  We study the phase transition from the persistence phase to the extinction
phase for the SIRS (susceptible/ infected/ refractory/ susceptible) model of
diseases spreading on the networks. We derive an analytical expression of the
probability for the descendants nodes to re-infect their ancestors nodes. We
find that, in the case of the recovery time $\tau_R$ is larger than the
infection time $\tau_I$, the infection will flow directionally from the
ancestors to the descendants however, the descendants will not able to reinfect
their ancestors during their infection time. This behavior leads us to deduce
that, for this case and when the infection rate $\lambda$ is high enough in
such that, any infected node on the network infects all of its neighbors during
its infection time, SIRS model on the network evolves to extinction state,
where all the nodes on the network become susceptible. Moreover, we assert
that, in order to the infection occurs repeatedly inside the network, the loops
on the network are necessary, which means the clustering coefficient will play
an important role for this model. Hence, unlike the other models such as SIS
model and SIR model, SIRS model has a two critical threshold which separate the
persistence phase from the extinction phase when $\tau_I<\tau_R$. That means,
for fixed values of $\tau_I$ and $\tau_R$ there are a two critical points for
infection rate $\lambda_1$ and $\lambda_2$, where epidemic persists in between
of those two points. We confirm those results numerically by the simulation of
regular one dimensional SIRS system.

Populations and Evolution;Physics and Society