Compute derivatives for the MYZ module macdonald

This implements a delay differential equation model for adult mosquito ecology and infection dynamics that is consistent with the model published by George Macdonald in 1952. A generalized version of this model, the MYZ module GeRM, was developed to handle exogenous forcing by weather and vector control. This model should be used only for educational purposes.

Variables:

• $M$: the density of adult mosquitoes

• $Y$: the density of infected adult mosquitoes

• $Z$: the density of infectious adult mosquitoes

Parameters and Terms:

• $\mathrm{\Lambda }$ or Lambda: the emergence rate of adult mosquitoes (from F_emerge)

• $f$ or f: the blood feeding rate

• $q$ or q: maturation rate

• $\tau$ or eip: the extrinsic incubation period

• $\mathrm{\Omega }$ or Omega: an adult mosquito demographic matrix, including mortality and migration

• $\mathrm{{\rm Y}}$ or Upsilon: survival and dispersal through the eip, $\mathrm{{\rm Y}}=e^{-\mathrm{\Omega }\tau }$

Dynamics: In the delay equation, we use the subscript to denote the lagged value of a variable or term: e.g., $M_{\tau }=M(t-\tau )$.

$$\begin{array}{rl}dM/dt& =\mathrm{\Lambda }-\mathrm{\Omega }\cdot M\\ dY/dt& =fq\kappa (M-Y)-\mathrm{\Omega }\cdot Y\\ dZ/dt& =\mathrm{{\rm Y}}\cdot (fq\kappa {)}_{\tau }(M_{\tau }-Y_{\tau })-\mathrm{\Omega }\cdot Y\end{array}$$

This model was included mainly for the historical interest. It has been updated to handle exogenous forcing by weather and vector control in the module GeRM