Delay distributions at the population level
Individual delays
If \(f(t)\) is our delay distribution then
\[ p(y_i) = f(y_i - x_i) \]
is the probability that secondary event of individual \(i\) happens at time \(y_i\) given its primary event happened at \(x_i\).
Population level counts
The expected number of individuals \(S_t\) that have their secondary event at time \(t\) can then be calculated as the sum of these probabilities
\[ S_t = \sum_i f_{t - x_i} \]
Note: If \(S_t\) is in discrete time steps then \(f_t\) needs to be a discrete probability distribution.
Population level counts
If the number of individuals \(P_{t'}\) that have their primary event at time \(t'\) then we can rewrite this as
\[ S_t = \sum_{t'} P_{t'} f_{t - t'} \]
This operation is called a (discrete) convolution of \(P\) with \(f\).
We can use convolutions with the delay distribution that applies at the individual level to determine population-level counts.
Example: infections to symptom onsets
Why use a convolution, not individual delays?
- we don’t always have individual data available
- we often model other processes at the population level (such as transmission) and so being able to model delays on the same scale is useful
- doing the computation at the population level requires fewer calculations (i.e. is faster)
- however, a downside is that we won’t have realistic uncertainty, especially if the number of individuals is small
What if \(f\) is continuous?
Having moved to the population level, we can’t estimate individual-level event times any more.
Instead, we discretise the distribution (remembering that it is double censored - as both events are censored).
This can be solved mathematically but in the session we will use simulation.
Your Turn
- Simulate convolutions with infection counts
- Discretise continuous distributions
- Estimate parameters numbers of infections from number of symptom onsets, using a convolution model
Delay distributions at the population level