H1N1 and the Logistic Equation
By Murray Bourne, 03 Oct 2009
The Northern Hemisphere is bracing for an outbreak of H1N1 flu this winter.
H1N1 virus [Image source]
According to the World Health Organization (WHO):
As of 27 September 2009, worldwide there have been more than 340,000 laboratory confirmed cases of pandemic influenza H1N1 in 2009 and over 4100 deaths reported to WHO.
The H1N1 virus is still in an exponential growth phase (it grows faster as time goes on). But populations don’t continue to increase exponentially forever. For example, we all know that rabbits breed very quickly, but if the supply of grass runs out, this will put a limit on their population growth.
Likewise with diseases, there is a limit to their growth. If the disease kills too many people, it will have nowhere to go and its growth will taper off over time and the number of new people affected will eventually drop.
Such a situation can be described by the Logistic Equation. This equation describes the case where a population initially grows in an exponential fashion, then tapers off to some constant value.
A simple form of the logistic equation is as follows (where P(t) is the population at time t and "e" is the constant e = 2.718 281 828...):
The graph of this logistic equation has an elongated “S” shape, as follows:
Notice that the early portion of the graph (up to t = 0) represents exponential growth, while from there, the growth tapers off and eventually the value of P becomes a constant value of 1.
Logistic equations result from solving certain Differential Equations (a topic in calculus).
The above model is too simple for discussing H1N1 (for starters, we can't have fractional populations). A more useful form of the logistic equation is:
The variables in the above equation are as follows:
P0 = population at time t = 0
K = final population after some (long) time, also called the "carrying capacity", which limits growth
r = initial growth rate
An example for the H1N1 outbreak might be as follows. Say there is a town with 1000 people and one day, 20 people wake up with H1N1. The virus eventually affects everyone in the town.
So P0 = 20 people affected, K = 1000 and we have (say) r = 0.2. Substituting these into the logistics equation and simplifying gives the following expression for the number of people affected by H1N1 at time t (in days):
Here is the graph of the situation:
We see that eventually everyone in the town is affected after about 50 days.
Mexico first noticed an outbreak of H1N1 in Mar 2009. The number of cases grew rapidly to 26 Apr, then rapidly declined.
If we take cumulative totals for the new cases (add each new number of cases to the total so far) and graph the result over the 53 worst days of the outbreak, we obtain the following.
At first, the growth is somewhat linear (up to day 30), then it resembles the logistic equation curve from then on.
You can see more on this topic here: Predicting the spread of AIDS using differential equations.
Logistic equations are also used when analyzing problems in neural networks, statistics, medicine, chemistry and physics.
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