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The study of recurrent epidemic outbreaks has been attracting great attention for decades, but its underlying mechanism is still under debate. Based on a large number of real data from different cities, we find that besides the seasonal periodic outbreaks of influenza, there are also non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors. To understand how the non-periodicity shows up, we present a network model of SIRS epidemic with both time-dependent infection rate and a small possibility of persistent epidemic seeds, representing the influences from the larger annual variation of environment and the infection generated spontaneously in nature, respectively. Our numerical simulations reveal that the model can reproduce the non-periodic outbreaks of recurrent epidemics with the main features of real influenza data. Further, we find that the recurrent outbreaks of epidemic depend not only on the infection rate but also on the density of susceptible agents, indicating that they are both the necessary conditions for the recurrent epidemic patterns with non-periodicity. A theoretical analysis based on Markov dynamics is presented to explain the numerical results. This finding may be of significance to the control of recurrent epidemics.

Non-periodic outbreaks of recurrent epidemics and its network modelling