Derivatives for the `SIP` Module (**X** Component)

Compute the derivatives for SIP compartmental model: $$\begin{array}{rrrrcc} dS/dt =& - (h +\xi) S &+ r I & + \eta P & + d{\cal H}(S) &+ B(t, H) \\ dI/dt =& (1-\rho) h S & - (r+\xi) I &&+ d{\cal H}(I) \\ dP/dt =& (\rho h +\xi)S & +\xi I & - \eta P &+ d{\cal H}(P) \end{array}$$ where $H = S+I+P$; $B(t, H)$ is the time-dependent birth rate; and the $d{\cal H}$ operator computes derivatives for the demographic model $\cal H$. The parameter $\xi$ can be modified by setting `mass_treat` to a non-zero function of time.