The most alarming fact is that infection is spreading……..the state and reaching villages and small towns.

The correct option is across.

Detailed Explanation:

In English grammar, prepositions of place and direction are used to describe how something moves or extends relative to a region, surface, or area.

Let's analyze the usage and meaning of the options:

1. across: Used as a preposition meaning 'from one side to the other' or 'everywhere in a particular region/country/state'. When talking about an infection, news, or trend spreading throughout an entire geographical region or area (like a state, country, or town), across is the correct preposition (e.g., "spreading across the state").

2. over: Typically implies covering a surface or moving above something, but is not the standard idiom for movement extending throughout a state or area in this context.

3. far: An adverb/adjective indicating distance, not a preposition of direction to connect to "the state".

4. from: Indicates a point of origin, but here the sentence requires expressing movement throughout the entire state, not starting point alone.

Therefore, inserting across completes the sentence correctly to mean that the infection is spreading everywhere throughout the state: "The most alarming fact is that infection is spreading across the state and reaching villages and small towns."