An Epidemic Model With Post-Contact Prophylaxis of Distributed Length II. Stability and Oscillations if Treatment is Fully Effective

Issue		Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb


Page(s)		267 - 293
DOI		10.1051/mmnp:2008052
Published online		23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 267-293
DOI: 10.1051/mmnp:2008052

An Epidemic Model With Post-Contact Prophylaxis of Distributed Length II. Stability and Oscillations if Treatment is Fully Effective

H. R. Thieme, A. Tridane and Y. Kuang

Department of Mathematics and Statistics, Arizona State University Tempe, AZ 85287-1804, U.S.A

h.thieme@asu.edu

Published online: 23 October 2008

Abstract
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.

Mathematics Subject Classification. 34K20, 37N25, 92C50, 92D25, 92D30

Key words: basic reproduction number -- standard incidence -- (class) age structure -- distributed time delay -- disease persistence -- global stability of endemic equilibria -- instability -- periodic solutions -- frequency domain

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