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Backward bifurcations in dengue transmission dynamics. (English) Zbl 1156.92036

Summary: A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number ${ℛ}_{0}$, is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making ${ℛ}_{0}$ less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community.

The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifolds, the extended model is also shown to undergo backward bifurcations. In both the original and the extended models, it is shown, using Lyapunov function theory and the LaSalle invariance principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.

##### MSC:
 92D30 Epidemiology 34C60 Qualitative investigation and simulation of models (ODE) 37N25 Dynamical systems in biology 92C60 Medical epidemiology 93A30 Mathematical modelling of systems
##### Keywords:
dengue; mosquitoes; equilibria; stability; bifurcation; vaccine