Appendix D — Odin notes

odin actually belongs to a series of 3 packages developed by researchers at Imperial College London that streamline infectious disease modelling tasks (see (FitzJohn et al., 2021) for details):

  1. odin: is a domain-specific language Definition A.1 for implementing state-space models Definition A.2 in R interface using C++, develop in 2016 and can model both continuous and discrete time.
  2. dust: is a framework for running discrete time stochastic models, it takes care of parallelisation, reproducible random number generation, execution on GPUs…
  3. mcstate: provides routines to fit the model using sequential Monte Carlo (particle filters) and prediction tasks.

D.1 odin

user expresses their problems as a set of mathematical relationships, modelled as assignments to form a directed acyclic graph (DAG). odin then sorts that graph, which is why users can therefore write their equations in any order.

For use with ordinary differential equations (ODEs), odin transpiles to C code or JavaScript.

Metapopulation models can be written in the form of an array. odin support arrays up to 8 dimensions with index follow this order: i, j, k, l, i5, i6, i7, i8 (source).

deriv() for continuous time,

update() is a function for dust, which is discrete time.

D.2 mcstate

D.2.1 Stochastic SIR model

The ODE of a deterministic SIR model is defined as:

\[\begin{align} \frac{dS}{dt} &= -\beta SI \\ \frac{dI}{dt} &= \beta SI - \gamma I \\ \frac{dR}{dt} &= \gamma I \end{align}\]

This model can be discretised in time steps of width \(dt\) using the following update equations for each time step:

\[\begin{align} S_{t + 1} &= S_t - n_{SI, t} \\ I_{t + 1} &= I_t + n_{SI, t} - n_{IR, t} \\ R_{t + 1} &= R_t + n_{IR, t} \end{align}\]

where the number of individuals moving between compartments are given by drawing from binomial distributions:

\[\begin{align} n_{SI, t} &\sim B(S_t, 1 - \exp(-\beta I_t)) \\ n_{IR, t} &\sim B(I_t, 1 - \exp(-\gamma)) \end{align}\]

The binomial distribution \(B(n, p)\) is used as there are \(n\) trials, one for each individual in the compartments. In a single time step, each p can be calculated as 1 dt eλ⋅ − where λ is the transition rate, as in a Poisson process time between events is exponentially distributed.