  Motivated by the need for novel robust approaches to modelling the Covid-19
epidemic, this paper treats a population of $N$ individuals as an inhomogeneous
random social network (IRSN). The nodes of the network represent different
types of individuals and the edges represent significant social relationships.
An epidemic is pictured as a contagion process that changes daily, triggered on
day $0$ by a seed infection introduced into the population. Individuals' social
behaviour and health status are assumed to be random, with probability
distributions that vary with their type. First a formulation and analysis is
given for the basic SI ("susceptible-infective") network contagion model, which
focusses on the cumulative number of people that have been infected. The main
result is an analytical formula valid in the large $N$ limit for the state of
the system on day $t$ in terms of the initial conditions. The formula involves
only one-dimensional integration. Next, more realistic SIR and SEIR network
models, including "removed" (R) and "exposed" (E) classes, are formulated.
These models also lead to analytical formulas that generalize the results for
the SI network model. The framework can be easily adapted for analysis of
different kinds of public health interventions, including vaccination, social
distancing and quarantine. The formulas can be implemented numerically by an
algorithm that efficiently incorporates the fast Fourier transform. Finally a
number of open questions and avenues of investigation are suggested, such as
the framework's relation to ordinary differential equation SIR models and agent
based contagion models that are more commonly used in real world epidemic
modelling.
