Leaving major cities during COVID-19 pandemic

My wife and I live in a major US city (Chicago) and are considering leaving for now to stay with family in a rural area a few hours away. Our reasons for this are mostly safety, seeing some of the reactions to toilet paper/groceries being low makes us both very concerned for what kind of civil unrest might happen in the coming weeks.

Our thoughts for why it would be good to go:

• Avoid possible looting/rioting/violence
• Emotional reasons (being with family, having more space)
• Easier to keep distance from people with a much smaller population around you

Possible reasons to stay where we are:

• Possibly spreading the virus (my wife has had a light cough for a few days, but no other symptoms)
• Health organizations asking us to stay put (although we would remain inside once we got to the new place)
• Maybe we're just overreacting and the chance of civil unrest right now is very low

I would like to know if there are any official recommendations on if this kind of travel is discouraged/forbidden, or even unofficial reasons why one option might be better than the other. Please let me know if this belongs in another network or if I should add/remove anything.

UPDATE:

Thanks for the advice, you've all made us reconsider and we'll be staying put to not risk spreading anything.