Epidemic Dynamics Under Pulse Population Exchange

Many epidemiological models assume either closed populations or continuous demographic turnover. We consider an intermediate setting in which exchange occurs through discrete pulses. In this paper, we introduce a population-conserving compartmental epidemiological model with pulse population exchange at a set of prescribed times. At each pulse, a fraction of the population in each compartment is replaced by a combination of susceptible and immunized individuals. In contrast to typical pulse birth or pulse vaccination models where only a single compartment undergoes a pulse population change, the pulses in this model have competing effects: removing infectious individuals reduces the force of infection, while introducing new susceptibles increases it. In addition, when some part of the incoming population is immunized, the net effect of pulses becomes state-dependent. After presenting the model, we prove both conservation of total population and non-negativity in each compartment. We then show that while the non-pulsed system has only the disease-free equilibrium, the demographic changes introduced by the pulses can produce two new types of solution: a pulse-periodic endemic equilibrium and a second wave of infection. We derive an exact expression for a post-pulse endemic equilibrium in terms of the integrals of the solutions between equispaced pulses using the Poincaré map. Next, we identify the exact threshold at which the pulse-induced reduction in the susceptible population changes sign. Above the threshold, the pulses in all compartments contribute to a net reduction in the force of infection. Below this threshold, competing effects take hold—incoming susceptible individuals replenish the susceptible compartment, while departing exposed and infectious individuals no longer contribute to secondary infections. Finally, using parameters motivated by recent outbreaks of disease on cruise ships, we characterize this threshold mechanism and investigate the effect of pulse start time on the solution trajectories.