The Logistic Equation

You may be familiar with exponential growth, where a population grows proportionately with the population's current size. This is modelled with the differential equation

y' = ky
This equation works adequately when the population in question can grow unchecked, but what happens when, such as in an epidemic, the total number of infections starts to approach the entire population? Clearly, the disease cannot spread exponentially, and so the growth must slow down.

In the exponential growth model, we assumed the population grows in proportion to its current size. In the model we are using for epidemics, called the "logistic growth model", we make other assumptions. Suppose the total population size is M and the infected population size is y(t). We assume that the total population behaves in such a way that everyone has an equal chance of encountering anyone else in a given time span. We assume that if an infected person meets an uninfected person, then the uninfected person becomes infected. Thus, the growth rate of the infected population, y', is proportional to the number of infected/non-infected contacts expected in a given time span. But this number is proportional to the product of the infected population and the non-infected population sizes. Therefore,

y' = k y (M-y)
This makes qualitative sense since if the number of infectees is small, the disease spreads slowly, while if the number of non-infectees is small, the disease must also spread slowly. We expect the disease to grow fastest when the number of infectees and the number of non-infectees are both large, facilitating contact between the two populations.

Graphs of functions satisfying

y' = k y (M-y)
start out being very flat, then they steepen, and then they end up flattening out when nearly the entire population is infected, or when y is near M.

The solution to the logistic equation is

y(t) = M/(1 + c exp(-kMt))
c = (M/y(0)) - 1

In the rumor spread simulation, we have M = 50, and y(0) = 1, so c = 49. What's left to be determined is the value of k, which depends on the radius of the balls and their starting velocity. Given a run of the simulation, how can you determine k? Since this simulation is somewhat random, the value of k can vary, even keeping the radius and starting velocity constant. Therefore, you should run the simulation several times to obtain an average value for k.

Questions to consider: If the radius is increased, do you suppose k increases or decreases? Why? If the starting velocity is increased, what happens to k?

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