In an infectious disease with a long infectious period (which can be the entire life for some diseases), the infectivity of individuals may change due to different reasons. For example, infected individuals may receive treatment and their level of infectivity can reduce depending on the efficacy of the treatment. Or, infected individuals may change their behaviour and reduce their activity once the disease is diagnosed, leading to a reduction of their infectivity. Treated individuals may stop getting treatment, and return to the infective class at a rate depending on how long they have been receiving treatment. In this thesis, a compartmental model consisting of three compartments (susceptibles, infectives and treated infectives) is formulated to study the effect of treatment on the transmission dynamics of a disease. Continuous and discrete treatment-age-structured models are derived and the asymptotic behaviour of the system is studied and the basic reproduction number is determined. / October 2016

Since the introduction of vaccines, many deaths due to various diseases including measles, have been drastically reduced. In Canada, there is a recommended vaccine schedule for all residents of the country; however, vaccine practises and immunisation schedules can vary from location to location as well as vary from country to country, leading to discrepancies in vaccine coverage and herd immunities. In addition, some anti-vaccination movements have been noted to persuade individuals into refusing vaccines, even in historically well immunised locations. In order to investigate the effect of varying vaccine coverage, a two patch metapopulation model for measles incorporating a single dose vaccine is formulated and studied. / October 2016

Avian influenza strains have been proven to be highly virulent in human populations, killing approximately 70 percent of infected individuals. Although the virus is able to spread across species from birds-to-humans, some strains, such as H5N1, have not been observed to spread from human-to-human. Pigs are capable of infection by both avian and human strains and seem to be likely candidates as intermediate hosts for co-infection of the inter-species strains. A co-infected pig potentially acts as a mixing vessel and could produce a new strain as a result of a recombination process. Humans could be immunologically naive to these new strains, hence making them super-strains. We propose an interacting host system (IHS) for such a process that considers three host species that interact by sharing strains; that is, a primary and secondary host species can both infect an intermediate host. When an intermediate host is co-infected with the strains from both the other hosts, co-infected individuals may become carriers of a super-strain back into the primary host population. The model is formulated as a classical susceptible-infectious-susceptible (SIS) model, where the primary and intermediate host species have a super-infection and co-infection with recombination structure, respectively. The intermediate host is coupled to the other host species at compartments of given infectious subclasses of the primary and secondary hosts. We use the model to give a new perspective for the trade-off hypothesis for disease virulence, by analyzing the behavior of a highly virulent super-strain. We give permanence conditions for a number of the subsystems of the IHS in terms of basic reproductive numbers of independent strains. We also simulate several relevant scenarios showing complicated dynamics and connections with epidemic forecasting.

This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes. The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.

Dengue (pronounced den'guee) Fever (DF) and Dengue Hemorrhagic Fever (DHF), collectively known as "dengue," are mosquito-borne, potentially mortal, flu-like viral diseases that affect humans worldwide. Transmitted to humans by the bite of an infected mosquito, dengue is caused by any one of four serotypes, or antigen-specific viruses. In this thesis, both the spatial and temporal dynamics of dengue transmission are investigated. Different chapters present new models while building on themes of previous chapters. In Chapter 2, we explore the temporal dynamics of dengue viral transmission by presenting and analyzing an ODE model that combines an SIR human host- with a multi-stage SI mosquito vector transmission system. In the case where the juvenile populations are at carrying capacity, juvenile mosquito mortality rates are sufficiently small to be absorbed by juvenile maturation rates, and no humans die from dengue, both the analysis and numerical simulations demonstrate that an epidemic will persist if the oviposition rate is greater than the adult mosquito death rate. In Chapter 3, we present and analyze a non-autonomous, non-linear ODE system that incorporates seasonality into the modeling of the transmission of the dengue virus. We derive conditions for the existence of a threshold parameter, the basic reproductive ratio, denoting the expected number of secondary cases produced by a typically infective individual. In Chapter 4, we present and analyze a non-linear system of coupled reaction-diffusion equations modeling the virus' spatial spread. In formulating our model, we seek to establish the existence of traveling wave solutions and to calculate spread rates for the spatial dissemination of the disease. We determine that the epidemic wave speed increases as average annual, and in our case, winter, temperatures increase. In Chapter 5, we present and analyze an ODE model that incorporates two serotypes of the dengue virus and allows for the possibility of both primary and secondary infections with each serotype. We obtain an analytical expression for the basic reproductive number, R_0, that defines it as the maximum of the reproduction numbers for each strain/serotype of the virus. In each chapter, numerical simulations are conducted to support the analytical conclusions.

Chlamydia, caused by the bacterium Chlamydia trachomatis, is one of the most important sexually-transmitted infections globally. In addition to accounting for millions of cases every year, the disease causes numerous irreversible complications such as chronic pelvic pain, infertility in females and pelvic inflammatory disease. This thesis presents a number of mathematical models, of the form of deterministic systems of non-linear differential equations, for gaining qualitative insight into the transmission dynamics and control of Chlamydia within an infected host (in vivo) and in a population. The models designed address numerous important issues relating to the transmission dynamics of Chlamydia trachomatis, such as the roles of immune response, sex structure, time delay (in modelling the latency period) and risk structure (i.e., risk of acquiring or transmitting infection). The in-host model is shown to have a globally-asymptotically stable Chlamydia-free equilibrium whenever a certain biological threshold is less than unity. It has a unique Chlamydia-present equilibrium when the threshold exceeds unity. Unlike the in-host model, the two-group (males and females) population-level model undergoes a backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon, which is shown to be caused by the re-infection of recovered individuals, makes the effort to eliminate the disease from the population more difficult. Extending the two-group model to incorporate risk structure shows that the backward bifurcation phenomenon persists even when recovered individuals do not acquire re-infection. In other words, it is shown that stratifying the sexually-active population in terms of risk of acquiring or transmitting infection guarantees the presence of backward bifurcation in the transmission dynamics of Chlamydia in a population. Finally, it is shown (via numerical simulations) that a future Chlamydia vaccine that boosts cell-mediated immune response will be more effective in curtailing Chlamydia burden in vivo than a vaccine that enhances humoral immune response. The population-level impact of various targeted treatment strategies, in controlling the spread of Chlamydia in a population, are compared. In particular, it is shown that the use of treatment could have positive or negative population-level impact (depending on the sign of a certain epidemiological threshold).

Chlamydia, caused by the bacterium Chlamydia trachomatis, is one of the most important sexually-transmitted infections globally. In addition to accounting for millions of cases every year, the disease causes numerous irreversible complications such as chronic pelvic pain, infertility in females and pelvic inflammatory disease. This thesis presents a number of mathematical models, of the form of deterministic systems of non-linear differential equations, for gaining qualitative insight into the transmission dynamics and control of Chlamydia within an infected host (in vivo) and in a population. The models designed address numerous important issues relating to the transmission dynamics of Chlamydia trachomatis, such as the roles of immune response, sex structure, time delay (in modelling the latency period) and risk structure (i.e., risk of acquiring or transmitting infection). The in-host model is shown to have a globally-asymptotically stable Chlamydia-free equilibrium whenever a certain biological threshold is less than unity. It has a unique Chlamydia-present equilibrium when the threshold exceeds unity. Unlike the in-host model, the two-group (males and females) population-level model undergoes a backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon, which is shown to be caused by the re-infection of recovered individuals, makes the effort to eliminate the disease from the population more difficult. Extending the two-group model to incorporate risk structure shows that the backward bifurcation phenomenon persists even when recovered individuals do not acquire re-infection. In other words, it is shown that stratifying the sexually-active population in terms of risk of acquiring or transmitting infection guarantees the presence of backward bifurcation in the transmission dynamics of Chlamydia in a population. Finally, it is shown (via numerical simulations) that a future Chlamydia vaccine that boosts cell-mediated immune response will be more effective in curtailing Chlamydia burden in vivo than a vaccine that enhances humoral immune response. The population-level impact of various targeted treatment strategies, in controlling the spread of Chlamydia in a population, are compared. In particular, it is shown that the use of treatment could have positive or negative population-level impact (depending on the sign of a certain epidemiological threshold).

Este trabalho, ao apresentar os conceitos básicos da Epidemiologia Matemática e da modelagem de populações por classes etárias, tem por objetivo desenvolver e implementar três modelos epidemiológicos de transmissão do dengue, a fim de avaliar teoricamente os efeitos da aplicação de inseticidas em populações de Aedes aegypti, em relação às epidemias de dengue. Uma variedade de métodos têm sido empregados no controle do vetor, sendo o Aedes aegypti a principal espécie envolvida na transmissão do dengue. A· aplicação de inseticidas de ultrabaixo volume (UL V) é uma das técnicas amplamente utilizada, particularmente durante epidemias. Tal técnica tem como objetivo matar os mosquitos adultos (adulticida). Há muita controvérsia em tomo destas aplicações, no que diz respeito ao impacto no controle da transmissão do dengue. Desta forma, através deste trabalho, procuramos observar a influência do uso de inseticida na dinâmica populacional do vetor transmissor e na dinâmica da epidemia, e analisar as circunstâncias em que o inseticida pode ser utilizado a fim de agir eficientemente no controle da transmissão do dengue. O emprego de larvicidas também é abordado, a fim de que se possa observar a influência deste na dinâmica populacional do vetor transmissor e na dinâmica da epidemia. Neste trabalho são detalhadas as hipóteses utilizadas na construção de cada modelo de transmissão do dengue apresentado. Apresentados os modelos, são considerados os aspectos relativos à implementação. Assim, mediante os aspectos teóricos envolvidos na modelagem e implementação, resultados numéricos são obtidos, através das simulações, as quais nos auxiliam a avaliar os efeitos da aplicação de inseticidas em populações de Aedes aegypti no controle da transmissão do dengue. / In this work the basic Mathematical Epidemiology ideas are presented and three models of Dengue Fever transrnission are developed in order to measure the effects of the use o f insecticides on the populations o f the mosquito Aedes aegypti that are related to the dengue epidemics. There is a good variety of control methods on the Aedes aegypti populations. The use of ultra-low volume (UL V) insecticides is widely employed specially during epidemics. The goal of such method is to eliminate a fraction of the adult mosquito population. There is a great deal o f controversy on the effectiveness o f insecticide use during a dengue epidemic. In this way we propose to investigate the impact o f UL V on the mosquito dynarnics and on the epidemics dynarnics as well in order to determine in which circumstances the use o f UL V can truly effective on the course o f a epidemic. On the same line, we also propose a study on the use of larvicides as a control method. In this study we detail the hypotheses that are used to construct the dynarnic models. Once the models are presented we consider the implementation details. The numerical results are obtained after various simulations which provide the data that allow us to measure the impact ofthe control technique on the dengue epidemics.

We introduce random fluctuations on contact and recovery rates in three basic deterministic models in mathematical epidemiology and obtain stochastic counterparts. This paper addresses qualitative and quantitative analysis of stochastic SIS model with disease deaths and demographic effects, and stochastic SIR models with/without disease deaths and demographic effects. We prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria. We also investigate numerical properties of these models and prove the convergence of the Balanced Implicit Method approximation to the analytic solution. We simulate the models with fairly realistic parameters to visualize our conclusions.

Este trabalho, ao apresentar os conceitos básicos da Epidemiologia Matemática e da modelagem de populações por classes etárias, tem por objetivo desenvolver e implementar três modelos epidemiológicos de transmissão do dengue, a fim de avaliar teoricamente os efeitos da aplicação de inseticidas em populações de Aedes aegypti, em relação às epidemias de dengue. Uma variedade de métodos têm sido empregados no controle do vetor, sendo o Aedes aegypti a principal espécie envolvida na transmissão do dengue. A· aplicação de inseticidas de ultrabaixo volume (UL V) é uma das técnicas amplamente utilizada, particularmente durante epidemias. Tal técnica tem como objetivo matar os mosquitos adultos (adulticida). Há muita controvérsia em tomo destas aplicações, no que diz respeito ao impacto no controle da transmissão do dengue. Desta forma, através deste trabalho, procuramos observar a influência do uso de inseticida na dinâmica populacional do vetor transmissor e na dinâmica da epidemia, e analisar as circunstâncias em que o inseticida pode ser utilizado a fim de agir eficientemente no controle da transmissão do dengue. O emprego de larvicidas também é abordado, a fim de que se possa observar a influência deste na dinâmica populacional do vetor transmissor e na dinâmica da epidemia. Neste trabalho são detalhadas as hipóteses utilizadas na construção de cada modelo de transmissão do dengue apresentado. Apresentados os modelos, são considerados os aspectos relativos à implementação. Assim, mediante os aspectos teóricos envolvidos na modelagem e implementação, resultados numéricos são obtidos, através das simulações, as quais nos auxiliam a avaliar os efeitos da aplicação de inseticidas em populações de Aedes aegypti no controle da transmissão do dengue. / In this work the basic Mathematical Epidemiology ideas are presented and three models of Dengue Fever transrnission are developed in order to measure the effects of the use o f insecticides on the populations o f the mosquito Aedes aegypti that are related to the dengue epidemics. There is a good variety of control methods on the Aedes aegypti populations. The use of ultra-low volume (UL V) insecticides is widely employed specially during epidemics. The goal of such method is to eliminate a fraction of the adult mosquito population. There is a great deal o f controversy on the effectiveness o f insecticide use during a dengue epidemic. In this way we propose to investigate the impact o f UL V on the mosquito dynarnics and on the epidemics dynarnics as well in order to determine in which circumstances the use o f UL V can truly effective on the course o f a epidemic. On the same line, we also propose a study on the use of larvicides as a control method. In this study we detail the hypotheses that are used to construct the dynarnic models. Once the models are presented we consider the implementation details. The numerical results are obtained after various simulations which provide the data that allow us to measure the impact ofthe control technique on the dengue epidemics.