## Contact network epidemiology: Mathematical methods of modeling a mutating pathogen on a two-type network

##### Abstract

With the threat of diseases like Sudden Acute Respiratory Syndrome (SARS)
and Avian Flu that can lead to global pandemics, it is important to be
able to understand how diseases spread through a population and predict
how many people will become infected. It is also important to learn how
preventative treatments affect disease spread. Public health officials must
prepare a different vaccine each year to deal with a different influenza strain.
Furthermore, it is important to be able to determine how effective a particular
vaccination strategy will be in the event of a limited vaccine supply.
The most common mathematical model of epidemic disease is based on
the assumption that the population is homogenously mixed. That is, every
member of the population is identical in how many other individuals he interacts with and every infected individual has the potential to spread disease
to every other individual. However, these assumptions are unrealistic. To
create a better model, the basic assumption of homogeneous mixing is removed. To do this, the population is modeled as a contact network. In a
contact network, the population is represented by dots connected by lines.
Each member is represented by a dot, or node, with disease-causing interactions between two members of a population represented by lines, or edges,
between two nodes. This models the structure that exists in human populations by allowing individuals in a population to infect only a limited number
of other individuals and allowing the number of contacts to vary between
individuals.
Much is already understood about how network structure affects disease
dynamics. This thesis uses the contact network model to study the impact
of network structure on the dynamics of a mutating pathogen. By distributing the contacts within the population in different ways, the effect of the
network's structure on the extent of the disease is observed. Given the distribution of contacts on the network and the probability that an individual
spreads the disease to a contact, the average sizes of a small outbreak (that
which spreads to only a few people) and a large epidemic (that which spreads
to a fixed proportion of the population, no matter the size) are calculated.
These calculations are computed for different contact distributions and for a
range of transmissibility values. Furthermore, these calculations are checked
against simulations of the disease spreading over contact networks. This
thesis also generalizes the contact network model to allow for both treated
and untreated individuals in a population. In this generalization, not only
does the number of contacts vary between individuals, but the probability of
transmission also differs between treated and untreated individuals.
This thesis shows that the contact network model with pathogen evolution is similar to the basic model. It also shows that the predictions made
by the model are supported by simulation in some cases but not in other.
Furthermore, it shows a contact network model that incorporates two different kinds of nodes. Lastly, it shows that this new model reduces to the basic
model under certain conditions.