A Stochastic Age-Structured SI Epidemics Model with Logarithmic Ornstein-Uhlenbeck Process

In this paper, a new stochastic age-structured SI epidemic model with logarithmic Ornstein-Uhlenbeck (OU) process is proposed. The study mainly focuses on the long-term behavior of solutions for the proposed stochastic system. For this, the existence and (global) asymptotic stability of the equilibria of the deterministic system without OU process are first discussed. Then the properties of extinction and asymptotic persistence of solutions for the stochastic system are investigated and the corresponding threshold conditions for them are obtained successfully. This research highlights the importance of considering randomness in the transmission of the epidemic, especially with complex disease mechanisms and random environmental factors. The findings here reveal that interestingly the reversion speed of the OU process impacts how quickly the system stabilizes, while the magnitude of random disturbances can alter the stability potentially and even cause a resurgence of an infectious disease which is expected to go extinct.