| Branching processes have served as a model for chemical reactions, biological growth processes and contagion (of disease, information or fads). Through this connection, these seemingly different physical processes share some common universalities that can be elucidated by analyzing the underlying branching process. In this thesis, we focus on branching processes as a model for infectious diseases spreading between individuals belonging to different populations. The distinction between populations can arise from species separation (as in the case of diseases which jump across species) or spatial separation (as in the case of disease spreading between farms, cities, urban centers, etc). A prominent example of the former is zoonoses -- infectious diseases that spill from animals to humans -- whose specific examples include Nipah virus, monkeypox, HIV and avian influenza. A prominent example of the latter is infectious diseases of animals such as foot and mouth disease and bovine tuberculosis that spread between farms or cattle herds. Another example of the latter is infectious diseases of humans such as H1N1 that spread from one city to another through migration of infectious hosts.;This thesis consists of three main chapters, an introduction and an appendix. The introduction gives a brief history of mathematics in modeling the spread of infectious diseases along with a detailed description of the most commonly used disease model -- the Susceptible-Infectious-Recovered (SIR) model. The introduction also describes how the stochastic formulation of the model reduces to a branching process in the limit of large population which is analyzed in detail. The second chapter describes a two species model of zoonoses with coupled SIR processes and proceeds into the calculation of statistics pertinent to cross species infection using multitype branching processes. The third chapter describes an SIR process driven by a Poisson process of infection spillovers. This is posed as a model of infectious diseases where a `reservoir' of infection exists that infects a susceptible host population at a constant rate. The final chapter of the thesis describes a general framework of modeling infectious diseases in a network of populations using multitype branching processes. |
