```
using DifferentialEquations
using DiffEqSensitivity
using Random
using Distributions
using Turing
using DataFrames
using StatsPlots
```

This post is exploring the use of Julia, Turing, and ODEs and largely expands on the work from Simon Frost’s awesome epirecipes.

Julia as a statistical programming language/environment has some really neat properties.

Speed, generally speed.

Turing also has some neat elements when it comes to probabilistic computing/programming. The key feature that interests me is the ability to switch between sampling approaches (e.g., Gibbs, Hamiltonian Monte Carlo, no u-turn sampling, sequential) while keeping the same syntax and model structure. While I absolutely love Stan and have spent many hours learning and the syntax (and it is actively developed, highly optimized, etc), I sometimes wish I could explore other sampling approaches due to strange posteriors (especially when it comes to times when SMC might be better).

This is a pretty vanilla post, in which we will first create some fake data from a known distribution and then fit said data. As always, we will bring in the required packages.

```
using DifferentialEquations
using DiffEqSensitivity
using Random
using Distributions
using Turing
using DataFrames
using StatsPlots
```

Now we can generate a simple three compartment model, with S (susceptibles), I, (infected), and R (recovered).

Additionally:

- On average 10 contacts are made per day
- The contact rate is 0.05 (probability of passing infection to any given contact)
- Those who are infected recover on average every 5 days
- Immunity lasts for a 180 days on average before people become susceptible again
- Population size of 6,000 individuals
- Standard mass-action approach

Something that looks like the following:

```
using Luxor
using MathTeXEngine
```

```
Drawing(200, 200, "sirs.png")
origin()
setline(10)
Luxor.arrow(Point(-40,0),Point(0,0))
Luxor.arrow(Point(-0,0),Point(40,0))
fontsize(15)
Luxor.text(L"S",Point(-40,20), halign = :center)
Luxor.text(L"I",Point(0,20), halign = :center)
Luxor.text(L"R",Point(40,20), halign = :center)
fontsize(12)
Luxor.text(L"β*c",Point(-20,10), halign = :center)
Luxor.text(L"γ",Point(20,10), halign = :center)
Luxor.text(L"δ",Point(0,-15), halign = :center)
loopx = 30
loopy = 40
adjx = 6
adjy = -2
Luxor.arrow(
Point(40,0) + Point(-adjx,adjy),
Point(40,0) + Point(-loopx,-loopy),
Point(-40,0)+ Point(loopx,-loopy),
Point(-40,0)+Point(adjx,adjy)
)
finish()
```

```
function sir_ode!(du,u,p,t)
(S,I,R,C) = u
(β,c,γ, δ) = p
N = S+I+R
infection = β*c*I*S/N
recovery = γ*I
wane = δ * R
@inbounds begin
du[1] = -infection + wane
du[2] = infection - recovery
du[3] = recovery - wane
du[4] = infection
end
nothing
end;
tmax = 365.0
tspan = (0.0,tmax)
obstimes = 1.0:1.0:tmax
u0 = [5990.0,10.0,0.0,0.0] # S,I.R,C
p = [0.05,10.0,0.20, 1.0/180]; # β, c, γ, δ
prob_ode = ODEProblem(sir_ode!,u0,tspan,p);
sol_ode = solve(prob_ode,
Tsit5(),
saveat = 1.0);
```

We can then visualize our expected cases:

```
C = Array(sol_ode)[4,:] # Cumulative cases
X = C[2:end] - C[1:(end-1)];
Random.seed!(1234)
Y = rand.(Poisson.(X));
bar(obstimes,Y,legend=false)
plot!(obstimes,X,legend=false)
```

Note the second wave of cases occuring as immunity begins to wane.

Now we build our model in Turing, using our prior defined ODE. Note that I included a check on the ODE output to try and catch values less than one, otherwise the sampler would reject the solution (because Poisson distributions cannot have a rate parameters less than 0).

```
@model bayes_sir(y) = begin
# Calculate number of timepoints
l = length(y)
i₀ ~ Uniform(0.0,.2)
β ~ Uniform(0, .1)
immunity ~ Uniform(90,365)
I = i₀*6000.0
u0=[6000.0-I,I,0.0,0.0]
p=[β,10.0,0.2, 1.0/immunity]
tspan = (0.0,float(l))
prob = ODEProblem(sir_ode!,
u0,
tspan,
p)
sol = solve(prob,
Tsit5(),
saveat = 1.0)
sol_C = Array(sol)[4,:] # Cumulative cases
sol_X = sol_C[2:end] - sol_C[1:(end-1)]
l = length(y)
for i in 1:l
y[i] ~ Poisson(ifelse(sol_X[i] <0 , 1e-6, sol_X[i] ))
end
end;
mod = bayes_sir(Y[1:200]);
ode_nuts = sample(mod, NUTS(0.65),10000);
```

`describe(ode_nuts)`

```
2-element Vector{ChainDataFrame}:
Summary Statistics (3 x 8)
Quantiles (3 x 6)
```

Now let’s see if we were able to recover our parameters:

```
plot(ode_nuts)
posterior = DataFrame(ode_nuts);
beta_hat = mean(posterior[!,:β])
immunity_hat = mean(posterior[!, :immunity])
println("Estimate for β: ", beta_hat)
println("Estimate for immunity duration: ", immunity_hat)
println("Estimate for initial proportion infected: ", mean(posterior[!, :i₀]))
```

```
Estimate for β: 0.04984659207646735
Estimate for immunity duration: 178.34561091700505
Estimate for initial proportion infected: 0.0016216537054861632
```

Not too bad! Now we can simulate the dynamics going forward:

```
function predict(y,chain)
# Length of data
l = length(y)
# Length of chain
m = length(chain)
# Choose random
idx = sample(1:m)
i₀ = chain[:i₀][idx]
β = chain[:β][idx]
immunity = chain[:immunity][idx]
I = i₀*6000.0
u0=[6000.0-I,I,0.0,0.0]
p=[β,10.0,0.2, 1.0/immunity]
tspan = (0.0,float(l))
prob = ODEProblem(sir_ode!,
u0,
tspan,
p)
sol = solve(prob,
Tsit5(),
saveat = 1.0)
out = Array(sol)
sol_X = [0.0; out[4,2:end] - out[4,1:(end-1)]]
hcat(sol_ode.t,out',sol_X)
end;
Xp = []
for i in 1:365
pred = predict(Y,ode_nuts)
push!(Xp,pred[2:end,6])
end
scatter(obstimes,Y,legend=false, ylims = [0,600])
plot!(obstimes,Xp,legend=false, color = "grey50", alpha = .2)
```

This shows a very intuitive approach towards compartmental modeling in a Bayesian framework using Julia.

BibTeX citation:

```
@online{dewitt2022,
author = {Michael DeWitt},
editor = {},
title = {Turing and {ODEs}},
date = {2022-08-13},
url = {https://michaeldewittjr.com/programming/2022-08-13-turing-and-odes},
langid = {en}
}
```

For attribution, please cite this work as:

Michael DeWitt. 2022. “Turing and ODEs.” August 13, 2022.
https://michaeldewittjr.com/programming/2022-08-13-turing-and-odes.

I am interested in generating some of the structures of Neisseria gonorrhoeae, the pathogen responsible for the sexually transmitted infection, gonorrhea. I can go over to the protein data bank and do a search and find some of the available structures.

```
ng_pdb <- "https://files.rcsb.org/download/4R1I.pdb"
download.file(ng_pdb, destfile = "4R1I.pdb")
ng_pdb <- readLines("4R1I.pdb")
```

Now for the magical part:

```
library("r3dmol")
m1 <- r3dmol( # Set up the initial viewer
viewer_spec = m_viewer_spec(
cartoonQuality = 10,
lowerZoomLimit = 50,
upperZoomLimit = 350
)
) %>%
m_add_model( # Add model to scene
data = ng_pdb,
format = "pdb"
) %>%
m_zoom_to() %>% # Zoom to encompass the whole scene
m_set_style( # Set style of structures
style = m_style_cartoon(
color = "#00cc96"
)
)%>%
m_set_style( # Set style of specific selection
sel = m_sel(ss = "s"), # (selecting by secondary)
style = m_style_cartoon(
color = "#636efa",
arrows = TRUE
)
) %>%
m_set_style( # Style the alpha helix
sel = m_sel(ss = "h"), # (selecting by alpha helix)
style = m_style_cartoon(
color = "#ff7f0e"
)
) %>%
m_rotate( # Rotate the scene by given angle on given axis
angle = 90,
axis = "y"
) %>%
m_spin()
m1
```