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

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:

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  is unreasonable to consider.In this case, saturated incidence is more suitable than bilinear incidence [11].
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][13][14][15][16][17][18][19].Tornatore et al. [18] studied an stochastic SIR model.They showed that under the condition 0 <  < min{ +  −  2 /2, 2}, the disease-free equilibrium is locally stable, but the authors do not discuss under which condition the disease will prevail.Concerning the transmission coefficient , Gray et al. [20] considered the stochastic SIS (susceptible-infective-susceptible) epidemic model with fluctuation.They proved threshold  0 which determines the extinction and persistence of () according to the fluctuation.Here,  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 , as in [20], where () is standard Brownian motion with (0) = 0 and  is the intensity.The stochastic version corresponding to the deterministic model ( 2) is the following: Throughout this paper, unless otherwise specified, let (Ω, {F  } ≥0 , ) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is right continuous and {F  } ≥0 contains all -null sets) and let () be the Brownian motion defined on the probability space.For simplicity, define

Existence and Uniqueness of the Nonnegative Solution
In this section, we will show that there is a unique positive solution of system (4).
Proof.Since the coefficients of system (4) are locally Lipschitz continuous, for any initial value ((0), (0), (0)) ∈  3 + , there is a unique local solution on [0,   ), where   is the explosion time.To show that this solution is global, we need to have   = ∞ a.s.To show that this solution is global, we need to have   = ∞ a.s.Let  0 ≥ 0 be sufficiently large so that ((0), (0), (0)) all lie in the interval [1/ 0 ,  0 ].For each integer  ≥  0 , define the stopping time where throughout this paper we set inf  = ∞ (as usual  denotes the empty set).Clearly,   is increasing as s. for all  ≥ 0. In other words, to complete the proof all we need to show is that  ∞ = ∞ a.s.If this statement is false, then there is a pair of constants  > 0 and  ∈ (0, 1) such that Hence there is an integer Besides, the total biomass () = () + () + () of model ( 4) satisfies the following equation: It is easy to know that, for all  <  ∞ , Define a  2 -function V : By using Itô's formula, we get where The remainder of the proof follows that in Li and Mao [21, Theorem 2.1].

Extinction
In this section, we discuss the conditions for the extinction of the disease.
Remark 3. We also notice that if / We have the following theorem by combing these arguments.
Then the disease tends to zero exponentially with probability one.
On the other hand, for the responding deterministic SIRS model,  0 = 1.7978 > 1; the disease will prevail.Using the method mentioned in [23], we provide the simulations shown in Figure 1 to support our results.

Persistence
e deterministic system e stochastic system e deterministic system e stochastic system e deterministic system e stochastic system 0.5  and then by Theorems 2, with any initial value ((0), (0), (0)) = (0.6, 0.4, 0.5) ∈ Γ, we get that the solution of system (4) obeys lim inf →+∞ ⟨ ()⟩ ≤ 0.2146 a.s. ( Using the method mentioned in [23], 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 .Change  to 0.06 to further illustrate the effect of the noise intensity  on this SIRS model.Notice that R0 = 1.7816 > 1 and  2 = (0.06) 2 ≤ ( − )/Λ = 0.4 Using the method mentioned in [14], 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.

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 R0 , and it is obvious that R0 <  0 when  ̸ = 0. We find when the noise is so small (in fact  2 < ( − )/Λ), the extinction and persistence are completely determined by R0 : if R0 < 1, the disease dies out; if R0 > 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  2 > max{(−)/Λ,  2 /2(++−)}, 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 and rewrite the condition R0 > 1 in Theorem 5 as follows:  < σ.