Citation
Mohammed Garba, Salisu
(2008)
Mathematical Modeling and Analysis of Dengue Transmission Dynamics.
PhD thesis, Universiti Putra Malaysia.
Abstract
The work in this thesis is based on the design and analysis of suitable
compartmental deterministic models for the transmission dynamics of dengue
fever in a population. A basic dengue model which allows transmission by
exposed humans and mosquitoes is developed and rigorously analysed. The
model, consisting of seven mutuallyexclusive compartments representing the
human and vector dynamics, has a locallyasymptotically stable (LAS) diseasefree
equilibrium (DFE) whenever a certain epidemiological threshold, known as
the basic reproduction number (Ro) 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 Ro 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. 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.
The model is extended to include the dynamics of two strains of dengue disease.
The extended model has a locallyasymptotically stable, diseasefree equilibrium
(DFE) whenever the maximum of the associated reproduction numbers of the two
strains (denoted by Ro) is less than unity. It is also shown, using a Lyapunov
function and LaSalle Invariance Principle, that the DFE of the model, in the
absence of dengueinduced mortality, is globallyasymptotically stable whenever
Ro<1. The two strains coexist if the reproduction number of each strain exceeds
unity (and are different). For the case when the two reproduction numbers
exceed unity but are equal, a continuum of coexistence equilibria exists. The
impact of crossimmunity is explored for the case when Ro >1. It is shown that the model can have infinitely many coexistence equilibria if infection with one
strain confers complete immunity against the other strain. However, if infection
with one strain has no effect on susceptibility to the other strain, the model can
have a unique coexistence equilibrium. It is shown that crossimmunity could
lead to disease elimination, competitive exclusion or coexistence of the strains.
Further, the effect of seasonality on dengue transmission dynamics is explored
using numerical simulations. It is shown that the oscillation pattern differs
between the strains, both in their subharmonic periods and the relative phase of
cycles, depending on the degree of the crossimmunity between the strains.
Finally, a deterministic model for monitoring the impact of treatment and vector
control strategy on the transmission dynamics of dengue in the human and
vector populations is formulated. In addition to having a locallyasymptotically
stable diseasefree equilibrium (DFE) whenever Ro is less than unity, it is shown,
using a Lyapunov function and LaSalle Invariance Principle, and using
comparison theorem that the DFE of both treatmentfree and treatment model, in
the absence of dengueinduced mortality, is globallyasymptotically stable
whenever Ro <1; each of the models has a unique endemic equilibrium whenever
its reproduction number exceed unity. Numerical simulations shows that, the
use of vector control strategies can result in the effective control of dengue in a
community by reducing the population of susceptible and exposed mosquitoes.
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