Abstract
The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.
Highlights
The spread of the vector-borne dengue fever in a human population has been studied in many papers: we mention a family of three nested multi-strain compartmental SIR type models analyzed in [1,2,3,4]
When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model
In [9] it was investigated how far the fast time scale of the mosquito epidemiology can be slaved by the slower human epidemiology, so that for the understanding of human disease data mainly the dynamics of the human time scale is essential because it will be only slightly perturbed by the mosquito dynamics
Summary
The spread of the vector-borne dengue fever in a human population has been studied in many papers: we mention a family of three nested multi-strain compartmental SIR type models analyzed in [1,2,3,4]. The analysis of [5] shows that an increased degree of personal protection and vector control are effective control measures in dengue-endemic areas This leads to the conclusion that in order to study the effects of vector-mosquito control an accurate vector population model together with a human-host population model describing the multiple-strain virus outbreaks and spread is required. In the vector-host epidemic model analyzed in this paper we couple a simplified dengue-fever model for the host population based on [3] with a simplified model for the mosquito vector population [9] This is done in order to gain insight into the effects of the vector dynamics on the host dynamics, especially when the time-scale of the vector population is much faster than that of the host. The resulting non-autonomous system is studied in the same way as the autonomous system whereby equilibria are replaced by limit cycles and eigenvalues by Floquet multipliers
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