15.3 Notes

15.3.1 Why use densities?

To show why we use densities, we present a simple example. If we write down an equation describing changes in the density of infected humans, \(X\), in a population with total human population density \(H\). We let \(V\) denote vectorial capacity, and \(b\) the fraction of infective bites that cause an infection, and we assume the force of infection is \(bVX/H\). The dynamics of infection are described by this simple equation:

\[\frac{dX}{dt} = bV\frac{X}{H}(H-X)-rX\] In this equation, prevalence is \(x = X/H.\) Following through with the change in variables, we can write down the equation for the change in prevalence:

\[\frac{dx}{dt} = \frac{1}{H^2} \left(H \frac{dX}{dt} - X \frac{dH}{dt} \right)\]

and with some rearranging, we get:

\[\frac{dx}{dt} = bVx(1-x)-rx -x \frac{dH}{dt}\]

The second equation is as simple as the first only if \(dH/dt=0\). Since we will want to deal with dynamical changes in host populations, we will avoid formulating base models that have proportions.