Stability analysis of a model of epidemic dynamics with nonlinear feedback producing recurrent infection waves

Complex dynamics characterizing human behavior in an epidemiological scenario can be modeled via a system of ordinary differential equations starting from a simple SIR (susceptible–infected–recovered) model. Here we propose a nonlinear mathematical model that describes the evolution in time of susceptible, infected and hospitalized individuals. A new variable that reflects the society’s “memory” of the severity of the epidemic is introduced, and this variable feeds back on the transmission rate of the disease. The nonlinear transmission rate reflects the fact that changes (e.g., an increase) in the number of hospitalized individuals can influence the behavior of society and individuals, which would affect (reduce) the probability of transmission. Differently from the standard SIR model, the nonlinear transmission rate may lead to complex dynamics with oscillatory solutions due to a Hopf bifurcation. Such oscillations correspond to recurrent infection waves. Using two parameter bifurcation diagrams we investigate the parameter space of the model. Finally, we report two examples on how the multiple infection waves present for the COVID-19 pandemic can be fitted by our model.