  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 small world network. We show the effects of all the
parameters associated with this model on small world network and we create the
full phase space. The results we obtained are consistent with those obtained in
Ref.[7] in terms of the existence of a phase transition from a fluctuating
endemic state to self-sustained oscillations in the size of the infected
subpopulation at a finite value of the disorder of the network. And also our
results assert that, that transition specifically occurs where the average
clusterization shifts from high to low. The effect of clustering coefficient on
SIRS model on the networks can be understood from the results obtained in Ref.
[9], which indicates the importance of existing the loops in the network, in
order to the disease to spread frequently throughout the nodes of network.
where, clusters tend to spread infection among close-knit neighborhoods. Hence,
when the loops are high inside the network, the reinfection occurs in the
network at many places and at different times, which looks like as a kind of
randomness in occurring the second period of infection. Whereas when the number
of loops are low, reinfection occurs at specific places and times on the
network, which looks like as a kind of regularity in occurring the second
period of infection.
