Diffusion and epidemics on networks

This thesis is about diffusion process on networks (in the particular case of diseasespreading), more precisely it looks on the impact of communities in the diffusion process on network. The main goal of this work is to find a parameter characterizing the diffusion given a particular form of the network. This parameter will be useful if we want to know if the diffusion will become an epidemic (i.e. reach the whole network) or not. The thesis is split into six chapters (first and last chapters are introduction and conclusion). Chapter two presents two classical epidemic models on single and homogeneous population. The main goal of this chapter is to give an intuitive notion of the reproductive number. Chapter three presents networks with two homogeneous communities on the same size (number of individuals). Chapter four presents a generalization of chapter three to several homogeneous communities. Chapter three and four are a study of the adjacent matrix of directed network. Chapter 5 use statistical arguments to study the diffusion in general networks which can be homogeneous of not.