Bi-virus Epidemics over Large-scale Networks: Emergent Dynamics and Qualitative Analysis

This Thesis studies bi-virus epidemics over large-scale networks. We set the rules of infection at the node
level and determine the dynamical law governing the evolution of the fraction of infected nodes that naturally
emerges in the limit of a large network. Then, we study the qualitative behavior of the fraction of infected
nodes under the fluid limit dynamics to determine if and when the epidemics develops into a pandemic state,
or leads to natural selection, with a dominant resilient virus strain. The Thesis is divided into two parts.
In the first part, we establish the fluid limit macroscopic dynamics of a multi-virus epidemics over
classes of non-complete networks as the number of nodes grows large. We assume peer-to-peer random rules
of infection in line with the Harris contact process. More specifically, The fluid limit ordinary differential
equation dynamics is cast as the weak limit (in the number of nodes) of the fraction of infected nodes over
time under the Skorokhod topology in the space of càdlàg sample paths. The microscopic model conforms
to a Susceptible-Infectious-Susceptible model. A node is either infected or it is healthy and prone to receive
infections. We prove the exact emergent dynamics for the class of complete-multipartite networks.
In the second part, we study the qualitative behavior of the fraction of infected nodes under the ordinary
differential equation limiting dynamics obtained in the first part. Namely, we characterize the attractors
– where the orbits of the differential equations converge to – and the corresponding basins of attraction. Due
to the coupled nonlinear high-dimension nature of the mean field dynamics, there is no natural Lyapunov
function to study their qualitative behavior. We establish their qualitative behavior, not by numerical simulations,
but by bounding the epidemics dynamics for generic graph networks by the epidemics dynamics
on two special regular networks – the inner and outer regular networks, for which we can carry out their
qualitative analysis.