Double Hopf bifurcation and chaotic dynamics in a periodically-forced SIR model

We perform a qualitative analysis of a periodically-forced SIR model which incorporates the disease transmission rate by direct contact with the natural viral source, in addition to the classic disease transmission rate between individuals. We direct this work towards two main topics: (i) in the absence of seasonality, the endemic equilibrium point (and unique) undergoes both supercritical and subcritical Hopf bifurcations. We identify a specific range of values for the disease transmission rate β, for which the system exhibits an attracting periodic solution while the equilibrium is unstable; (ii) in the presence of seasonality, we prove via torus-breakdown theory that the system exhibits strange attractors (observable chaos). These findings reveal that small changes in parameters can generate complex epidemic dynamics, becoming very difficult to control. All findings are derived analytically and supported by numerical simulations.