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Mathematical Medicine and Biology Advance Access originally published online on March 18, 2009
Mathematical Medicine and Biology 2009 26(3):225-239; doi:10.1093/imammb/dqp006
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate

Andrei Korobeinikov{dagger}

Laboratory of Nonlinear Science and Computation, Research Institute for Electronic Science, Hokkaido University, Sapporo 060–0812, Japan

{dagger} Email: andrei.korobeinikov{at}ul.ie

Received on November 6, 2006. Revised on February 16, 2007. Accepted on April 29, 2007.

We consider two models for the spread of an infection with a free-living infective stage, where parasite reproduction and virulence (parasite-induced mortality) depend on the parasite dose to which the host is exposed and are given by unspecified non-linear functions of the number of the free pathogen particles, and the incidence rate is non-linear. We study the impact of these non-linearities with the focus on the global properties of these models. We consider a very general form of the non-linearities: we assume that the virulence and the parasite reproduction rates are given by unspecified non-linear functions of the number of the free pathogen particles and that the incidence rate is an unspecified function of the number of susceptible hosts and free pathogen particles; all these functions are constrained by a few biologically feasible conditions. We construct Lyapunov functions that enable us to find biologically realistic conditions which are sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number, this equilibrium state can be either positive, where parasite endemically persists, or infection free.

Keywords: non-linear incidence; direct Lyapunov method; Lyapunov function; non-linear virulence; equilibrium state; global stability; dose effect; microparasites; pathogen

Present address: MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland.


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