1.4 Malaria Population Dynamics
He recognized that prevalence could change, and that what set its value was a balance between two processes. The quantitative logic looks like this:
\[ \left[ \begin{array}{rcl} \mbox{Fraction Infected Today} &=& \mbox{Fraction Infected Yesterday} \;\; -\\ && \mbox{Cleared Infections} \;\; + \\ && \mbox{Fraction Uninfected that Got Infected}\\ \end{array} \right] \]
This description ignores malaria importation, infected people who died, infected people who emigrated, and infected people who immigrated. The model is neither comprehensive nor perfect. It was a starting point.
For short-lived mosquitoes, we assume they are much more likely to die than to clear infections. So for mosquitoes, the process is slightly different:
\[ \left[ \begin{array}{rcl} \mbox{Fraction Infected Today} &=& \mbox{Fraction Infected that Survived} \;\; -\\ && \mbox{Fraction Uninfected that Got Infected}\\ \end{array} \right] \]
What’s important here is that infectious mosquitoes are infecting people when they bite, and infectious people are infecting mosquitoes when the mosquito blood feeds.
In planning and evaluating malaria control, it was important to develop some kind of expectations about where and when to do malaria control, what types of malaria control are likely to work best, and so on.
Despite all the focus on malaria, transmission Ross’s first mathematical model of malaria was about mosquito populations.
In 1911, Ross formulated a model (his second) as a system of two ordinary differential equations. The model appeared in the \(2^{nd}\) edition of The Prevention of Malaria [1], and it also appeared in Nature [32] in this form:
\[ \begin{array}{rl} dz/dt &= k' z' (p-z) + q z \\ dz'/dt &= k z (p'-z') + q' z' \\ \end{array} \]