\affil{Centre for Health Informatics, Computing, and Statistics,\\Lancaster University}
\begin{document}
\maketitle
\section{Introduction}
\section{State transition models}
Epidemic models belong to a particular class of discrete-space modules known as State Transition Models (STM). An STM characterises an infectious disease process as a directed graph with nodes representing mutually exclusive epidemiological states into which individuals within a population exist at any particular time, and edges representing possible transitions between the nodes. Figure \ref{fig:SIR-model0} provides the concrete example of the ``SIR'' model. Here, at any particular time individuals (or epidemiological units) within a population are classified into \emph{susceptible} to the disease, \emph{infected} by the disease (and infectious to others), and \emph{removed} from the population either by death or recovery with long-lasting complete immunity to further infection. Individuals are then assumed to progress from susceptible to infected, and infected to removed in a sequential fashion.
\begin{figure}
\centering
\begin{tikzpicture}
\node[draw, state] (S) {$\mathcal{S}$};
\node[draw, state, right of=S] (I) {$\mathcal{I}$};
\node[draw, state, right of=I] (R) {$\mathcal{R}$};
\draw[->, thick] (S) to node (SI) {} (I);
\draw[->, thick] (I) to node (IR) {} (R);
\end{tikzpicture}
\caption{\label{fig:SIR-model0}The SIR epidemic model.}
\end{figure}
Considering Figure \ref{fig:SIR-model0} as a general directed graph, we see immediately that it is acyclic, with the $\mathcal{S}$ and $\mathcal{R}$ nodes representing \emph{source} and \emph{sink} nodes respectively. The graph is also linear, with each individual following (eventually) the same transition trajectory. However, in general we are not confined to acyclic or linear STMs. Indeed, suppose we have a non-fatal infectious disease (such as influenza) for which a vaccine exists, and immunity wanes post-infection. A plausible model for such a disease process is posited in Figure \ref{fig:SVIRS-model0} demonstrating a branch ($\mathcal{S}\rightarrow\mathcal{I},\mathcal{V}$), coalescence ($\mathcal{S},\mathcal{V}\rightarrow\mathcal{I}$), and a cycle ($\mathcal{R}\rightarrow\mathcal{S}$).
\begin{figure}
\centering
\begin{tikzpicture}
\node[draw, state] (S) {$\mathcal{S}$};
\node[draw, state, right of=S] (I) {$\mathcal{I}$};
To complete the STM, we further need to define how quickly individuals transition from one state to another. From a statistical perspective, it is natural to consider this in terms of time periods, or \emph{sojourns}, spent in each state before transitioning along an edge. This view allows us to consider both Markov models, in which individuals spend an exponentially distributed sojourn in each state, and non-Markovian models where sojourns may be distributed according to non-memoryless distributions such as the Gamma or Weibull. <now talk about incorporating infectious feedback into the system>.