"Spreads" means to move or extend over an area, and "over" is the correct preposition to indicate the distribution across the district.
- "Expands" is not typically used for diseases, and "from" refers to the origin, not the spread.
- "Goes" and "in" are not appropriate for describing the spread of a disease.
"Spreads over" is a common expression for something (like a disease) affecting a large area.