Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Dynamic Analysis and Extinction Thresholds in a Stochastic SEIRS Model with Standard and Saturated Incidence Rates
Saravanan, S. and Monica, C.
Corresponding Email: monica.c@vit.ac.in
Received date: 21 August 2024
Accepted date: 25 April 2025
Abstract:
This study examines a stochastic SEIRS epidemic model featuring bilinear standard and saturated incidence rates, providing a more precise depiction of the interactions between susceptible and infected individuals in the population. In order to establish the validity of our mathematical strategy, we first construct a pertinent Lyapunov function to demonstrate the existence and uniqueness of a positive global solution for this SEIRS model. Next, we explore the dynamic behavior of the stochastic SEIRS model, with a specific focus on the conditions that lead to certain long-term behaviors in the system. It is worth noting that the condition $\mathscr{R}^S_0 > 1$ is enough to establish the presence of an ergodic stationary distribution, which signifies the stable persistence of the disease within the population. On the other hand, we show that the condition $\widehat{\mathscr{R}}_0^S < 1$ is essential for the disease to be eradicated, offering valuable insights into strategies for eliminating the disease. In order to validate our theoretical findings, we present a set of numerical simulations that serve to illustrate and verify our results. These simulations provide a concise and scholarly view of the theoretical constructs, emphasizing the practicality of our model in addressing real-world epidemics.
Keywords: extinction; Lyapunov function; stationary distribution; standard incidence rate; saturated incidence rate; stochastic SEIRS epidemic model
