Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites

T. Dhirasakdanon; H. R. Thieme

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 6, page 109-138
  • ISSN: 0973-5348

Abstract

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

How to cite

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Dhirasakdanon, T., and Thieme, H. R.. "Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites." Mathematical Modelling of Natural Phenomena 5.6 (2010): 109-138. <http://eudml.org/doc/197678>.

@article{Dhirasakdanon2010,
abstract = {For an SI type endemic model with one host and two parasite strains, we study the stability of the endemic coexistence equilibrium, where the host and both parasite strains are present. Our model, which is a system of three ordinary differential equations, assumes complete cross-protection between the parasite strains and reduced fertility and increased mortality of infected hosts. It also assumes that one parasite strain is exclusively vertically transmitted and cannot persists just by itself. We give several sufficient conditions for the equilibrium to be locally asymptotically stable. One of them is that the horizontal transmission is of density-dependent (mass-action) type. If the horizontal transmission is of frequency-dependent (standard) type, we show that, under certain conditions, the equilibrium can be unstable and undamped oscillations can occur. We support and extend our analytical results by numerical simulations and by two-dimensional plots of stability regions for various pairs of parameters.},
author = {Dhirasakdanon, T., Thieme, H. R.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {coexistence of parasite strains; disease incidence; SI endemic model; Routh-Hurwitz conditions; undamped oscillations; vertical transmission; horizontal transmission; disease-related fertility reduction},
language = {eng},
month = {4},
number = {6},
pages = {109-138},
publisher = {EDP Sciences},
title = {Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites},
url = {http://eudml.org/doc/197678},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Dhirasakdanon, T.
AU - Thieme, H. R.
TI - Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/4//
PB - EDP Sciences
VL - 5
IS - 6
SP - 109
EP - 138
AB - For an SI type endemic model with one host and two parasite strains, we study the stability of the endemic coexistence equilibrium, where the host and both parasite strains are present. Our model, which is a system of three ordinary differential equations, assumes complete cross-protection between the parasite strains and reduced fertility and increased mortality of infected hosts. It also assumes that one parasite strain is exclusively vertically transmitted and cannot persists just by itself. We give several sufficient conditions for the equilibrium to be locally asymptotically stable. One of them is that the horizontal transmission is of density-dependent (mass-action) type. If the horizontal transmission is of frequency-dependent (standard) type, we show that, under certain conditions, the equilibrium can be unstable and undamped oscillations can occur. We support and extend our analytical results by numerical simulations and by two-dimensional plots of stability regions for various pairs of parameters.
LA - eng
KW - coexistence of parasite strains; disease incidence; SI endemic model; Routh-Hurwitz conditions; undamped oscillations; vertical transmission; horizontal transmission; disease-related fertility reduction
UR - http://eudml.org/doc/197678
ER -

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