An analytical approach to modelling epidemics on networks : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics at Massey University, Albany, New Zealand
A signi cant amount of e ort has been directed at understanding how the struc-
ture of a contact network can impact the spread of an infection through a population.
This thesis is focused on obtaining tractable analytic results to aid our understand-
ing of how infections spread through contact networks and to contribute to the
existing body of research that is aimed at determining exact epidemic results on -
nite networks. We use SIR (Susceptible-Infected-Recovered) and SIS (Susceptible-
Infected-Susceptible) models to investigate the impact network topology has on the
spread of an infection through a population.
For an SIR model, the probability mass functions of the nal epidemic size are
derived for eight small networks of di erent topological structure. Results from the
small networks are used to illustrate how it is possible to describe how an infection
spreads through a larger network, namely a line of triangles network. The key here
is to correctly decompose the larger network into an appropriate assemblage of small
networks so that the results are exact.
We use Markov Chain theory to derive results for an SIS model on eight small
networks such as the expected time to absorption, the expected number of times each
individual is infected and the cumulative incidence of the epidemic. An algorithm
to derive the transition matrix for any small network structure is presented, from
which, in theory, all other results for the SIS model can be obtained using Markov
Chain theory. In theory, this algorithm is applicable to networks of any size, however
in practice it is too computationally intensive to be practical for larger networks than
those presented in this thesis.
We give examples for both types of model and illustrate how to parameterise the
small networks to investigate the spread of in
uenza, measles, rabies and chlamydia
through a small community or population.