abstract: Extraordinary medical advances have led to significant reductions in the burden of infectious diseases in humans. However, infectious diseases still account for more than 13 million annual deaths. This large burden is partly due to some pathogens having found suitable conditions to emerge and spread in denser and more connected host populations, and others having evolved to escape the pressures imposed by the rampant use of antimicrobials. It is then critical to improve our understanding of how diseases spread in these modern landscapes, characterized by new host population structures and socio-economic environments, as well as containment measures such as the deployment of drugs. Thus, the motivation of this dissertation is two-fold. First, we study, using both data-driven and modeling approaches, the the spread of infectious diseases in urban areas. As a case study, we use confirmed-cases data on sexually transmitted diseases (STDs) in the United States to assess the conduciveness of population size of urban areas and their socio-economic characteristics as predictors of STD incidence. We find that the scaling of STD incidence in cities is superlinear, and that the percent of African-Americans residing in cities largely determines these statistical patterns. Since disparities in access to health care are often exacerbated in urban areas, within this project we also develop two modeling frameworks to study the effect of health care disparities on epidemic outcomes. Discrepant results between the two approaches indicate that knowledge of the shape of the recovery period distribution, not just its mean and variance, is key for assessing the epidemiological impact of inequalities. The second project proposes to study, from a modeling perspective, the spread of drug resistance in human populations featuring vital dynamics, stochasticity and contact structure. We derive effective treatment regimes that minimize both the overall disease burden and the spread of resistance. Additionally, targeted treatment in structured host populations may lead to higher levels of drug resistance, and if drug-resistant strains are compensated, they can spread widely even when the wild-type strain is below its epidemic threshold. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2014

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.

No description available.

For decades, dengue virus has been a cause of major public health concern in Costa Rica, due to its landscape and climatic conditions that favor the circumstances in which the vector, Aedes aegypti, thrives. The emergence and introduction throughout tropical and subtropical countries of the chikungunya virus, as of 2014, challenged Costa Rican health authorities to provide a correct diagnosis since it is also transmitted by the same vector and infected hosts may share similar symptoms. We study the 2015–2016 dengue and chikungunya outbreaks in Costa Rica while establishing how point estimates of epidemic parameters for both diseases compare to one another. Longitudinal weekly incidence reports of these outbreaks signal likely misdiagnosis of infected individuals: underreporting of chikungunya cases, while overreporting cases of dengue. Our comparative analysis is formulated with a single-outbreak deterministic model that features an undiagnosed class. Additionally, we also used a genetic algorithm in the context of weighted least squares to calculate point estimates of key model parameters and initial conditions, while formally quantifying misdiagnosis.

<p>Infectious diseases spreading in a human population can occasionally exhibit sudden transitions in their qualitative dynamics. Previous work has been very successful in predicting such transitions in New York City's measles incidence rates using the standard SIR model (susceptible, infected, recovered). This work relied on a dataset spanning 45 years, which we have extended to 93 years (1891-1984). We continue previous research in transition analysis on this larger dataset, and compare resonant and transient periods predicted to exist in NYC's measles incidence rates with those observed through a continuous wavelet transform of the data. We find good agreement between SIR predictions and observation, and in particular note the likely existence of previously unobserved hysteresis early in our new time-series.</p> / Master of Science (MSc)

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.

<div>Wildlife diseases are an increasingly growing concern for public health managers, conservation biologists, and society at large. These diseases may be zoonotic -- infective wildlife are able to spread pathogens to human populations. Animal or plant species of conservation concern may also be threatened with extinction or extirpation due to the spread of novel pathogens into their native ranges. In this thesis, I develop some mathematical methods for understanding the dynamics of vector-borne diseases in wildlife populations which include several elements of host and vector biology. </div><div><br></div><div>We consider systems where a vector-borne pathogen is transmitted to a host population wherein individuals either die to disease or recover, remaining chronically infective. Both ordinary differential equations (ODE) and individual based (IBM) models of such systems are formulated then applied to a specific system of wildlife disease: avian malaria in Hawaiian honeycreeper populations -- where some species endure disease-induced mortality rates exceeding 90\%. The ODE model predicts that conventional management methods cannot fully stop pathogen transmission.</div><div><br></div><div>Vector dispersal and reproductive biology may also play a large role in the transmission of vector-borne diseases in forested environments. Using an IBM which models dispersal and mosquito reproductive biology, we predict that reducing larval habitat at low elevations is much more effective than at higher elevations. The ODE model is extended to include distinct populations of sensitive and tolerant hosts. We find that the form which interaction between the hosts takes has a significant impact on model predictions.</div>

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.

In this study we are interested on the following 4-dimensional system of stochastic differential equations.dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4 dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2 dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3 dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4 with variance parameters σ_i≥0 and constants α,β,η,γ,μ ζ≥0. This system may be used to model the dynamics of susceptible, exposed, infected and recovering individuals subject to a present virus with state-dependent random transitions. Our main goal is to prove the existence of a bounded, unique, strong (pathwise), global solution to this system, and to discuss asymptotic stochastic and moment stability of the two equilibrium points, namely the disease free and the endemic equilibria. In this model, as suggested by our advisor, diffusion coefficients can be any local Lipschitz continuous functions on bounded domain D={(S,E,I,R)∈R_+^4:00 of maximum carrying capacity and W_i are independent and identical Wiener processes defined on a complete probability space (Ω,F,{F_t }_(t≥0),P). At the end we carry out some simulations to illustrate our results.

The stochastic SEIR(S) model with random total population is given by the system of stochastic differential equations:dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4+σ_5 S(K-N)dW_5\\ dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2+σ_5 E(K-N)dW_5 \\ dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3+σ_5 I(K-N)dW_5 \\ dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4+σ_5 R(K-N)dW_5, where σ_i>0 and constants α, β, η, γ, ζ, μ≥0. K represents the maximum carrying capacity for the total population and W_k=(W_k (t))_(t≥0) are independent, standard Wiener processes on a complete probability space (Ω,F,(F_t )_(t≥0),P). The SDE for the total population N=S+E+I+R has the form dN(t)=μ(K-N)dt+σ_5 N(K-N)dW_5 on D_0=(0,K). The goal of our study is to prove the existence of unique, Markovian, continuous time solutions on the 4D prism D={(S,E,I,R)∈R_+^4:0≤S, E,I,R≤K, S+E+I+R≤K}. Then using the method of Lyapunov functions we prove the asymptotic stochastic and moment stability of disease-free and endemic equilibria. Finally, we use numerical simulations to illustrate our results.