I wanted to separate these ideas from the other ones I just did because they involve adding new layers of strategy, not just adding to the strategic depth using current elements.

1. Show the mutated virus on the map. Make it clear which city the virus mutated in and have the mutated virus be a different color on the map. The mutated virus would spread to cities just like the original virus. Additionally, the mutated part of the virus should get bonus "mutation spread" action to spread the mutation to existing cities with the lower form of the virus (this could be blocked by quarantines) making the whole city change to the mutated form of the virus. It might even spread faster or intesify quicker (perhaps using the "mutation spread" actions or just additional mutation only spread and intensify actions). You would have to get mutated samples to research mutated traits.

2. One of the biggest things about epidemics is airline travel... and the disease popping up all the way on the other side of the map. Now, obviously you can't have hundreds of airline connections to other cities, but you could abstract that to 0, 1, 2, or 3 airline connections per city.

For gameplay purposes I think all airline connections should be single direction. In other words, they represent outbound flights only as a gameplay abstraction and cities don't necessarily have to have reciprocal outbound flights. There may be an airline connection from London to Moscow, but Moscow doesn't have one back to london but to other cities. Though I would make a couple cities on the map have reciprocal connections.

How it would work on the user inferface:
The airline connections would not be shown on the map unless you hover over the city, then, while you were hovering over the city, it would give you a nice arcing connections that showed you the air travel connections, Perhaps with a highlight movement on the arc or an arrow so you can tell the direction of the connection is outbound. You could have a symbol of an aircraft on or near the city circle if the city had air travel connections. (Make that aircraft light up red too if the city is infected and make it green if air traffic is quarantined) This will give a visual indicator to players if they need to hover over to check the airline connections. Additionally, when you hover over a city it should dim all the other cities that are not adjacent by land or reached by air. If a city is quarantined or air traffic is grounded then the appropriate cities would be dimmed as well because the quarantine makes them unreachable.

Also, some of the existing intercontinental connections would be changed into airline connections if it made sense.

This would jive with sniper__prime's suggestion to shutdown either the land or air travel when you quarantine (which would shut down incoming AND outgoing routes). Keep in mind, some cities in the world rely heavily on tourism and air travel for daily business so shutting down air travel might cost you per turn (as the country will demand that you reimberse them for their losses) and having an office in that location could help offset the loss of income (like you lose income per turn for shutting down airline travel, but it's free if you have an office there because offices represent diplomacy and the ability to convince the city/country that you are working on their behalf so shutting down air travel could be free or reduced cost where you have an office.

Also airline travel routes could be used to help your characters get around the map unless you close the airline travel from that city.

This would all bring a whole new dynamic to the disease spread and give the disease it's own "movement" to hop to the other side of the world.

Final thoughts:
I think both of these ideas I think are intuitive enough that they will still keep the game accessible while expanding the games strategic depth, and adding increased intenseness to the feeling of combating the virus, while also staying grounded in the reality of how diseases work and spread.