Improving transit compartment models of delayed processes throughout mathematical biology

From microbiology to pharmacology and epidemiology, transit compartment models are ubiquitous throughout mathematical biology. These models are simple to use, direct to implement in existing software packages, and easy to explain to collaborators. These transit compartments represent a delayed process  and are mathematically equivalent to an Erlang distributed delay differential equation (DDE). However, using these  Erlang DDEs imposes an artificial relationship between the mean and variance of the delayed process, which may not be biologically realistic, and severely increases computational cost when fitting models to data.

However, replacing the Erlang delay distribution by the more general gamma distribution removes the artificial relationship between the mean and variance of the delayed process, but there are no existing numerical techniques for gamma distributed DDEs.  I'll present a consistent numerical method for gamma distributed delay differential equations, use this numerical method to illustrate some of the problems induced by using transit compartment models, and  present an alternative approach to model delayed processes that retains the benefits of transit compartment models but removes the artificial relationship between mean and variance. Through an example from chemotherapy induced toxicity, I'll show how this alternative method can lead to more robust model predictions while significantly decreasing computational cost.