Spreading of periodic diseases and synchronization phenomena on networks

In this paper, we investigate numerically the Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model on an exponential network generated by a preferential attachment procedure. The discrete SIRS model considers two main parameters: the duration τ 0 of the complete infection–recovery cycle and the duration τ I of infection. A permanent source of infection I 0 has also been introduced in order to avoid the vanishing of the disease in the SIRS model. The fraction of infected agents is found to oscillate with a period T ≥ τ 0 . Simulations reveal that the average fraction of infected agents depends on I 0 and τ I / τ 0 . A maximum of synchronization of infected agents, i.e. a maximum amplitude of periodic spreading oscillations, is found to occur when the ratio τ I / τ 0 is slightly smaller than 1 / 2 . The model is in agreement with the general observation that an outbreak corresponds to high τ I / τ 0 values.