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Mathematical Modelling of Natural Phenomena (1)

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Stability of the Endemic Coexistence Equilibrium for One Hostand Two Parasites
T. Dhirasakdanon, H. R. Thieme
Journal:

Mathematical Modelling of Natural Phenomena / Volume 5 / Issue 6 / 2010

Published online by Cambridge University Press:

08 April 2010, pp. 109-138

Print publication:

2010
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For an SI type endemic model with one host and two parasite strains, we study thestability of the endemic coexistence equilibrium, where the host and both parasite strainsare present. Our model, which is a system of three ordinary differential equations,assumes complete cross-protection between the parasite strains and reduced fertility andincreased mortality of infected hosts. It also assumes that one parasite strain isexclusively vertically transmitted and cannot persists just by itself. We give severalsufficient conditions for the equilibrium to be locally asymptotically stable. One of themis that the horizontal transmission is of density-dependent (mass-action) type. If thehorizontal transmission is of frequency-dependent (standard) type, we show that, undercertain conditions, the equilibrium can be unstable and undamped oscillations can occur.We support and extend our analytical results by numerical simulations and bytwo-dimensional plots of stability regions for various pairs of parameters.