# The asymptotic behavior of a stochastic SIS epidemic model with vaccination

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## Abstract

In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number \(R_{0}\). When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of \(R_{0}\). Moreover, we get the same exponential stability and the convergence if \(R_{0}<1\). When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if \(R_{0}>1\). Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations.

## Keywords

stochastic SIS epidemic model vaccination exponential stability persistent in mean Lyapunov function## MSC

34F05 37H10 60H10 92D25 92D30## 1 Introduction

Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence. Significant progress has been made in the theory and application of epidemiology modeling by mathematical research. In a simple epidemic model, there is generally a threshold, \(R_{0}\). If \(R_{0} \leq1\), the disease-free equilibrium is a unique equilibrium in this type of epidemic model and it is globally asymptotically stable; if \(R_{0}>1\), this type of model has also a unique endemic equilibrium, which is globally asymptotically stable. Therefore, the threshold \(R_{0}\) determines the extinction and persistence of the epidemic.

Controlling infectious diseases has been an increasingly complex issue in recent years. Vaccination is an important strategy for the elimination of infectious diseases [1, 2, 3]. The vaccination enables the vaccinated to acquire a permanent or temporary immunity. When the immunity is temporary, the immunity can be lost after a period of time. It is used in many references [4, 5, 6, 7] where one assumes the process of losing immunity is in the exponential form.

*t*, \(I(t)\) denotes the number of infected individuals at time

*t*and \(V(t)\) denotes the number of members who are immune to an infection as the result of vaccination at time

*t*. The parameters in the model are summarized in the following list:

*A*: the constant input of new members into the population per unit time;

*q*: the fraction of vaccinated new-borns;

*β*: transmission coefficient between compartments *S* and *I*;

*μ*: the natural death rate of the *S*, *I*, *V* compartments;

*p*: the proportionality coefficient of vaccinated cases for the susceptible;

*γ*: the recovery rate of infectious individuals;

*ε*: the rate of losing their immunity for vaccinated individuals;

*α*: the disease-caused death rate of infectious individuals.

All parameter values are assumed to be nonnegative and \(\mu,A> 0\).

However, the deterministic approach has some limitations in the mathematical modeling of the transmission of an infectious disease and it is quite difficult to predict the future dynamics of a system accurately. This happens due to the fact that deterministic models do not incorporate the effect of a fluctuating environment. Stochastic differential equation models play a significant role in various branches of applied sciences including infectious dynamics, as they provide some additional degree of realism compared to their deterministic counterpart. In reality, parameters involved in the modeling approach of ecological systems are not absolute constants, and they always fluctuate around some average value due to continuous fluctuations in the environment. As a result, parameters in the model never attain a fixed value with the advancement of time and rather exhibit a continuous oscillation around some average values. Many authors have introduced parameters of random perturbation into epidemic models and have studied their dynamics.

*β*are considered in [8, 9, 10, 11, 12]. Tornatore

*et al.*in [9] studied the following stochastic SIR model with no delay:

*β*. When \(R_{0}\leq1\), one deduced the globally asymptotic stability of the disease-free equilibrium \(P_{0}\), which meant the disease would die out. When \(R_{0}>1\) and the perturbation was small, by measuring through the difference between the solution and the endemic equilibrium of the deterministic model in time average, they derived that the disease would persist. However, the authors did not consider the case when the perturbation would be large. Besides, Gray

*et al.*in [12] discussed the following stochastic SIS model:

Jing Fu *et al.* introduced stochasticity into a multigroup SIS model in [25]. They presented the sufficient condition for the exponential extinction of the disease and proved that the noises significantly raise the threshold of a deterministic system. In the case of persistence, they proved that there exists an invariant distribution which is ergodic.

In [26], Golmankhaneh *et al.* applied the homotopy analysis method (HAM) successfully for solving second-order random differential equations, homogeneous or inhomogeneous. Expectation and variance of the approximate solutions were computed. Several numerical examples were presented to show the ability and efficiency of this method. Jafarian *et al.* in [27] solved linear second kind Fredholm and Volterra integral equations systems by applying the Bernstein polynomials expansion method. Illustrative examples were provided to demonstrate the preciseness and effectiveness of the proposed technique.

*α*are small, they showed that there is a stationary distribution and it is ergodic as \(R_{0} > 1\) and the asymptotic behavior of the solution around the disease-free equilibrium \(P_{0}\) as \(R_{0} \leq1\). Here \(R_{0}\) is the reproductive number of the deterministic model.

*β*, as in [12],

This paper is organized as follows. In Section 2, we show there is a unique positive solution of system (1.3) for any positive initial value. In Section 3, we show that the positive solution of system (1.3) converges to \(P_{0}\) exponentially as the perturbation is large. In Section 4, we get the same exponential stability and the convergence when \(R_{0}<1\). On the other hand we investigate the asymptotic behavior of the solution of the system (1.3) according to \(R_{0} > 1\) although the solution of system (1.3) does not converge to \(P^{*}\). When the perturbation is not large, we consider the disease to persist. Moreover, we show system (1.3) is persistent in the mean. The key to our analysis is choosing appropriate Lyapunov functions. In Section 5, we make simulations to conform our analytical results. In Section 6, we give a short conclusion. Finally, in order to be self-contained, we have an Appendix containing a lemma used in the previous sections.

## 2 Existence and uniqueness of positive solution

To investigate the dynamical behavior, the first concern is whether the solution has a global existence. Moreover, for a population dynamics model, whether the value is positive is also considered. Hence in this section we first show that the solution of system (1.3) is global and positive. As we know, in order to get a stochastic differential equation which has a unique global (*i.e.* no explosion in finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (*cf.* [20]). However, the coefficients of system (1.3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (1.3) may explode at a finite time. In this section, using the Lyapunov analysis method (mentioned in [11, 12, 13]), we show the solution of system (1.3) is positive and global.

### Theorem 2.1

*There is a unique solution* \((S(t),I(t),V(t))\) *of system* (1.3) *on* \(t\geqslant0\) *for any initial value* \((S(0),I(0),V(0))\in\mathbb{R}_{+}^{3}\), *and the solution will remain in* \(\mathbb{R}_{+}^{3}\) *with probability* 1, *namely*, \((S(t),I(t),V(t))\in\mathbb{R}_{+}^{3}\) *for all* \(t\geqslant0\) *almost surely*.

### Proof

*k*,

*k*or \(1/k\), and hence \(W(S(\tau_{k},\omega),I(\tau _{k},\omega),V(\tau_{k},\omega))\) is not less than

### Remark 2.1

From now on, we always assume that \((S(0),I(0),V(0))\in\Gamma^{*}\).

## 3 Exponential stability in large perturbation

In this section, we show that the large perturbation forces the number of infected cases to zero exponentially regardless of the magnitude of \(R_{0}\).

### Theorem 3.1

*Let*\((S(t),I(t),V(t))\)

*be the solution of system*(1.3)

*with initial value*\((S(0),I(0), V(0))\in \Gamma^{\ast}\).

*If*\(\sigma^{2}>{\beta^{2}}/{2(\mu+\gamma+\alpha)}\),

*then*

*Moreover*,

*namely*, \(I(t)\)

*tends to zero exponentially a*.

*s*.

*i*.

*e*.,

*the disease dies out with probability*1).

### Proof

*t*and dividing

*t*on the both sides, we have

### Remark 3.1

When the perturbation is very large, the number of infected decays exponentially to zero. It makes sense in the point that the extinction of epidemics can be caused by the occurrence of a large perturbation.

## 4 Asymptotic behavior around \(P_{0}\) and \(P^{*}\)

When studying epidemic dynamical systems, we are interested in two problems. One is when the disease will die out, and the other is when the disease will persist. In the section, we shall investigate the two problems according to the threshold \(R_{0}\).

### 4.1 Asymptotic behavior around \(P_{0}\)

When \(R_{0}\leqslant1\), \(P_{0}\) of system (1.1) is globally stable, which means the disease will die out. We shall show in Theorem 4.1 below, the same exponential stability of system (1.3), obtained in Theorem 3.1, continues valid when \(R_{0} < 1\). Moreover, in this case, we prove the solution to (1.3) converges to \(P_{0}\), a.s.

### Theorem 4.1

*Let*\((S(t),I(t),V(t))\)

*be the solution of system*(1.3)

*with initial value*\((S(0),I(0), V(0))\in\Gamma ^{\ast}\).

*If*\(R_{0}<1\),

*then*

*Moreover*,

### Proof

### 4.2 Asymptotic behavior around \(P^{*}\)

In the deterministic models, the second problem is usually solved by showing that the endemic equilibrium to the corresponding model is a global attractor or is globally asymptotically stable under some conditions. But there is no endemic equilibrium of system (1.3). How can one measure whether the disease will persist? In this section, we show that the difference between the solution of system (1.3) and \(P^{*}\) is small if white noise is weak, reflecting that the disease is prevalent.

### Theorem 4.2

*Assume*\(R_{0}>1\).

*Let*\((S(t),I(t),V(t))\)

*be the solution of system*(1.3)

*with initial value*\((S(0),I(0),V(0))\in\Gamma^{*}\).

*If*\(\alpha^{2}<4\mu(\mu+\alpha )(1+{2\mu}/{\varepsilon})\),

*then*

*where*\(P^{*}=(S^{*},I^{*},V^{*})\)

*is the endemic equilibrium of system*(1.1),

*ρ*

*is positive constant satisfied*\({\alpha}/{4(\mu+\alpha )(1+{2\mu}/{\varepsilon})}<\rho<{\mu}/{\alpha}\).

### Proof

*L*be the generating operator of system (1.3). Then we get

*ρ*is positive constant to be specified later and Young’s inequality is used. If

*ρ*such that \({\alpha}/{4(\mu+\alpha)(1+{2\mu}/{\varepsilon})}<\rho<{\mu }/{\alpha}\). This implies

*t*yields

### Remark 4.1

*C*is a positive constant. Although the solution of system (1.3) does not have stability as the deterministic system, we can think there is approximate stability, provided \(\|\sigma \|^{2}\) is sufficiently small. In this situation, we consider the disease to persist.

From the result of Theorem 4.2, we conclude system (1.3) is persistent, which also reflects that the disease is prevalent. Chen *et al.* in [21] proposed the definition of persistence in the mean for the deterministic system. Here, we also use this definition for the stochastic system.

### Definition 4.1

### Theorem 4.3

*Assume*\(R_{0}>1\).

*Let*\((S(t),I(t),V(t))\)

*be the solution of system*(1.3)

*with initial value*\((S(0),I(0),V(0))\in\Gamma^{\ast}\).

*If*\(\alpha^{2}<4\mu(\mu+\alpha )(1+{2\mu}/{\varepsilon})\)

*and*

*where*

*ρ*

*is a positive constant satisfying*\({\alpha}/{4(\mu+\alpha)(1+{2\mu }/{\varepsilon})}<\rho<{\mu}/{\alpha}\),

*then system*(1.3)

*is persistent in the mean*.

### Proof

*i.e.*

## 5 Numerical simulations

## 6 Conclusions

This paper studies the extinction and persistence in the mean of a stochastic SIS epidemic model with vaccination. When the perturbation is very large, the number of infected decays exponentially to zero, which means the disease will die out. A similar conclusion to the system (1.2) is obtained in Theorem 4.3 in [12], but there one simplified system (1.2) into a single equation without vaccination and \(\alpha=0\). The present paper is the first attempt, to the best of our knowledge, of such a study of a large perturbation that surpasses the effect of \(R_{0}\) as a threshold value in a high-dimensional system. Theorem 4.1 showed \(P_{0}\) is exponentially stable if \(R_{0}<1\). The result is better than the asymptotic stability of \(P_{0}\) in Theorem 3.1 in [11]. From Theorems 4.2 and 4.3, when the perturbation and the disease-related death rate are small, if \(R_{0}>1\), we see that the disease will persist in the mean. In such a case, \(R_{0}\) plays a role similar to the threshold of the deterministic model. Hence, a large perturbation surpasses the effect of \(R_{0}\) as a threshold value, and a small perturbation retains some role of \(R_{0}\) in a stochastic sense. Also, it is interesting to study the threshold \(\tilde{R}_{0}\) of a stochastic SIR model, and these investigations are in progress.

## Notes

### Acknowledgements

The work was supported by Program for NSFC of China (No: 11371085,11426060), and the Fundamental Research Funds for the Central Universities (No: 15CX08011A), the Scientific and Technological Research Project of Jilin Province’s Education Department (No: 2012244) and the Education Science Research Project of Jilin Province (No: GH150104).

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