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 analytically demonstrate that, in the case when the recovery time τR is larger than the infection time τI , the infection flows on a one direction, i. e. from an infected node to its susceptible neighbor but, that neighbor can not reinfect that node during its infection time. This behavior leads us to deduce that, when the infection rate λ is high enough such that, each infected node on the network infects all its neighbors during its infection time, SIRS model on the network evolves to extinction state, where all nodes on the network become susceptible. Moreover, we assert that, existing the loops on the network are necessary in this case in order to, the infection occurs repeatedly within the nodes on the network. In this case, unlike the other models such as SIS model and SIR model, SIRS model will have two critical thresholds separate the persistence phase from extinction phase. That means, for fixed values of τI and τR there are two critical points of infection rate λ1 and λ2, epidemic persists in between of those two points. We also prove that, in between of these points, there is a value of λ in which the system settles down to stable state with highest infected nodes. We confirm these results numerically by a simulation of regular a one dimension system .