In epidemiology, the two-stage catalytic model gives the observed cumulative prevalence $y$ of a disease (often termed seroprevalence, measured with an antibody test) for individuals of age $a$; where $\lambda$ is the per unit time probability of infection, and $\omega$ is the per unit time probability that an infected individual will revert from testing positive to negative (Muench 1959).
In this model it is assumed that individuals can only be infected once, death from infection is negligible, and that $\lambda$ does not vary by age.
\begin{equation} \frac{dy}{da} = \lambda e^{-\lambda a}-\omega y \end{equation}
Given the condition $y(0)=0$, and if $\omega>\lambda$; this can be solved as:
\begin{equation} y(a) = \frac{\lambda}{\omega-\lambda} \big(e^{-\lambda a} - e^{-\omega a}\big) \end{equation}
My question - how can we write this expression where the parameter $\lambda$ can vary as a function of time $\lambda(t)$ in discrete time with yearly time steps for $a$ and $t$?
As an additional note, I know that for a simpler model where the diagnostic never reverts from positive to negative (i.e., $\omega=0$):
\begin{equation} y(a) = 1-e^{-\lambda a} \end{equation}
If $\lambda$ varies by time $t$ in yearly discrete steps, then the expression becomes:
\begin{equation} y(a,t) = 1-\exp\left(-\sum_{i=t-a+1}^{t} \lambda_i\right)\end{equation}