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Stability and Hopf bifurcation for an epidemic disease model with delay. (English) Zbl 1165.34048

The authors consider the following predator-prey system with disease in the prey

$\begin{array}{cc}\hfill \stackrel{˙}{S}\left(t\right)& =rS\left(t\right)\left(1-\frac{S\left(t\right)+I\left(t\right)}{K}\right)-\beta S\left(t\right)I\left(t\right),\hfill \\ \hfill \stackrel{˙}{I}\left(t\right)& =\beta S\left(t\right)I\left(t\right)-cI\left(t\right)-pI\left(t\right)y\left(t\right),\hfill \\ \hfill \stackrel{˙}{y}\left(t\right)& =-dy\left(t\right)+kpI\left(t-\tau \right)y\left(t-\tau \right),\hfill \end{array}$

where $S\left(t\right),I\left(t\right),y\left(t\right)$ denote the susceptible prey, the infected prey and the predator population at time $t$, respectively; $r>0$ is the intrinsic growth rate of the prey, $K>0$ is the carrying capacity of the prey, $\beta >0$ is the transmission coefficient, $d>0$ is the death rate of the predator, $c>0$ is the death rate of the infected prey, $k>0$ is the conversing rate of the predator by consuming prey, $\tau >0$ is a time delay due to the gestation of predator.

By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium and the existence of Hopf bifurcations are established. By using the normal form theory and center manifold argument, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions.

MSC:
 34K60 Qualitative investigation and simulation of models 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional differential equations 92D30 Epidemiology 34K17 Transformation and reduction of functional-differential equations and systems; normal forms 34K19 Invariant manifolds (functional-differential equations) 34K13 Periodic solutions of functional differential equations