Steady state of SIS (suspected-infected-suspected) reaction-diffusion model

Considering the following SIS reaction diffusion model:

\[\left\{\begin{array}{l} S_{t} = d_{S} S_{x x}-\beta(x) \frac{S I}{S+I}+\gamma(x) I=0, \quad 0<x<1 \\ I_{t} = d_{I} I_{x x}+\beta(x) \frac{S I}{S+I}-\gamma(x) I=0, \quad 0<x<1 \\ S_{x}=I_{x}=0, \quad x=0,1, \end{array}\right.\]

where $\int_{0}^{1} S(x,t)+I(x,t)dx = 1$. $S(x,t)$ and $I(x,t)$ denote the density of susceptible and infected populations at location $x$ and time $t$, $d_{S}$ and $d_{I}$ represent the diffusion coefficients for susceptible and infected individuals, and $\beta(x)$, $\gamma(x)$ are transmission and recovery rates at $x$, respectively.

We want to solve the steady state problem (same notations for convenience):

\[\left\{\begin{array}{l} d_{S} S_{x x}-\beta(x) \frac{S I}{S+I}+\gamma(x) I=0, \quad 0<x<1 \\ d_{I} I_{x x}+\beta(x) \frac{S I}{S+I}-\gamma(x) I=0, \quad 0<x<1 \\ S_{x}=I_{x}=0, \quad x=0,1, \end{array}\right.\]

where $\int_{0}^{1} S(x)+I(x)dx = 1$.

Note here elliptic problem has condition $\int_{0}^{1} S(x)+I(x)dx = 1$.

using DifferentialEquations, ModelingToolkit, MethodOfLines, DomainSets, Plots

# Parameters, variables, and derivatives
@parameters t x
@parameters dS dI brn ϵ
@variables S(..) I(..)
Dt = Differential(t)
Dx = Differential(x)
Dxx = Differential(x)^2

# Define functions
function γ(x)
    y = x + 1.0
    return y
end

function ratio(x, brn, ϵ)
    y = brn + ϵ * sin(2 * pi * x)
    return y
end

# 1D PDE and boundary conditions
eq = [Dt(S(t, x)) ~ dS * Dxx(S(t, x)) - ratio(x, brn, ϵ) * γ(x) * S(t, x) * I(t, x) / (S(t, x) + I(t, x)) + γ(x) * I(t, x),
    Dt(I(t, x)) ~ dI * Dxx(I(t, x)) + ratio(x, brn, ϵ) * γ(x) * S(t, x) * I(t, x) / (S(t, x) + I(t, x)) - γ(x) * I(t, x)]
bcs = [S(0, x) ~ 0.9 + 0.1 * sin(2 * pi * x),
    I(0, x) ~ 0.1 + 0.1 * cos(2 * pi * x),
    Dx(S(t, 0)) ~ 0.0,
    Dx(S(t, 1)) ~ 0.0,
    Dx(I(t, 0)) ~ 0.0,
    Dx(I(t, 1)) ~ 0.0]

# Space and time domains
domains = [t ∈ Interval(0.0, 10.0),
    x ∈ Interval(0.0, 1.0)]

# PDE system
@named pdesys = PDESystem(eq, bcs, domains, [t, x], [S(t, x), I(t, x)], [dS => 0.5, dI => 0.1, brn => 3, ϵ => 0.1])

# Method of lines discretization
# Need a small dx here for accuracy
dx = 0.01
order = 2
discretization = MOLFiniteDifference([x => dx], t)

# Convert the PDE problem into an ODE problem
prob = discretize(pdesys, discretization);

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 10.0)
u0: 198-element Vector{Float64}:
 0.9062790519529313
 0.9125333233564304
 0.9187381314585725
 0.9248689887164855
 0.9309016994374948
 0.9368124552684678
 0.9425779291565073
 0.9481753674101716
 0.9535826794978997
 0.9587785252292473
 ⋮
 0.18443279255020154
 0.18763066800438638
 0.190482705246602
 0.19297764858882513
 0.19510565162951538
 0.1968583161128631
 0.19822872507286887
 0.1992114701314478
 0.19980267284282716

Solving time-dependent SIS epidemic model

# Solving SIS reaction diffusion model
sol = solve(prob, Tsit5(), saveat=0.2);

# Retriving the results
discrete_x = sol[x]
discrete_t = sol[t]
S_solution = sol[S(t, x)]
I_solution = sol[I(t, x)]

p = surface(discrete_x, discrete_t, S_solution)
display(p)

Solving steady state problem

Change the elliptic problem to steady state problem of reaction diffusion equation.

See more solvers in Steady State Solvers · DifferentialEquations.jl

steadystateprob = SteadyStateProblem(prob)
steadystate = solve(steadystateprob, DynamicSS(Tsit5()))

retcode: Failure
u: 198-element Vector{Float64}:
 0.3317836646210077
 0.3317732755346262
 0.3317891113928477
 0.33175474345802314
 0.33179088720150746
 0.33173373143278306
 0.33179111779052045
 0.33171244226430213
 0.33179191509999756
 0.3316930492696892
 ⋮
 0.6560941213703403
 0.6561639796904541
 0.6562347277323771
 0.6563041441210961
 0.6563698199305195
 0.6564293278188778
 0.6564800894273904
 0.656519489448634
 0.656544799858111

The effect of human mobility on endemic size

Set the endemic size $f(d_{S},d_{I}) = \int_{0}^{1}I(x;d_{S},d_{I}).$

function episize!(dS, dI)
    newprob = remake(prob, p=[dS, dI, 3, 0.1])
    steadystateprob = SteadyStateProblem(newprob)
    state = solve(steadystateprob, DynamicSS(Tsit5()))
    y = sum(state[100:end]) / 99
    return y
end
episize!(exp(1.0),exp(0.5))

0.6656214868928503

References:

Allen L J S, Bolker B M, Lou Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model[J]. Discrete & Continuous Dynamical Systems, 2008, 21(1): 1.