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Abstract
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 manifold, the extended model is also shown to undergo backward bifurcation.
In both the original and the extended models, it is shown, using Lyapunov function
theory and 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.