Extinction and persistence of a stochastic SIRV epidemic model with nonlinear incidence rate

In this paper, a stochastic SIRV epidemic model with general nonlinear incidence and vaccination is investigated. The value of our study lies in two aspects. Mathematically, with the help of Lyapunov function method and stochastic analysis theory, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. In other words, neglecting random perturbations overestimates the ability of the disease to spread. The numerical simulations are given to illustrate the main theoretical results.

In fact, our real life is full of randomness and stochasticity. For human disease related epidemics, the nature of epidemic growth and spread is random due to the unpredictability in person to person contacts. Because of environmental noises, the deterministic approach has some limitations in the mathematical modeling transmission of an infectious disease, and several authors began to consider the effect of white noise on the epidemic systems. In order to improve the understanding of the difference of random environmental fluctuations, many scholars have introduced white noise in deterministic models .
There are different approaches used in the literature to introduce random perturbations into population models, both from a mathematical and biological perspective. One is to perturb the positive endemic equilibria in order to make the equilibria of deterministic models robust. In this situation, the essence of the investigation using the approach is to check if the asymptotic stability of the positive equilibria of deterministic models can be preserved. The other important approach is with parameter perturbation. Many literature works on this approach can be found, for example, [25][26][27][28][29]. In epidemic models, the natural death rate d and the disease transmission parameter β are two of the key parameters to disease transmission. May in [37] pointed out that all the parameters involved in the population model exhibit random fluctuation as the factors controlling them are not constant. And in the real situation, the natural death rate d and the disease transmission parameter β always fluctuate around some average value due to continuous fluctuation in the environment. In this sense, d can seem as a random variabled, β changes to a random variableβ. More precisely, each infected individual makes -d dt = -d dt + σ dB(t), β dt = β dt + σ dB(t) potentially infectious contacts with each other individual in [t, t + dt).
In recent years, the stochastic SIV and SIRV type epidemic models have been extensively studied, and many important results have been established, see, for example, articles [18][19][20][21][22][23][24][40][41][42][43] and the references cited therein. We easily see that most of these research works aim at the models with bilinear incidence, and there exists some research on the models with special nonlinear incidences (see [19][20][21]). Particularly in [20], the authors studied a class of stochastic SIVS epidemic models with nonlinear saturated incidence: A threshold valueR 0 is identified. It is shown that ifR 0 < 1, then the disease in the stochastic model is extinct, and ifR 0 > 1, then any solution with positive initial value in R 3 + is permanent in the mean. In [21], the authors studied a class of stochastic SIVR epidemic models where vaccination is included and such that the immunity is permanent, respectively: The sufficient conditions on the exponential stability in mean square of disease-free equilibrium are obtained.
Motivated by the above work, in this paper, we consider the following deterministic SIRV epidemic model with nonlinear incidence rate and disease-induced mortality: (1.1) In model (1.1), the basic reproduction number R 0 = β ∂f (S 0 ,0) ∂I μ+γ +α is a threshold which completely determines the persistence or extinction of the disease. It is shown that, if R 0 ≤ 1, then the disease-free equilibrium E 0 is globally asymptotically stable, and if R 0 > 1, then model (1.1) has a unique endemic equilibrium E * which is globally asymptotically stable. Now, we assume that the random effects of the environment make the transmission coefficient β of the disease in deterministic model (1.1) generate random disturbance. That is, β → β + σḂ(t), where B(t) is a one-dimensional standard Brownian motion defined on some probability space. Thus, model (1.1) will become the following stochastic SIRV epidemic model with nonlinear incidence rate: The biological meaning of all parameters in (1.2) is the same as that in system (1) in [18]. All parameter values are assumed to be nonnegative and λ, μ > 0. The portion (0 ≤ ≤ 1) is vaccinated, whereas the rest (1 -) remains in the susceptible class.
As well as we know, in modeling the dynamics of epidemic systems the incidence rate is an important substance. In practice 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, [30][31][32][33][34][35][38][39][40][41][42][43][44]). However, we see that stochastic epidemic models with nonlinear incidence and vaccination are barely studied. Therefore, our first question is: Can we also establish a series of similar results on the extinction (i.e., disease-free) or persistence (i.e., endemic) of the disease for the stochastic SIRV epidemic model with nonlinear incidence and vaccination?
The main focus of this article is to investigate how environment fluctuations affect disease dynamics through studying the global dynamics of an SIRV model with nonlinear incidence in both the deterministic and the corresponding stochastic version. The organization of this paper is as follows. In Sect. 2, we give some useful lemmas and fundamental assumption for general nonlinear incidence functions. In Sect. 3, the results on the extinction of the disease with probability one are stated and proved. In Sect. 4, we prove that the disease is persistent under one condition. In Sect. 5, the numerical simulations are presented. Finally, in Sect. 6, a conclusion is given.

Preliminaries
For any integrable function g(t) defined for t ≥ 0, we denote g t = 1 t t 0 g(s) ds. The initial condition for model (1.2) is given by where (S 0 , I 0 , R 0 , V 0 ) ∈ R 4 + . Moreover, for a nonlinear function f (S, I) in model (1.2), we always introduce the following assumptions (see [34]).
(H) The function f (S, I) is two-order continuously differentiable for any S, I ≥ 0 with S + I > 0, strictly monotone increasing for S ≥ 0 for any fixed I > 0, and monotone in-creasing for I > 0 for any fixed S ≥ 0. Moreover, the functionξ (S, I) f (S,I) I is bounded and monotone decreasing for I > 0 for any fixed S ≥ 0, and f (0, I) = f (S, 0) = 0 for all S, I > 0.

Lemma 2.2 For any initial value
has a unique solution (S(t), I(t), R(t), V (t)) with initial condition (2.1) defined for all t ≥ 0, and the solution remains in R 4 + with probability one for any t ≥ 0. Furthermore, Proof Since the coefficients of model (1.2) are locally Lipschitz continuous, by the fundamental theory of stochastic differential equations, for any initial value (S 0 , where τ e is the explosion time (see [36]). Consequently, Therefore, we further have, for any t ∈ [0, τ e ), To show that the solution is global, we only need to prove that τ e = ∞ a.s. Let k 0 ≥ 0 be large enough such that (S 0 , For each integer k ≥ k 0 , define the following stopping time: Throughout this paper, we set inf Ø=∞, where Ø denotes the empty set. It is clear that τ k is increasing as k → ∞. Set τ ∞ = lim k→∞ τ k , then τ ∞ ≤ τ e a.s. Namely, we need to show that τ ∞ =∞ a.s. Assume that there exist a pair of constants T > 0 and ∈ (0, 1) such that P{τ ∞ ≤ T} > . Then there is an integer k 1 ≥ k 0 such that, for all k ≥ k 1 , Define a C 2 -function as follows: The nonnegativity of V (t) can be seen from u -1 -log u ≥ 0 for u ≥ 0. Using Itô's formula (see [45]), we obtain where Clearly, we further have Integrating (2.6) from 0 to T ∧ τ k and then taking expectations, we can obtain Set k = {τ k ≤ T} for k ≥ k 1 , then P{ k } ≥ by (2.5). Noticing that, for every ω ∈ k , there is at least one of S(τ k , ω), In view of (2.7) and (2.8), we have Therefore, we must have τ ∞ = ∞ a.s.. Furthermore, since (2.3) and (2.4) hold for all t ∈ [0, ∞), we can obtain . This competes the proof.
The proof of Lemma 2.2 shows that is globally attractive and positive invariant with respect to model (1.2) with probability one. Therefore, in the following discussions we can assume that the initial value (S 0 , I 0 , R 0 , V 0 ) ∈ for any solution (S(t), Furthermore, Proof From (2.2) and model (1.2), we can obtain and (2.14) Combining with (2.12), (2.13), and (2.14), we can obtain where   Next, we prove (2.10), from (2.2) and model (1.2) we further have Consequently, Therefore, Thus, we finally obtain (2.10). This completes the proof.

Extinction of the disease
Define the parameter where we can easily see that R 0 is the basic reproduction number of deterministic model (1.1). On the extinction of the disease in probability for model (1.2) we have the following result. Proof Using Itô's formula to ln I(t), and then integrating from 0 to t for any t > 0, we can obtain

S(s), I(s) dB(s). (3.2)
If condition (a) holds, we further have S(s), I(s) dB(s), By the mean value theorem, we havẽ where ξ is situated between S 0 and S. Since ∂ζ (S) and if S ≤ S 0 , then ξ ∈ (S, S 0 ), we also have

Permanence in the mean
On the permanence in the mean with probability one for model (1.2) we have the following result.

f S(s), I(s) dB(s).
Taking t → ∞ and the strong law of large numbers, we further have Secondly, from (4.10) and integrating the third equation of model (1.2) from 0 to t, we can obtain Lastly, integrating the last equation of model (1.2), we obtain Therefore, taking t → ∞, we finally have lim inf This completes the proof.

Numerical simulation
In this section we analyze the stochastic behavior of model (1.2) by means of the numerical simulations in order to make readers understand our results better. The numerical simulation method can be found in [32]. The corresponding discretization system of model (1.2) is given as follows: where ξ k (k = 1, 2, . . .) are the Gaussian random variables which follow the standard normal distribution N(0, 1).  Fig. 1.a), it is clear that the endemic equilibrium E * is also globally asymptotically stable if only the basic reproduction number R 0 is greater than one. For the corresponding stochastic model (1.2), we haveR 0 = 0.975 < 1, which is the case of Theorem 3.1. From the numerical simulations, we see that the disease will die out (see Fig. 1.b).

Conclusion
Environmental noises have a critical influence on the development of an epidemic. In this paper, we study the dynamics of a stochastic SIRV model with general nonlinear incidence rate. We assume that the stochastic perturbation is a white noise type which perturbs the disease transmission coefficient β. This is a well-established way of introducing stochastic environmental noise into biologically realistic population dynamic models that were used in [44,46].
The value of our study lies in two aspects: Mathematically, we show that the global dynamics of deterministic model (1.1) can be governed by its reproduction number R 0 , while the dynamics of its stochastic version (1.2) is seen to be governed by R 0 . In addition, we have provided the analytic results on the existence of the global positive solution, the extinction (i.e.,disease-free) or persistence (i.e.,endemic) of the disease for stochastic model (1.2).
Epidemiologically, we summarize our main findings as follows: 1. Noise can suppress the disease outbreak: Theorem 3.1 (a) indicates that the extinction of the disease in stochastic model (1.2) occurs if the basic reproduction number R 0 = βζ (S 0 ) μ+γ +α -σ 2 (ζ (S 0 )) 2 2(μ+γ +α) < 1. We show that deterministic model (1.1) admits a unique endemic equilibrium E * which is globally asymptotically stable if its basic reproduction number R 0 > 1. Notice that R 0 < R 0 , and it is possible that R 0 < 1 < R 0 . This is the case when deterministic model (1.1) has an endemic (see Fig. 1a) while stochastic model (1.2) has disease extinction with probability one (see Fig. 1b). This implies that large noise intensities can inhibit the spread of a disease, which means the random perturbations can change the disease dynamics.
2. The effects of the intensity of noise level: From part (b) of Theorem 3.2, under large noise intensity case, i.e., the condition σ 2 > β 2 2(μ+γ +α) holds, the disease will become extinct exponentially. In other words, in the case of sufficiently large noise, we should use the stochastic model rather than the deterministic model to describe the population dynamics (see Fig. 1.c). One can know that if R 0 > 1, model(1.1) admits a globally stable endemic equilibrium E * . In this case, when the noise intensity σ is small enough to imply that R 0 > 1 from Furthermore, from Theorems 3.1, 4.1 and numerical simulation results (e.g., Figs. 1-4), we can conclude that, when the intensity of noise is small, the stochastic model preserves the property of the global stability. In this case, we can ignore noise and use the deterministic model to approximate the population dynamics. However, the large intensity of noise can force the solution of model (1.2) to oscillate strongly around the disease-free or endemic points, or the extinction. In these cases, we cannot ignore the effect of noise and, therefore, we cannot use the deterministic model but the stochastic model to describe the disease dynamics.
Some interesting topics deserve further investigations. SDEs are being increasingly used in a wide range of areas, for example, finance and biology. There has recently been a large explosion in the number of papers using SDEs to model how diseases spread. However, these papers introduce stochasticity in a different way by parameter perturbation, which is appropriate if one of the parameters is a random variable. Another way to introduce stochasticity into deterministic models is telegraph noise where the parameters switch from one set to another according to a Markov switching process. Therefore, we may study a stochastic version of model (1.2) including Markovian switching into all parameters. These studies are in progress.