MMinfinity

The 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, \(h_\bday(a)\)). 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 $$\frac{d\zeta_0}{da}= -h_\bday(a) \zeta_0 + r \zeta_1$$ and for \(i\geq 1\) $$\frac{d\zeta_i}{da}= h_\bday(a) \left( \zeta_{i-1} - \zeta_i \right) - ri \zeta_i + r(i+1)\zeta_{i+1}$$