A Markovian regime-switching stochastic hybrid time-delayed epidemic model with vaccination

Abstract

This paper considers a stochastic hybrid time-delayed epidemic model with vaccination perturbed by both white noise and telegraph noise in the form of Markovian switching. Using Lyapunov method, we show existence and uniqueness of the global positive solution. Under some suitable conditions, we derive extinction with and without effect of delay. Moreover, the asymptotic bounds on solutions is also discussed. Finally, we carry out the numerical simulations, including two examples based on real life diseases to confirm our theoretical results.

Introduction

Mathematical deterministic epidemic models have played a crucial role in understanding, predicting and controlling real-life disease outbreaks. However, in the natural world, epidemic models are always affected by the environmental noise. So, it is necessary to reveal how the environmental noise affects epidemic models. Stochastic models can acquire more real benefits and can predict the dynamics of the system accurately. In this paper, we consider a stochastic epidemic model, which regards the deterministic transmission rate $\beta$ to change randomly by $\beta +\sigma dB\left(t\right)/dt$, where $\sigma$ is the standard deviation of white noise and $B\left(t\right)$ is a Brownian motion defined on a complete probability space $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t⩾0},\mathbb{P})$ with a filtration ${\left\{{\mathcal{F}}_t\right\}}_{t⩾0}$ satisfying the usual conditions. In addition to white noise, epidemic models are also subject to telegraph noise which switches the system from one environmental state to another (Boukanjime et al., 2020, Greenhalgh et al., 2016, Lahrouz and Settati, 2013, Liu et al., 2018, Luo and Mao, 2007). Therefore, the state transition can be modelled using a continuous time Markov chain ${\left(r\left(t\right)\right)}_{t⩾0}$ that takes the values in the finite state space $\mathbb{S}=\{1,2,\dots ,N\}$ (see e.g. Mao and Yuan, 2006, Yin and Zhu, 2010 and the references therein). Various types of work have been dedicated for the SIR model (Anderson and May, 1991, Hethcote, 2000), which consists of system with three compartments: susceptible individuals ($S$), infective individuals ($I$) and recovered individuals ($R$). However, as far as we know, there is no research on the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate (Beddington, 1975, DeAngelis et al., 1975) and effect of delay (El Fatini and Boukanjime, 2020, Liu et al., 2016) under regime switching in the existing literature. So, introducing Beddington–DeAngelis functional response and motivated by previous works, we are interested to study a stochastic SIR model perturbed by both white noise and telegraph noise in the form of Markovian switching asfollows

$$\left\{\begin{array}{c}dS\left(t\right)=[A\left(r\left(t\right)\right)-\frac{\beta \left(r\left(t\right)\right)S\left(t\right)I\left(t\right)}{1+{\alpha }_1\left(r\left(t\right)\right)S\left(t\right)+{\alpha }_2\left(r\left(t\right)\right)I\left(t\right)}\hfill \\ -(\mu \left(r\left(t\right)\right)+v\left(r\left(t\right)\right))S\left(t\right)\hfill \\ +v\left(r\left(t\right)\right)S(t-\tau )e^{-\mu \left(r\left(t\right)\right)\tau }]dt\hfill \\ -\frac{\sigma \left(r\left(t\right)\right)S\left(t\right)I\left(t\right)}{1+{\alpha }_1\left(r\left(t\right)\right)S\left(t\right)+{\alpha }_2\left(r\left(t\right)\right)I\left(t\right)}dB\left(t\right),\hfill \\ dI\left(t\right)=[\frac{\beta \left(r\left(t\right)\right)S\left(t\right)I\left(t\right)}{1+{\alpha }_1\left(r\left(t\right)\right)S\left(t\right)+{\alpha }_2\left(r\left(t\right)\right)I\left(t\right)}\hfill \\ -(\mu \left(r\left(t\right)\right)+{\mu }_1\left(r\left(t\right)\right)+\lambda \left(r\left(t\right)\right))I\left(t\right)]dt\hfill \\ +\frac{\sigma \left(r\left(t\right)\right)S\left(t\right)I\left(t\right)}{1+{\alpha }_1\left(r\left(t\right)\right)S\left(t\right)+{\alpha }_2\left(r\left(t\right)\right)I\left(t\right)}dB\left(t\right),\hfill \\ dR\left(t\right)=[v\left(r\left(t\right)\right)S\left(t\right)-v\left(r\left(t\right)\right)S(t-\tau )e^{-\mu \left(r\left(t\right)\right)\tau }\hfill \\ +\lambda \left(r\left(t\right)\right)I\left(t\right)-\mu \left(r\left(t\right)\right)R\left(t\right)]dt,\hfill \end{array}\right.$$

where $A$ denotes a constant input of new members into the population, $\mu$ represents the natural death rate, ${\mu }_1$ denotes the disease caused death rate of infectious individuals, $v$ stands for the proportional coefficient of vaccinated for the susceptible. The parameter $\beta$ is the transmission coefficient between compartments $S$ and $I$, $\lambda$ describes the recovery rate of infected people, $\tau >0$ is a time delay representing the validity period of the vaccination. The parameters ${\alpha }_1$ and ${\alpha }_2$ are the saturation factors measuring the psychological or inhibitory effect.

We arrange the rest of this paper as follows. Section 2 introduces some preliminaries that will be used in our following analysis. In Section 3, we exhibit the existence and uniqueness of the global positive solution. Section 4 gives sufficient conditions for the extinction of disease with and without effect of delay. Section 5 is about exploring the asymptotic bounds on solutions. The paper is concluded with numerical simulations to illustrate the main results.