add call out and more detail on convolution

Author: amanda-minter

episodes/disease-burden.Rmd

@@ -220,9 +220,20 @@ hosp <- new_cases$new_infections * ihr
 
 #### 2. Calculate the estimated number of new hospitalisations using the onset to admission distribution
 
-We convolve the expected number of hospitalisations (hosp) with the distribution of onset to admission times to obtain the estimated number of new hospitalisations
+To estimate the number of new hospitalisations we use a method called convolution. 
 
+::::::::::::::::::::::::::::::::::::: callout
+### What is convolution?
 
+Briefly, convolution is a mathematical process which combines two variables by seeing how much they overlap each other (see this [Wolfram article](https://mathworld.wolfram.com/Convolution.html) for some mathematical detail). There are different methods to perform convolution, the built in R function `convolve()` uses the Fast Fourier Transform. 
+
+::::::::::::::::::::::::::::::::::::::::::::::::
+
+The function `convolve()` requires inputs of two vectors which will be convolved and `type`. Here we will specify `type = "open"`, this fills the vectors with 0s to ensure they are the same length. 
+
+The inputs to be convolved are the expected number of hospitalisations (`hosp`) and the density values of the density distribution of onset to admission times. We will calculate the density for the minimum possible value (0 days) up to the tail of the distribution (here defined as the 99.9th quantile).
+
+Convolution requires one of the inputs to be reversed, in our case we will reverse the density distribution of onset to admission times. 
 
 ```{r}
 # define tail of the delay distribution
@@ -237,7 +248,7 @@ hospitalisations <- convolve(hosp,
 
 #### 3. Calculate the estimated number of discharges
 
-Using the same approach as above, we convolve the hospitalisations with the distribution of admission to discharge times to obtain the estimated number of new discharges
+Using the same approach as above, we convolve the hospitalisations with the distribution of admission to discharge times to obtain the estimated number of new discharges.
 
 
 ```{r}