SIS Epidemic Model Birth-and-Death Markov Chain Approach

We are interested in describing the infected size of SIS Epidemic model using Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) is defined within a population constant $M$; kept by replacing each death with newborn healthy individual. life span individual modelled an exponential distribution parameter $\alpha$; and disease spreads Poisson process rate $\lambda_{I}$. $\lambda_{I}=\beta I(1-\frac{I}{M}) $ similar to instantaneous logistic growth model. analysis focused on outbreak, where reproduction number $R=\frac{\beta}{\alpha} greater than one. As methodology, we use both numerical analytical approaches. relies stationary for Birth Death approach creates sample path simulations into order show dynamics, relationship between $R$. $M$ becomes large, some stable statistical characteristics can be deduced. And shown analytically follow normal mean $(1-\frac{1}{R}) M$ Variance $\frac{M}{R} $.