For a compartmental disease model, like the SIR model, the sojourn times in the compartments are inherently Exponentially distributed.  An interesting property of the Exponential distribution is that the probability of an individual leaving a compartment in a small time step is completely independent of the time the individual has spent in the compartment.

To show why this is the case, let’s calculate the probability that an infected individual in an SIR model will recover between times t to t+delta_t (refer to this as event “A”), given that the individual has been infected up until time t (refer to this as event “B”).  Baye’s theorem tells us that

P(A|B) = P(B|A)P(A)/P(B)

It should be easy to see that P(B|A)=1 because A occurring necessarily implies that B must have occurred too. The exponential probability distribution underlying the sojourn time in the infectious compartment is f(t) = gamma exp(-gamma*t), where gamma is the average recovery rate. We can easily calculate, by integrating the probability distribution, that

P(A) = (1-exp(-gamma*delta_tt))*exp(-gamma*t), 
and 
P(B) = exp(-gamma*t).

Thus

P(A|B) = 1-exp(-gamma*delta_t)

Notice that this does not depend on time!  The probability of leaving the infectious state is constant in time, and does not depend on how long an individual has been infectious.  This is know as the “memoryless” property of the exponential distribution.

More realistic probability distributions for the infectious stage (like the Gamma distribution) are not memoryless; the probability of leaving a class in some time step depends on how long the individual has so far sojourned in that class.