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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

S ˙(t)=rS(t)1-S(t)+I(t) K-βS(t)I(t),I ˙(t)=βS(t)I(t)-cI(t)-pI(t)y(t),y ˙(t)=-dy(t)+kpI(t-τ)y(t-τ),

where S(t),I(t),y(t) 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, β>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, τ>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:
34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
92D30Epidemiology
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
34K13Periodic solutions of functional differential equations