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We're continuing with the very simple and artificial example of disease spread, as an example of exponential growth. The simple system we have been looking at does not just apply to simple epidemics. It may also model other living organisms under certain situations. Instead of using I, let's use N to keep this in mind.
Remember the fundamental features of the model: the population consists of N(t) things at some time t. Each thing gives rise to a new things (and is removed), each time step. The model is N(t+1)=a × N(t). Every time step, we're multiplying the old value N(t) by a to get the new value N(t+1); the new value is just proportional to the old one. Each individual at time t becomes a new individuals at the next time period; the growth rate per capita is constant. The solution is N(t)=atN(0). This is called exponential (or geometric) growth. Thus, constant per capita growth rates lead to exponential growth.
Exponential growth can lead to big numbers in a hurry.
Let's do a little thought experiment to drive the point home.
In the old science fiction horror movie
Island of Terror,
some scientists accidentally create some silicate monsters that
appear to be dividing (and doubling) every six hours. Taking some
the movie, let's say each one takes up 0.1 cubic meter, and that they
really will somehow double every 6 hours. We'll take our time step to
be 6 hours. Here is the question: at that rate, how long until the
would consist of nothing but silicate monsters?
We know from our analysis earlier that
the number of the monsters at step t would have to be
N(t)=2t × N(0). If the Earth is
about 12764 km in diameter, then its volume is about 1012
cubic kilometers (in round numbers), or about 1021
cubic meters. So the whole world could hold about 1022
of the monsters. How many time steps to get from one to this number?
We need to solve the equation 2t=
1022 for t. Taking logarithms, we have
log(2) = 22
log10, so that
t is about 73 time steps, or during the 18th day. That's bad.
What is important about this example? Starting from a single organism, doubling every 6 hours, we find that extremely large numbers can result from exponential growth fairly quickly. Unlimited exponential growth is not possible.
Try this one: suppose that on average, one Ixodes scapularis tick, (a vector of Lyme disease) has 1000 offspring, and suppose that they all live to adulthood every generation and reproduce to the same degree. Suppose an adult tick takes up 0.5 square centimeters of land. How many years until the entire land mass of earth would be covered with ticks?
Or this one, a bit closer to home: suppose that at the end of each year, the human population is 4 percent larger than it was at the end of the previous year. Suppose a person needs 0.2 square meters to stand. How long until the entire land mass of earth would be covered with people?
It was realized centuries ago that constant per-capita growth rates lead to exponential growth, and that exponential growth will quickly outstrip any available resources. Malthus argued that the exponential growth of humans would come to a stop due to famine (while one may disagree on whether famine will be what stops exponential growth, there is no doubt that something will stop it). The inventors of the theory of evolution by natural selection, Darwin and Wallace, understood that the impossibility of indefinite exponential growth imples that for many creatures, most offspring cannot survive; this realization helped motivate the evolutionary theories.
Thus, not only does exponential growth illustrate the basic principles of mathematical modeling, but also it is a concept of great theoretical and historical significance.
As an aside, exponential growth might be important in your life. Suppose that you set aside some money and that you receive a fixed percent interest per year at the end of the year (call the interest rate i). If you have P(t) dollars at time t, then you have P(t+1)=(1+i) × P(t) at the end of the year. This is just the same difference equation for exponential growth, with a=1+i. Suppose you start with P(0)=$2000 and receive 5 percent How much do you have after 10, 20, 30, 40, 50 years? This is just another example of exponential growth. Of course, in reality you'd have to take taxes and inflation into account.
Just as the number or biomass of organisms cannot undergo unlimited exponential growth, neither can an epidemic. Mathematical epidemic models take into account the fact that the resource base (the number of susceptibles) is depleted as an epidemic progresses, and we'll discuss this in further topics.
We next look at exponential decay.