Epidemiology Problems - Page 9

Applications of Systems of Differential Equations

Epidemiology: The Spread of Disease

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Your final diagram should look like this:

Look at the picture carefully! Does it make sense? Remember that from our previous analysis, we know that each trajectory in our picture starts at the right extreme, and heads to the left.

What information can be gleaned from following each of these trajectories? Do they have anything in common? Don't they all reach a maximum value at the same horizontal position, namely at about S = 300?

Notice then that if we look at this diagram through the eyes of an epidemiologist we might take note of the fact that any trajectory formed from an initial point to the right of S = 300 will lead at first to a rise in the number of infected people. Let's say that again, only louder: If you start out with more than 300 susceptible people, then the number of infected people will increase. Epidemiologists say that under these conditions there is an epidemic. (Yes, that's the real, non-media-hyped meaning of the word.) A disease is epidemic if the number of infected people is rising over time.

On the other hand, what if we start out with an initial point that lies to the left of S = 300? Then the trajectory heads downwards towards the horizontal axis, i.e. the number of infected people decreases along with the number of susceptible people. This means that people are recovering faster from the illness than they are being recruited from the susceptible population. This is a non-epidemic situation. It looks like all of our trajectories make it "over the hump" and eventually enter this mode.

Final Remarks

Health officials consider models like the one we have just studied when making decisions about health policies such as vaccination programs, etc. By the way, don't go away from this lab thinking that there's anything special about the number 300 in the spread of disease. This number will vary from model to model. If you'd really like to see why the number 300 came about in our model, we can go and discuss it if you like. Or you can give it a skip if you lack the thirst for knowledge you should have as a college student.

We have just done one example case of a very simple model. Researchers in this field consider far more complicated models that may even take into account some statistical theory. Their powerful modeling techniques in general lead to far more accurate predictions than our little model is capable of.

Don't make fun of our model too much, though. In several studies of the spread of disease in closed social groups, such as boarding schools, it has been known to do extremely well in making predictions about how the contagion might evolve.

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