This model makes the following assumptions:

- Before GdnHCl is added to the culture medium, all cells are [PSI+] and the number of prions per cell varies as a Poisson random variable with mean .
- At cell division, the prions in the cell are divided independently and randomly with equal probability between the two offspring cells.
- Cells reproduce independently, according to an age-dependent branching process.

Let denote the probability
density function (p.d.f.) of generation time (the random time between successive cell divisions)
and let
denote
the number of renewals in an equilibrium renewal process with inter-occurrence time p.d.f.
.
Morgan *et al.*(2003) show that the probablity
that a randomly
selected cell at time *t* is [PSI+]
is given by