Epidemic Threshold for the SIRS Model on 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 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.