiable function; f'(S) [greater than or equal to] 0 and f" (S) [less than or equal to] 0 for all S [greater than or equal to] 0, and f(0) = 0.

It is well known that the basic reproduction number for model (2) is defined by [R.sub.0] = [beta]f([S.sub.0])g (0)/([d.sub.I] + [gamma]), where [S.sub.0] = [(1 - q)[d.sub.R] + [epsilon]][LAMBDA]/([d.sub.s]([d.sub.R] + [epsilon]) + p[d.sub.R]). Applying the Lyapunov function method and the theory of persistence for dynamical systems, we can prove that, when [R.sub.0] < 1, model (2) has a globally asymptotically stable disease-free equilibrium [E.sub.0] = ([S.sub.0],0,[R.sub.0]) and, when [R.sub.0] > 1, model (2) has a unique endemic equilibrium [E.sup.*] ([S.sup.*], [I.sup.*], [R.sup.*]) and disease I is permanent.

In this paper, we extend model (1) to more general cases. As in , taking into account the effect of randomly fluctuating environment, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in parameters [beta], [d.sub.s], [d.sub.I], and [d.sub.R] in model (2) change to random variables [mathematical expression not reproducible] such that

[mathematical expression not reproducible]. (3)

Accordingly, model (2) becomes

[mathematical expression not reproducible]. (4)

By the central limit theorem, the error term [error.sub.i] dt (0 [less than or equal to] i [less than or equal to] 3) may be approximated by a normal distribution with zero mean and variance [[sigma].sup.2.sub.i]dt (0 [less than or equal to] i [less than or equal to] 3), respectively. That is, [error.sub.i] dt = [??](0,[[sigma].sup.2.sub.i]dt). Since these [error.sub.i] dt may correlate with each other, we represent them by l-dimensional Brownian motion B(t) = ([B.sub.1](t), ..., [B.sub.l]()) as follows:

[error.sub.i]dt = [t.summation over (j=1)][[sigma].sub.ij]d[B.sub.j](t), 0 [less than or equal to] i [less than or equal to] 3, (5)

where [[sigma].sub.ij] are all nonnegative real numbers. Therefore, model (4) is characterized by the following Ito stochastic differential equation:

[mathematical expression not reproducible]. (6)

Model (6) in the special case where f (S) = S, g(I) = I, and p = q = 0 has been investigated by Yang and Mao in  and in the special case where [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l) and p = q = 0 also has been discussed in . It is well known that, in a stochastic epidemic model, the dynamical behaviors, like the extinction, persistence, stationary distribution, and stability of the model, are the most interesting topics. Therefore, in this paper, as an important extension and improvement of the results given in [11,25], we aim to discuss the dynamical behaviors of model (6). Particularly, we will explore the stochastic extinction and persistence in the mean of disease with probability one and the existence of stationary distribution.

This paper is organized as follows. In Section 2, we introduce some preliminaries to be used in later sections. In Section 3, we establish the threshold condition for stochastic extinction of disease with probability one of model (6). In Section 4, we deduce the threshold conditions for the disease being stochastically persistent and permanent in the mean with probability one. In Section 5, we discuss the existence of the stationary distribution of model (6) under some sufficient conditions. In Section 6, the numerical simulations are presented to illustrate the main results obtained in this paper and some conjectures are further proposed. Finally, in Section 7, a brief conclusion is given.

2. Preliminaries

Through this paper, we let ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) be a complete probability space with a filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] satisfying the usual conditions (that is, it is right continuous and increasing while [F.sub.0] contains all P-null sets). In this paper, we always assume that stochastic model (6) is defined on probability space ([OMEGA], F,[{[F.sub.t]}.sub.t[greater than or equal to]0],P). Furthermore, we denote [R.sup.3.sub.+] = {(x,y,z) : x > 0, y > 0,z > 0}, [[sigma].sup.2.sub.i] = [[summation].sup.l.sub.j=1] [[sigma].sup.2.sub.ij] 0 [less than or equal to] i [less than or equal to] 3, and [[sigma].sup.2] = [[summation].sup.3.sub.i=0][[sigma].sup.2.sub.i].

Firstly, on the existence and uniqueness of global positive solution for model (4) we have the following result.

Lemma 1. Assume that ([H.sub.1]) and ([H.sub.2]) hold; then, for any initial value (S(0),I(0),R(0)) [member of] [R.sup.3.sub.+], model (6) has a unique solution (S(t), I(t), R(t)) defined for all t [greater than or equal to] 0, and the solution will remain in [R.sup.3.sub+] with probability one.

This lemma can be proved by using a similar argument as in the proof of Theorem 3.1 given in . We hence omit it here.

Lemma 2. Assume that ([H.sub.1]) and ([H.sub.2]) hold and let (S(t),I(t),R(t)) be the solution of model (6) with initial value (S(0), 1(0), R(0)) [member of] [R.sup.3.sub.+]. Then lim [sup.sub.t[right arrow][infinity]] (S(i) + I(t) + R(t)) < [infinity] a.s. Moreover, let h(x, y, z) be any continuous function defined on [R.sup.3.sub.+]; then for each 1 [less than or equal to] j [less than or equal to] I we have

[mathematical expression not reproducible]. (7)

Proof. Let N(t) = S(t) + I(t) + R(t); then we have from model (6)

[mathematical expression not reproducible], (8)

where [mu] = min {[d.sub.s],[d.sub.I],[d.sub.R]} and P(t) = [[summation].sup.l.sub.j=1]([[sigma].sub.1j]S(t)+[[sigma].sub.2j]I(t)+ [[sigma].sub.3j]R(t))d[B.sub.j](t). By the comparison theorem of stochastic differential equations, we further have

N(t) [less than or equal to] N (0) [e.sup.-[mu]t] + [LAMBDA]/[mu](l - [e.sup.- [mu]t]) - Q(t), (9)

where

[mathematical expression not reproducible]. (10)

Define X(t) = N(0) + A(t) - U(t) - Q(t), where A(t) = ([LAMBDA]/[mu])(1 - [e.sup.-[mu]t]) and U(t) = N(0)(1 - [e.sup.-[mu]t]). It is clear that from Lemma 1 and (9) X(t) is nonnegative for t [greater than or equal to] 0, and A(t) and U(t) are continuous adapted increasing processes for t [greater than or equal to] 0 and A(0) = U(0) = 0. Therefore, by Theorem 3.9 given in , we obtain that [lim.sub.t[right arrow][infinity]] X(t) < [infinity] a.s. exists. From (9), we further have

[mathematical expression not reproducible]. (11)

Denote

[M.sub.j](t) = [[integral].sup.t.sub.0]h(S(s), I(s),R(s))d[B.sub.j](s). (12)

By (11), we have

[mathematical expression not reproducible]. (13)

Hence, the strong law of large number (see [27, 28]) implies [lim.sub.t[right arrow][infinity]](1/t)[M.sub.j](t) = 0 a.s. This completes the proof.

For any function h(t) defined on [R.sub.+0] = [0, +[infinity]), we denote the average value on [0, t] by <h(t)> = (1/t) [[integral].sup.t.sub.0]h(s)ds.

Lemma 3. Assume that ([H.sub.1]) and ([H.sub.2]) hold. Let (S(t),I(t), R(t)) be any positive solution of model (6); then

[mathematical expression not reproducible], (14)

where function [phi](t) is defined for all t [greater than or equal to] 0 satisfying [lim.sub.t[right arrow][infinity]][phi](t) = 0.

Proof. Taking integration from 0 to t for model (6), we get

[mathematical expression not reproducible]. (15)

Hence, we have

[mathematical expression not reproducible]. (16)

With a simple calculation from (16) we can easily obtain formula (14) with which [phi](t) is defined by

[mathematical expression not reproducible]. (17)

By Lemma 2, we further have [lim.sub.t[right arrow][infinity]][phi](t) = 0 a.s.

Lemma 4. Assume that ([H.sub.1]) and ([H.sub.2]) hold and [[sigma].sub.ij] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l). Then, for any solution (S(t),I(t),R(t)) of system (6) with (S(0), 1(0), R(0)) [member of] [R.sup.3.sub.+], one has

[mathematical expression not reproducible], (18)

where [bar.S] = [LAMBDA]/[mu]. Furthermore, the region

[GAMMA] = {(S,I,R):S > 0,I > 0,R > 0,S + I + R [less than or equal to] [bar.S] a.s.} (19)

is positive invariant with probability one for model (6), where [mu] = min{[d.sub.S], [d.sub.I], [d.sub.R]}.

In fact, for N(t) = S(t) + I(t) + R(t), from model (6) we have

[mathematical expression not reproducible]. (20)

This implies that (18) holds, and set T is positive invariant with probability one for model (6).

Lemma 5. Assume that ([H.sub.1]) and ([H.sub.2]) hold, [[sigma].sub.ij] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l), [d.sub.s] = [d.sub.R], and [d.sub.I] = [d.sub.S] + [alpha] with constant [alpha] [greater than or equal to] 0. Then, for any solution (S(t), I(t), R(t)) of model (6) with (S(0),I(0),R(0)) [member of] [R.sup.3.sub.+], one has

[mathematical expression not reproducible], (21)

where

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible]. (23)

Proof. Since

dN(t) = ([LAMBDA] - [d.sub.s]N(t) - [alpha]l(t)) dt, a.s., (24)

then

[mathematical expression not reproducible], (25)

where N(0) = S(0) + 1(0) + R(0). From the third equation of model (6) we have

[mathematical expression not reproducible]. (26)

Therefore,

[mathematical expression not reproducible]. (27)

[mathematical expression not reproducible]. (28)

Solving y(t), we obtain

[mathematical expression not reproducible]. (29)

Substituting (29) into (27), we immediately obtain (21)-(23). This completes the proof.

Remark 6. When [d.sub.s] [not equal to] [d.sub.R] in model (6), whether we can also establish a similar result as in Lemma 5 still is an interesting open problem.

Consider the following n-dimensional stochastic differential equation:

dx(t) = b(x)dt + [m.summation over (r=1)][[sigma].sub.r](x)d[B.sub.r](t), (30)

where x = ([x.sub.1],[x.sub.2], ...,[x.sub.n]), [[sigma].sub.r](x) = ([[sigma].sup.1.sub.r](x),[[sigma].sup.2.sub.r](x),..., [[sigma].sup.n.sub.r](x)), and [B.sub.r](t) (1 [less than or equal to] r [less than or equal to] m) are standard Brownian motions defined on the above probability space. The diffusion matrix is defined by

[mathematical expression not reproducible]. (31)

For any second-order continuously differentiable function V(x), we define

[mathematical expression not reproducible]. (32)

The following lemma gives a criterion for the existence of stationary distribution in terms of Lyapunov function.

Lemma 7 (see ). Assume that there is a bounded open subset D in [R.sup.n] with a regular (i.e., smooth) boundary such that

(i) there exist some i = 1,2, ..., n and positive constant [eta] > 0 such that [a.sub.ii](x) [greater than or equal to] [eta] for all x [member of] D;

(ii) there exists a nonnegative function V(x) : [D.sup.c] [right arrow] R such that V(x) is second-order continuously differentiate function and that, for some [theta]s > 0, LV(x) [less than or equal to] -[theta] for all x [member of] [D.sup.c], where [D.sup.c] = [R.sup.n] \ D.

Then (30) has a unique stationary distribution [pi]. That is, if function f is integrable with respect to the measure [pi], then for all [x.sub.0] [member of] [R.sup.n]

[mathematical expression not reproducible]. (33)

To study the permanence in mean with probability one of model (6) we need the following result on the stochastic integrable inequality.

Lemma 8 (see ). Assume that functions Y [member of] C([R.sub.+] x [OMEGA],[R.sub.+]) and Z [member of] C([R.sub.+] [OMEGA],[R.sub.+]) satisfy [lim.sub.t[right arrow][infinity]](Z(t)/t) = 0 a.s. If there is T > 0 such that

ln Y(t) [greater than or equal to] [v.sub.0]t - v [[integral].supt.sub.0]Y(s)ds + Z(t) a.s., (34)

for all t [greater than or equal to] T, then

[mathematical expression not reproducible]. (35)

3. Extinction of Disease

For the convenience of following statements, we denote

[mathematical expression not reproducible]. (36)

We further define a threshold value

[mathematical expression not reproducible]. (37)

Theorem 9. Assume that ([H.sub.1]) and ([H.sub.2]) hold. If one of the following conditions holds:

(a) [[??].sub.0] < 1 and [mathematical expression not reproducible]

(b) [[sigma].sub.0] > 0 and [mathematical expression not reproducible]

then, for any initial value (S(0), 1(0), R(0)) [member of] [R.sup.3.sub.+], one has

[mathematical expression not reproducible]. (38)

That is, disease I is stochastically extinct exponentially with probability one. Moreover,

[mathematical expression not reproducible]. (39)

Proof. Applying Ito's formula to model (6) leads to

[mathematical expression not reproducible], (40)

where x = (S, I) and

[mathematical expression not reproducible]. (41)

Assume that condition (b) holds. Since

[mathematical expression not reproducible], (42)

then from (40)

[mathematical expression not reproducible]. (43)

By Lemma 2, we have

[mathematical expression not reproducible]. (44)

Therefore,

[mathematical expression not reproducible]. (45)

Assume that condition (a) holds. Choose constant [epsilon] > 0 such that [beta] + [[summation].sup.l.sub.j=1][[sigma].sub.0j][[sigma].sub.2j] [greater than or equal to]g'(0)f([epsilon])[[sigma].sup.2.sub.0]

[mathematical expression not reproducible]. (46)

When [[sigma].sup.2.sub.0] = 0, which implies [[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] I), we have from (46)

[mathematical expression not reproducible]. (47)

Since

[mathematical expression not reproducible], (48)

where [xi] [member of] (S, [S.sub.0]), from ([H.sub.2]), we can obtain f'([xi])(S - [S.sub.0]) [less than or equal to] f'([S.sub.0])(S - [S.sub.0]). Hence, we have

f(S) [less than or equal to] f([S.sub.0]) + f'([S.sub.0])(S - [S.sub.0]). (49)

According to (14), (40), and (49), we have

[mathematical expression not reproducible]. (50)

Hence, from (44) and Lemma 3, we finally have

[mathematical expression not reproducible]. (51)

When [[sigma].sup.2.sub.0] [not equal] 0, from (40) and (46) we have

[mathematical expression not reproducible]. (52)

Define a function

[mathematical expression not reproducible]. (53)

Clearly, F(u) is a monotone increasing for u [member of] [0, ([beta] + [[summation].sup.l.sub.j=1][[sigma].sub.0j][[sigma].sub.2j])/2f([epsilon])[[sigm a].sup.2.sub.0]] and monotone decreasing for u [member of] [([beta] + [[summation].sup.l.sub.j=1][[sigma].sub.0j][[sigma].sub.2j])/2f([epsilon])([[sig ma].sup.2.sub.0],[infinity]). With condition [beta] + [[summation].sup.l.sub.j=1][[sigma].sub.0j][[sigma].sub.2j] [greater than or equal to] g'(0)f([epsilon])[[sigma].sup.2.sub.0], that is, g(I)/I [less than or equal to] g'(0) < ([beta] + [[summation].sup.l.sub.j=1][[sigma].sub.0j][[sigma].sub.2j])/2f([epsilon])[[sigm a].sup.2.sub.0], we have

[mathematical expression not reproducible]. (54)

Hence, by (14) and (49), we have

[mathematical expression not reproducible]. (55)

Choose [epsilon] = [S.sub.0]; from (44) and Lemma 3, we finally have

[mathematical expression not reproducible]. (56)

From (45), (51), and (56), it follows that (38) holds.

Since [lim.sub.t[right arrow][infinity]] I(t) = 0 a.s., by (14) of Lemma 3 and the last equation of (15), we further obtain

[mathematical expression not reproducible]. (57)

This completes the proof.

Remark 10. Condition (b) in Theorem 9 can be rewritten in the following form:

[mathematical expression not reproducible]. (58)

It is clear that

[mathematical expression not reproducible]. (59)

Therefore, when condition (b) holds, from (58) we also have

[mathematical expression not reproducible]. (60)

Remark 11. From Remark 10 above, we see that in Theorem 9 if condition (a) holds, then we directly have [[??].sub.0] < 1, and if condition (b) holds, then we also have [[??].sub.0] <1. Therefore, an interesting open problem is whether we can establish the extinction of disease I with probability one for model (6) only when [[??].sub.0] < 1.

4. Stochastic Persistence in the Mean

In this section, we discuss the stochastic persistence and permanence in the mean with probability one for model (6) only for the following two special cases: (1)[[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] l) and (2)[[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l). Furthermore, we also assume that in model (6) function f(S) = S.

4.1. Case [[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] I). When f(S) = S and [[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] l) in model (6), we have

[mathematical expression not reproducible]. (61)

Theorem 12. Assume that ([H.sub.1]) holds, f(S) = S, and [[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] l). If [[??].sub.0] > 1; then disease I in model (6) is stochastically persistent in the mean; that is,

[mathematical expression not reproducible]. (62)

Proof. Let (S(t), I(t), R(t)) be any positive solution of model (6). Lemma 2 implies that there is a constant [M.sub.0] > 0 such that S(t) + I(t) + R(t) [less than or equal to] [M.sub.0] a.s. for all [greater than or equal to] 0. Define a Lyapunov function

U(I) = [[integral].sup.I(t).sub.I(0)] 1/g(I)dI. (63)

Using Ito's formula to model (6) leads to

[mathematical expression not reproducible]. (64)

From ([H.sub.1]), which implies that g'(I) [less than or equal to] g(I)/I [less than or equal to] g'(0), we have

[mathematical expression not reproducible], (65)

where M = [mathematical expression not reproducible] {I/g(I)}. Since [mathematical expression not reproducible](I/g(I)) = 1/g'(0), then 0< M < [infinity]. Substituting (65) into (64) and then integrating from 0 to t [greater than or equal to] 0, we get

[mathematical expression not reproducible], (66)

where [M.sub.j](t) = [[integral].sup.t.sub.0](I(r)/g(I(r)))d[B.sub.j](r). From Lemma 2, we have

[mathematical expression not reproducible]. (67)

Define a function G(I) as follows. When I >0, G(I) = I/g(I), and when I = 0, G(0) = [lim.sub.I[right arrow]0](I/g(I)) = 1/g'(0). Then G(I) is continuous for I [greater than or equal to] 0 and differentiable for I >0. Applying Lagrange's mean value theorem to G(I) - G(0), we have from (66)

[mathematical expression not reproducible], (68)

Substituting (14) into (68), it follows that

[mathematical expression not reproducible]. (69)

Since

[mathematical expression not reproducible], (70)

we have

[mathematical expression not reproducible], (71)

where

[mathematical expression not reproducible]. (72)

From (67) and Lemma 3 we have [lim.sub.t [right arrow] [infinity]] [THETA](t) = 0. Finally, by Lemma 8, we obtain

[mathematical expression not reproducible], (73)

where

[mathematical expression not reproducible]. (74)

This completes the proof.

Remark 13. In the proof of Theorem 12, we easily see that three constants [mathematical expression not reproducible], and [I.sup.*] given in (74) are dependent on every solution (S(t),I(t),R(t)) of model (6). This shows that in Theorem 12 we only obtain the stochastic persistence in the mean of the disease.

4.2. Case [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l). When f(S) = S and [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l) in model (6), we have

[mathematical expression not reproducible]. (75)

In order to obtain the stochastic permanence in the mean with probability one for model (6), we need to introduce a new threshold value

[mathematical expression not reproducible]. (76)

Obviously, we have [[bar.R].sub.0] [less than or equal to] [[??].sub.0].

Theorem 14. Assume that ([H.sub.1]) holds, f(S) = S, and [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l). If [[bar.R].sub.0] > 1, then disease I in model (6) is stochastically permanent in the mean, that is,

[mathematical expression not reproducible], (77)

where function G(I) is defined in Theorem 12 above.

Proof. Let U(I) = [[integral].sup.I(t).sub.I(0)]\(1/g(I))dI; using Ito's formula to model (6) and (18) leads to

[mathematical expression not reproducible]. (78)

Similarly to above proof of Theorem 12, we have

[mathematical expression not reproducible]. (79)

Substituting (14) into (79) yields

[mathematical expression not reproducible]. (80)

Since by (18)

[mathematical expression not reproducible], (81)

(80) can be rewritten as

[mathematical expression not reproducible], (82)

where

[mathematical expression not reproducible]. (83)

By Lemmas 2 and 3, it follows that [lim.sub.t[right arrow][infinity]][PHI](t) = 0. Therefore, taking t [right arrow] [infinity] in (82) it follows that (77) holds. This completes the proof.

Using Lemma 5, we can establish the following result which shows that [[??].sub.0] can be a threshold value for the stochastic permanence of disease I in the mean for a more special case of model (6): [d.sub.S] = [d.sub.R] and [d.sub.I] = [d.sub.S] + [alpha] with constant [alpha] [greater than or equal to] 0.

Theorem 15. Assume that ([H.sub.1]) holds, f(S) = S, [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] l), [d.sub.S] = [d.sub.R], and [d.sub.I] = [d.sub.S] + [alpha] with constant [alpha] [greater than or equal to] 0. If [[??].sub.0] > 1; then disease I in model (6) is stochastically permanent in the mean; that is,

[mathematical expression not reproducible], (84)

where function G(I) is defined in above Theorem 12 and

[mathematical expression not reproducible]. (85)

Proof. Firstly, when [d.sub.S] = [d.sub.R] and [d.sub.I] = [d.sub.S] + [alpha], then threshold value [[??].sub.0] becomes

[mathematical expression not reproducible], (86)

where [S.sub.0] = [NABLA][(1 - q)[d.sub.S] + [epsilon]]/[d.sub.S]([d.sub.S] + [epsilon] + p).

Let U(I) = [[integral].sup.I(t).sub.I(0)](1/g(I))dI; similarly to above proof of Theorem 12, we have

[mathematical expression not reproducible]. (87)

Since [S.sup.2] = [S.sup.2.sub.0] + 2[S.sub.0](S - [S.sub.0]) + [(S - [S.sub.0]).sup.2], we further have

[mathematical expression not reproducible]. (88)

Substituting (14) and (21) of Lemma 5 into (88), using inequality [(a + b).sup.2] [less than or equal to] 2([a.sup.2] + [b.sup.2]), it follows that

[mathematical expression not reproducible]. (89)

From expression (22) of H(t), we easily have [lim.sub.t[right arrow][infinity]]<[H.sup.2](t)> = 0. By (18), without loss of generality, we can assume that S(t) + I(t) + R(t) [less than or equal to] [NABLA]/[d.sub.S] a.s. for all t [greater than or equal to] 0. Hence,

[mathematical expression not reproducible]. (90)

By computing, we obtain

[mathematical expression not reproducible]. (91)

Therefore, we finally have

<[G.sup.2](t)> [less than or equal to] [M.sub.0]<I(t)>. (92)

From (81), (89), and (92) we further obtain

[mathematical expression not reproducible], (93)

where

[mathematical expression not reproducible]. (94)

By Lemmas 2 and 3, it follows that [lim.sub.t[right arrow][infinity]][PHI](t) = 0. Therefore, taking t [right arrow] [infinity] in (93) it follows that (84) holds. This completes the proof.

5. Stationary Distribution

In this section, we discuss the stationary distribution of model (6) by using Lyapunov function method. We firstly define the diffusion matrix A(x) = h(x)[h.sup.T](x), where x = (S, I, R),

[mathematical expression not reproducible]. (95)

Furthermore, we denote by [a.sub.ii](x)(i = 1,2,3) the diagonal elements of matrix A(x). We have [a.sub.ii](x) = [[summation].sup.l.sub.j=1][h.sup.2.sub.ij](x).

Theorem 16. Assume that ([H.sub.1]) holds, f(S) = S, and there is a constant [rho] > 0 such that [a.sub.ii](x) > [rho], for any x [member of] [R.sup.3.sub.+] and i = 1,2,3, [gamma] > p, [d.sub.I] > [d.sub.S], and [gamma]([d.sub.S] + [d.sub.R]) > p([d.sub.I] + [d.sub.R]). If [R.sub.0] > 1 and

[mathematical expression not reproducible], (96)

where

[mathematical expression not reproducible], (97)

and ([S.sup.*],[I.sup.*],[R.sup.*]) is the unique endemic equilibrium of model (2), then model (6) has a unique stationary distribution.

Proof. We here use the Lyapunov function method to prove this theorem. The proof given here is similar to Theorem 5.1 in . But, due to nonlinear function g(I), the Lyapunov function structured in the following is different from that given in .

By Lemma 7, it suffices to find a nonnegative Lyapunov function V(x) and compact set K [subset] [R.sup.3.sub.+] such that LV(x) [less than or equal to] -C for some C >0 and x [member of] [R.sup.3.sub.+]/K.

Denote x = (S, I, R) [member of] [R.sup.3.sub.+]. Define the function

[V.sub.1](x) = 1/2[(R - [R.sup.*]).sup.2]. (98)

Calculating [LV.sub.1] (x), we have

[mathematical expression not reproducible]. (99)

Define the function

[V.sub.2](x) = I - [I.sup.*] - [I.sup.*] ln I/[I.sup.*]. (100)

Calculating [LV.sub.2](x), we have

[mathematical expression not reproducible]. (101)

Define the function

[V.sub.3](x) = 1/2[(S + I - [S.sup.*] - [I.sup.*]).sup.2]. (102)

Calculating [LV.sub.3](x), we get

[mathematical expression not reproducible]. (103)

Define the function

[V.sub.4](x) = 1/2[(S + I + R - [S.sup.*] - [I.sup.*] - [R.sup.*]).sup.2]. (104)

Calculating [LV.sub.4](x), we get

[mathematical expression not reproducible]. (105)

Define the Lyapunov function for model (6) as follows:

[mathematical expression not reproducible]. (106)

Then from (99), (101), (103), and (105) we have

[mathematical expression not reproducible]. (107)

If condition (96) holds, then the surface

[mathematical expression not reproducible] (108)

lies in the interior of [R.sup.3.sub.+]. Hence, we can easily obtain that there exists a constant C > 0 and a compact set K of [R.sup.3.sub.+] such that, for any x [member of] [R.sup.3.sub.+]/K,

LV(x) [less than or equal to] -C. (109)

Therefore, model (6) has a unique stationary distribution. This completes the proof.

Remark 17. In fact, the variances of errors usually should be small enough to justify their validity of real data; otherwise, the data may not be considered as a good one. It is clear that when [[sigma].sub.ij] are very small, condition (96) is always satisfied.

6. Numerical Examples

To verify the theoretical results in this paper, we next give numerical simulations of model (6).

Throughout the following numerical simulations, we choose I = 2 and g(I) = 1/(1 + [omega][I.sup.2]), where w is a positive constant. It is easy to verify that assumption ([H.sub.1]) holds. By Milstein's higher-order method [29, 30], we drive the corresponding discretization equations of model (6):

[mathematical expression not reproducible]. (110)

Here, [[xi].sub.ji] (i = 1, 2, ..., j = 1, ..., l) are N(0,1)--distributed independent Gaussian random variables and [DELTA]t > 0 is time increment.

Example 1. In model (6), we take f(S) = S/(1 + 0.2S), [LAMBDA] = 1.85, q = 0.52, [beta] = 0.52, p = 0.24, [epsilon] = 0.2, [gamma] = 0.3, [omega] = 2, [d.sub.S] = 0.4, [d.sub.I] = 0.21, [d.sub.R] = 0.3, [[sigma].sub.01] = 0.15, [[sigma].sub.02] = 0.99, [[sigma].sub.11] = 0.23, [[sigma].sub.12] = 0.17, [[sigma].sub.21] = 0.14, [[sigma].sub.22] = 0.72, [[sigma].sub.31] = 0.47, and [[sigma].sub.32] = 0.93. By computing, we obtain [[??].sub.0] = 0.8939 < 1, [[sigma].sup.2.sub.0]f([S.sub.0])g'(0) - ([beta] + [[summation].sup.2.sub.j=1][[sigma].sub.0j][[sigma].sub.2j]) = 0.3442 > 0, and ([beta] + [[summation].sup.2.sub.j=1][[sigma].sub.0j][[sigma].sub.2j])2[[sigma].sup.2.sub. 0] -([d.sub.I] + [gamma] + (1/2)[[sigma].sup.2.sub.2]) = 0.005 > 0. This shows that conditions (a) and (b) of Theorem 9 do not hold. The numerical simulations (see Figure 1) suggest that disease I(t) of model (6) is still stochastically extinct with probability one. Therefore, as an improvement of Theorem 9, we have the following interesting conjecture.

Conjecture 2. Assume ([H.sub.1]) holds. The disease I(t) in model (6) is stochastically extinct with probability one only when [[??].sub.0] < 1 holds.

Example 3. In model (6), we take f(S) = S/(1 + 1.5S), [LAMBDA] = 3, q = 0.2, [beta] = 2.1, p = 0.3, [epsilon] = 0.8, [gamma] = 0.1, [omega] = 2, [d.sub.S] = 0.5, [d.sub.I] = 0.8, [d.sub.R] = 0.4, [[sigma].sub.01] = 0.8, [[sigma].sub.02] = 1.2, [[sigma].sub.11] = 0.3, [[sigma].sub.12] = 0.75, [[sigma].sub.21] = 0.45, [[sigma].sub.22] = 0.8, [[sigma].sub.31] = 0.8, and [[sigma].sub.32] = 0.3. By computing, we obtain [[??].sub.0] = 1.3554 > 1. From the numerical simulations given in Figure 2, it is shown that disease I(t) of model (6) is not only stochastically persistent in the mean but also stochastically persistent with probability one. Therefore, as an improvement of Theorem 12, we have the following interesting conjecture.

Conjecture 4. Assume ([H.sub.1]) holds. The disease I(t) in model (6) is stochastically persistent in the mean only when [[??].sub.0] > 1.

Example 5. In model (6), we take f(S) = S/(1+0.1S), [LAMBDA] = 1.2, q = 0.5, [beta] = 1.5, p = 0.9, [epsilon] = 1.1, [gamma] = 0.9, [omega] = 2, [d.sub.S] = 0.6, [d.sub.I] = 0.35, [d.sub.R] = 0.4, [[sigma].sub.01] = 0.4, [[sigma].sub.02] = 0.2, [[sigma].sub.11] = 0.1, [[sigma].sub.12] = 0.45, [[sigma].sub.21] = 0.2, [[sigma].sub.22] = 0.1, [[sigma].sub.31] = 0.2, and [[sigma].sub.32] = 0.3. By computing, we obtain [[bar.R].sub.0] = 0.8687 < 1 and [[??].sub.0] = 1.2931 > 1. The numerical simulations given in Figure 3 show that disease I(t) of model (6) is still stochastically permanent in the mean. Therefore, combining Theorem 12 and Theorem 14, we can obtain the following interesting conjecture about the stochastic permanence in the mean of disease I(t).

Conjecture 6. Assume ([H.sub.1]) holds. The disease I(t) in model (6) is stochastically permanent in the mean only when [[??].sub.0] > 1.

Example 7. In model (6), we take f(S) = S, [LAMBDA] = 0.67, q = 0.02, [beta] = 1.7, p = 0.05, [epsilon] = 3, [gamma] = 0.99, [omega] = 4, [d.sub.S] = 0.29, [d.sub.I] = 0.53, [d.sub.R] = 0.39, [[sigma].sub.01] = 0.025, [[sigma].sub.02] = 0.02, [[sigma].sub.11] = 0.0121, [[sigma].sub.12] = 0.01, [[sigma].sub.21] = 0, [[sigma].sub.22] = 0, [[sigma].sub.31] = 0.02, and [[sigma].sub.32] = 0.01. By computing, we obtain that the basic reproduction number for deterministic model (2) is [R.sub.0] = 2.5279 > 1 and the unique endemic equilibrium of model (2) is ([S.sup.*], [I.sup.*], [R.sup.*]) = (1.4230,0.3845, 0.1372). Furthermore, we can verify that there is a constant [rho] > 0 such that [a.sub.ii](x) > p for any x [epsilon] [R.sup.3.sub.+] (i = 1,2, 3), [d.sub.I] - [d.sub.S] = 0.24 > 0, [gamma] - p = 0.94 > 0, [gamma]([d.sub.S] + [d.sub.R])- p([d.sub.I] + [d.sub.R]) = 0.6272 > 0, and

[mathematical expression not reproducible]. (111)

That is, all conditions in Theorem 16 are satisfied. The stationary distributions about the susceptible, infected, and removed individuals obtained through the numerical simulations are reported in Figure 4, which shows that after some initial transients the population densities fluctuate around the deterministic steady-state values [S.sup.*] = 1.4230, [I.sup.*] = 0.3845, and [R.sup.*] = 0.1372.

Example 8. In model (6), we take f(S) = S/(1+0.4S), [LAMBDA] = 2.5, q = 0,5, [beta] = 1.4, p = 0,7, [epsilon] = 0.9, [gamma] = 0.51, [omega] = 1.89, [d.sub.S] = 0.7, [d.sub.I] = 0.45, [d.sub.R] = 0.58, [[sigma].sub.01] = 0.4, [[sigma].sub.02] = 0.2, [[sigma].sub.11] = 0.21, [[sigma].sub.12] = 0.1, [[sigma].sub.21] = 0.1, [[sigma].sub.22] = 0.24, [[sigma].sub.31] = 0.2, and [[sigma].sub.32] = 0.1. By computing, we obtain that the basic reproduction number for deterministic model (2) is [R.sub.0] = 1.6484 > 1 and the unique endemic equilibrium of model

(2) is ([S.sup.*],[I.sup.*],[R.sup.*]) = (1.7242,0.5082,1.8352). Furthermore, we can verify that there is not a constant [rho] > 0 such that

[a.sub.ii](x) > p for any x [member of] [R.sup.3.sub.+] and i = 1,2,3, [d.sub.I] - [d.sub.S] = -0.25 < 0, [gamma] - p = -0.19 < 0,[gamma]([d.sub.S] + [d.sub.R])-p([d.sub.I] + [d.sub.R]) = -0.0682 < 0 and

[mathematical expression not reproducible]. (112)

That is, the conditions in Theorem 16 are not satisfied. However, we obtain that threshold value [[??].sub.0] = 2.7192 > 1. The numerical simulations given in Figure 5 show the stationary distributions about the susceptible, infected, and removed individuals. Therefore, we can obtain the following interesting conjecture about the stationary distribution for model (6), as described in the conclusion part.

Conjecture 9. Assume (Hi) holds. Model (6) has a unique stationary distribution only when [[??].sub.0] > 1.

7. Conclusion

In this paper, as an extension of the results given in [11, 25], we investigated the dynamical behaviors for a stochastic SIRS epidemic model (6) with nonlinear incidence and vaccination. In model (6), the disease transmission coefficient [beta] and the removal rates [d.sub.S], [d.sub.I], and [d.sub.R] are affected by noise. Some new basic properties of model (6) are found in Lemmas 2, 3, and 5. Applying these lemmas, we established a series of new threshold value criteria on the stochastic extinction, persistence in the mean, and permanence in the mean of the disease with probability one. Furthermore, by using the Lyapunov function method, a sufficient condition on the existence of unique stationary distribution for model (6) is also obtained.

The stochastic persistence and permanence in the mean of the disease for model (6) are established in this paper only for the special cases: f(S) [equivalent to] S and (1) [[sigma].sub.0j] = 0 (1 [less than or equal to] j [less than or equal to] I) or (2) [[sigma].sub.1j] = [[sigma].sub.2j] = [[sigma].sub.3j] = 0 (1 [less than or equal to] j [less than or equal to] I). However, for the general model (6), particularly, f(S) [not equal] S and ([[sigma].sub.1j], [[sigma].sub.2j], [[sigma].sub.3j]) = (0,0,0) (1 [less than or equal to] j [less than or equal to] I), whether we also can establish similar results still is an interesting open problem.

In fact, under the above case, from the proofs of Theorems 12 and 14, we can see that an important question is to deal with terms [beta]f(S(t)) and [f.sup.2](S(t))g'(I(t)). If we may get

[mathematical expression not reproducible], (113)

where [v.sub.1] and [v.sub.2] are two positive constants; then the following perfect result may be established.

Assume that ([H.sub.1]) holds. If [[??].sub.0] > 1,then disease I in model (6) is stochastically persistent in the mean; that is,

[mathematical expression not reproducible]. (114)

Another important open problem is about the existence of stationary distribution of model (6), that is, whether we can establish a similar result as in Theorem 16 when f(S) is a nonlinear function. The best perfect result on the stationary distribution is to prove that model (6) possesses a unique stationary distribution only when threshold value [[??].sub.0] > 1.But this is a very difficult open problem.

However, the numerical examples given in Section 6 propose some affirmative answer for above open problems.

http://dx.doi.org/10.1155/2017/7294761

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research is supported by the Natural Science Foundation of Xinjiang (Grant no. 2016D03022) and the National Natural Science Foundation of China (Grant nos. 11401512,11271312, and 11660176).

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Lei Wang, (1) Zhidong Teng, (2) Tingting Tang, (2) and Zhiming Li (2)

(1) Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China

(2) College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Zhidong Teng; zhidong_teng@sina.com

Received 30 July 2016; Accepted 21 November 2016; Published 16 January 2017

Caption: FIGURE 1: The path of lit) for the stochastic model (6) with parameters in Example 1, compared to the corresponding deterministic model. (a) is trajectories of the solution I(t) with the initial value I(0) = 0.05 and (b) with the initial value I(0) = 0.5. The disease of model (6) is extinct with probability one.

Caption: FIGURE 2: The paths of I(t) and (1/t) [[integral].sup.t.sub.0] I(s)ds for the stochastic model (6) with parameters in Example 3, (a) with the initial value I(0) = 0.05 and (b) with the initial value I(0) = 0.5.

Caption: FIGURE 3: The paths of I(t) and (1/t) [[integral].sup.t.sub.0]I(s)ds for the stochastic model (6) with parameters in Example 5, (a) with the initial value I(0) =0.05 and (b) with the initial value I(0) = 0.5.

Caption: FIGURE 4: The solution of stochastic model (6) and its histogram with parameters in Example 7.

Caption: FIGURE 5: The solution of stochastic model (6) and its histogram with parameters in Example 8.
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