Analysis of a stochastic SEIS epidemic model motivated by Black-Karasinski process: Probability density function

This paper examines a stochastic SEIS epidemic model motivated by Black-Karasinski process. First, it is shown that Black-Karasinski process is a both biologically and mathematically reasonable assumption compared with existing stochastic modeling methods. By analyzing the diffusion structure of the model and solving the relevant Kolmogorov-Fokker-Planck equation, a complete characterization for explicitly approximating the stationary density function near some quasi-positive equilibria is provided. Then for the deterministic model, the basic reproduction number and related asymptotic stability are studied. Finally, several numerical examples are given to substantiate our theoretical findings.