Stochastic epidemic models with a backward bifurcation
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Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042
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Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4
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Received:
01 March 2006
Accepted:
29 June 2018
Published:
01 May 2006
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MSC :
92D30.
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Two new stochastic epidemic models, a continuous-time Markov chain
model and a stochastic differential equation model, are formulated. These are
based on a deterministic model that includes vaccination and is applicable
to pertussis. For some parameter values, the deterministic model exhibits
a backward bifurcation if the vaccine is imperfect. Thus
a region of bistability exists in a subset of parameter space.
The dynamics of the stochastic epidemic models are investigated in this region
of bistability, and compared with those of the deterministic model. In this region the probability
distribution associated with the infective population exhibits bimodality with
one mode at the disease-free equilibrium and the other at the larger endemic
equilibrium. For population sizes $N\geq 1000$, the deterministic and stochastic models agree,
but
for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.
Citation: Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation[J]. Mathematical Biosciences and Engineering, 2006, 3(3): 445-458. doi: 10.3934/mbe.2006.3.445
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Abstract
Two new stochastic epidemic models, a continuous-time Markov chain
model and a stochastic differential equation model, are formulated. These are
based on a deterministic model that includes vaccination and is applicable
to pertussis. For some parameter values, the deterministic model exhibits
a backward bifurcation if the vaccine is imperfect. Thus
a region of bistability exists in a subset of parameter space.
The dynamics of the stochastic epidemic models are investigated in this region
of bistability, and compared with those of the deterministic model. In this region the probability
distribution associated with the infective population exhibits bimodality with
one mode at the disease-free equilibrium and the other at the larger endemic
equilibrium. For population sizes $N\geq 1000$, the deterministic and stochastic models agree,
but
for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.
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