We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible–infectious–recovered–susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible–Infectious–Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results.
Greenhalgh, D., Liang, Y., & Mao, X. (2016). Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching. Physica A: Statistical Mechanics and Its Applications, 462, 684–704. https://doi.org/10.1016/j.physa.2016.06.125