This thesis discusses limit cycles and behavior of rumor models. The first part presents the deterministic Daley-Kendall model (DK) with arrivals and departures and comparison of the Susceptibles, Infectives and Removed (SIR) model and the DK model. The second result is a part of the qualitative analysis and the general behavior of extension of the Daley-Kendall model. Here we discuss how the halting rate of spreaders causes the model to change from a stable equilibrium or a stable limit cycle. In the third part we carry out model validation and use both synthetic data and real data sets and fit them to the numerical solutions of the extended Daley-Kendall model. Finally, we find the parameter estimates and standard errors. In this way we shall be able to decide whether the numerical solutions quantifying the relationships between the variables obtained from the qualitative analysis can be accepted as the best description of the data. We discuss sensitivity analysis results and traditional sensitivity functions.

Leishmaniasis is a parasitic disease that affects humans and animals in more than 98 countries across the globe placing more than 1 billion people at risk for the disease and killing more than 20,000 people per year. In the United States the disease is enzootic within the hunting dog population and vertical transmission has been identified as the primary route of transmission in this population. In Brazil the disease is endemic in the human population and enzootic in the dog population with vector and vertical transmission having been reported. In many diseases reports have found there is increased disease severity when an individual is co-infected with another organism. Case reports have suggested this may also occur with tick borne diseases and leishmaniosis in dogs but there is limited longitudinal data to support this relationship. Even less is known and understood regarding the risk factors and basic reproduction number, number of secondary cases one infected individual can cause in a susceptible population, of leishmaniosis in regards to vertical transmission. The goal of the work presented in this thesis is to address host factors related to the transmission of L. infantum and the way in which co-infections affect the progression of the disease both in the U.S. and in Brazil. Understanding the risk factors associated with the transmission of the parasite Leishmania infantum, the causative agent of the disease, are necessary to controlling and potentially elimination the disease. Utilizing a large prospective cohort and both active and passive surveillance it was identified that leishmaniosis can be maintained in a population via vertical transmission at prevalence rates similar to other endemic countries, 20%. With this knowledge an additional study examining a longitudinal cohort and assessing the impact of tick borne disease co-infections upon disease transmission was performed. It was identified that dogs exposed to three or more tick borne diseases were 11x more likely to progress to clinical disease (Adjusted RR: 11.64 95% CI: 1.22-110.99 p-value: 0.03) than dogs with no tick borne disease exposures. Furthermore, dogs with Leishmania and tick borne disease were 5x more likely to die within the study (RR: 4.85 95% CI: 1.65-14.24 p-value: 0.0051). When examining this relationship in a cross-sectional study in Brazil it was found that dogs with multiple tick borne disease exposures had 1.68x greater risk of being positive for Leishmania (Adjusted RR: 1.68 95% CI: 1.09-2.61 p-value: 0.019). Using a retrospective cohort of dogs and information regarding their dam’s diagnostic status near the time of pregnancy risk factors associated with vertical transmission and the basic reproduction number were calculated. It was found that dogs who were born to dams that were ever diagnostically positive for exposure and/or infection with L. infantum were 13.84x more likely become positive for L. infantum within their lifetime (RR: 13.84 95% CI: 3.54-54.20 p-value < 0.0001). The basic reproduction number for vertically transmitted L. infantum within this cohort was 4.16. The results of these studies suggest that leishmaniosis can be maintained in a population through vertical transmission. Furthermore, the studies show the risk factors associated with vertical transmission relate to the mother’s diagnostic status at time of pregnancy. The results of the co-infection studies highlight the importance of tick prevention in order to reduce disease progression. With increased disease severity associated with increased transmission to potential vectors these studies underline the need for immunotherapies and prevention measures to reduce disease progression in order to reduce transmission. Furthermore, these studies highlight the need for public health control and prevention programs to address vertical transmission if elimination of the disease is to ever be successful.

Seasonal and non-seasonal Susceptible-Exposed-Infective-Recovered-Susceptible (SEIRS) models are formulated and analyzed. It is proved that the disease-free steady state of the non-seasonal model is locally asymptotically stable if Rv < 1, and disease invades if Rv > 1. For the seasonal SEIRS model, it is shown that the disease-free periodic solution is locally asymptotically stable when R̅v < 1, and I(t) is persistent with sustained oscillations when R̅v > 1. Numerical simulations indicate that the orbit representing I(t) decays when R̅v < 1 < Rv. The seasonal SEIRS model with routine and pulse vaccination is simulated, and results depict an unsustained decrease in the maximum of prevalence of infectives upon the introduction of routine vaccination and a sustained decrease as pulse vaccination is introduced in the population. Mortality data of pneumonia and influenza is collected and analyzed. A decomposition of the data is analyzed, trend and seasonality effects ascertained, and a forecasting strategy proposed.

O fenômeno do aumento global da temperatura é uma realidade inquestionável. Tendo em vista tal cenário, acredita-se que haverá uma expansão geográfica (migração de populações humanas) e um aumento na incidência de infecções tropicais. No entanto, a tendência de aumento da severidade destas infecções como função do aumento da temperatura ainda é desconhecida. Suponha que duas cepas de um dado parasita estejam competindo pelo mesmo hospedeiro. É possível mostrar que, em geral, a cepa com uma estratégia evolu- cionária estável, isto é, aquela que vence a competição, é aquela com maior valor de reprodutibilidade basal. Queremos saber quais combinações de temperatura ambiental T e virulência V maximizam Ro(T, V). Para isto calculamos o plano tangente ao ponto máximo (ou a uma região de máximo) e analisamos as respectivas curvas de nível. Para tanto, calculamos o seguinte sistema de equações diferenciais: ?Ro/?T = 0 ; ?Ro/?V = 0 (1). Agora, consideremos o caso de uma infecção transmitida por um vetor. De- monstramos que, neste caso, o aumento na Virulência do parasita está associada ao aumento na Temperatura. Esta hipótese é embasada por evidências empíricas de dengue hemorrágica em Singapura que vem aumentando sua virulência à medida em que há um aumento observado da temperatura local nos últimos anos. / The phenomenon of global increase of the temperature is reality unquestionable. In this case, it is expected that the increase in the global temperature will lead to an expansion of the geographical spread and to an increase in the incidence of tropical infections. However, the trend in severity of those infections as a function of the increase in the temperature is still unknown. Suppose that two strains of a given parasite are competing for the same host. It is possible to demonstrate that, in general, the strain with an evolutionary stable strategy, that is, the one that wins the competition, is the one with the highest value of R 0. We want to know which combination of environmental temperature T and virulence V maximizes R 0( V ). For this we calculate the tangent plane to the maximum point, that is ?Ro/?T=0 ; ?Ro/?V=0 (2) Now, let us consider the case of a vector-borne infection. We demonstrate, in this case, that the increase in temperature is associated with an increase in the parasite virulence. This hypothesis is supported by empirical evidence from dengue hemorrhagic fever in Singapore, which is increasing its virulence along with the increase in the local temperature observed in the last years.

Les infections à Papillomavirus Humain (HPV) sont des infections sexuellement transmissibles très fréquentes. La persistance de ces infections est un facteur causal du cancer du col de l’utérus et est aussi à l’origine d’autres cancers de la zone ano-génitale et de verrues génitales chez les femmes et chez les hommes. Depuis l’introduction de deux vaccins bivalent et quadrivalent permettant de prévenir certains types d’HPV, de nombreux modèles mathématiques ont été développés afin d’estimer l’impact potentiel de différentes stratégies de vaccination. L’objectif de ce travail de thèse a été d’estimer l’impact potentiel de la vaccination en France sur l’incidence de certains cancers liés à l’HPV, notamment le cancer du col de l’utérus et le cancer anal chez les femmes françaises ; ainsi que sur la prévalence des infections à HPV 6/11/16/18. Différents modèles dynamiques de type déterministe ont été développés. Ils sont représentés par des systèmes d’équations différentielles ordinaires. Une étude théorique du comportement asymptotique d’un premier modèle comportant peu de strates a été réalisée. Le nombre de reproduction de base R0 et le nombre de reproduction avec vaccination Rv ont été estimés. Des modèles plus complexes ont intégré une structure d’âge et de comportement sexuel. Les modélisations réalisées permettent de conclure à l’impact important de la vaccination sur la prévalence des infections à HPV et sur l’incidence des cancers du col de l’utérus et de la zone anale chez les femmes françaises dans un délai de quelques décennies, si l’on prend en compte les taux de vaccination observés en France au début de la campagne de vaccination / Human Papillomavirus infection (HPV) is the most frequent sexually transmitted disease. Epidemiological studies have established a causal relationship between HPV infections and occurence of cervical cancer. These infections have also been incriminated in anogenital cancers and anogenital warts among women and men. Since the introduction of bivalent and quadrivalent vaccines which offer protection against some HPV genotypes, many mathematical models have been developed in order to assess the potential impact of vaccine strategies. The aim of this thesis work was to assess the potential impact of HPV vaccination in France on the incidence of some cancers linked with HPV, particularly cervical cancer and anal cancer in French women, and on the prevalence of HPV 6/11/16/18 infections. Different deterministic dynamic models have been developped. They are represented by systems of ordinary differential equations. A theoretical analysis of the asymptotic behavior for a first model with few strata is realized. The basic reproduction number R0 and the vaccinated reproduction number Rv are assessed. More complex models taking into account age and sexual behavior have been developed. Using vaccination rates observed in France at the launch of the vaccination campaign, our modeling shows the large impact of vaccination on HPV prevalences, on cervical cancer and anal cancer incidences among French women within a few decades

O fenômeno do aumento global da temperatura é uma realidade inquestionável. Tendo em vista tal cenário, acredita-se que haverá uma expansão geográfica (migração de populações humanas) e um aumento na incidência de infecções tropicais. No entanto, a tendência de aumento da severidade destas infecções como função do aumento da temperatura ainda é desconhecida. Suponha que duas cepas de um dado parasita estejam competindo pelo mesmo hospedeiro. É possível mostrar que, em geral, a cepa com uma estratégia evolu- cionária estável, isto é, aquela que vence a competição, é aquela com maior valor de reprodutibilidade basal. Queremos saber quais combinações de temperatura ambiental T e virulência V maximizam Ro(T, V). Para isto calculamos o plano tangente ao ponto máximo (ou a uma região de máximo) e analisamos as respectivas curvas de nível. Para tanto, calculamos o seguinte sistema de equações diferenciais: ?Ro/?T = 0 ; ?Ro/?V = 0 (1). Agora, consideremos o caso de uma infecção transmitida por um vetor. De- monstramos que, neste caso, o aumento na Virulência do parasita está associada ao aumento na Temperatura. Esta hipótese é embasada por evidências empíricas de dengue hemorrágica em Singapura que vem aumentando sua virulência à medida em que há um aumento observado da temperatura local nos últimos anos. / The phenomenon of global increase of the temperature is reality unquestionable. In this case, it is expected that the increase in the global temperature will lead to an expansion of the geographical spread and to an increase in the incidence of tropical infections. However, the trend in severity of those infections as a function of the increase in the temperature is still unknown. Suppose that two strains of a given parasite are competing for the same host. It is possible to demonstrate that, in general, the strain with an evolutionary stable strategy, that is, the one that wins the competition, is the one with the highest value of R 0. We want to know which combination of environmental temperature T and virulence V maximizes R 0( V ). For this we calculate the tangent plane to the maximum point, that is ?Ro/?T=0 ; ?Ro/?V=0 (2) Now, let us consider the case of a vector-borne infection. We demonstrate, in this case, that the increase in temperature is associated with an increase in the parasite virulence. This hypothesis is supported by empirical evidence from dengue hemorrhagic fever in Singapore, which is increasing its virulence along with the increase in the local temperature observed in the last years.

Ce travail de thèse traite de la modélisation et du contrôle des maladies infectieuses à l’aide des automates cellulaires. Nous nous sommes d’abord focalisés sur l’étude d’un modèle de type SEIR. Nous avons pu monter d’une part qu’un voisinage fixe pouvait entrainer une sous-évaluation de l’incidence et de la prévalence et d’autre part que sa structure a un impact direct sur la structure de la distribution de la maladie. Nous nous sommes intéressés également la propagation des maladies vectorielles à travers un modèle de type SIRS-SI multi-hôtes dans un environnement hétérogène.Les hôtes y étaient caractérisés par leur niveau de compétence et l’environnement par la variation du taux de reproduction et de mortalité. Son application à la maladie de Chagas, nous a permis de montrer que l’hétérogénéité de l’habitat et la diversité des hôtes contribuaient à faire baisser l’infection. Cependant l’un des principaux résultats de notre travail à été la formulation du nombre de reproduction spatiale grâce à deux matrices qui représentent les coefficients d’interactions entre les différentes cellules du réseau. / This PhD thesis considers the general problem of epidemiological modelling and control using cellular automata approach.We first focused on the study of the SEIR model. On the one hand, we have shown that the traditionnal neighborhood contribute to underestimate the incidence and prevalence of infection disease. On the other hand, it appeared that the spatial distribution of the cells in the lattice have a real impact on the disease spreading. The second study concerns the transmission of the vector-borne disease in heterogeneous landscape with host community. We considered a SIRS-SI with various level of competence at witch the environnment heterogeneity has been characterized by the variation of the birth flow and the death rate. We simulated the Chagas disease spreading and shown that the heterogeneity of habitat and host diversity contribute to decrease the infection. One of the most important results of our work, was the proposition of the spatial reproduction number expression based on two matrices that represent the interaction factors between the cells in the lattice.

>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).

A pandemic subtype of influenza A sometimes replaces (e.g., in 1918, 1957, 1968) but sometimes coexists (e.g., in 1977) with the previous seasonal subtype. This research aims to determine a condition for replacement or coexistence of influenza subtypes. We formulate a hybrid model for the dynamics of influenza A epidemics taking into account cross-immunity of influenza strains depending on the most recent seasonal infection. A combination of theoretical and numerical analyses shows that for very strong cross-immunity between seasonal and pandemic subtypes, the pandemic cannot invade, whereas for strong and weak cross-immunity there is coexistence, and for intermediate levels of cross-immunity the pandemic may replace the seasonal subtype. Cross-immunity between seasonal strains is also a key factor of our model because it has a major influence on the final size of seasonal epidemics, and on the distribution of susceptibility in the population. To determine this cross-immunity, we design a novel statistical method, which uses a theoretical model and clinical data on attack rates and vaccine efficacy among school children for two seasons after the 1968 A/H3N2 pandemic. This model incorporates the distribution of susceptibility and the dependence of cross-immunity on the antigenic distance of drifted strains. We find that the cross-immunity between an influenza strain and the mutant that causes the next epidemic is 88%. Our method also gives an estimated value 2.15 for the basic reproduction number of the 1968 pandemic influenza. Our hybrid model agrees qualitatively with the observed subtype replacement or coexistence in 1957, 1968 and 1977. However, our model with the homogeneous mixing assumption significantly over estimates the pandemic attack rate. Thus, we modify the model to incorporate heterogeneity in the contact rate of individuals. Using the determined values of cross-immunity and the basic reproduction number, this modification lowers the pandemic attack rate slightly, but it is still higher than the observed attack rates. / Graduate

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.