SIS-xde (Susceptible-Infected-Susceptible) Human Model • ramp.xds

The SIS (Susceptible-Infected-Susceptible) human xde model model fulfills the generic interface of the human population component. It is the simplest model of endemic diseases in humans.

We subdivide a population into susceptible (\(S\)) and infected and infectious (\(I\)) individuals, where the total population is \(H = S+I.\) We assume the force of infection (\(h\), FoI) is linearly proportional to the EIR: \(h = b \times EIR.\) In its general form, with births (\(B(H)\)) and deaths (at the per-capita rate \(\mu\)), the generalized SIS_xde dynamics are:

\[ \begin{array}{rl} \dot{S} &= -h S + rI + B(H) -\mu S\\ \dot{I} &= h S - rI - \mu I \end{array} \]

If there is no demographic change, the SIS-xde model can be rewritten as a single equation:

\[ \dot{I} = h (H-I) - rI \] Even in this simplified form, we are assuming that a population could be stratified, such that the variables and parameter are all vectors with length nStrata.

Equilibrium Solutions

A typical situation when using this model is that \(H\) (total population size by strata) and \(X\) (number of infectious persons by strata) are known from census and survey data. Then it is of interest to find the value of \(EIR\) (Entomological Inoculation Rate) which leads to that prevalence at equilibrium.

\[ 0 = h \cdot (H-I) - rI \]

\[ \bar I = H \frac{h}{h+r} \]

\[ \bar S = H - \bar I \]

Example

library(ramp.xds)
library(deSolve)
library(viridisLite)

Here we run a simple example with 3 population strata at equilibrium. We use ramp.xds::make_parameters_X_SIS_xde to set up parameters. Please note that this only runs the human population component and that most users should read our fully worked example to run a full simulation.

We use the null (constant) model of human demography (\(H\) constant for all time).

The Long Way

To set up systems of differential equations, we must set the values of all our parameters.

nStrata <- 3
H <- c(100, 500, 250)
residence <- rep(1,3) 
nPatches=1
nHabitats=1
membership=1
params <- make_xds_object_template("ode", "cohort", nPatches, membership, residence)

b <- rep(0.55, nStrata) 
c <- rep(0.15, nStrata) 
r <- rep(1/200, nStrata) 
Xo = list(b=b, c=c, r=r)
class(Xo) <- "SIS"

foi = c(1:3)/365 
eir <- foi/b 
xde_steady_state_X(foi, H, Xo)-> ss
ss
#> $S
#> [1]  64.60177 238.56209  94.55959
#> 
#> $I
#> [1]  35.39823 261.43791 155.44041

MYZo = list(MYZm = eir*H)

Xo$S=ss$S
Xo$I=ss$I

params <- setup_Xpar("SIS", params, 1, Xo)
params <- setup_Xinits(params, H, 1, Xo)
params <- setup_Hpar_static(params, 1)
params <- setup_MYZpar("trivial", params, 1)
params <- setup_Lpar("trivial", params, 1)
params <- setup_Linits(params, 1)
params <- make_indices(params)

F_season = function(t){0*t+1}
F_trend = function(t){0*t+1}
F_age = function(a){0*a+1}

params$EIRpar = list() 
params$EIRpar$eir <- as.vector(eir)
params$EIRpar$scale <- 1 
params$EIRpar$F_season <- F_season
params$EIRpar$F_trend <- F_trend
params$EIRpar$F_age <- F_age

params = make_indices(params)

Xo$S=H 
Xo$I=H*0
params = setup_Xinits(params, H, 1, Xopts = Xo)
y0 <- get_inits(params)
y0$X
#> $H
#> [1] 100 500 250
#> 
#> $I
#> [1] 0 0 0

params <-xds_solve_cohort(params) 
out1 <- params$outputs$orbits

clrs = turbo(5)
XH <- out1$XH[[1]] 
age <- out1$age


plot(age, XH$true_pr[,1], col = clrs[1], ylim = c(0,1), type = "l")
lines(age, XH$true_pr[,2], col = clrs[2])
lines(age, XH$true_pr[,3], col =clrs[5])

Using Setup

We have developed utilities for setting up models. We pass the parameter values and initial values as lists:

xds_setup_cohort(eir, Xname="SIS", HPop=H, Xopts = Xo) -> test_SIS_xde

xds_solve_cohort(test_SIS_xde)-> test_SIS_xde 
test_SIS_xde$outputs$orbits$XH[[1]] -> XH2
sum((XH$true_pr-XH2$true_pr)^2)
#> [1] 0