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Parasite Densities in Simple Infections

Simple_Infections.Rmd

In the following, we translate information about the distribution of the multiplicity of infection (MoI) and the age of infection (AoI) into probability distributions describing parasite densities. To do so:

  • We develop a function to model mean logged parasite densities as a function of the AoI: \(\mu(\alpha)\):

    • the variance of the distribution is a function of the mean, \(\sigma = F_\sigma(\mu)\);

    • the relationship between the AoI and \(\mu\) can be modified by immunity;

  • We develop a modified version of the \(beta\) distribution function family to model the density distribution of parasites with a given mean; \(P(\mu)\);

  • We compute the parasite density distributions for a function \(P(\alpha) = P_\mu(\alpha))\);

  • We develop a probability distribution function family to model the density distribution of parasites in a human cohort of age \(a\), \(P_\tau(a):\)

\[ P_\tau(a) \sim f_P(\xi; a, \tau) = \beta'(\xi; \mu(A_\tau(a)), \hat b(a)) \] The software is modular, making it possible to extend the code by developing other function families.

Mean, logged parasite densities, \(\mu(\alpha)\)

In pf.memory, a S3 function family was developed to model the relationship between the age of infection (AoI) for a parasite infection, \(\alpha\), and the mean, \(\mu(\alpha).\) The function alpha2mu.base has the following form:

\[\begin{equation} \mu(\alpha) = \begin{cases} \text{NA} \ & \mbox{if } 0 \leq \alpha < 7 \\[10pt] % l + (b - l) \frac{{\textstyle \alpha}}{{\textstyle D}} & \mbox{if } 7 \leq \alpha \leq D \\[10pt] l + (b - l) \frac{{\textstyle \alpha}}{{\textstyle \delta}} & \mbox{if } 7 \leq \alpha \leq \delta \\[10pt] l + (b - l) e^{-s_\alpha (\alpha-\delta)} & \mbox{if } \alpha \geq \delta\\ \end{cases} \label{mualpha} \end{equation}\]

In pf.memory, there are functions to set up the parameters for each case, with over-writable defaults:

base_pars = par_alpha2mu_base()
base_pars1 = par_alpha2mu_base(tildeb=11, Sa=0.0045)
aalpha = 7:366
mu_alpha =  alpha2mu(aalpha, 0, base_pars)
mu_alpha1 =  alpha2mu(aalpha, 0, base_pars1)

We plot the two functions here: with the default parameters (in black), and the other parameters (in dark blue):

\(P_\mu\)

We have developed a function family to compute parasite densities as a function of the mean of logged parasite densities, \(\mu\)

par(mfrow = c(2,2))
xx = seq(0, 13, by=0.1)
plot(xx, dDensityPmu(xx,9), type = "l", xlab = expression(x), ylab = "dDensityPmu", main = "dDensityPmu")
lines(xx, dDensityPmu(xx, 7), col = "darkred")
pp = seq(0,13, by = 0.01)
plot(pp, pDensityPmu(pp,9), type = "l", xlab = expression(x), ylab = "pDensityPmu", main = "pDensityPmu")
lines(xx, pDensityPmu(xx, 7), col = "darkred")
hist(rDensityPmu(1000, 9), main = "rDensityPmu", xlim = c(0,13))
qq = seq(0.01,.99, length.out=100)
plot(qq, qDensityPmu(qq,9), type = "l", xlab = expression(x), ylab = "qDensityPmu", main = "qDensityPmu")

\(P(\xi, \alpha)\)

We must specify a function to model exposure:

At age five, the distribution of the age of infection looks like this:

To compute parasite density distributions, we need to construct a mesh over \(\xi.\) The function dDensityPa computes the parasite distributions as a function of the host age, given a function describing the history of exposure.

meshX= seq(0, 13, by=.1)
Pmu = dDensityPa(meshX,5*365, foiP3)

Now, parasite densities in simple infections have the given distribution of mean logged parasite densities.