# The Threshold of a Stochastic SIRS Model with Vertical Transmission and Saturated Incidence.

1. Introduction

Mathematical models can describe the progress of a disease and predict the trend of the disease and they can provide the theoretical basis for people to undertake prevention strategies. At present, researches have constructed a series of mathematical models [1-10], including SIRS model [5, 9,10]. They divided people into susceptible, S, infective, I, and removed, R, categories and one of the most famous SIRS epidemic models is the following:

[mathematical expression not reproducible] (1)

Here, [LAMBDA] is the birth rate, [beta] is the transmission rate, b and [mu] are the birth rate and natural death rate, respectively, of S, I, and R individuals, a is the death rate of the disease, and [gamma] is the recovery rate of the infective individuals. Assume [mu] > b.

Medical research has shown that the herpes virus will be in the form of mother-to-child transmission (vertical transmission) to the baby. In addition, since susceptible individuals in contact with every infective individual are limited, we see that when the number of the susceptible individuals is large, the bilinear incidence [beta]SI is unreasonable to consider. In this case, saturated incidence is more suitable than bilinear incidence .

In this paper, the transmission rate is chosen as the saturated incidence rate [beta]SI/(1 + [alpha]I), and the SIRS model is described as the following:

[mathematical expression not reproducible] (2)

In system (2), q is vertical transmission rate, [R.sub.0] = [beta][LAMBDA]/([mu]-b)([mu] = [gamma] + a - qb) is the threshold which determines whether the disease will die out or persist, and there always is a disease-free equilibrium [P.sup.0] = ([LAMBDA]/([mu] - b), 0, 0); see [5, 9,10] and the references therein. When [R.sub.0] < 1, the disease-free equilibrium [P.sup.0] is globally asymptotically stable; when [R.sub.0] > 1, [P.sup.0] is unstable and there is an endemic equilibrium [mathematical expression not reproducible], which is globally stable under a sufficient condition.

However, all parameters in system (2) are affected by environmental noise, so it is of benefit to use a stochastic model. Stochastic models are more realistic compared to deterministic models. Many stochastic models for epidemic populations have been studied [12-19]. Tornatore et al.  studied an stochastic SIR model. They showed that under the condition 0 < [beta] < min{[gamma] + [mu] - [[sigma].sup.2]/2, 2[mu]}, the disease-free equilibrium is locally stable, but the authors do not discuss under which condition the disease will prevail. Concerning the transmission coefficient p, Gray et al.  considered the stochastic SIS (susceptible-infective-susceptible) epidemic model with fluctuation. They proved threshold [R.sub.0] which determines the extinction and persistence of I(t) according to the fluctuation. Here, [R.sub.0] is the threshold of the deterministic model; however, it is more difficult to get the threshold of the stochastic model.

We consider certain stochastic environmental factors and assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the parameter [beta], as in ,

[beta] [right arrow] [beta] + [??](t), (3)

where B(t) is standard Brownian motion with B(0) = 0 and [sigma] is the intensity. The stochastic version corresponding to the deterministic model (2) is the following:

[mathematical expression not reproducible] (4)

Throughout this paper, unless otherwise specified, let ([OMEGA], [{[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 (i.e., it is right continuous and [{[F.sub.t]}.sub.t[greater than or equal to]0] contains all P-null sets) and let B(t) be the Brownian motion defined on the probability space.

For simplicity, define

<x(t)> = [1/t] [[integral].sup.t.sub.0] x(r)dr. (5)

2. Existence and Uniqueness of the Nonnegative Solution

In this section, we will show that there is a unique positive solution of system (4).

Theorem 1. For any initial value (S(0), 1(0), R(0)) in [R.sup.3.sub.+], there is a unique solution (S(t), I(t), R(t)) of system (4) on t [greater than or equal to] 0, and the solution will remain in [R.sup.3.sub.+] with probability 1.

Proof. Since the coefficients of system (4) are locally Lipschitz continuous, for any initial value (S(0), I(0), R(0)) [member of] [R.sup.3.sub.+], there is a unique local solution on [0, [[tau].sub.e]), where [[tau].sub.e] is the explosion time. To show that this solution is global, we need to have [[tau].sub.e] = [infinity] a.s. To show that this solution is global, we need to have [[tau].sub.e] = [infinity] a.s. Let [m.sub.0] [greater than or equal to] 0 be sufficiently large so that (S(0), 1(0), R(0)) all lie in the interval [1/[m.sub.0], [m.sub.0]]. For each integer m [greater than or equal to] [m.sub.0], define the stopping time

[mathematical expression not reproducible] (6)

where throughout this paper we set inf [phi] = [infinity] (as usual [phi] denotes the empty set). Clearly, [[tau].sub.m] is increasing as m [right arrow] [infinity]. Set [[tau].sub.[infinity]] = [lim.sub.t[right arrow]+[infinity]] [[tau].sub.[infinity]] [less than or equal to] [[tau].sub.e] a.s. If we can show that [[tau].sub.[infinity]] = [infinity] a.s., then [[tau].sub.[infinity]] = [infinity] and (S(0), I(0), R(0)) [member of] [R.sup.3.sub.+] a.s. for all t [greater than or equal to] 0. In other words, to complete the proof all we need to show is that [[tau].sub.[infinity]] = [infinity] a.s. If this statement is false, then there is a pair of constants T > 0 and [epsilon] [member of] (0,1) such that P{[[tau].sub.[infinity]] [less than or equal to] T} > [epsilon].

Hence there is an integer [m.sub.1] [greater than or equal to] [m.sub.0] such that

P{[[tau].sub.[infinity]] [less than or equal to] T} > [epsilon], [for all]m [greater than or equal to] [m.sub.1]. (7)

Besides, the total biomass N(t) = S(t) + I(t) + R(t) of model (4) satisfies the following equation:

dN (t) = [[LAMBDA] - ([mu] - b)N- aI] dt. (8)

It is easy to know that, for all t < [[tau].sub.[infinity]],

[mathematical expression not reproducible] (9)

Define a [C.sup.2]-function V : [R.sup.3.sub.+] [right arrow] [[bar.R].sub.+] by

V(S, I, R) = -log [S/[K.sub.1]] - log [I/[K.sub.1]] - log [R/[K.sub.1]]. (10)

By using Ito's formula, we get

dV = LVdt + [sigma] (S-I)/1 + [sigma]I, (11)

where

[mathematical expression not reproducible] (12)

The remainder of the proof follows that in Li and Mao [21, Theorem 2.1].

Since there is a positive solution (S(t), I(t), R(t)) [greater than or equal to] 0 of system (4) for any given initial value (S(0), 1(0), R(0)) [member of] [R.sup.3.sub.+] and

[mathematical expression not reproducible] (13)

is an invariant set , then from now on, we can assume that (S(0), I(0), R(0)) [member of] [GAMMA].

3. Extinction

In this section, we discuss the conditions for the extinction of the disease.

Theorem 2. Let (S(t), I(t), R(t)) be the solution of system (4) with initial value (S(0), I(0), R(0)) [member of] [GAMMA]. If [[??].sub.0] < 1 and [[sigma].sup.2] [greater than or equal to] [beta](u - b)/[LAMBDA] then

[mathematical expression not reproducible]. (14)

Namely, I(t) tends to zero exponentially a.s. where

[mathematical expression not reproducible]. (15)

Proof. Notice that S/(1 + [alpha]I) [member of] (0([LAMBDA]/([mu] - b))]; that is, [[sigma].sup.2] [less than or equal to] [beta])([mu] - b)/[LAMBDA] the quadratic function

f (z) = - [[[sigma].sup.2]/2] [z.sup.2] + [beta]z - ([mu] + a + [gamma] - qb) (16)

gets its maximum value [f.sub.max] on the interval [0, [LAMBDA]/(u - b)] at z = [LAMBDA]/([mu]- b), where

[mathematical expression not reproducible] (17)

It follows from the monotonicity of the function [beta]S/(1 + [alpha]I) on [0, [LAMBDA]([mu] - b)] that when [[sigma].sup.2] [less than or equal to] [beta](u - b)/[LAMBDA], we get

[mathematical expression not reproducible] (18)

Applying Ito's formula to system (4) leads to

[mathematical expression not reproducible] (19)

Integrating this from 0 to t and diving t on both sides, we have

[mathematical expression not reproducible] (20)

where M(t) := [[integral].sup.t.sub.0] ([sigma]S(r)/(1 + [alpha]I(r)))dB(r), which is local martingale and M(0) = 0. Moreover,

[mathematical expression not reproducible] (21)

According to the large number theorem for martingales (see, e.g., ), we obtain

[mathematical expression not reproducible]. (22)

Then

[mathematical expression not reproducible]. (23)

The proof of Theorem 2 is completed.

Remark 3. We also notice that if [beta]/[[sigma].sup.2] [member of] (0([GAMMA]/(u - b))], that is, [[sigma].sup.2] [greater than or equal to] [beta]([mu] - b)/[LAMBDA], the quadratic function f(z) obtains the maximum value [[beta].sup.2]/2[[sigma].sup.2] - ([mu] + [gamma] + a - qb) at z = [beta]/[[sigma].sup.2]. The proof of Theorem 2 proceeded; we have

[mathematical expression not reproducible] (24)

if [[sigma].sup.2] > [[beta].sup.2]/2([mu]+ a + [gamma] - qb). It means that the disease will always die out if the noise is large enough such that [[sigma].sup.2] > max{[beta]([mu] - b)/[GAMMA], [[beta].sup.2]/2 ([mu] + a + [gamma] - qb)}.

Remark 4. From Theorem 2, we can get that the disease will die out if [[??].sub.0] < 1 and the white noise is not large such that [[sigma].sup.2] [less than or equal to] [beta] ([mu] - b)/[LAMBDA]. Meanwhile if white noise is large enough such that [[[sigma].sup.2] > max{[beta]([mu]-b)/[LAMBDA], [[beta].sup.2]/2([mu] + a + [gamma] - qb)} is satisfied, then the disease will also die out. Moreover, we notice [[??].sub.0] is smaller than the threshold of the corresponding deterministic model. The following examples illustrate this result more explicitly.

From Theorem 2, it is obvious that under some conditions

[mathematical expression not reproducible] (25)

which implies

[mathematical expression not reproducible]. (26)

Next, according to system (4), we get

[mathematical expression not reproducible] (27)

which implies that

[mathematical expression not reproducible] (28)

Together with (13), we have

[mathematical expression not reproducible]. (29)

According to (26) and the last equation of system (4), we obtain

[mathematical expression not reproducible]. (30)

Therefore, by (29), we have

[mathematical expression not reproducible] (31)

We have the following theorem by combing these arguments.

Theorem 5. Let (S(t), I(t), R(t)) be the solution of system (4) with initial value (S(0), I(0), R(0)) [member of] [GAMMA]. If

(i) [mathematical expression not reproducible]

(ii) [[sigma].sup.2] > max{[beta]([mu] - b)/[LAMBDA], [[beta].sup.2]/2([mu] + a + [gamma] - qb)}

then

[mathematical expression not reproducible] (32)

Example 6. We assume that the unit of time is one day and the population sizes are measured in units of 1 million. The parameters in system (4) are chosen as follows:

[mathematical expression not reproducible] (33)

Note that

[[??].sub.0] = 0.6742 < 1, (34)

[[sigma].sup.2] = 0.25 [less than or equal to] [beta]([mu] - b)/[LAMBDA] = 0.4 (35)

and then by Theorem 5(i), the solution (S(t), I(t), R(t)) of system (4) obeys

[mathematical expression not reproducible] (36)

with any initial value (S(0), I(0), R(0)) = (0.6, 0.4, 0.5) [member of] [GAMMA]. Then the disease tends to zero exponentially with probability one.

On the other hand, for the responding deterministic SIRS model, [R.sub.0] = 1.7978 > 1; the disease will prevail. Using the method mentioned in , we provide the simulations shown in Figure 1 to support our results.

Example 7. We choose the same values of the parameters of (33) except for the value of [[sigma]. Because [[sigma].sup.2] > [beta])/[LAMBDA] = 0.4, [[sigma].sup.2] > [[beta].sup.2]/2([mu] + [gamma] + a - qb) = 0.3595, we let [sigma] = 0.7; then by Theorem 5(ii), the solution of system (4) obeys [mathematical expression not reproducible] lim [sup.sub.t[right arrow]t[infinity]] (logI(t)/t) [less than or equal to] - ([micro] + [gamma] + a - qb) + [[beta].sup.2]/2[[sigma].sup.2] = -0.2369 < 0 a.s., in which case the disease will also die out. Note that [R.sub.0] = 1.2584 > 1, so that the disease for the corresponding system will prevail too (see Figure 2).

4. Persistence

Definition 8. System (4) is said to be persistent in the mean if

[mathematical expression not reproducible] (37)

Theorem 9. If [[??].sub.0] > 1, then for any initial value (S(0), 1(0), R(0)) [member of] [GAMMA], the solution (S(t), I(t), R(t)) of system (4) has the following property:

[mathematical expression not reproducible] (38)

Proof. An integration of system (4) yields

[mathematical expression not reproducible] (39)

According to (39), we have

[mathematical expression not reproducible] (40)

After simple computing

[mathematical expression not reproducible] (41)

where [mathematical expression not reproducible].

Let X = (([mu] + [epsilon])([mu] + a + [gamma] - qb) - [gamma](b + [epsilon]))/([mu] + [epsilon])([mu] - b).

By Ito's formula, we obtain

[mathematical expression not reproducible] (42)

and then

[mathematical expression not reproducible] (43)

Then the inequality can be rewritten as

[mathematical expression not reproducible] (44)

By (29), then taking the limit inferior of both sides (44) leads to

[mathematical expression not reproducible]. (45)

This completes the proof of Theorem 9.

Example 10. Choose the parameters in system (4) as follows:

[mathematical expression not reproducible] (46)

Note that

[mathematical expression not reproducible] (47)

and then by Theorems 2, with any initial value (S(0), 1(0), R(0)) = (0.6,0.4,0.5) [member of] [GAMMA], we get that the solution of system (4) obeys

[mathematical expression not reproducible]. (48)

Using the method mentioned in , we give the simulations shown in Figure 3, and we find the disease is persistent.

Example 11. We consider the same values of the parameters of (46) except a. Change [sigma] to 0.06 to further illustrate the effect of the noise intensity [sigma] on this SIRS model. Notice that [[??].sub.0] = 1.7816 > 1 and [[sigma].sup.2] = [(0.06).sup.2] [less than or equal to] [beta]([mu] - b)/[LAMBDA] = 0.4 Using the method mentioned in , we give the simulations shown in Figure 4 to support our results. And we find the fluctuation of the solution of system (4) is becoming weaker compared with the picture in Figure 3.

5. Conclusion

In this paper, we study a stochastic SIRS model with vertical transmission and saturated incidence. We find that the extinction or persistence of the disease is mainly determined by the value of [[??].sub.0], and it is obvious that [[??].sub.0] < [R.sub.0] when [sigma] [not equal to] 0. We find when the noise is so small (in fact [[sigma].sup.2] < [beta]([mu] - b)/[LAMBDA]), the extinction and persistence are completely determined by [mathematical expression not reproducible] < 1, the disease dies out; if [[??].sub.0] > 1, the disease prevails. So the noise is adverse to the survival of the disease. We also find that if the noise is so large that [[sigma].sup.2] > max{[beta]([mu]-b)/[LAMBDA], [[beta].sup.2]/ 2([mu]+ a + [gamma] - qb)}, the disease prevails too.

To understand how the noise effects the survival of the disease better, the conditions in Theorem 2 can be written as

[mathematical expression not reproducible] (49)

and rewrite the condition [[??].sub.0] > 1 in Theorem 5 as follows: [sigma] < [??].

We obtain the main results as follows:

(a) If [R.sub.0] < 1, the disease will always die out without considering the existence of the noise according to (15).

(b) If [R.sub.0] > 1, the disease will persist for the deterministic model (2) while for the stochastic model (4), the disease will persist when [sigma] < [??] and die out when [??] < [sigma] [less than or equal to] (1/[LAMBDA]) [square root of ([beta][LAMBDA]([mu]- b))]. The existence of noise determines the extinction of the disease.

(c) If [mathematical expression not reproducible], the disease will always die out.

From (b) we get that when [R.sub.0] > 1, [??] = [sigma] is the threshold which determines the extinction and persistence of the disease for stochastic system (4). Meanwhile, we can obtain from cases (b) and (c) that if

[mathematical expression not reproducible] (50)

then disease will die out when [sigma] > [??]; meanwhile if

[mathematical expression not reproducible] (51)

the disease will die out only for [mathematical expression not reproducible].

https://doi.org/10.1155/2017/5620301

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The research was supported by the Project for Science and Technology in Shaoguan (2014CX/K231) and the High-Level Talents Project of Guangdong Province Colleges and Universities (2013-178).

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Chunjuan Zhu, Guangzhao Zeng, and Yufeng Sun

School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China

Correspondence should be addressed to Yufeng Sun; suny2001@sina.com

Received 27 January 2016; Accepted 15 November 2016; Published 28 February 2017

Academic Editor: Manuel De la Sen

Caption: Figure 1: Simulations of the path (S(t), I(t), R(t)) for the corresponding deterministic system (2) and stochastic system (4) with initial value (S(0), 1(0), R(0)) = (0.6, 0.4, 0.5), [sigma] = 0.5.

Caption: Figure 2: Simulations of the path S(t), I(t), R(t) for the corresponding deterministic system (2) and stochastic system (4) with initial value (S(0), 1(0), R(0)) = (0.6, 0.4, 0.5), [sigma] = 0.7.

Caption: Figure 3: Simulations of the path S(t), I(t), R(t) for the corresponding deterministic system (2) and stochastic system (4) with initial value (S(0), 1(0), R(0)) = (0.6, 0.4, 0.5), [sigma] = 0.2.

Caption: Figure 4: Simulations of the path S(t), I(t), R(t) for the corresponding deterministic system (2) and stochastic system (4) with initial value (S(0), 1(0), R(0)) = (0.6, 0.4, 0.5), [sigma] = 0.06.
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