Mathematical Analysis of the Role of Movement in the Spread of Tuberculosis
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Date
2013-09-19
Authors
Soliman, Iman
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Abstract
Tuberculosis (TB) is an infectious respiratory disease caused by the bacterium Mycobacterium tuberculosis. TB is the second largest cause of mortality by infectious diseases and is a challenging disease to control. It spreads easily among people via droplets propagated by an infectious person. Treatment against TB has been available since the 1950s; however, various problems with treatment have led to the emergence of drug-resistance in TB bacteria, which further complicates disease control.
Furthermore, TB is a disease that predominantly affects poor countries or countries with high population densities. With the generalization of travel and migration in the second half of the twentieth century, individuals infected in such countries are likely to move to or spend some time in richer countries, making TB a worldwide problem.
In this thesis, we consider the role of population movement in the spread of tuberculosis by studying two different models. The first one is an extension to a spatialized context of a simple existing mathematical model for the spread of TB. We establish that, similarly to the original model, the equilibrium without disease is globally asymptotically stable when the basic reproduction number $\R_0$ is less than one. In the case that $\R_0>1$, we prove that the system is uniformly persistent. The second model considers the spread of drug-resistant TB in a population, then between connected populations. We establish that a backward bifurcation can occur and that the coupled system has more types of equilibria than the systems in isolation. Finally, we consider a general class of models including the previous two in isolation and after coupling. We investigate which dynamical properties of the isolated models are preserved when coupling the models through movement. Some new results are provided in that direction.
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Keywords
Tuberculosis, Movement, Mathematical, Analysis, Epidemiology, Resistant Tuberculosis, Metapopulation