Stochastic Dynamics of an Epidemic-Behavior Model

We incorporate stochastic fluctuations into an epidemic-behavior coevolution model and study its threshold dynamics and stationary distributions. By introducing the stochastic basic reproduction number R₀^σ, the behavioral threshold d, and two potential infection proportions R_P^σ,1 and R_P^σ,2, we establish conditions for disease extinction and persistence. For persistent dynamics, we derive explicit stationary densities and associated mean infection levels on the two behavioral boundaries. When the dynamics do not converge to either boundary, we establish a unique ergodic stationary distribution and derive a local bivariate log-normal approximation near the quasi-equilibrium. Comparison with the deterministic counterpart reveals that stochastic perturbations reduce the basic reproduction number as well as the two potential infection proportions, indicating that stochasticity may induce disease extinction, while widespread adoption of self-protective behaviors is required only when disease transmission is strong. Numerical simulations are performed to illustrate the theoretical results, and model fitting to COVID-19 data further demonstrates the relevance of the proposed framework to real epidemic scenarios.