A novel SVIR epidemic model with jumps for understanding the dynamics of the spread of dual diseases

The emergence of multi-disease epidemics presents an escalating threat to global health. In response to this serious challenge, we present an innovative stochastic susceptible-vaccinated-infected-recovered epidemic model that addresses the dynamics of two diseases alongside intricate vaccination strategies. Our novel model undergoes a comprehensive exploration through both theoretical and numerical analyses. The stopping time concept, along with appropriate Lyapunov functions, allows us to explore the possibility of a globally positive solution. Through the derivation of reproduction numbers associated with the stochastic model, we establish criteria for the potential extinction of the diseases. The conditions under which one or both diseases may persist are explained. In the numerical aspect, we derive a computational scheme based on the Milstein method. The scheme will not only substantiate the theoretical results but also facilitate the examination of the impact of parameters on disease dynamics. Through examples and simulations, we have a crucial impact of varying parameters on the system's behavior.

Stationary distribution and extinction of a stochastic HIV/AIDS model with nonlinear incidence rate

This paper studies a stochastic HIV/AIDS model with nonlinear incidence rate. In the model, the infection rate coefficient and the natural death rates are affected by white noise, and infected people are affected by an intervention strategy. We derive the conditions of extinction and permanence for the stochastic HIV/AIDS model, that is, if $ R_0^s < 1, $ HIV/AIDS will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; if $ R_0^s > 1 $, HIV/AIDS will be persistent almost surely and there exists a unique stationary distribution. The conclusions are verified by numerical simulation.