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Exponential probability distribution.

In reality, most likely time to leave the compartment is not at t=0!

Take, as an example, an SIR model (show equations)

The inherent assumption is that the sojourn time in the infectious stage is exponentially distributed (show proof); thus on average people recover after time t=1/gamma, but the most likely time to recover is at time t=0 after being infected!

But, say that we know, from observational studies of many people, that the probability of recovering after days 0, 1, 2, 3, etc looks like this: (show plot)

Obviously, the most likely day to recover is not on day 0!

Many probability distributions that look like “bumps”, defined in the range t=[0,inf] can be more or less adequately parameterized as a Gamma distribution.

Two parameters, shape factor (k) and scale factor (theta).  If shape factor an integer, then known as the Erlang distribution.  The mean of the distribution is mu=k*theta, and the variance is sigma^2=k*theta^2.

Now, conveniently, it happens that the Erlang distribution is the distribution of the sum of k exponentially distributed random variables with mean theta. (give R code)

This property provides a convenient trick to incorporate realistic sojourn probability distributions into a compartmental model.

The weighted average of that distribution is, and the weighted variance is. Doing the algebra yields that the shape factor is k~3, and theta~4.5

Why incorporate gamma distribution… does it make a difference to epidemic curve?