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Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. (English) Zbl 1204.92059

Summary: We consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for the Bovine Respiratory Syncytial Virus (BRSV) amongst a cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for \(R _{0} < 1\), where \(R _{0}\) is the basic reproductive number, and a unique supercritical endemic equilibrium for \(R _{0} > 1\). In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcations and then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcations can occur in the three-stage model for small \(b\), where \(b\) is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for \(b\) small and for larger values of \(b\).

MSC:

92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations

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