Modeling and analysis of vector-borne diseases on complex networks

Abstract

Vector-borne diseases not only cause devastating economic losses, they also significantly impact human health in terms of morbidity and mortality. From an economical and humane point of view, mitigation and control of vector-borne diseases are essential. Studying dynamics of vector-borne disease transmission is a challenging task because vector-borne diseases show complex dynamics impacted by a wide range of ecological factors. Understanding these factors is important for the development of mitigation and control strategies. Mathematical models have been commonly used to translate assumptions concerning biological (medical, demographical, behavioral, immunological) aspects into mathematics, linking biological processes of transmission and dynamics of infection at population level. Mathematical analysis translates results back into biology. Classical deterministic epidemic models do not consider spatial variation, assuming space is homogeneous. Spatial spread of vector-borne diseases observed many times highlights the necessity of incorporating spatial dynamics into mathematical models. Heterogeneous demography, geography, and ecology in various regions may result in different epidemiological characteristics. Network approach is commonly used to study spatial evolution of communicable diseases transmitted among connected populations. In this dissertation, the spread of vector-borne diseases in time and space, is studied to understand factors that contribute to disease evolution. Network-based models have been developed to capture different features of disease transmission in various environments. Network nodes represent geographical locations, and the weights represent the level of contact between regional pairings. Two competent vector populations, Aedes mosquitoes and Culex mosquitoes, and two host populations, cattle and humans were considered. The deterministic model was applied to the 2010 Rift Valley fever outbreak in three provinces of South Africa. Trends and timing of the outbreak in animals and humans were reproduced. The deterministic model with stochastic parameters was applied to hypothetical Rift Valley fever outbreak on a large network in Texas, the United States. The role of starting location and size of initial infection in Rift Valley fever virus spread were studied under various scenarios on a large-scale network. The reproduction number, defined as the number of secondary infections produced by one infected individual in a completely susceptible population, is typically considered an epidemic threshold of determining whether a disease can persist in a population. Extinction thresholds for corresponding Continuous-time Markov chain model is used to predict whether a disease can perish in a stochastic setting. The network level reproduction number for diseases vertically and horizontally transmitted among multiple species on heterogeneous networks was derived to predict whether a disease can invade the whole system in a deterministic setting. The complexity of computing the reproduction number is reduced because the expression of the reproduction number is the spectral radius of a matrix whose size is smaller than the original next generation matrix. The expression of the reproduction number may have a wide range of applications to many vector-borne diseases. Reproduction numbers can vary from below one to above one or from above one to below one by changing movement rates in different scenarios. The observations provide guidelines on executing movement bans in case of an epidemic. To compute the extinction threshold, corresponding Markov chain process is approximated near disease free equilibrium. The extinction threshold for Continuous-time Markov chain model was analytically connected to the reproduction number under some assumptions. Numerical simulation results agree with analytical results without assumptions, proposing a mathematical problem of proving the existence of the relationships in general. The distance of the extinction threshold were shown to be closer to one than the reproduction number. Consistent trends of probability of extinction varying with disease parameters observed through numerical simulations provide novel insights into disease mitigation, control, and elimination.