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

thesis

posted on 01.09.2014 by Augusto Santos#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

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.

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.

### History

#### Date

01/09/2014#### Degree Type

Dissertation#### Department

Electrical and Computer Engineering#### Degree Name

- Doctor of Philosophy (PhD)