Prevalence

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.

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

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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:

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