Modeling, analysis and numerical method for HIV-TB co-infection with TB treatment in Ethiopia
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Authors
Abdella Arega Tessema
Issue Date
2018-07
Type
Thesis
Language
en
Keywords
HIV , TB , Nonstandard finite difference methods , Basic reproduction number , Stability , Co-infection
Alternative Title
Abstract
In this thesis, a mathematical model for HIV and TB co-infection with TB
treatment among populations of Ethiopia is developed and analyzed. The
TB model includes an age of infection. We compute the basic reproduction
numbers RTB and RH for TB and HIV respectively, and the overall repro-
duction number R for the system. We find that if R < 1 and R > 1; then
the disease-free and the endemic equilibria are locally asymptotically stable,
respectively. Otherwise these equilibria are unstable. The TB-only endemic
equilibrium is locally asymptotically stable if RTB > 1, and RH < 1. How-
ever, the symmetric condition, RTB < 1 and RH > 1, does not necessarily
guarantee the stability of the HIV-only equilibrium, but it is possible that
TB can coexist with HIV when RH > 1: As a result, we assess the impact of
TB treatment on the prevalence of TB and HIV co-infection.
To derive and formulate the nonlinear differential equations models for HIV and TB co-infection that accounts for treatment, we formulate and analyze
the HIV only sub models, the TB-only sub models and the full models of HIV
and TB combined. The TB-only sub model includes both ODEs and PDEs
in order to describe the variable infectiousness and e ect of TB treatment
during the infectious period.
To analyse and solve the three models, we construct robust methods, namely
the numerical nonstandard definite difference methods (NSFDMs). Moreover,
we improve the order of convergence of these methods in their applications
to solve the model of HIV and TB co-infection with TB treatment at the
population level in Ethiopia. The methods developed in this thesis work
and show convergence, especially for individuals with small tolerance either
to the disease free or the endemic equilibria for first order mixed ODE and PDE as we observed in our models.
Description
Citation
Abdella Arega Tessema (2018) Modellling, analysis and numerical method for HIV-TB co-infection with TB treatment in Ethiopia, University of South Africa, Pretoria, <http://hdl.handle.net/10500/24851>