Network worms that scan random computers have caused billions of dollars in damage to enterprises across the Internet. Earlier research has concentrated on using epidemiological models to predict the number of computers a worm will infect and how long it takes to do so. In this research, one possible approach is outlined for predicting the spatial flow of a worm within the local area network (LAN). The approach in this research is based on the application of mathematical models and variables inherent in plant epidemiology. In particular, spatial autocorrelation has been identified as a candidate variable that helps predict the spread of a worm over a LAN. This research describes the application of spatial autocorrelation to the geography and topology of the LAN and describes the methods used to determine spatial autocorrelation. Also discussed is the data collection process and methods used to extract pertinent information. Data collection and analyses are applied to the spread of three historical network worms on the Virginia Tech campus and the results are described. Spatial autocorrelation exists in the spread of network worms across the Virginia Tech campus when the geographic aspect is considered. If a new network worm were to start spreading across Virginia Tech's campus, spatial autocorrelation would facilitate tracking the geographical locations of the spread. In addition if an infection with a known value of spatial autocorrelation is detected, the characteristics of the worm can be identified without a complete analysis. / Master of Science

Doctor of Philosophy / Department of Electrical and Computer Engineering / Caterina Scoglio / 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.

abstract: Rabies disease remains enzootic among raccoons, skunks, foxes and bats in the United States. It is of primary concern for public-health agencies to control spatial spread of rabies in wildlife and its potential spillover infection of domestic animals and humans. Rabies is invariably fatal in wildlife if untreated, with a non-negligible incubation period. Understanding how this latency affects spatial spread of rabies in wildlife is the concern of chapter 2 and 3. Chapter 1 deals with the background of mathematical models for rabies and lists main objectives. In chapter 2, a reaction-diffusion susceptible-exposed-infected (SEI) model and a delayed diffusive susceptible-infected (SI) model are constructed to describe the same epidemic process -- rabies spread in foxes. For the delayed diffusive model a non-local infection term with delay is resulted from modeling the dispersal during incubation stage. Comparison is made regarding minimum traveling wave speeds of the two models, which are verified using numerical experiments. In chapter 3, starting with two Kermack and McKendrick's models where infectivity, death rate and diffusion rate of infected individuals can depend on the age of infection, the asymptotic speed of spread $c^\ast$ for the cumulated force of infection can be analyzed. For the special case of fixed incubation period, the asymptotic speed of spread is governed by the same integral equation for both models. Although explicit solutions for $c^\ast$ are difficult to obtain, assuming that diffusion coefficient of incubating animals is small, $c^\ast$ can be estimated in terms of model parameter values. Chapter 4 considers the implementation of realistic landscape in simulation of rabies spread in skunks and bats in northeast Texas. The Finite Element Method (FEM) is adopted because the irregular shapes of realistic landscape naturally lead to unstructured grids in the spatial domain. This implementation leads to a more accurate description of skunk rabies cases distributions. / Dissertation/Thesis / Ph.D. Mathematics 2013