In this thesis, we propose a statistical model to predict disease dispersal in dynamic networks. We model the process of disease spreading using discrete time Markov chain. In this case, the vector of probability of infection is the state vector and every element of the state vector is a continuous variable between zero and one. In discrete time Markov chains, state probability vectors in each time step depends on state probability vector in the previous time step and one step transition probability matrix. The transition probability matrix can be time variant or time invariant. If this matrix’s elements are functions of elements of vector state probability in previous step, the corresponding Markov chain is non linear dynamical system. However, if those elements are independent of vector state probability, the corresponding Markov chain is a linear dynamical system.

We especially focus on the dispersal of soybean rust. In our problem, we have a network of US counties and we aim at predicting that which counties are more likely to get infected by soybean rust during a year based on observations of soybean rust up to that time as well as corresponding observations to previous years. Other data such as soybean and kudzu densities in each county, daily wind data, and distance between counties helps us to build the model.

The rapid growth in the number of Internet users in recent years has led malware generators to exploit this potential to attack computer users around the word. Internet users are frequent targets of malicious software every day. The ability of malware to exploit the infrastructures of networks for propagation determines how detrimental they can be to the network’s security. Malicious software can make large outbreaks if they are able to exploit the structure of the Internet and interactions between users to propagate.

Epidemics typically start with some initial infected nodes. Infected nodes can cause their

healthy neighbors to become infected with some probability. With time and in some cases with external intervention, infected nodes can be cured and go back to a healthy state. The study of epidemic dispersals on networks aims at explaining how epidemics evolve and spread in networks. One of the most interesting questions regarding an epidemic spread in a network is whether the epidemic dies out or results in a massive outbreak. Epidemic threshold is a parameter that addresses this question by considering both the network topology and epidemic strength.