Abstract: This paper studies a class of two-disease epidemic models that incorporate both SIS and SIRS transmission mechanisms. First,the local asymptotic stability of the four local equilibrium points in the deterministic model is rigorously established,and the global stability of the disease-free equilibrium is proved under the condition that the basic reproduction numbers satisfy R₁<1 and R₂<1. Subsequently, the corresponding stochastic model is formulated,for which the existence and uniqueness of a global positive solution are proved. Furthermore,sufficient conditions for disease extinction and persistence in the mean are derived. Specifically,when the stochastic reproduction number, the disease is shown to become extinct,whereas if, the disease is persistent in the mean. Finally, theoretical results are validated through numerical simulations.

Key words: double epidemic, epidemic model, extinction, persistence in mean, Lyapunov function