Age of Infection • pf.memory

suppressMessages(library(pf.memory))

library(deSolve)
library(knitr)
library(viridisLite)
library(viridis)

Since $A_\tau(a)$ is a density function and $m_\tau(a)$ its mass, a PDF for the age of infection is:

$A_\tau(a) \sim f_A(\alpha; a, \tau) = \frac{z_\tau(\alpha,a)}{m_\tau(a)}$

The CDF for $A_\tau(a)$ is:

$F_A(a) \sim \int_0^\alpha f_A(\alpha; a, \tau) d\alpha$

Functions in pf.memory compute the density, distribution and random generation for the distribution of $A_\tau(a)$ for any integrable function $h_\tau(a)$

The Density Function

The following shows what the distribution of the AoI looks like in a cohort of age 2.

plot(a2years, dAoI(a2years, max(a2years), foiP3), type ="l", xlab = "Parasite Age", ylab = expression(F[A](alpha, a, tau)), ylim = c(0,0.009))

Here, we plot the same distributions for 12 cohorts at age 2, each born one month apart.

plot(a2years, dAoI(a2years, max(a2years), foiP3), type = "n", xlab = "Parasite Age", ylab = expression(F[A](alpha, a, tau)), ylim = c(0,0.013))
for(i in 1:12)
  lines(a2years, dAoI(a2years,max(a2years),foiP3, tau=30*i), col = clrs[i])

The Distribution Function

plot(a2years, cumsum(dAoI(a2years, max(a2years), foiP3)), type = "l", xlab = "Parasite Cohort Age", ylab = expression(1-F[X](alpha, a, tau)), lwd=2)
lines(a2years, pAoI(a2years, max(a2years), foiP3), col = "red", lwd=2, lty =2)

par(mar = c(5,4,1,2))

plot(a2years, pAoI(a2years, max(a2years), foiP3), type = "n", xlab =  expression(list(alpha, " Parasite Age")), ylab = expression(F[X](alpha, a, tau)))

for(i in 1:12)
  lines(a2years, pAoI(a2years,max(a2years), foiP3, tau=30*i), col = clrs[i])

Random Numbers

par(mar = c(5,4,1,2))
a5years = 0:(5*365)
rhx = rAoI(10000, 5*365, foiP3)
hist(rhx, breaks = c(0:c(5*365)), right=F, probability=T, main = "", xlab = expression(alpha), border = grey(0.5))
cdf1 = pAoI(a5years, 5*365, foiP3)
pdf1 = dAoI(a5years, 5*365, foiP3) 
lines(a5years, pdf1, col="red", lwd=2)

par(mar = c(5,4,1,2))
plot(ecdf(rhx), xlim = c(0,1825), cex=0.2, main = "", xlab = expression(alpha))
lines(a5years, cdf1, col = "red", lty = 2, lwd=2)

Moments

Let $x$ denote the first moment of of $A_\tau(a)$ :

$x_\tau(a) = \left< A_\tau(a) \right> = \int_0^\infty \alpha \frac{z_\tau(\alpha, a)} {m_\tau(a)}$

Similarly, we let $x_\tau(a)[n]$ denote the higher order moments of $A_\tau(a)$ :

$x_{n}(a, \tau) = \int_0^\infty \alpha^n \frac{z_\tau(\alpha, a)} {m_\tau(a)}$

aa = seq(1, 5*365, by = 5) 
moment1 = momentAoI(aa, foiP3)
moment2 = momentAoI(aa, foiP3, n=2)
moment3 = momentAoI(aa, foiP3, n=3)

The first three moments of the AoY plotted over time. In the top plot, we’ve also plotted the $n^{th}$ root of the $n^{th}$ moment.

par(mfrow = c(3,1), mar = c(0.5, 4, 0.5, 2))
plot(aa, moment1, type = "l", xlab = "", ylab = expression(E(A)), lwd=2, xaxt="n", ylim = range((moment3)^(1/3)) ) 
lines(aa, sqrt(moment2), col = "darkgreen")
lines(aa, (moment3)^(1/3), col = "purple")
plot(aa, moment2, type = "l", xlab = "", ylab = expression(E(A^2)), lwd=2, xaxt="n", col = "darkgreen") 
plot(aa, moment3, type = "l", xlab = "", ylab = expression(E(A^2)), lwd=2, col = "purple") 
mtext("Age (in Days)", 1, 3)