Critical Threshold for SIRS Model on Small World Networks

Abstract

Abstract 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. Our results also assert 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.