Aron & May
Basic Dynamics
If you start with the Ross-Macdonald model and try to start adding complexity, it’s pretty easy to get stuck. One of the first things we would like to do is start varying some of the quantities over time. If we did this, we might find ourselves stuck in a mess. Instead of trying to explain all that now, we’ll present a different version of the equations that is much easier to work with. we give credit to Joan Aron and Bob May [1].
Aron & May
The main difference in the model by Aron & May is that mode variables are population densities.
The variables are:
\(H, X\) describe the density of humans and infected humans, respectively;
\(M, Y, Z\) describe the density of mosquitoes, infected mosquitoes, and infectious mosquitoes, respectively; and \(m=M/H;\) \(y=Y/H\) and \(z=Z/H\)
\(\Lambda(t)\) is the emergence rate of adult mosquitoes;
We introduce one new parameter describing the fraction of mosquitoes that would become infected after blood feeding on an infectious human, \(c.\) Keeping all other parameters the same, we can write a system of differential equations:
\[ \begin{array}{rl} \dot M &= \Lambda(t) -g M \\ \dot Y &= a cx (M-Y) - g Y \\ \dot Z &= e^{-g\tau} a c x_\tau (M_\tau-Y_\tau) - g Z \\ \dot X &= mabz (H-X) - r X\\ \end{array} \] Note that we used \(x\) and \(z\) in the equation to keep the equations looking tidy. Also, note that if \(\Lambda(t)\) is constant over time, and assuming \(H\) is also constant, then \(M\) reaches a steady state \[\bar M = \frac \Lambda g\] If we set the initial condition to \(M(0) = \bar M\), then we can derive Macdonald’s equation above by simply changing variables.