Derivatives for the queuing model $M/M/\infty$

This queuing model $M/M/\infty$ tracks the MoI in a cohort of humans as it ages. It assumes a time- and age-dependent hazard rate for infection, called the force of infection (FoI, $h_\tau(a)$). Infections do not affect each other, and each one clears independently at the rate $r$.

Let $\zeta_i$ the fraction of the population with MoI = i, then $$\frac{d\zeta_0}{da}= -h_\tau(a) \zeta_0 + r \zeta_1$$ and for $i\geq 1$ $$\frac{d\zeta_i}{da}= h_\tau(a) \left( \zeta_{i-1} - \zeta_i \right) - ri \zeta_i + r(i+1)\zeta_{i+1}$$

This function computes the derivatives in a form that can be used by deSolve::ode.