The Symbiotic Contact Process on Non-Lattice Structures

The Symbiotic Contact Process (SCP) can be thought of as a two type generalisation of the Contact Process which can be used to model the spread of two symbiotic diseases. The SCP is an interacting particle system defined on a graph, which can be characterised by the interactions between sites. Each site can either be infected with type A, type B, both, or neither. Infections of either type at a given site occur at a rate of lambda multiplied by the number of neighbours infected by that type. Recoveries of either type at a given site occur at rate 1 if only one type is present, or at a lower rate mu if both types are present, hence the symbiotic name. Both the Contact Process and the Symbiotic Contact Process have two critical infection rates on a tree (random or otherwise) that may or may not coincide, one determining weak survival, and the other strong survival. Here, weak survival refers to the event where at least one A infection and at least one B infection is present at all times. Strong survival is the event that the root of the tree is infected with both A and B infections at the same time infinitely often. 

In this thesis we will investigate the persistence of the process on star graphs. This persistence will be used to prove bounds on the critical values for SCP on k regular trees as well as Galton-Watson trees. Finally, we discuss the asymptotic decay of the survival probability of the SCP on Galton-Watson trees, when lambda^2/mu is very small.