# SIR package

Simple SIR (susceptible-infectious-recovered) compartmental model for an infectious disease outbreak.

This package is open source software.

## Getting started

To build, use:

birch build


To run, use:

birch sample --config config/sir.json


## Details

The model on which this is based is described in Murray et. al. (2018).

The parameter model is given by:

\begin{align} \lambda &\sim \mathrm{Gamma}(2,5) \\ \delta &\sim \mathrm{Beta}(2,2) \\ \gamma &\sim \mathrm{Beta}(2,2), \end{align}

where $\lambda$ is a rate of interaction in the population, $\delta$ the probability of infection when a susceptible individual interacts with an infectious individual, and $\gamma$ the daily recovery probability.

The initial model for time $t = 0$ is:

\begin{align} s_0 &= 760 \\ i_0 &= 3 \\ r_0 &= 0. \end{align}

The transition model for time $t$ is:

\begin{align} \tau_t &\sim \mathrm{Binomial}\left(s_{t-1}, 1 - \exp\left(\frac{-\lambda i_{t-1} }{s_{t-1} + i_{t-1} + r_{t-1}}\right) \right) \\ \Delta i_t &\sim \mathrm{Binomial}(\tau_t, \delta) \\ \Delta r_t &\sim \mathrm{Binomial}(i_{t-1}, \gamma), \end{align}

where $\tau_t$ is the number of interactions between infectious and susceptible individuals, $\Delta i_t$ the number of newly infected individuals, and $\Delta r_t$ the number of newly recovered individuals.

Population counts are then updated:

\begin{align} s_t &= s_{t-1} - \Delta i_t \\ i_t &= i_{t-1} + \Delta i_t - \Delta r_t \\ r_t &= r_{t-1} + \Delta r_t. \end{align}

## Acknowledgements

This package contains an influenza data set from Anonymous (1978), prepared in JSON format for Birch.

## References

1. Anonymous (1978). Influenza in a boarding school. British Medical Journal. 1:587.

2. L.M. Murray, D. Lundén, J. Kudlicka, D. Broman, and T.B. Schön (2018). Delayed Sampling and Automatic Rao--Blackwellization of Probabilistic Programs.