Pandemic: The Board Game

This is one of the most challenging tasks in standard Pandemic with 4+ Epidemics. Fortunately, you can change the difficulty to only have 2 Epidemics and help yourself tremendously.

The Medic is Strongly Recommended based on their ability to clear cubes of cured diseases quickly . I would also recommend the Dispatcher to be able to move the Medic 4 additional moves each round when you are attempting to Eradicate diseases.
Recommended Characters
Researcher/Scientist/Medic (+Dispatcher if you're struggling to clear cubes)

Strategy
To have all 4 diseases Eradicated when you win the game, you have to either cure the 4th disease with no cubes at all on the board, or cure the 4th disease with the Medic occupying the same space as the last cubes on the board (they will get cleared once the disease is cured). Your goal throughout should be to eradicate diseases as you go, leaving yourself less work at the end where you might run out of time. As you get down to 2 or 1 diseases left to cure/eradicate, try and remember which cards are in the infection pile so you can locate your characters strategically for easy treating of cubes as they come on the board.

Forecast Exploit
Thanks to Taro Korya San for identifying this exploit. If you have the Forecast event card, you can play it to see the top cards of the Infection Deck. Before finishing the action, quit out to the main menu. You can then reload the game and you'll have the Forecast event back in your hand and you'll know the next cards coming up and can plan movements accordingly. Be aware, once you draw an Epidemic, the Infection Deck gets shuffled and you'll have to check the top cards again.

The technique that worked for me was using the scientist and medic.

The scientist stayed within 3 moves of atlanta so she could cure the first 3 quickly and I had the medic move around eradicating them as quickly as possible. I wasn't swapping cards so used the medics to fly him when necessary.

When it came to the 4th, I saved my forecast card and was able to use it so that only cities with eradicated viruses would show for a couple of turns, which gave me enough time to get rid of all of the last virus before the scientist cured it. By chance the blue was the last one I had, which meant I was able to use the scientist to help treat before curing it.

To be honest there is a bit of chance involved as well, but this worked for me the first time I tried it.

Hi

thanks to thanatos8285 for the idea of the dispatcher and the medic.

it is with these 2 characters that I managed to do it.
(I put the medic first and dispatcher the second)
We easily get to 3 eradicated diseases.
Many times I could have arrived at 4 but I missed 1 round lol.
I don't really have a technique to give.
It's a bit depending on what's going on in the game.

to have it I treated the cities with the medic. I used the skill of the dispatcher to easily move the medicine once the illnesses had healed.
I also used the dispatcher to join the medic or have the medic join the dispatcher.

concerning the cards I gave to the one with the most of the same suit.
As well thought of setting up centers to cure illnesses it saves time by avoiding returning to Atlanta.

I hope my comment will help you get it.

I did this on the Virulent Strain challenge mode, with a team of 2, first up was the researcher and second was the scientist.

I eradicated the first 3 viruses as soon as I had the cures.

The last one is the tricky one you need to remove all the virus blocks before you cure it, I made sure both team's had enough cards to cure the virus and once the last virus block was removed, I played the One Quiet night card, and used the Government Grant card, to cure the virus on the city I was currently on.

You do have to be lucky with the cards, it took me 3 attempts, the first couple I ran out of cards and was forced to cure the virus before I'd cleared all the virus blocks.
I complete the game like this a couple more times when I was doing the Are You Not Entertained? Achievement.