Assume you've use a accelerated failure time model to find that the transition of subjects from State A to State B is log-normally distributed with parameters $\mu$ = X and $\sigma$ = Y.
This now needs to be used in a differential equation model as the rate at which subjects move from State A to State B. However, just using $\mu$ as the average probability of moving from A to B results in an exponentially distributed waiting time, not a log-normally distributed time.
I know you can however use a series of sequential independent exponential distributions to obtain an overall gamma distributed waiting time. For example, if it takes 2 days to move from A to B, four equally spaced exponential distributions results in an overall gamma distributed waiting time with $\kappa$ = 4 and $\theta$ = 2.
The question is, is there a gamma distribution whose shape and scale parameters approximate a log-normal? I know if I used many exponential distributions for a high $\kappa$ the central limit theorem allows the gamma distribution to approximate a log-normal, but I'm not sure if there's a way to obtain a log-normal waiting time from that. Essentially, is there some $\kappa$ and some $\theta$ that yields something approximating a log-normal with $\mu$ = X and $\sigma$ = Y?