Facebook is (like) an infectious disease

The table below gives very rough data (from here) about the growth of the website Facebook.com from December 2004 to February 2009.

Date Months since December 2004 Active Facebook users
December 2004 0 nearly 1 million
December 2005 12 more than 5.5 million
November 2006 23 more than 12 million
April 2007 28 20 million
October 2007 34 over 50 million
August 2008 44 over 100 million
January 2009 49 over 150 million
February 2009 50 over 175 million

If you plot this information, you will notice that as time went on, the number of active users grew faster and faster. In other words, the larger the number of users, the faster that number grew. This makes sense, since the more people joined Facebook, the more other people were likely to do so. In this way, Facebook membership resembled the early exponential growth of an epidemic.

In fact, if we choose two of the data points, we can find an exponential function that fits them. For example, if we choose u(0) = 1 and u(50) = 175, we can find that the simple function

fits these points. The graph below shows the data given above along with this exponential function. Even though the function was created based only on the number of active users at 0 months and at 50 months, it fits most of the rest of the rough data surprisingly well! According to this function, the number of active Facebook users grew by about 10.882% each month.

Now, the table below gives the remaining available data about the growth of Facebook. If we compare these numbers with what the function given above would predict, it is clear that the exponential growth of Facebook has started to slow. This may be because most of the people worldwide who are willing and able to join the website have already done so. As an analogy, Facebook membership is like an infectious disease that is running out of potential victims.

Date Months since December 2004 Active Facebook users
April 2009 52 over 200 million
July 2009 55 over 250 million
September 2009 57 over 300 million
December 2009 60 over 350 million
February 2010 62 over 400 million
July 2010 67 over 500 million

A logistic function is a relatively simple model that scientists often use for the long-term growth of a population, or the spread of a disease. For example, the logistic function

seems to be a very good model for the recorded growth of Facebook so far. The graph of this function and the data is shown below. It will be interesting to see whether the function remains accurate in the coming months. If so, when will Facebook reach 750 million active users?

The following figure shows the Facebook data and exponential model function. Try adjusting the function by dragging the two red points it fits. Can you create an exponential function that fits the data reasonably well?

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