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An epidemic model of a vector-borne disease with direct transmission and time delay. (English) Zbl 1146.34059
Authors’ abstract: This paper considers an epidemic model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The incidence term is assumed to be of the bilinear mass-action form. We include both a baseline ODE version of the model, and, a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction number ${R}_{0}$. If ${R}_{0}⩽1$, the disease-free equilibrium is globally stable and the disease dies out. If ${R}_{0}>1$, a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. The delay in the differential-delay model accounts for the incubation time the vectors need to become infectious. We study the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 92D30 Epidemiology