A series of pulses vaccination in SIR model – Understanding periodic orbits and irregular trajectories

Abstract

AbstractWhen a Susceptible-Infective-Recovered (SIR) model with a constant contact rate is used to describe the dynamics of directly transmitted infections, oscillations, which decay exponentially with time, are obtained. Due to damped oscillations, intermittent vaccination schemes can be designed in order to reduce or even eliminate the infection. A simple intermittent vaccination can be described by a series of pulses, i.e., a proportion of susceptible individuals is vaccinated intermittently at every fixed period of time. Analysis of the model is done by numerical simulations in order to determine the trajectories in the phase space. It is observed that as the proportion of vaccinated individuals increases, closed orbits with multiple cycles appear, even irregular trajectories arise occasionally. These results can be understood by comparing with bifurcations occurring in a discrete logistic model describing a single population. Further, bifurcations occurring in epidemiological models that use periodic functions to mimic seasonal variations in the disease transmission are discussed.