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

2014-09-01T00:00:00Z (GMT) by Augusto Santos
This Thesis studies bi-virus epidemics over large-scale networks. We set the rules of infection at the node<br>level and determine the dynamical law governing the evolution of the fraction of infected nodes that naturally<br>emerges in the limit of a large network. Then, we study the qualitative behavior of the fraction of infected<br>nodes under the fluid limit dynamics to determine if and when the epidemics develops into a pandemic state,<br>or leads to natural selection, with a dominant resilient virus strain. The Thesis is divided into two parts.<br>In the first part, we establish the fluid limit macroscopic dynamics of a multi-virus epidemics over<br>classes of non-complete networks as the number of nodes grows large. We assume peer-to-peer random rules<br>of infection in line with the Harris contact process. More specifically, The fluid limit ordinary differential<br>equation dynamics is cast as the weak limit (in the number of nodes) of the fraction of infected nodes over<br>time under the Skorokhod topology in the space of càdlàg sample paths. The microscopic model conforms<br>to a Susceptible-Infectious-Susceptible model. A node is either infected or it is healthy and prone to receive<br>infections. We prove the exact emergent dynamics for the class of complete-multipartite networks.<br>In the second part, we study the qualitative behavior of the fraction of infected nodes under the ordinary<br>differential equation limiting dynamics obtained in the first part. Namely, we characterize the attractors<br>– where the orbits of the differential equations converge to – and the corresponding basins of attraction. Due<br>to the coupled nonlinear high-dimension nature of the mean field dynamics, there is no natural Lyapunov<br>function to study their qualitative behavior. We establish their qualitative behavior, not by numerical simulations,<br>but by bounding the epidemics dynamics for generic graph networks by the epidemics dynamics<br>on two special regular networks – the inner and outer regular networks, for which we can carry out their<br>qualitative analysis.