Deterministic and stochastic SAIU epidemic models with general incidence rate

Abstract

In this study, we develop a SAIU epidemic model with general incidence rate in deterministic and stochastic systems. To begin with, the existence and local asymptotically stability of equilibriums are discussed in deterministic system. Considering the interference of environmental noise on the spread of infectious diseases, we use logarithmic Ornstein–Uhlenbeck process to describe the random phenomenon. Moreover, we show the existence of stationary distribution in stochastic system by constructing several Lyapunov functions and give sufficient conditions for the extinction of the disease. Furthermore, we investigate the exact local expression of the density function of the stochastic model around the unique local equilibrium. In the end, we employ numerical simulations to support theoretical results and compare deterministic and stochastic environments.