Prevalence

prevalence.Rmd

library(knitr)
library(deSolve)
library(pf.memory)

George Maccdonald published a model of superinfection in 1950¹. In presenting the model, Macdonald was trying to solve a problem: if super-infection was possible, then how long would it take until a person would clear all of the parasites to become uninfected again? Macdonald proposed a solution, but there was a problem with the mathematical formulation (An essay by Paul Fine, from 1975, is recommended reading)²: the model that Macdonald described didn’t match the function he used. In a model formulated as part of the Garki project (Dietz K, et al., 1974)³, the model allowed for superinfection and proposed a useful approximating model. One solution came from taking a hybrid modeling approach (Nåsell I, 1985)⁴, which we have discussed in a vignette about the Multiplicity of Infection (MoI).

The model Macdonald described was also formulted in queuing theory, where the model is called $M/M/\infty$. In this model, each new infection increases the MoI by one; the transition rate to a higher MoI is the force of infection (FoI), denoted $h$. Each parasite can clear at some rate, $r$.

The following diagram shows how the fraction of the population with MoI $=i$, denoted $\zeta_i$, changes over time.

$$\begin{equation} \begin{array}{ccccccccc} \zeta_0 & {h\atop \longrightarrow} \atop {\longleftarrow \atop r} & \zeta_1 & {h\atop \longrightarrow} \atop {\longleftarrow \atop {2r}} & \zeta_2 & {h \atop \longrightarrow} \atop {\longleftarrow \atop {3r}} & \zeta_3 & {h \atop \longrightarrow} \atop {\longleftarrow \atop {4r}}& \ldots \end{array} \end{equation}$$

While the infinite system of differential equations can be formulated and solved numerically, Nåsell showed that the system can be described by a simple equation. Let $m$ denote the mean MoI.

$$\frac{dm}{dt} = h-rm$$

He also showed that the distribution of the MoI would converge to a Poisson distribution:

$$M(t) \sim f_M(\zeta; m) = \mbox{Pois}(m(t))$$