Age of Infection (AoI)

A Probabilistic Synthesis of Malaria Epidemiology

library(ramp.falciparum)
library(deSolve)
library(knitr)

Warning: package 'knitr' was built under R version 4.3.3

The Random Variables Approach

Let $z(\alpha ,a)$ denote the density of parasite cohorts of age $\alpha$ in a host cohort of age $a$. We assume infections clear at the constant rate $r$. Since infections in $M/M/\mathrm{\infty }$ are independent, we can track the dynamics for the AoI of all parasite cohorts with the equation,

$$\frac{\partial z}{\partial a}+\frac{\partial z}{\partial \alpha }=-rz,$$

with the boundary condition $z_{\tau }(a,0)=h_{\tau }(a).$ We note that the age of the host birth cohort sets an upper limit for the parasite cohort, so $0<\alpha <a$. The solution, which describes density of infections of age $\alpha$ in a host cohort of age $a$, is given by the formula:

$$z_{\tau }(\alpha ,a|h)=h_{\tau }(a-\alpha )e^{-r\alpha }.$$

From these equations, we derive random variables describing the MoI, the AoI, and the AoY, noting that the mean MoI is:

$$m_{\tau }(a|h)={\int }_0^a{z}_{\tau }(\alpha ,a|h)d\alpha$$

In ramp.falciparum, this is computed with the function meanMoI

moi = meanMoI(aa, foiP3, hhat=5/365)

plot(aa, moi, type = "l", ylab = expression(m[tau](a)), xlab = "a - cohort age (in days)")

The three give the same answers, up to slight differences introduced by the numerical methods:

c(mean(abs(moi - hybrid$m)) < 1e-9,
mean(abs(MMinf$m- hybrid$m)) < 1e-10,
max(abs(moi - hybrid$m)) < 1e-8,
max(abs(MMinf$m- hybrid$m))< 1e-10)