The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

Philosophiae Doctor - PhD / The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model. / South Africa

In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graph-theoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phase-lock oscillations.

Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.

<p>The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models.&nbsp / Comparative numerical results are also provided for each model.</p>

<p>The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models.&nbsp / Comparative numerical results are also provided for each model.</p>

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.