An epidemic model with post-contact prophylaxis of distributed length II. Stability and oscillations if treatment is fully effective

A possible control strategy against the spread of an infectious disease is the treatment with antimicrobials that are given prophylactically to those that had contact with an infective person. The treatment continues until recovery or until it becomes obvious that there was no infection in the first place. The model considers susceptible, treated uninfected exposed, treated infected, (untreated) infectious, and recovered individuals. The overly optimistic assumptions are made that treated uninfected individuals are not susceptible and treated infected individuals are not infectious. Since treatment lengths are considered that have an arbitrary distribution, the model system consists of ordinary differential and integral equations. We study the impact of the treatment length distribution on the large-time behavior of the model solutions, namely whether the solutions converge to an equilibrium or whether they are driven into undamped oscillations.