An epidemic model with post-contact prophylaxis of distributed length i. thresholds for disease persistence and extinction

A possible control strategy against the spread of an infectious disease is the treatment with antimicrobials that are given prophylactically to those who had a 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. Since treatment lengths have an arbitrary distribution, the model system consists of ordinary differential and integral equations. A sharp threshold condition is established in terms of a basic replacement ratio (disease reproduction number) that separates disease extinction from uniform disease persistence. We use results from dynamical systems persistence theory proving the existence of a global compact attractor along the way. Existence and multiplicity of endemic equilibria are also studied.