I was not having any success with SUMPRODUCT so I switched to array.
I should have persisted with SUMPRODUCT since it seems this standard formula is OK... 
=SUMPRODUCT((A1:H1="sick")*(B1:I1<>"sick"))+(H1="sick")*(I1="sick")+(H1<>"sick")*(I1="sick")
or, if the blank column between the formula and the row of entries is used... 
=SUMPRODUCT((A1:I1="sick")*(B1:J1<>"sick"))