Threshold dynamics of a stochastic general SIRS epidemic model with migration

Area of Science:

• Mathematical epidemiology
• Stochastic modeling
• Disease dynamics

Background:

• SIRS epidemic models are crucial for understanding disease transmission.
• Stochasticity and constant immigration influence epidemic dynamics.
• General incidence rates add complexity to disease spread.

Purpose of the Study:

• To investigate a stochastic SIRS epidemic model with constant immigration and general incidence.
• To determine the role of the stochastic threshold $R_0^S$ in predicting disease extinction or persistence.
• To establish conditions for the stationary distribution of the disease.

Main Methods:

• Development of a stochastic SIRS epidemic model.
• Analysis of the model's dynamical behavior using the stochastic threshold $R_0^S$.
• Determination of conditions for disease extinction, persistence, and stationary distribution.
• Validation through numerical simulations.

Main Results:

• The stochastic threshold $R_0^S$ accurately predicts disease dynamics.
• If $R_0^S < 1$, disease extinction is certain under specific conditions.
• If $R_0^S > 1$, the disease can persist, with conditions for stationary distribution identified.

Conclusions:

• The stochastic threshold $R_0^S$ is a key determinant in the long-term behavior of the SIRS model.
• The model provides insights into disease persistence and extinction in stochastic environments.
• Numerical simulations confirm the theoretical predictions regarding disease dynamics.