library(ramp.falciparum)
library(deSolve)
library(knitr)Hybrid Models
Modeling Superinfection & the MoI with Hybrid Models
While \(M/M/\infty\) is an infinite system of differential equations, the function solveMMinfinity solves a finite system of differential equations. The maximum MoI is chosen to be large enough that it doesn’t affect the results.
The Hybrid Model
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. If the initial distribution was Poisson, then it would remain Poisson forever.
hybrid = solve_dm(5/365, foiP3, Amax=1095)with(hybrid, plot(time, m, type = "l", ylab = expression(m[tau](a)), xlab = "a - cohort age (in days)"))
Prevalence
With all this information at hand, we are now in a position to formulate a model for super-infection.
We note that since \(m\) is the mean of a Poisson, the prevalence of infection can be computed directly from \(m\). It is the complement of the zero term from a Poisson: \[p(t) = 1-\mbox{Pois}(\zeta =0; m(t)) = 1 - e^{-m(t)}\] The change in prevalence is thus described by the equation:
\[ \frac{dp}{dt} = h(1-p)-R(m)p \]
While this is interesting, and while it solves the problem of modeling clearance under superinfection, the equation depends on \(m\). Since we can compute \(p(t)\) directly from \(m(t)\), the equation for prevalence is completely redundant.
On the other hand, if we wanted a model for prevalence that did not use \(m\), we solve, \(m = -\ln(1-p),\) substitute, and rewrite the equation:
\[ \frac{dp}{dt} = h(1-p) + r \ln (1-p) \frac{(1-p) }{p} \]