experience a disease

Yes, in the sense of "have the experience of" :

There is no research to explore differences between how girls and boys experience arthritis.
Despite the fact that one third of the population will experience cancer in their lifetime ...
One of these factors can double your risks to experience heart disease ...
Individuals whose primary relatives have been affected with this condition are more likely to experience Crohn's Disease.
... and their babies will be unlikely to experience measles before 18 months.
People who experience Herpes frequently may take continuous treatment to try reduce the occurrence

All examples from Google.

Thanks a lot