A Note on the Extinction of a Stochastic Differential Equation SIS Epidemic Model

TL;DR

This paper provides a simpler and more direct proof of the extinction conjecture for a stochastic differential equation SIS epidemic model, originally proposed by Gray et al. and proved by Xu.

Contribution

It introduces a new, more straightforward proof of the extinction conjecture for the stochastic SIS model, improving understanding of the model's behavior.

Findings

The proof confirms the extinction behavior of the stochastic SIS model.

The new proof simplifies the mathematical approach to the conjecture.

Supports the validity of the extinction conjecture in stochastic epidemic models.

Abstract

The aim of this note is to give a new proof to the conjecture, proposed by Gray et al. (2011), on the extinction of a stochastic differential equation SIS epidemic model. The conjecture was proved by Xu (2017). Our proof is much more direct and simpler.

Equations

$$\{\begin{array}{rl}& dS(t)=[\mu N-\beta S(t)I(t)+\gamma I(t)-\mu S(t)]dt-\sigma S(t)I(t)dB(t),\\ & dI(t)=[\beta S(t)I(t)-(\mu +\gamma )I(t)]dt+\sigma S(t)I(t)dB(t).\end{array}$$

$$\begin{array}{r}dI(t)=[(\beta N-\mu -\gamma )I(t)-\beta I^2(t)]dt+\sigma (N-I(t))I(t)dB(t).\end{array}$$

$$\begin{array}{r}(i)R_0^S<1and{\sigma }^2\le \frac{\beta }{N}or(ii){\sigma }^2>\frac{\beta }{N}\vee \frac{{\beta }^2}{2(\mu +\beta )},\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\mathrm{limsup}}\frac{\log I(t)}{t}\le (\mu +\gamma )(R_0^S-1)a.s.\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\mathrm{limsup}}\frac{\log I(t)}{t}\le -\frac{{\sigma }^2}{2}{\left(\frac{\mu +\gamma }{\beta }\right)}^{2}a.s.\end{array}$$

$$\begin{array}{rl}I(t)-I(0)=& (\beta N-\mu -\gamma ){\int }_0^{t}I(s)ds-\beta {\int }_0^t{I}^2(s)ds\\ & +\sigma {\int }_0^t(N-I(s))I(s)dB(s),\end{array}$$

$$\begin{array}{r}⟨I^2(t)⟩=\left(N-\frac{\mu +\gamma }{\beta }\right)⟨I(t)⟩+\psi (t),\end{array}$$

$$\begin{array}{r}⟨I(t){⟩}^2\le \left(N-\frac{\mu +\gamma }{\beta }\right)⟨I(t)⟩+\psi (t).\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\lim }\frac{\sigma }{\beta t}{\int }_0^t(N-I(s))I(s)dB(s)=0a.s.\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\lim }\psi (t)=0a.s.\end{array}$$

$$\begin{split}\Big{(}\mathop{\lim\sup}\limits_{t\rightarrow+\infty}\langle I(t)\rangle\Big{)}^{2}\leq\left(N-\frac{\mu+\gamma}{\beta}\right)\mathop{\lim\sup}\limits_{t\rightarrow+\infty}\langle I(t)\rangle~{}~{}a.s.\end{split}$$

$$\begin{array}{rl}d\log I(t)=& \beta N-\mu -\gamma -\beta I(t)-\frac{1}{2}{\sigma }^2(N-I(t){)}^2+\sigma (N-I(t))dB(t)\\ =& \beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2+({\sigma }^{2}N-\beta )I(t)-\frac{1}{2}{\sigma }^2{I}^2(t)+\sigma (N-I(t))dB(t).\end{array}$$

$$\begin{array}{rl}\frac{\log I(t)-\log I(0)}{t}=& \beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2+({\sigma }^{2}N-\beta )⟨I(t)⟩-\frac{1}{2}{\sigma }^2⟨I^2(t)⟩\\ & +\frac{\sigma }{t}{\int }_0^t(N-I(s))dB(s).\end{array}$$

$$\begin{array}{r}\frac{\log I(t)}{t}=\left(\beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2\right)+\left[\frac{{\sigma }^2}{2}\left(N+\frac{\mu +\gamma }{\beta }\right)-\beta \right]⟨I(t)⟩+\Psi (t),\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\lim }\frac{\sigma }{t}{\int }_0^t(N-I(s))dB(s)=0a.s.\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\lim }\Psi (t)=0a.s.\end{array}$$

$$\begin{array}{r}\underset{t\to +\infty }{\mathrm{limsup}}\frac{\log I(t)}{t}\le \beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2=(\mu +\gamma )(R_0^S-1)a.s.\end{array}$$

$$\begin{array}{rl}\underset{t\to +\infty }{\mathrm{limsup}}\frac{\log I(t)}{t}& =\left(\beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2\right)+\left[\frac{{\sigma }^2}{2}\left(N+\frac{\mu +\gamma }{\beta }\right)-\beta \right]\underset{t\to +\infty }{\mathrm{limsup}}⟨I(t)⟩\\ & \le \left(\beta N-\mu -\gamma -\frac{1}{2}{\sigma }^2{N}^2\right)+\left[\frac{{\sigma }^2}{2}\left(N+\frac{\mu +\gamma }{\beta }\right)-\beta \right]\left(N-\frac{\mu +\gamma }{\beta }\right)\\ & =-\frac{{\sigma }^2}{2}{\left(\frac{\mu +\gamma }{\beta }\right)}^{2}a.s.\end{array}$$

Full text

Keywords. Stochastic differential equation; Epidemic model; Extinction.

2010 MSC. 34F05, 37H10, 60H10, 92D25, 92D30.

1 Introduction

Epidemic models are inevitably affected by environmental white noise which is an important component in realism. In fact, the presence of a noise source can modify the behavior of corresponding deterministic evolution of the system (cf.[1, 2, 3, 4, 5]). For this reason, many stochastic epidemic models have been developed, see for instance [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and references therein. However, compared to deterministic systems, it is extremely difficult to give the basic reproduction number of stochastic systems. Recently, Gray et al. [5] studied the following stochastic SIS system:

[TABLE]

Here $S$ and $I$ denote the susceptible and infected fractions of the population, respectively. $\mu$ is the per capita death rate, and $\gamma$ is the rate at which infected individuals become cured, $\beta$ is the disease transmission coefficient. $B(t)$ is the Brownian motion with $B(0)=0$ and with the intensity $\sigma^{2}>0$ . Given that $S(t)+I(t)\equiv N$, they simplified system (1.1) into the following single equation:

[TABLE]

They firstly showed that for any given initial value $I(0)=I_{0}\in(0,N)$, system (1.2) has a unique global solution $I(t)\in(0,N)$ for all $t\geq 0$ with probability one. Furthermore, they studied the extinction and the persistence of system (1.2), and proved that if either

[TABLE]

where $R_{0}^{S}=\frac{\beta N}{\mu+\gamma}-\frac{\sigma^{2}N^{2}}{2(\mu+\gamma)}$, then the disease will die out, whereas if $R^{S}_{0}>1$, then the disease will persist. Naturally, they proposed the following conjecture (see Conjecture 8.1 in [5]):

Conjecture  If $R_{0}^{S}<1$ and $\frac{\beta^{2}}{2(\mu+\gamma)}\geq\sigma^{2}>\frac{\beta}{N}$, then the disease will die out.

Recently, Xu [17] proved this conjecture by using Feller’s test (cf. Appendix A in [17]). The aim of the present paper is to give a new proof of the conjecture, which is much more direct and simpler. Our main result is as follows.

Theorem 1.1**.**

Assume that $I(t)$ is the solution of system (1.2) with the initial value $I(0)=I_{0}\in(0,N)$. We have

(i)  If either $\frac{\sigma^{2}}{2}\left(N+\frac{\mu+\gamma}{\beta}\right)\leq\beta$ or $N\leq\frac{\mu+\gamma}{\beta}$, then

[TABLE]

(ii)  If $\frac{\sigma^{2}}{2}\left(N+\frac{\mu+\gamma}{\beta}\right)>\beta$ and $N>\frac{\mu+\gamma}{\beta}$, then

[TABLE]

Namely, $I(t)$ goes to zero exponentially a.s. if $R_{0}^{S}<1$. In other words, the disease will die out with probability one.

2 Proof of Theorem 1.1

For convenience we introduce the notation: $\langle x(t)\rangle=\frac{1}{t}\int_{0}^{t}x(s)ds.$

The following lemma will play an important role in the proof of Theorem 1.1.

Lemma 2.1**.**

Assume that $I(t)$ is the solution of system (1.2) with the initial value $I(0)=I_{0}\in(0,N)$. We have

(i)  If $N\leq\frac{\mu+\gamma}{\beta}$, then $\lim\limits_{t\rightarrow+\infty}\langle I(t)\rangle=0~{}~{}a.s.$

(ii)  If $N>\frac{\mu+\gamma}{\beta}$, then $\mathop{\lim\sup}\limits_{t\rightarrow+\infty}\langle I(t)\rangle\leq N-\frac{\mu+\gamma}{\beta}~{}~{}a.s.$

Proof.

Integrating (1.2) from [math] to $t$ yields

[TABLE]

and dividing $t$ on both sides of (2.1) gives

[TABLE]

where $\psi(t)=\frac{I(0)-I(t)}{\beta t}+\frac{\sigma}{\beta t}\int_{0}^{t}(N-I(s))I(s)dB(s)$. By the Hölder inequality, we have

[TABLE]

By the large number theorem for martingales (cf. [5, 2]), we have

[TABLE]

This leads to

[TABLE]

If $N\leq\frac{\mu+\gamma}{\beta}$, then the conclusion (i) follows from (2.3) and (2.4) immediately.

If $N>\frac{\mu+\gamma}{\beta}$, we deduce from (2.3) and (2.4) that

[TABLE]

This implies the conclusion (ii), and ends the proof of Lemma 2.1. ∎

Now we can give the proof of Theorem 1.1.

Proof of Theorem 1.1  By the Itô formula, we deduce from (1.2) that

[TABLE]

Integrating (2.5) from [math] to $t$ and dividing $t$ on both sides of the resulting equation, we obtain

[TABLE]

Substituting (2.2) into (2.6) yields

[TABLE]

where $\Psi(t)=\frac{\log I(0)}{t}+\frac{\sigma}{t}\int_{0}^{t}(N-I(s))dB(s)-\frac{\sigma^{2}}{2}\psi(t).$ By the large number theorem for martingales (cf. [5, 2]), we have

[TABLE]

This, together with (2.4), leads to

[TABLE]

If either $\frac{\sigma^{2}}{2}\left(N+\frac{\mu+\gamma}{\beta}\right)\leq\beta$ or $N\leq\frac{\mu+\gamma}{\beta}$, by Lemma 2.1 (i) and (2.8), we derive from (2.7) that

[TABLE]

If $\frac{\sigma^{2}}{2}\left(N+\frac{\mu+\gamma}{\beta}\right)>\beta$ and $N>\frac{\mu+\gamma}{\beta}$, by (2.8) and Lemma 2.1 (ii), we derive from (2.7) that

[TABLE]

The proof of Theorem 1.1 is completed.

Acknowledgments

The research was supported by the NSFC (11571062), the Program for Liaoning Innovative Talents in University (LR2016004), and the Fundamental Research Fund for the Central Universities (DMU).