Network behavioral-feedback SIR epidemic model

We propose a network behavioral-feedback Susceptible–Infected–Recovered (SIR) epidemic model in which the interaction matrix describing the infection rates across subpopulations depends in feedback on the current epidemic state. This model captures both heterogeneities in individuals’ mixing, contact frequency, aptitude to contract and spread the infection, and endogenous behavioral responses such as voluntary social distancing or the adoption of self-protective measures. We first show that the set of equilibrium points coincides with the disease-free states and we analyze their stability. Then, we illustrate through several examples how the shape of the set of stable equilibrium points depends on the structure of the interaction matrix, providing insights for the design of effective control strategies. Finally, we analyze the transient dynamic behavior of the epidemics, showing that, in a special case of rank-1 interaction matrices, there always exists an aggregate infection curve that exhibits a unimodal behavior: this result extends the ones on unimodality of the infection curve already known in the literature of epidemic models and paves the way for future control applications.