You don't get the flu by having friends. On the screen for the flu there are 5 levels. If you don't have it yet you will see a number such as 0/100 100 is the number to reach to get level one. You get one count by having a friend visit you, and tapping on a location, then you tap the shield that appears. When you reach the 100 you will get a message that the flu has arrived in your town. You will then start working on level 2 which is reached at 200, and when you reach that you will get another message that the flu has hit your town. Continue till 1000 to get level 5.
This is why it is so important to visit all your friends each day in this event as the only way they progress is by being visited. There is no requirement that they have the flu to give it to you, its strictly by count. Hope this helps.