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