Dwell times and the memoryless property
The math and examples above describe memoryless processes. This means that the probability of an individual transitioning to a new state does not depend on how long they have already been in their current state.
The dynamical systems perspective allows us to see that the duration of time individuals spend in each state, called the dwell time, follows an exponential distribution characterized by the rate \(\lambda\). The mean dwell time is
Therefore, you may see these transitions characterized by the mean dwell time instead of the rate.
While some epidemiological processes are well characterized by exponentially-distributed dwell times, others are not. For example, the latent stage of a disease may be better described by a log-normal, gamma, or Weibull distribution. Non-exponential dwell times are due to underlying bio-physical processes that are not memoryless.
Agent-based models can easily accommodate non-exponential dwell times by allowing the time each agent spends in a state to be drawn directly from non-exponential distributions. Here, we typically move away from describing the process in terms of rates and transition probabilities to instead describe the parameters of the (non-exponential) dwell time distributions. Compartment models can also accommodate non-exponential dwell times by using a more complex set of equations, but this is beyond the scope of this guide.