Epidemic threshold for the SIRS model on the networks

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 for the probability of the descendants to reinfect their ancestors. We find that, in the case of the recovery time τR is larger than the infection time τI, the infection will flow directionally from the ancestors to descendants however, the descendants will be unable to reinfect their ancestors during the time of their illness. This behavior leads us to deduce that, if the infection rate λ is high enough so that any infected node on the network infects all its neighbors during its infection time, then, the 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 disease spreads frequently throughout the nodes of the networks, the loops on the network are necessary, which means the clustering coefficient will play an important role in this model. Hence, unlike the other models such as SIS model and SIR model, the SIRS model has two critical thresholds which separate the persistence phase from the extinction phase when τI<τR, i.e. for fixed values of τI and τR there are two critical points for infection rate λ1 and λ2, where epidemic persists in between of those two points. We confirm those results numerically with the simulation of a regular one dimensional SIRS system and small world network.