3.3 Equations

Variables

The model has three variables:

• \(x\) is the fraction of humans who are infected;

• \(y\) is the fraction of adult female mosquitoes who are infected.

• \(z\) is the fraction of adult female mosquitoes who are infective.

Parameters

The model has several parameters:

• \(g\): mosquitoes die at a constant rate, \(g\). This is equivalent to assuming that the mosquito lifespan is exponentially distributed with a mean \(1/g\). Macdonald’s equation use \(p\), the fraction surviving one day, \(p = e^{-g}\).

• \(\tau\): it takes \(\tau\) days for parasites to mature and reach the salivary glands, called the EIP. The fraction surviving the EIP is \(p^\tau = e^{-g \tau}\).

• \(m\): there are \(m\) mosquitoes per human;

• \(a\): mosquitoes blood feed on humans at the rate \(a\);

• \(b\): a fraction of bites by infectious mosquitoes causes an infection;

• \(r\): human malaria infections clear at the rate \(r\);

• \(c\): a fraction of bites on infected humans infect a mosquito, denoted \(c\)

In this formulation of the model, we ignore the delay for the EIP but we count the mortality:

\[ \begin{array}{rl} dx/dt &= b m a z (1-x) - r x \\ dy/dt &= c a x (1-y) - g y \\ dz/dt &= e^{-g \tau} c a x_\tau (1-y_\tau) - g z \\ \end{array} \]

this.