# Julia Implementation of Host SEIR + Vector SEI

Author: Carl A. B. Pearson @pearsonca

Date: 2018-10-02

This version considers multiple hosts and multiple vectors, which further complicates indexing. Now the indexing scheme is:

• all of host $i$ compartments ($S_H^i$, $E_H^i$, etc), for each $i \in 1\ldots N$
• all of vector $j$ compartments ($S_V^j$, $E_V^j$, etc), for each $j \in 1\ldots M$

With this approach, we can again re-use the solutions for single host and single vector, with slight modifications:

H_comps = 4
V_comps = 3

# in sub functions, du / u are the particular relevant slices only
function F1H(du, u, p, t, βslice, I_V, N_H)
S_H, E_H, I_H, R_H = u

# host dynamics
host_infection = sum(βslice .* I_V)*S_H/N_H
host_mortality = p.μ_H .* u # include S_H, so easier to remove mortality
host_births = sum(host_mortality)
host_progression = p.σ_H*E_H
recovery = p.λ*I_H

du[1] = -host_infection + host_births
du[2] = host_infection - host_progression
du[3] = host_progression - recovery
du[4] = recovery
du[1:end] -= host_mortality
end

# in sub functions, du / u are the particular relevant slices only
function F1V(du, u, p, t, βslice, I_H, N_H)
S_V, E_V, I_V = u
vec_infection = sum(βslice .* I_H)*S_V/N_H
vec_mortality = p.μ_V .* u # include S_V, so easier to remove mortality
vec_births = sum(vec_mortality)
vec_progression = p.σ_V*E_V

du[1] = -vec_infection + vec_births
du[2] = vec_infection - vec_progression
du[3] = vec_progression
du[1:end] -= vec_mortality
end

function F(du,u,p,t)
dH = @view(du[1:(p.nHosts*H_comps)])
dV = @view(du[(p.nHosts*H_comps+1):end])
Hs = @view(u[1:(p.nHosts*H_comps)])
Vs = @view(u[(p.nHosts*H_comps+1):end])

I_Vs = @view(Vs[3:V_comps:V_comps*p.nVecs])
I_Hs = @view(Hs[3:H_comps:H_comps*p.nHosts])

for host in 0:(p.nHosts-1)
slice = (1:H_comps).+(H_comps*host)
F1H(@view(dH[slice]), @view(Hs[slice]), p.host[host+1], t, @view(p.β[host+1,:]), I_Vs, p.N_H)
end
for vec in 0:(p.nVecs-1)
slice = (1:V_comps).+(V_comps*vec)
F1V(@view(dV[slice]), @view(Vs[slice]), p.vec[vec+1], t, @view(p.β[:,vec+1]), I_Hs, p.N_H)
end
end

F (generic function with 1 method)


First, state initial conditions. This code generates them randomly for convenience, though they could be assigned based on data, desired parameter space, or algorithmically as part of a fitting process:

nH = 2
nV = 2
srand(0)

S_Hs = ones(nH) .* 100.0
E_Hs = zeros(nH)
I_Hs = shuffle(vcat(zeros(nH-1),[1.0]))
R_Hs = zeros(nH)
host0 = reshape(hcat(S_Hs,E_Hs,I_Hs,R_Hs)', nH*H_comps, 1)

S_Vs = ones(nV) .* 1000.0
E_Vs = zeros(nV)
I_Vs = zeros(nV)
vec0 = reshape(hcat(S_Vs,E_Vs,I_Vs)', nV*V_comps, 1)

u0 = vcat(host0, vec0)

14×1 Array{Float64,2}:
100.0
0.0
1.0
0.0
100.0
0.0
0.0
0.0
1000.0
0.0
0.0
1000.0
0.0
0.0


Now, generate dynamic parameters. Again: this code generates them randomly for convenience, though they could be assigned based on data, desired parameter space, or algorithmically as part of a fitting process:

srand(1)

μs = 1 ./ (rand(nH) .* 360)
σs = 1 ./ (rand(nH) .* 6)
μVs = 1 ./ (rand(nV) .* 60)
σVs = 1 ./ (rand(nV) .* 14)

λs = 1 ./ (rand(nH) .* 28)
βs = rand(nH*nV) ./ 10.0

using NamedTuples
# nb: in >= Julia v0.7, can eliminate this import
#  and the @NT syntax
p = @NT(
nHosts = nH, nVecs = nV,
N_H = sum(host0),
β = reshape(βs,nH,nV), # information in hosts (rows) by vectors (cols)
vec  = [@NT(μ_V=μVs[j], σ_V=σVs[j]) for j in 1:nV],
host = [@NT(μ_H=μs[i], σ_H=σs[i], λ=λs[i]) for i in 1:nH]
# just building up a random collection of params for demonstration
)

(nHosts = 2, nVecs = 2, N_H = 201.0, β = [0.0555751 0.0424718; 0.0437108 0.0773223], vec = NamedTuples._NT_μ__V_σ__V{Float64,Float64}[(μ_V = 0.0341102, σ_V = 0.0750366), (μ_V = 0.0790008, σ_V = 0.0714354)], host = NamedTuples._NT_μ__H_σ__H_λ{Float64,Float64,Float64}[(μ_H = 0.0117686, σ_H = 0.53298, λ = 0.141914), (μ_H = 0.00801628, σ_H = 21.0723, λ = 0.0361969)])


Now these values can be used with the ODE solver:

using DifferentialEquations
using IterableTables, DataFrames

tspan = (0.0, 365.0)
prob = ODEProblem(F, u0, tspan, p)
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8,saveat=linspace(0,365,365*10+1))

retcode: Success
Interpolation: 1st order linear
t: 3651-element Array{Float64,1}:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
⋮
363.9
364.0
364.1
364.2
364.3
364.4
364.5
364.6
364.7
364.8
364.9
365.0
u: 3651-element Array{Array{Float64,2},1}:
[100.0; 0.0; … ; 0.0; 0.0]
[100.001; 1.45562e-7; … ; 0.0208111; 7.45142e-5]
[100.002; 1.1395e-6; … ; 0.0409938; 0.000294274]
[100.004; 3.76354e-6; … ; 0.0605622; 0.000653715]
[100.005; 8.73081e-6; … ; 0.0795303; 0.00114742]
[100.006; 1.66901e-5; … ; 0.0979121; 0.00177011]
[100.007; 2.82298e-5; … ; 0.115721; 0.00251664]
[100.008; 4.38818e-5; … ; 0.13297; 0.00338201]
[100.009; 6.41252e-5; … ; 0.149673; 0.00436136]
[100.01; 8.93894e-5; … ; 0.165843; 0.00544994]
[100.012; 0.000120058; … ; 0.181491; 0.00664314]
[100.013; 0.000156469; … ; 0.196631; 0.00793648]
[100.014; 0.000198924; … ; 0.211274; 0.00932559]
⋮
[28.4073; 1.55928; … ; 40.3947; 36.2825]
[28.4077; 1.55936; … ; 40.3969; 36.2844]
[28.4082; 1.55943; … ; 40.3992; 36.2863]
[28.4086; 1.5595; … ; 40.4014; 36.2883]
[28.409; 1.55958; … ; 40.4036; 36.2902]
[28.4094; 1.55965; … ; 40.4058; 36.2921]
[28.4098; 1.55972; … ; 40.4081; 36.294]
[28.4101; 1.55979; … ; 40.4103; 36.296]
[28.4105; 1.55987; … ; 40.4125; 36.2979]
[28.4109; 1.55994; … ; 40.4147; 36.2998]
[28.4113; 1.56001; … ; 40.4169; 36.3018]
[28.4116; 1.56009; … ; 40.4191; 36.3037]

# rename!(df, Dict(:timestamp => :t,
#  :value1 => :S_H, :value2 => :E_H, :value3 => :I_H, :value4 => :R_H,
#  :value5 => :S_V, :value6 => :E_V, :value7 => :I_V
# ))
# mlt[:host] = contains.(string.(mlt[:variable]),"H"); # tag which entries are host vs vector
# df
df = DataFrame(sol)
mlt = melt(df,:timestamp) # convert results into long format for plotting
mlt[:index] = parse.(Int,replace.(string.(mlt[:variable]),r"[^\d]+"=>""))
namekey = hcat(
reshape(["$(compartment)_H$species" for compartment in ["S","E","I","R"], species in 1:nH],1,:),
reshape(["$(compartment)_V$species" for compartment in ["S","E","I"], species in 1:nV],1,:)
)

mlt[:name] = namekey[mlt[:index]]
mlt[:facet] = replace.(string.(mlt[:name]),r"\w+_"=>"")
mlt[:compartment] = replace.(string.(mlt[:name]),r"_\w+"=>"")
mlt

variablevaluetimestampindexnamefacetcompartment
1value1100.00.01S_H1H1S
2value1100.00117601846260.11S_H1H1S
3value1100.00234977846130.21S_H1H1S
4value1100.003520433593410.31S_H1H1S
5value1100.00468716635490.41S_H1H1S
6value1100.005849187462960.51S_H1H1S
7value1100.007005735190840.61S_H1H1S
8value1100.008156074707330.71S_H1H1S
9value1100.009299497453650.81S_H1H1S
10value1100.010435320490540.91S_H1H1S
11value1100.011562885893941.01S_H1H1S
12value1100.012681560163341.11S_H1H1S
13value1100.01379073358661.21S_H1H1S
14value1100.014889819708931.31S_H1H1S
15value1100.015978254713521.41S_H1H1S
16value1100.01705549689531.51S_H1H1S
17value1100.01812102608851.61S_H1H1S
18value1100.019174343135871.71S_H1H1S
19value1100.020214969359331.81S_H1H1S
20value1100.021242446033341.91S_H1H1S
21value1100.022256333881292.01S_H1H1S
22value1100.023256212575152.11S_H1H1S
23value1100.024241680237922.21S_H1H1S
24value1100.025212352973112.31S_H1H1S
25value1100.026167864383612.41S_H1H1S
26value1100.027107865113212.51S_H1H1S
27value1100.028032022396232.61S_H1H1S
28value1100.028940019602892.71S_H1H1S
29value1100.029831555815972.81S_H1H1S
30value1100.030706345391982.91S_H1H1S

Now that we have a solution, we want to view what is happening in host vs mosquito population:

using RCall
# current version RCall supports better transfers, which would simplify this mess
# but requires Julia v >= 0.7
vals = mlt[:value]
tstamps = mlt[:timestamp]
fcts = mlt[:facet]
comps = mlt[:compartment]
@rput vals tstamps fcts comps
R"
library(ggplot2)
suppressPackageStartupMessages(library(data.table))
dt <- data.table(t=tstamps, y=vals, species=fcts, compartment=comps)
ggplot(dt) + aes(x=t, y=y, color=compartment) + facet_grid(species ~ ., scale = 'free_y') +
theme_minimal() +
geom_line()
"


RCall.RObject{RCall.VecSxp}