Various implementations of the classical SIR model in Julia

Try the notebooks out in Binder:

GitHub Markdown doesn't parse equations, so here's a description of the underlying SIR model.

• The ordinary differential equation model considers:
  • Susceptible, S, with initial condition S(0)=990
  • Infected, I, with initial condition, I(0)=10
  • Recovered, R, with initial condition R(0)=10
  • Total population, N=S+I+R=1000
• Susceptible individuals make contacts with others at rate c (=10.0), with the probability of a contact with an infectious person being I/N. With probability β (=0.05), an infected person will infect a susceptible given a contact.
• Infected individuals recover at a per-capita rate γ (=0.25).

There are two types of parameterization commonly used in this project; the 'standard' version, that considers the number of individuals in the S, I, and R groups, and an alternative version, in which the dynamics of transmission (βSI/N) and recovery (γI) are modelled directly, with S, I, and R being calculated based on these dynamics and the initial conditions for S, I and R.

The above process can be represented in different kinds of ways:

• Ordinary differential equation using DifferentialEquations.jl
• Ordinary differential equation using ModelingToolkit.jl
• Ordinary differential equation with composition using AlgebraicDynamics.jl
• Stochastic differential equation using DifferentialEquations.jl
• Stochastic differential equation using StochasticDiffEq.jl
• Stochastic differential equation using Bridge.jl
• Linear noise approximation (LNA) to the stochastic differential equation
• Multivariate birth process reparameterisation of the stochastic differential equation
• ODEs of means, variances, etc. through moment closure
• Function map
• Function map using DynamicalSystems.jl
• Stochastic Markov model
• Stochastic Markov model using Soss.jl
• Jump process (Gillespie) using DifferentialEquations.jl
• Jump process (Gillespie) using reaction networks from Catalyst.jl
• Reaction network conversion to ODEs, SDEs and jump process using ModelingToolkit
• Petri net model to ODEs, SDEs, and jump process using Petri.jl
• Petri net model to ODEs, SDEs, and jump process using AlgebraicPetri.jl
• Jump process (Gillespie) using Gillespie.jl
• Discrete event simulation using SimJulia
• Agent-based model using base Julia as well as using DifferentialEquations
• Agent-based model using Agents.jl

We usually do not observe the trajectory of susceptible, infected, and recovered individuals. Rather, we often obtain data in terms of new cases aggregated over a particular timescale (e.g. a day or a week).

• Simulated data using ODEs

• Changing parameter values at fixed times e.g. lockdown in an SIR model
• Preventing negative populations in stochastic differential equations

In addition to the above examples of simulation, there are also examples of inference of the parameters of the model using counts of new cases. Although these are toy examples, they provide the building blocks for more complex situations.

• Point estimates of parameters of the ODE system using Optim.jl and DiffEqParamEstim.jl
• Bayesian estimates of parameters of the ODE system using Approximate Bayesian Computation
• Bayesian estimates of parameters of the ODE system using Turing.jl
• Bayesian estimates of parameters of the ODE system using NestedSamplers.jl

• Fixed (rather than exponential) distribution of infectious period using a delay differential equation and DelayDiffEq.jl
• An Erlang distribution for the infectious period using the method of stages is illustrated in this notebook using AlgebraicDynamics.jl

Note that the implementations and choice of parameters may be suboptimal, and are intended to illustrate more-or-less the same underlying biological process with different mathematical representations. Additional optimizations may be obtained e.g. by using StaticArrays.

I've also tried to transform parameterisations in discrete time as closely as possible to their continuous counterparts. Please see the great work by Linda Allen for how these different representations compare.

Thanks to Weave.jl, Julia Markdown files (in tutorials/) are converted into multiple formats.

• Jupyter notebooks
• GitHub Markdown
• HTML
• Julia script

git clone https://github.com/epirecipes/sir-julia
cd sir-julia

Then launch julia and run the following.

cd(@__DIR__)
import IJulia
IJulia.notebook(;dir="notebook")

Plans for new examples are typically posted on the Issues page.

To add an example, make a new subdirectory in the tutorials directory, and add a Julia Markdown (.jmd) document to it. Set the beginning to something like the following:

# Agent-based model using Agents.jl
Simon Frost (@sdwfrost), 2020-04-27

Suggested sections:

• Introduction
• Libraries
• Utility functions
• Transitions
• Time domain
• Initial conditions
• Parameter values
• Random number seed
• Running the model
• Post-processing
• Plotting
• Benchmarking

Change to the root directory of the repository and run the following from within Julia.

include("build.jl")
weave_all() # or e.g. weave_folder("abm") for an individual tutorial

If additional packages are added, then these need to be added to build_project_toml.jl, which when run, will regenerate Project.toml.

Examples use the following libraries (see the Project.toml file for a full list of dependencies):

• The DifferentialEquations.jl ecosystem for many of the examples
• SimJulia for discrete event simulations
• Agents.jl for agent-based models
• Gillespie.jl for the Doob-Gillespie process
• Petri.jl for the Petri net models
• AlgebraicPetri.jl for a category theory based modeling framework for creating Petri net models
• Turing.jl for inference using probabilistic programs
• NestedSamplers.jl for nested sampling
• GpABC for inference using Approximate Bayesian Computation
• Soss.jl for Markov models
• MomentClosure.jl for moment closure
• Bridge.jl for stochastic differential equations

Parts of the code were taken from @ChrisRackauckas DiffEqTutorials, which comes highly recommended.