Rates used in counting processes

Some processes are naturally described by counting the number of events that occur in a fixed time interval, without regard for the number at risk. These processes are often modeled using Poisson processes. The "rate" parameter of a Poisson process is often called an intensity or arrival rate, but sometimes just "rate". Here, the term refers to the expected number of events per unit time, not a state-dependent transition rate like above.

Counting processes described by rates are used in epidemiological models. For example, in a simulation of a sexually transmitted disease, each relationship may have a "coital act rate" describing the frequency of acts between partners. The expected number of acts is the coital rate multiplied by the length of the time interval, \(\lambda dt\).

The formula for the expected number of events, \(\lambda dt\), is equivalent to the linear approximation of the transition probability we discussed above. However, in this case, it is not an approximation and often can/should exceed 1.0 (it's an expected number of events, not a probability).