Periodic solution and stationary distribution of a stochastic epidemic model with two different epidemics and different epidemiological frameworks

In epidemiology, more than one infectious disease can pose a risk to the host population. This area of study has attracted researchers in recent times. In this article, we have considered an epidemic model that incorporates two different transmission techniques, namely SIR and SIRS. The considered deterministic model has been perturbed stochastically at transmission rates. The analysis has been done for the resulting stochastic model. Firstly, we explore the existence of the positive $T$-periodic solution for the stochastic system. We determine that the non-autonomous periodic version of the system with white noise has a positive periodic solution using the Lyapunov function and Khasminskii theory. In addition, we analyze the positive recurrence of the system. The results obtained in this article give the idea that reducing the white noise in the stochastic model is critical for observing positive $T$-periodic solutions and positive recurrence. Finally, we present some examples and perform numerical simulations to validate the theoretical results established in this study.