Multiple peaks in network SIR epidemic models

We study a susceptible-infected-recovered (SIR) epidemic model on a network of interacting subpopulations and analyze the dynamical behavior of the fraction of infected agents in each node of the network. In contrast to the classical scalar epidemic SIR model, in which the fraction of infected is known to have a unimodal behavior (decreasing over time or initially increasing until reaching a peak and then decreasing), we show the possible occurrence of new multimodal behaviors in the network SIR model. We focus on the special case of rank-1 network matrices, which model subpopulations of homogeneously mixing agents with different interaction levels. We provide an upper bound on the number of changes of monotonicity of the fraction of infected at the single node level and provide sufficient conditions under which such multimodal behavior occurs. We then conduct a numerical analysis, revealing that, in case of more general network matrices, the dynamics may exhibit complex behaviors with multiple peaks of infection in each node