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

When 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. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This study did not receive any funding ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable. Yes All data produced in the present work are contained in the manuscript