Epidemiology Problems - Page 5

Applications of Systems of Differential Equations

Epidemiology: The Spread of Disease

(continued from last page...)

Your infecteds vs. susceptibles graph should look like this:

Phase Plot

Remembering that the horizontal axis represents susceptibles and the vertical axis represents infecteds, which direction would you expect the movement to be around this "orbit"? (Actually, since the parametric path does not form a closed loop in the picture it would be better if we referred to it as a trajectory.)

There are two ways of looking at the picture in order to answer this question:

  • The realistic approach prompts us to reason that the number of susceptibles must be decreasing over time, as they become infected and move out of the susceptible population. This would mean that we should travel the trajectory from right to left.

  • The initial condition condition was S(0) = 600, I(0) = 50, or more briefly (600, 50). Clearly this point lies at the extreme right end of the trajectory, so the only direction we could possibly travel along it is from right to left.

Obviously both lines of reasoning lead us to the same conclusion, and we might annotate the above graph as follows, where the blue arrows indicate the direction of increasing time:

Annotated Phase Plot

The fact that the trajectory eventually ends up hitting the horizontal axis isn't surprising, is it?

Also, note the horizontal value (the number of susceptibles) at which the number of infecteds reaches its peak. About S = 300, right? Keep this number in the back of your mind for future comparison.

 

Now let's move on and give you some work to do...

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