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. / Mathematical Sciences / Ph. D. (Applied Mathematics)

The last two decades have seen several large-scale epidemics of international impact, including human, animal and plant epidemics. Policy makers face health challenges that require epidemic predictions based on limited information. There is therefore a pressing need to construct models that allow us to frame all available information to predict an emerging outbreak and to control it in a timely manner. The aim of this thesis is to develop an early-warning modelling approach that can predict emerging disease outbreaks. Based on Bayesian techniques ideally suited to combine information from different sources into a single modelling and estimation framework, I developed a suite of approaches to epidemiological data that can deal with data from different sources and of varying quality. The SEIR model, particle filter algorithm and a number of influenza-related datasets were utilised to examine various models and methodologies to predict influenza outbreaks. The data included a combination of consultations and diagnosed influenza-like illness (ILI) cases for five influenza seasons. I showed that for the pandemic season, different proxies lead to similar behaviour of the effective reproduction number. For influenza datasets, there exists a strong relationship between consultations and diagnosed datasets, especially when considering time-dependent models. Individual parameters for different influenza seasons provided similar values, thereby offering an opportunity to utilise such information in future outbreaks. Moreover, my findings showed that when the temperature drops below 14°C, this triggers the first substantial rise in the number of ILI cases, highlighting that temperature data is an important signal to trigger the start of the influenza epidemic. Further probing was carried out among Maltese citizens and estimates on the under-reporting rate of the seasonal influenza were established. Based on these findings, a new epidemiological model and framework were developed, providing accurate real-time forecasts with a clear early warning signal to the influenza outbreak. This research utilised a combination of novel data sources to predict influenza outbreaks. Such information is beneficial for health authorities to plan health strategies and control epidemics.

Em epidemiologia matemática, muitos modelos de propagação de doenças infecciosas em populações têm sido analisados matematicamente e aplicados para doenças específicas. Neste trabalho um modelo de propagação de dengue é analisado considerando-se diferentes hipóteses sobre o tamanho da população humana. Mais precisamente, estamos interessados em verificar o impacto das variações populacionais a longo prazo no cálculo do parâmetro Ro e no equilíbrio endêmico. Vamos discutir algumas ideias que nortearam o processo de definição do parâmetro Ro a partir da construção do Operador de Próxima Geração. Através de um estudo qualitativo do modelo matemático, obtivemos que o equilíbrio livre de doença é globalmente assintoticamente estável se Ro é menor ou igual a 1 e instável se Ro>1. Para Ro>1, a estabilidade global do equilíbrio endêmico é provada usando um critério geral para estabilidade orbital de órbitas periódicas associadas a sistemas autônomos não lineares de altas ordens e resultados da teoria de sistemas competitivos para equações diferenciais ordinárias. Também foi desenvolvida uma análise de sensibilidade do Ro e do equilíbrio endêmico com relação aos parâmetros do modelo de propagação. Diversos cenários foram simulados a partir dos índices de sensibilidade obtidos nesta análise. Os resultados demonstram que, de forma geral, o parâmetro Ro e o equilíbrio endêmico apresentam considerável sensibilidade a taxa de picadas do vetor e a taxa de mortalidade do vetor. / In mathematical epidemiology many models of spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. In this work a dengue propagation model is analyzed considering different assumptions about the size of the human population. More precisely, we are interested to verify the impact of population long-term variations in the calculation of the parameter Ro and endemic equilibrium. We will discuss some ideas that guided the parameter setting process Ro from the construction of the Next Generation Operator. Through a qualitative study of the mathematical model, we found that the disease-free equilibrium is globally asymptotically stable if Ro is less or equal than 1 and unstable if Ro> 1. For Ro> 1 the global stability of the endemic equilibrium is proved using a general criterion for orbital stability of periodic orbits associated with nonlinear autonomous systems of higher orders and results of the theory of competitive systems for ordinary differential equations. Also a sensitivity analysis of the Ro and the endemic equilibrium with respect to the parameters of the propagation model was developed. Several scenarios were simulated from the sensitivity index obtained in this analysis. The results demonstrate that in general the parameter Ro and the endemic equilibrium are the most sensitive to the vector biting rate and the vector mortality rate.

O número de reprodutibilidade basal, R_0, é definido como o número esperado de casos secundários de uma doença produzidos por um indivíduo infectado em uma população suscetível durante seu período de infecciosidade. Tem-se que, para R_0 < 1 a doença não consegue se manter na população, e para R_0 >1 a doença irá se estabelecer. O cálculo do valor de R_0 pode ser feito de diversas maneiras, como por exemplo: a partir da análise de estabilidade de um modelo compartimental, através da matriz de próxima geração, da fase final de uma epidemia, entre outros. Neste trabalho foi estudado o cálculo de R_0 a partir da fase inicial de crescimento de um surto, em que ao fazer este cálculo não é suposto crescimento exponencial da doença, o que é proposto implicitamente na maior parte dos estudos. Foram estudadas as técnicas propostas por Nishiura, Ross-Macdonald e White e Pagano. O objetivo deste estudo foi comparar essas técnicas e avaliar como cada técnica estima o valor do número de reprodutibilidade basal, aplicando-as a doenças causadas por vetores, neste caso em particular foram utilizados dados de dengue. Foram utilizados dados da cidade de Ribeirão Preto nos períodos de 2009-2010 e 2010-2011, em ambos os casos a cidade apresentou um surto epidêmico. Os resultados apresentados pelos três métodos são numericamente diferentes. Pode-se concluir que todos os métodos acertam na previsão de que a dengue irá se propagar na cidade estudada, o que é verdade para os casos estudados, e que apesar de serem numericamente diferentes a análise semanal dos dados mostra que os valores calculados apresentam um mesmo padrão ao longo do tempo / The basic reproduction number,R_0, is defined as the expected number of secondary cases of a disease produced by a single infection in a susceptible population. If R_0 < 1 the disease cannot establish in the population, and if R_0 > 1 we expect the disease spread in the population. The value of R_0 can be estimated in several ways, for example, with the stability analysis of a compartmental model, through the matrix of next generation, using the final phase of an epidemic, etc. In this work we studied methods for estimating R_0 from the initial growth phase of the outbreak, without assuming exponential growth of cases, which is suggested in most studies. We used the methods proposed by Nishiura, Ross-Macdonald and White and Pagano. The objective of this work was to compare these techniques and to evaluate how these technique estimate the value of the basic reproduction number, applying them to diseases caused by vectors. In this particular case we used data of dengue. We used data from the city of Ribeirão Preto in the periods of 2009-2010 and 2010-2011, in both cases the city had an outbreak. The results obtained by the three methods are numerically different. We can conclude that all methods are correct in the sense that dengue will spread in the city studied, what is true for the cases studied, although they are numerically different. Weekly analysis of the data show that the estimated values have a same pattern over time

>Magister Scientiae - MSc / Herbal medicines have been an important part of health and wellness for hundreds of years. Recently the World Health Organization estimated that 80% of people worldwide rely on herbal medicines. Herbs contain many substances that are good for protecting the body and are therefore used in the treatment of various illnesses. Along with traditional medicines, herbs are often used in the treatment of chronic diseases such as rheumatism, migraine, chronic fatigue, asthma, eczema, and irritable bowel syndrome, among others. Herbal medicines are also applied in certain traditional communities as treatment against infectious diseases such as flu, malaria, measles, and even human immunodeficiency virus HIV-infection. Approximately 34 million people are currently infected with the human immunodeficiency virus (HIV) and 2.5 million newly infected. Therefore, HIV has become one of the major public health problems worldwide. It is important to understand the impact of herbal medicines used on HIV/AIDS. Mathematical models enable us to make predictions about the qualitative behaviour of disease outbreaks and evaluation of the impact of prevention or intervention strategies. In this dissertation we explore mathematical models for studying the effect of usage of herbal medicines on HIV. In particular we analyze a mathematical model for population dynamics of HIV/AIDS. The latter will include the impact of herbal medicines and traditional healing methods. The HIV model exhibits two steady states; a trivial steady state (HIV-infection free population) and a non-trivial steady state (persistence of HIV infection). We investigate the local asymptotic stability of the deterministic epidemic model and similar properties in terms of the basic reproduction number. Furthermore, we investigate for optimal control strategies. We study a stochastic version of the deterministic model by introducing white noise and show that this model has a unique global positive solution. We also study computationally the stochastic stability of the white noise perturbation model. Finally, qualitative results are illustrated by means of numerical simulations. Some articles from the literature that feature prominently in this dissertation are [14] of Cai et al, [10] of Bhunu et al., [86] of Van den Driessche and Watmough, [64] of Naresh et al., Through the study in this dissertation, we have prepared a research paper [1], jointly with the supervisors to be submitted for publication in an accredited journal. The author of this dissertation also contributed to the research paper [2], which close to completion. 1. Abdulaziz Y.A. Mukhtar, Peter J. Witbooi and Gail D. Hughes. A mathematical model for population dynamics of HIV with ARV and herbal medicine. 2. P.J. Witbooi, T. Seatlhodi, A.Y.A. Mukhtar, E. Mwambene. Mathematical modeling of HIV/AIDS with recruitment of infecteds.

La maladie de Newcastle (MN) grève lourdement les productions aviaires malgaches, essentielles à l'alimentation et à l'économie familiales. La MN est une dominante pathologique en l'absence de vaccination généralisée. L'objectif de cette thèse est la modélisation, la validation et l'analyse mathématique de modèles de transmission du virus de la MN (VMN) dans les systèmes avicoles villageois en général et à Madagascar en particulier. Nous proposons de nouveaux modèles basés sur les connaissances actuelles de l'histoire naturelle de la transmission du VMN. Ainsi, nous présentons deux modèles mathématiques à compartiments de la transmission du VMN dans une population de poules : un premier modèle avec transmission environnementale et un deuxième modèle où la vaccination contre la maladie est prise en compte. Nous présentons une analyse complète de la stabilité de ces modèles à l'aide des techniques de Lyapunov suivant la valeur du taux de reproduction de base R0. Le travail s'est appuyé sur des enquêtes de terrain pour comprendre les pratiques de vaccination actuelles à Madagascar. / Newcastle disease (ND) severely harms Malagasy bird productions, mainly uses to food and family economy. ND is a pathological dominant without general vaccination. The objective of this thesis is modelling the transmission of ND virus (NDV) in smallholder chicken farms in general and, Madagascar in particular. We propose new models based on the state of art and the epidemiology currently known from the transmission of the NDV. Thus, we present two models of the transmission of NDV: a first model with environmental transmission and a second model in which imperfect vaccination of chickens is considered. We present a thorough analysis of the stability of the models using the Lyapunov techniques and obtain the basic reproduction ratio R0. This work is based on field surveys to understand the current vaccination practices in Madagascar.

Estuda-se o espalhamento de doenças contagiosas utilizando modelos suscetível-infectado-recuperado (SIR) representados por equações diferenciais ordinárias (EDOs) e por autômatos celulares probabilistas (ACPs) conectados por redes aleatórias. Cada indivíduo (célula) do reticulado do ACP sofre a influência de outros, sendo que a probabilidade de ocorrer interação com os mais próximos é maior. Efetuam-se simulações para investigar como a propagação da doença é afetada pela topologia de acoplamento da população. Comparam-se os resultados numéricos obtidos com o modelo baseado em ACPs aleatoriamente conectados com os resultados obtidos com o modelo descrito por EDOs. Conclui-se que considerar a estrutura topológica da população pode dificultar a caracterização da doença, a partir da observação da evolução temporal do número de infectados. Conclui-se também que isolar alguns infectados causa o mesmo efeito do que isolar muitos suscetíveis. Além disso, analisa-se uma estratégia de vacinação com base em teoria dos jogos. Nesse jogo, o governo tenta minimizar os gastos para controlar a epidemia. Como resultado, o governo realiza campanhas quase-periódicas de vacinação. / The spreading of contagious diseases is studied by using susceptible-infected-recovered (SIR) models represented by ordinary differential equations (ODE) and by probabilistic cellular automata (PCA) connected by random networks. Each individual (cell) of the PCA lattice experiences the influence of others, where the probability of occurring interaction with the nearest ones is higher. Simulations for investigating how the disease propagation is affected by the coupling topology of the population are performed. The numerical results obtained with the model based on randomly connected PCA are compared to the results obtained with the model described by ODE. It is concluded that considering the topological structure of the population can pose difficulties for characterizing the disease, from the observation of the time evolution of the number of infected individuals. It is also concluded that isolating a few infected subjects can cause the same effect than isolating many susceptible individuals. Furthermore, a vaccination strategy based on game theory is analyzed. In this game, the government tries to minimize the expenses for controlling the epidemic. As consequence, the government implements quasi-periodic vaccination campaigns.

The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and non-periodic environments. The classical SEIRS model, with standard incidence function, is, first of all, extended to incorporate multiple infectious stages. Using Lyapunov function theory and LaSalle's Invariance Principle, the disease-free equilibrium (DFE) of the resulting SEIⁿRS model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, this model has a unique endemic equilibrium point (EEP), which is shown (using a non-linear Lyapunov function of Goh-Volterra type) to be globally-asymptotically stable for a special case. The SEIⁿRS model is further extended to incorporate arbitrary number of latent stages. A notable feature of the resulting SE^mIⁿRS model is that it uses gamma distribution assumptions for the average waiting times in the latent (m) and infectious (n) stages. Like in the case of the SEIⁿRS model, the SE^mIⁿRS model also has a globally-asymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globally-stable for a special case) when the threshold exceeds unity. The SE^mIⁿRS model is further extended to incorporate the effect of periodicity on the disease transmission dynamics. The resulting non-autonomous SE^mIⁿRS model is shown to have a globally-stable disease-free solution when the associated reproduction ratio is less than unity. Furthermore, the non-autonomous model has at least one positive (non-trivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the non-autonomous model, the disease will always persist in the population whenever the reproduction ratio is greater than unity. One of the main mathematical contributions of this thesis is that it shows that adding multiple latent and infectious stages, gamma distribution assumptions (for the average waiting times in these stages) and periodicity to the classical SEIRS model (with standard incidence) does not alter the main qualitative dynamics (pertaining to the persistence or elimination of the disease from the population) of the SEIRS model.

The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and non-periodic environments. The classical SEIRS model, with standard incidence function, is, first of all, extended to incorporate multiple infectious stages. Using Lyapunov function theory and LaSalle's Invariance Principle, the disease-free equilibrium (DFE) of the resulting SEIⁿRS model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, this model has a unique endemic equilibrium point (EEP), which is shown (using a non-linear Lyapunov function of Goh-Volterra type) to be globally-asymptotically stable for a special case. The SEIⁿRS model is further extended to incorporate arbitrary number of latent stages. A notable feature of the resulting SE^mIⁿRS model is that it uses gamma distribution assumptions for the average waiting times in the latent (m) and infectious (n) stages. Like in the case of the SEIⁿRS model, the SE^mIⁿRS model also has a globally-asymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globally-stable for a special case) when the threshold exceeds unity. The SE^mIⁿRS model is further extended to incorporate the effect of periodicity on the disease transmission dynamics. The resulting non-autonomous SE^mIⁿRS model is shown to have a globally-stable disease-free solution when the associated reproduction ratio is less than unity. Furthermore, the non-autonomous model has at least one positive (non-trivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the non-autonomous model, the disease will always persist in the population whenever the reproduction ratio is greater than unity. One of the main mathematical contributions of this thesis is that it shows that adding multiple latent and infectious stages, gamma distribution assumptions (for the average waiting times in these stages) and periodicity to the classical SEIRS model (with standard incidence) does not alter the main qualitative dynamics (pertaining to the persistence or elimination of the disease from the population) of the SEIRS model.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The aim of this work is to study the temporal and spatiotemporal evolution of a threedimensional system that describes a predator-prey dynamics, where the predator population can develop an infectious disease. Thus, the predators are split into two subpopulations: susceptible predators and infected predators. The rate at which susceptible become infected is described by a Holling type II functional response giving saturation when the number of susceptible predators increases. We assume that the disease develops only in the predators population and that all are born susceptible, ie, there is no vertical transmission. In the temporal evolution system, described by ordinary di�erential equations, we analyze the asymptotic behavior of the model, describing the necessary conditions for the occurrence of qualitative changes, relating them to the basic reproduction number of predators and the basic reproduction number of the disease. In numerical simulations these changes are graphically described, from the variation of the parameters that determine the predation efficiency of the infected predator and the mortality rate of susceptible and infected predators. Starting from the same local dynamics, we include spatial variation and consider movement by difusion to the population, obtaining a system described by partial diferential equations in which we can observe in addition to the temporal evolution of the spatial evolution of the system, or as populations are distributed spatially over time, when and how invasions occur in the domain. The temporal evolution of the system exhibits complex dynamics such as stable equilibrium, limit cycles, periodic oscillations and aperiodicity. The same dynamics are found in reaction-difusion system, considering that every point of the space represented by x displays a local dynamic . Spatially, invasions were observed in the form of wave fronts, making populations evenly distributed over time. / O objetivo central deste trabalho é estudar a evolução temporal e espaço-temporal do sistema tridimensional que descreve uma dinâmica presa-predador, onde a população de predadores pode desenvolver uma doença infecciosa. Desta forma, os predadores são divididos em duas subpopulações: predadores suscetí- veis e predadores infectados. A taxa com que os suscetíveis se tornam infectados é dada por uma resposta funcional tipo II, que exibe uma saturação conforme o número de predadores suscetíveis aumenta. Assumimos que a doença se desenvolve apenas na população de predadores e que todos nascem suscetíveis, ou seja, não há transmissão vertical. No sistema de evolução temporal, descrito por equações diferenciais ordinárias, analisamos o comportamento assintótico do modelo, descrevendo as condições necessárias para a ocorrência de mudanças qualitativas, relacionando-as ao número de reprodução básico dos predadores e ao número de reprodução básico da doença. Nas simulações numéricas essas mudanças são descritas gra�camente, a partir da variação dos parâmetros que determinam a e�ciência de predação do predador infectado e a taxa de mortalidade de predadores suscetíveis e infectados. Partindo da mesma dinâmica local, incluímos a variação espacial e consideramos movimenta ção por difusão para as populações, obtendo um sistema descrito por equações diferenciais parciais, com o qual podemos observar, além da evolução temporal, a evolução espacial do sistema, ou seja, como as populações se distribuem espacialmente com o passar do tempo, quando e como ocorrem as invasões do domínio. A evolução temporal do sistema exibe dinâmicas complexas, como equilíbrios estáveis, ciclos limites, oscilações periódicas e aperiodicidade. As mesmas dinâmicas são encontradas no sistema de reação-difusão, considerando-se que cada ponto do espaço, representado por x, exibe uma dinâmica local. Espacialmente, foram observadas invasões em forma de frentes de ondas, tornando as populações homogeneamente distribuídas com o passar do tempo.