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Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics
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Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics
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Journal Article
Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics
2020
Overview
Background
The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued.
Results
A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number
R
0
. Then the stability of the equilibria is given in terms of
R
0
, and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed.
Conclusion
It is proven that the disease-free equilibrium and endemic equilibrium are stable if
R
0
<
1
and
R
0
>
1
, respectively. Also, the disease-free equilibrium is globally stable when
R
0
≤
1
. The system has a transcritical bifurcation when
R
0
=
1
and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.
Publisher
BioMed Central,BioMed Central Ltd,Springer Nature B.V,BMC
