  Compartment models with delay terms are widely used across a range of
disciplines. The motivation to include delay terms varies across different
contexts. In epidemiological and pharmacokinetic models, the delays are often
used to represent an incubation period. In this work, we derive a compartment
model with delay terms from an underlying non-Markov stochastic process. Delay
terms arise when waiting times are drawn from a delay exponential distribution.
This stochastic process approach allows us to preserve the physicality of the
model, gaining understanding into the conditions under which delay terms can
arise. By providing the conditions under which the delay exponential function
is a probability distribution, we establish a critical value for the delay
terms. An exact stochastic simulation method is introduced for the generalized
model, enabling us to utilize the simulation in scenarios where intrinsic
stochasticity is significant, such as when the population size is small. We
illustrate the applications of the model and validate our simulation algorithm
on examples drawn from epidemiology and pharmacokinetics.
