If on Sheet2 you have a list of the conditions in column A, then you can get the number of days for each condition in column B:
1 Condition Total days
2 Headache 8
3 Flu 32
4 Injury 16
5 Hospital 25
In cell B2 enter this formula:
Note the $ signs
Now drag the formula down alongside the conditions in column A.
The $ signs are required to ensure that the formula continues to refer to the same range of cells on sheet 1.
Change this range to match the range used in columns Q and S on Sheet1 (the ranges must be the same number of rows).
The range can include empty cells, so that there is some space for adding additional staff, or you could use named ranges to make changes to the ranges used on Sheet1 easier to manage.
If your source data is in a worksheet with a name containing spaces then the name must be enclosed in single quotes like this:
=SUMIF('Source Data'!$S$2:$S$14,A2,'Source Data'!$Q$2:$Q$14)