Wednesday, February 24, 2010

exponential vs additive growth

02.24.10. Given ample nutrients, warmth, and space, bacteria divide regularly with division times on the order of an hour, and are thus a canonical model for exponential growth (see 10.17.08 post). The proliferation of viruses, on the other hand, is host-dependent, and therefore an additive model might be more appropriate. The rhinovirus, for example, a causative agent of the common cold, can be present in as many as 1 million copies per milliliter of nasal mucus (Web MD). Since we each produce about 1 liter of mucus per day (Dr. Oz), this amounts to approximately 1 billion viruses per cold victim. Assuming that an average of one person per day in a local community contracts a cold, the number of viruses grows additively: 1 billion, 2 billion, 3 billion, etc. Students were given this information and posed the question "Will there be more bacteria or viruses, and when?". Answering the question led students to recall (1) the conversion between hours and days, (2) the representation of repeated multiplication as exponentiation, and (3) the use of scientific notation and exponential notation, both on paper and in the calculator. As demonstrated by the plot above, exponential growth is much more rapid than linear growth, and under these models the bacterial population handily overtakes the viral population in about 30 hours.