Dynamics of a Variable Perturbed Double Epidemic Stochastic Model With Saturated Incidence Rates

ABSTRACT Recently, there have been many communicable diseases reported worldwide. Studying a variety of these diseases is therefore necessary, particularly double‐ and multi‐epidemics with fluctuating environmental factors. This paper addresses the existence of the positive solution of an epidemic model with two separate hypotheses, namely, the susceptible‐infected‐recovered (SIR) and susceptible‐infected‐recovered‐susceptible (SIRS) transmission mechanisms, to focus on this particular case. In this instance, white noise describes the surrounding fluctuations, and a saturated incidence rate carries the disease(s) that are being transmitted. Sufficient conditions are investigated for the disease(s) to persist and eventually go extinct. An ergodic stationary distribution is established under specific sufficient conditions for the considered scenario. Lastly, a few numerical examples are provided to support the theory developed.