abstract: The 2009-10 influenza and the 2014-15 Ebola pandemics brought once again urgency to an old question: What are the limits on prediction and what can be proposed that is useful in the face of an epidemic outbreak? This thesis looks first at the impact that limited access to vaccine stockpiles may have on a single influenza outbreak. The purpose is to highlight the challenges faced by populations embedded in inadequate health systems and to identify and assess ways of ameliorating the impact of resource limitations on public health policy. Age-specific per capita constraint rates play an important role on the dynamics of communicable diseases and, influenza is, of course, no exception. Yet the challenges associated with estimating age-specific contact rates have not been decisively met. And so, this thesis attempts to connect contact theory with age-specific contact data in the context of influenza outbreaks in practical ways. In mathematical epidemiology, proportionate mixing is used as the preferred theoretical mixing structure and so, the frame of discussion of this dissertation follows this specific theoretical framework. The questions that drive this dissertation, in the context of influenza dynamics, proportionate mixing, and control, are: I. What is the role of age-aggregation on the dynamics of a single outbreak? Or simply speaking, does the number and length of the age-classes used to model a population make a significant difference on quantitative predictions? II. What would the age-specific optimal influenza vaccination policies be? Or, what are the age-specific vaccination policies needed to control an outbreak in the presence of limited or unlimited vaccine stockpiles? Intertwined with the above questions are issues of resilience and uncertainty including, whether or not data collected on mixing (by social scientists) can be used effectively to address both questions in the context of influenza and proportionate mixing. The objective is to provide answers to these questions by assessing the role of aggregation (number and length of age classes) and model robustness (does the aggregation scheme selected makes a difference on influenza dynamics and control) via comparisons between purely data-driven model and proportionate mixing models. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2016

Malaria is one of the most devastating infectious diseases, with nearly half of the worlds population currently at risk of infection. Although mathematical models have made significant contributions towards the control and elimination of malaria, it continues to evade control. This thesis focuses on two aspects of malaria that complicate dynamics, helping it persist. The basic reproductive number is one of the most important epidemiological quantities as it provides a foundation for control and elimination. Recently, it has been suggested that R0 should be modified to account for the effects of finite host population on a single disease-generation. In chapter 2, we analytically calculate these finite-population reproductive numbers for both vector-borne and directly transmitted diseases with homogeneous transmission. We find simple, generalizable formula and show that when the population is small, control and elimination may be easier than predicted by R0. In chapter 3, we extend the results of chapter 2 and find expressions for the finite- population reproductive numbers for directly transmitted diseases with different types of heterogeneity in transmission. We also outline a framework for discussing the different types of heterogeneity in transmission. We show that although the effects of heterogeneity in a small population are complex, the implications for control are simple: when R0 is large relative to the size of the population, control or elimination is made easier by heterogeneity. Another basic question in malaria modeling is the effects of immunity on the population- level dynamics of malaria. In chapter 4, we explore the possibility that clinical immunity can cause bistable malaria dynamics. This has important implications for control: in areas with bistable malaria, if malaria could be eliminated until clinical immunity wanes, it would not be able to invade. We built a simple, analytically tractable model of malaria transmission and solved it to find a criterion for when we expect bistability to occur. Additionally, we review what is known about about the parameters underlying the model and highlighted key clinical immunity parameters for which little is known. Building on these results, in chapter 5, we fit the model developed in chapter 4 to incidence data from Kericho, Kenya and estimate key clinical immunity parameters to better understand the role clinical immunity plays in malaria transmission. Finally, in chapter 6, we summarize the key results and discuss the broader implications of these findings on future malaria control. / Thesis / Doctor of Philosophy (PhD)

No description available.

Visceral leishmaniasis (VL) is a serious neglected tropical disease that is endemic in 98 countries and presents a significant public health risk. The epidemiology of VL is complex. In the Americas, it is a zoonotic disease that is caused by a parasite and transmitted among humans and dogs through the bite of an infected sand fly vector. The infection also can be transmitted vertically from mother to child during pregnancy. Infected individuals can be classified as asymptomatic or symptomatic; both classes can transmit infection. In part due to its complexity, VL transmission dynamics are not fully understood. Stochastic compartmental epidemic models are a powerful set of tools that can be used to study these transmission dynamics. Past compartmental models for VL have been developed in a deterministic framework to accommodate complexity while remaining computationally tractable. In this work, we propose stochastic compartmental models for VL, which are simpler than their deterministic counterparts, but also have several advantages. Notably, this framework allows us to: (1) define a probability of infection transmission between two individuals, (2) obtain both parameter estimates and corresponding uncertainty measures, and (3) employ formal model comparisons. In this dissertation, we develop both population level and individual level Bayesian compartmental models to study both vector and vertical VL transmission dynamics. As part of this model development, we introduce a compartmental model that allows for two infectious classes. We also derive source specific reproductive numbers to quantify the contributions of different species and infectious classes to maintaining infection in a population. Finally, we propose a formal model comparison method for Bayesian models with high-dimensional discrete parameter spaces. These models, reproductive numbers, and model comparison method are explored in the context of simulations and real VL data from Brazil and the United States.

Malaria is an infectious disease, transmitted between humans through mosquito bites, that kills about two million people a year. We derive and analyze a mathematical model to better understand the transmission and spread of this disease. Our main goal is to use this model to compare intervention strategies for malaria control for two representative areas of high and low transmission. We model malaria using ordinary differential equations. We analyze the existence and stability of disease-free and endemic (malaria persisting in the population) equilibria. Key to our analysis is the definition of a reproductive number, R₀ (the number of new infections caused by one individual in an otherwise fully susceptible population through the duration of the infectious period). We prove the loss of stability of the disease-free equilibrium as R0 increases through R₀ = 1. Using global bifurcation theory developed by Rabinowitz, we show the bifurcation of endemic equilibria at R₀ = 1. This bifurcation can be either supercritical (leading to stable endemic equilibria for R₀ > 1) or subcritical (leading to stable endemic equilibria for R₀ < 1 in the presence of hysteresis). We compile two reasonable sets of values for the parameters in the model: for areas of high and low transmission. We compute sensitivity indices of R₀ and the endemic equilibrium to the parameters around the baseline values. R₀ is most sensitive to the mosquito biting rate in both high and low transmission areas. The fraction of infectious humans at the endemic equilibrium is most sensitive to the mosquito biting rate in low transmission areas, and to the human recovery rate in high transmission areas. This sensitivity analysis allows us to compare the effectiveness of different control strategies. According to our model, the most effective methods for malaria control are the use of insecticide-treated bed nets and the prompt diagnosis and treatment of infected individuals.

Magister Scientiae - MSc / We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 < 1. This is the case of a disease-free state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method. These are the most useful references: [G. Zaman, Y.H Kang, II. H. Jung. BioSystems 93, (2008), 240 − 249], [K. Hattaf, N. Yousfi. The Journal of Advanced Studies in Biology, Vol. 1(8), (2008), 383 − 390.], [Lenhart, J.T. Workman. Optimal Control and Applied to Biological Models. Chapman and Hall/CRC, (2007).], [P. Van den Driessche, J. Watmough. Math. Biosci., 7, (2005)], and [J. Wu, G. R¨ost. Mathematical Biosciences and Engineering, Vol 5(2), (2008), 389 − 391]. / South Africa

The question “who infected whom” is a perennial one in the study of infectious disease dynamics. To understand characteristics of infectious diseases such as how many people will one case produce over the course of infection (the reproductive number), how much time between the infection of two connected cases (the generation interval), and what factors are associated with transmission, one must ascertain who infected whom. The current best practices for linking cases are contact investigations and pathogen whole genome sequencing (WGS). However, these data sources cannot perfectly link cases, are expensive to obtain, and are often not available for all cases in a study. This lack of discriminatory data limits the use of established methods in many existing infectious disease datasets. We developed a method to estimate the relative probability of direct transmission between any two infectious disease cases. We used a subset of cases that have pathogen WGS or contact investigation data to train a model and then used demographic, spatial, clinical, and temporal data to predict the relative transmission probabilities for all case-pairs using a simple machine learning algorithm called naive Bayes. We adapted existing methods to estimate the reproductive number and generation interval to use these probabilities. Finally, we explored the associations between various covariates and transmission and how they related to the associations between covariates and pathogen genetic relatedness. We applied these methods to a tuberculosis outbreak in Hamburg, Germany and to surveillance data in Massachusetts, USA. Through simulations we found that our estimated transmission probabilities accurately classified pairs as links and nonlinks and were able to accurately estimate the reproductive number and the generation interval. We also found that the association between covariates and genetic relatedness captures the direction but not absolute magnitude of the association between covariates and transmission, but the bias was improved by using effect estimates from the naive Bayes algorithm. The methods developed in this dissertation can be used to explore transmission dynamics and estimate infectious disease parameters in established datasets where this was not previously feasible because of a lack of highly discriminatory information, and therefore expand our understanding of many infectious diseases.

When an outbreak of an infectious disease occurs, public health officials need to understand the dynamics of disease transmission in order to launch an effective response. Two quantities that are often used to describe transmission are the basic reproductive number and the distribution of the serial interval. The basic reproductive number, R0, is the average number of secondary cases a primary case will infect, assuming a completely susceptible population. The serial interval (SI) provides a measure of temporality, and is defined as the time between symptom onset between a primary case and its secondary case. Investigators typically collect outbreak data in the form of an epidemic curve that displays the number of cases by each day (or other time scale) of the outbreak. Occasionally the epidemic curve data is more expansive and includes demographic or other information. A contact trace sample may also be collected, which is based on a sample of the cases that have their contact patterns traced to determine the timing and sequence of transmission. In addition, numerous large scale social mixing surveys have been administered in recent years to collect information about contact patterns and infection rates among different age groups. These are readily available and are sometimes used to account for population heterogeneity. In this dissertation, we modify the methods presented in White and Pagano (2008) to account for additional data beyond the epidemic curve to estimate R0 and SI. We present two approaches that incorporate these data through the use of a Bayesian framework. First, we consider informing the prior distribution of the SI with contact trace data and examine implications of combining data that are in conflict. The second approach extends the first approach to account for heterogeneity in the estimation of R0. We derive a modification to the White and Pagano likelihood function and utilize social mixing surveys to inform the prior distributions of R0. Both approaches are assessed through a simulation study and are compared to alternative approaches, and are applied to real outbreak data from the 2003 SARS outbreak in Hong Kong and Singapore, and the influenza A(H1N1)2009pdm outbreak in South Africa.

Significant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. In-fact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune defense system after it has infected the host. In this dissertation we develop an SIR model with variable infection age for the transmission of a pathogen that can mutate in the host to produce a second infectious mutant strain. We assume that there is a period of temporary immunity in the model. A temporary immunity period along with variable infection age leads to an integro-differential-difference model. Previous efforts on incorporating delays in epidemic models have mainly concentrated on inclusion of latency periods (this assumes that the force of infection at a present time is determined by the number of infectives in the past). We begin with reviewing some basic models. These basic models are the building blocks for the later, more detailed models. Next we consider the model for mutation of pathogen and discuss its implications. Finally, we improve this model for mutation of pathogen by incorporating delay induced by temporary immunity. We examine the influence of delay as we establish the existence, and derive the explicit forms of disease-free, boundary and endemic equilibriums. We will also investigate the local stability of each of these equilibriums. The possibility of Hopf bifurcation using delay as the bifurcation parameter is studied using both analytical and numerical solutions.

<p>We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 &lt / 1. This is the case of a disease-free&nbsp / state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic&nbsp / and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We&nbsp / use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on&nbsp / vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious&nbsp / disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method.&nbsp / These are the most useful references: [G. Zaman, Y.H Kang, II. H. Jung. BioSystems 93,&nbsp / (2008), 240 &minus / 249], [K. Hattaf, N. Yousfi. The Journal of Advanced Studies in Biology, Vol. 1(8), (2008), 383 &minus / 390.], [Lenhart, J.T. Workman. Optimal Control and Applied to Biological Models.&nbsp / Chapman and Hall/CRC, (2007).], [P. Van den Driessche, J. Watmough. Math. Biosci., 7,&nbsp / (2005)], and [J. Wu, G. R&uml / ost. Mathematical Biosciences and Engineering, Vol 5(2), (2008), 389 &minus / 391].</p>