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_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 <- make_Xpar("SIS", params, 1, Xo)
params <- make_Xinits(params, H, 1, Xo)
params <- setup_Hpar_static(params, 1)
params <- make_MYZpar("trivial", params, 1)
params <- make_Lpar("trivial", params, 1)
params <- make_Linits(params, 1)
params <- make_indices(params)

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

params$EIRpar = list() 
params$EIRpar$eir <- as.vector(eir)
params$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 = make_Xinits(params, H, 1, Xopts = Xo)
y0 <- get_inits(params)
y0$X
#> $S
#> [1] 100 500 250
#> 
#> $I
#> [1] 0 0 0

out1 <- xds_solve_cohort(params)$outputs$cohort

clrs = turbo(5)
with(out1,{
  plot(age, true_pr[,1], col = clrs[1], ylim = c(0,1), type = "l")
  lines(age, true_pr[,2], col = clrs[2])
  lines(age, 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)$outputs$cohort -> out2
sum((out2$XH[[1]]$true_pr-out1$XH[[1]]$true_pr)^2)
#> [1] 0