Dynamical Behavior of a Stochastic Epidemiological Model: Stationary Distribution and Extinction of a SIRS Model with Stochastic Perturbations

TL;DR

This work addresses the long-term dynamics of a stochastic SIRS model subject to environmental noise. It formulates the system as Itô SDEs with multiplicative noise and proves global existence and positivity of solutions, establishing a well-posed stochastic framework. It then shows that a stationary distribution exists under small noise via Khasminskii-type criteria and proves ergodicity near the endemic equilibrium when $\mathscr{R}_0>1$, while deriving a noise-induced extinction threshold: if $\bigl(\alpha+\mu+\gamma\bigr) + \frac{\sigma_2^2}{2} > \frac{\beta^2}{2\sigma_4^2}$, the infectious class decays to zero almost surely. Numerical simulations using the Milstein scheme corroborate the theory, demonstrating extinction in high-noise regimes and persistence/ergodic behavior in low-noise settings. Overall, the results highlight how stochastic fluctuations can suppress disease transmission even when the deterministic model would predict persistence, informing public health risk assessment under uncertainty.

Abstract

This paper deals with a new epidemiological model of SIRS with stochastic perturbations. The primary objective is to establish the existence of a unique non-negative nonlocal solution. Using the basic reproduction number $\mathscr{R}_0$ derived from the associated deterministic model, we demonstrate the existence of a stationary distribution in the stochastic model. In addition, we study the fluctuation of the unique solution of the deterministic problem around the disease-free equilibrium under certain conditions. In particular, we reveal scenarios where random effects induce disease extinction, contrary to the persistence predicted by the deterministic model. The theoretical insights are complemented by numerical simulations, which provide further validation of our findings.

Figures (8)

• Figure 1: Transmission dynamics of the disease for $\mathscr{R}_0 < 1$, with $\sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 = 0$.
• Figure 2: Transmission dynamics of the disease for $\mathscr{R}_0 > 1$, with $\sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 = 0$.
• Figure 3: Transmission dynamics of the disease for $\mathscr{R}_0 < 1$. Group ($a$): $\sigma_1 = \sigma_2 = \sigma_3 = 0$ and $\sigma_4 = 0.01$. Group ($b$): $\sigma_1 = \sigma_2 = \sigma_3 = 0$ and $\sigma_4 = 0.03$.
• Figure 4: Transmission dynamics of the disease for $\mathscr{R}_0 > 1$. Group ($a$): $\sigma_1 = \sigma_2 = \sigma_3 = 0$ and $\sigma_4 = 0.01$. Group ($b$): $\sigma_1 = \sigma_2 = \sigma_3 = 0$ and $\sigma_4 = 0.03$.
• Figure 5: Transmission dynamics of the disease for $\mathscr{R}_0 < 1$. Group ($a$): $\sigma_1 = 0.01$, $\sigma_2 = 0.02$, $\sigma_3 = 0.03$ and $\sigma_4 = 0$. Group ($b$): $\sigma_1 = 0.03$, $\sigma_2 = 0.02$, $\sigma_3 = 0.01$ and $\sigma_4 = 0$.

Theorems & Definitions (16)

• Theorem 1
• proof
• Lemma 1: Liu2012
• Remark 1
• Theorem 2
• proof
• Remark 2
• Theorem 3
• proof
• Theorem 4