This is a hybrid model which tracks the mean multiplicity of infection (superinfection) in two compartments. The first, \(m1\) is all infections, and the second \(m2\) are apparent (patent) infections. Therefore \(m2\) is “nested” within \(m1\). It is a “hybrid” model in the sense of Nåsell (1985).
Differential Equations
The equations are as follows:
\[ \dot{m_{1}} = h - r_{1}m_{1} \] \[ \dot{m_{2}} = h - r_{2}m_{2} \] Where \(h = b EIR\), is the force of infection. Prevalence can be calculated from these MoI values by:
\[ x_{1} = 1-e^{-m_{1}} \] \[ x_{2} = 1-e^{-m_{2}} \] The net infectious probability to mosquitoes is therefore given by:
\[ x = c_{2}x_{2} + c_{1}(x_{1}-x_{2}) \]
Where \(c_{1}\) is the infectiousness of inapparent infections, and \(c_{2}\) is the infectiousness of patent infections.
Equilibrium solutions
One way to proceed is assume that \(m_{2}\) is known, as it models the MoI of patent (observable) infections. Then we have:
\[ h = r_{2}/m_{2} \] \[ m_{1} = h/r_{1} \] We can use this to calculate the net infectious probability, and then \(\kappa = \beta^{\top} \cdot x\), allowing the equilibrium solutions of this model to feed into the other components.
Example
Here we run a simple example with 3 population strata at equilibrium.
We use ramp.xds::make_parameters_X_hMoI_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). While \(H\) does not appear in the equations above, it must be specified as the model of bloodfeeding (\(\beta\)) relies on \(H\) to compute consistent values.
The Long Way
Here, we build a model step-by-step.
nStrata <- 3
H <- c(100, 500, 250)
residence = rep(1,3)
params <- make_xds_object_template("ode", "cohort", 1, 1, residence)
b <- 0.55
c1 <- 0.05
c2 <- 0.25
r1 <- 1/250
r2 <- 1/50
m2 <- rlnorm(3, .2, .5)
h <- r2*m2
m1 <- h/r1
Xo = list(b = b, c1 = c1, c2 = c2, r1 = r1, r2 = r2, m1=m1, m2=m2)
eir <- h/b
params = setup_Xpar("hMoI", params, 1, Xo)
params = setup_Xinits(params, H, 1, Xo)
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$F_age <- F_age
params$EIRpar$season_par <- list()
params$EIRpar$trend_par<- list()
params$EIRpar$age_par<- list()
params = make_indices(params)
y0 <- get_inits(params)
y0
#> $L
#> list()
#>
#> $MYZ
#> NULL
#>
#> $X
#> $X$m1
#> [1] 3.032587 6.938575 1.805443
#>
#> $X$m2
#> [1] 0.6065175 1.3877150 0.3610886
params <- xds_solve_cohort(params)
out1 <- params$outputs$orbits
XH <- out1$XH[[1]]
age <- out1$age
clrs = turbo(5)
plot(age, XH$m1[,1], col = clrs[1], ylim = range(XH$m1), type = "l")
lines(age, XH$m1[,2], col = clrs[2])
lines(age, XH$m1[,3], col =clrs[5])
lines(age, XH$m2[,1], col = clrs[1], lty = 2)
lines(age, XH$m2[,2], col = clrs[2], lty=2)
lines(age, XH$m2[,3], col =clrs[5], lty=2)
Using Setup
xds_setup_cohort(eir, Xname="hMoI", HPop=H, Xopts = Xo) ->test_hMoI
xds_solve_cohort(test_hMoI)-> test_hMoI
XH2 <- params$outputs$orbits$XH[[1]]
sum((XH$m1-XH$m1)^2)
#> [1] 0