A mathematical study of the influence of media on the asymptotic dynamics of diseases

This study explores the asymptotic behavior of a stochastic epidemic model that accounts for the impact of media coverage. The initial focus lies on determining the conditions leading to the exponential extinction of the disease. Additionally, we investigate the weak convergence of the probability distribution of the stochastic process $N\left(t\right)$, representing the total population, to a one-dimensional stochastic process with density calculated through the Fokker–Planck equation. Subsequently, we demonstrate the persistent nature of the disease and utilize Has’minskii theory to establish the presence of a unique ergodic stationary distribution for our stochastic epidemic model. Finally, numerical simulations are conducted to validate the theoretical findings.