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Uncertainty propagation applied to ordinary differential equation model using MonteCarloMeasurements.jl

Simon Frost (@sdwfrost), 2022-02-14

Introduction

The classical ODE version of the SIR model is:

  • Deterministic
  • Continuous in time
  • Continuous in state

One elegant approach to investigating how uncertainty in parameters propagates to the output involves the use of MonteCarloMeasurements.jl. Parameter values and initial conditions can be represented by a set of Particles, which can accommodate arbitrary patterns of variation. These can be included in Diffe solvers in the same way as other, simpler types such as Float64. This approach is well suited to non-linear patterns of uncertainty propagation.

Libraries

using DifferentialEquations
using OrdinaryDiffEq
using Distributions
using MonteCarloMeasurements
using StatsBase
using Plots

Transitions

The following function provides the derivatives of the model, which it changes in-place. State variables and parameters are unpacked from u and p.

function sir_ode!(du,u,p,t)
    (S,I,R) = u
    (β,c,γ) = p
    N = S+I+R
    @inbounds begin
        du[1] = -β*c*I/N*S
        du[2] = β*c*I/N*S - γ*I
        du[3] = γ*I
    end
    nothing
end;

Time domain

We set the timespan for simulations, tspan, initial conditions, u0, and parameter values, p. We will set the maximum time to be high, as we will be using a callback in order to stop the integration early.

δt = 1.0
tmax = 40.0
tspan = (0.0,tmax);

Initial conditions and parameter values

We first set fixed parameters, in this case, the total population size, N.

N = 1000.0;

We then generate a random sample of parameter values as well as the initial number of infected individuals. Rather than a full factorial design, we use LatinHypercubeSample from the QuasiMonteCarlo.jl package. We specify lower (lb) and upper (ub) bounds for each parameter.

n_samples = 1000; # Number of samples
p = [Particles(n_samples,Uniform(0.01,0.1)),
      Particles(n_samples,Uniform(5,20.0)),
      Particles(n_samples,Uniform(0.1,1.0))]
3-element Vector{MonteCarloMeasurements.Particles{Float64, 1000}}:
  0.055 ± 0.026
 12.5 ± 4.3
  0.55 ± 0.26

I₀=Particles(n_samples,Uniform(1.0,50.0))
u0 = [N-I₀,I₀,0.0]
3-element Vector{MonteCarloMeasurements.Particles{Float64, 1000}}:
 974.0 ± 14.0
  25.5 ± 14.0
   0.0

Running the model

prob_ode = ODEProblem(sir_ode!,u0,tspan,p);
sol_ode = solve(prob_ode, Tsit5(), dt=δt);

Post-processing

Here are the (uncertain) states at time t=20.0.

s20 = sol_ode(20.0)
3-element Vector{MonteCarloMeasurements.Particles{Float64, 1000}}:
 588.0 ± 370.0
  28.0 ± 54.0
 384.0 ± 350.0

These states are markedly different from Gaussian distributions, as the parameter set has combinations that lead to outbreaks (R₀>1) as well as to fade-out.

l = @layout [a b c]
binwidth = 50
pl1 = histogram(s20[1],bins=0:binwidth:N, title="S(20)", xlabel="S", ylabel="Frequency", color=:blue)
pl2 = histogram(s20[2],bins=0:binwidth:N, title="I(20)", xlabel="I", ylabel="Frequency", color=:red)
pl3 = histogram(s20[3],bins=0:binwidth:N, title="R(20)", xlabel="R", ylabel="Frequency", color=:green)
plot(pl1,pl2,pl3,layout=l,legend=false)

Further processing of the output can be performed after converting to Arrays, e.g. Kendalls rank correlation of β, c, γ, and I₀ against S(20), I(20), and R(20).

corkendall(hcat(Array(p),Array(I₀)),Array(s20))
4×3 Matrix{Float64}:
 -0.432496   0.207059    0.455307
 -0.272136   0.0889489   0.291528
  0.387788  -0.607596   -0.323688
 -0.124452  -0.0522002   0.143972