Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity

Area of Science:

• Mathematical epidemiology
• Stochastic modeling
• Dynamical systems theory

Background:

• Previous models like susceptible-infected-recovered-susceptible (SIRS) lack realistic quarantine and stochasticity.
• Temporary immunity after recovery is a key factor in disease dynamics.
• Logistic growth influences population dynamics in epidemic models.

Purpose of the Study:

• To analyze a stochastic susceptible-infected-quarantined-recovered-susceptible (SIQRS) epidemic model with temporary immunity.
• To determine the conditions for disease persistence and the existence of a unique ergodic stationary distribution.
• To derive the explicit expression for the stationary distribution's probability density function.

Main Methods:

• Application of Khas'minskii theory and Lyapunov function approach.
• Construction of a critical value related to the basic reproduction number.
• Development of solving theories for a four-dimensional Fokker-Planck equation.
• Analysis of stochastic differential equations and ergodic theory.

Main Results:

• A critical value $$ is established, determining disease persistence.
• A unique ergodic stationary distribution exists when $$ .
• The explicit density function of the stationary distribution is obtained under $$ .
• Numerical simulations validate the theoretical findings.

Conclusions:

• The existence of a unique stationary distribution reveals disease persistence dynamics.
• The model provides insights into the interplay of quarantine, temporary immunity, and stochasticity.
• Findings contribute to understanding disease spread in both epidemiological and statistical contexts.