Mathematics & Science Learning Center
Computer Laboratory

Applications of Systems of Differential Equations

Epidemiology: The Spread of Disease

(continued from last page...)

Where Did the 300 Come From?

Let's recall our model's system of differential equations for a moment:

Now, we're trying to investigate the reason why all of the solution trajectories of this system's phase plane diagram have a maximum number of infecteds at S=300. The key phrase here is maximum number of infecteds. Think back to your first calculus course. How do you find the maximum value of a function?

I hope you answered "Analyze the points where it's derivative is zero (or undefined)," or words to that effect.

So to find the maximum number of infecteds, we want to work with the derivative of the number of infecteds. That would be I′, right? But we have a formula for I′, don't we? Yes, look at the system, it's:

I′ = 0.001 S I - 0.3 I

And we want to find out where this would be zero. For starters, we should probably factor the right hand side. After all, there's an I in both terms. Doing this we get:

I′ = I(0.001 S - 0.3)

If we want to know where I′ = 0, then we should set:

I(0.001 S - 0.3) = 0

Either one of the factors being zero could make this true, so:

I = 0

or:

0.001 S - 0.3 = 0

The first of these doesn't make much of a maximum. In fact, since it's impossible to have a negative number of infected people, I'd say that this is pretty much a minimum. Our maximum must come from the other equation, (if we have one at all.) Solving the last equation for S we get:

S = 0.3/0.001

or

S = 300

Well, how about that! Looks familiar, right? Anyway, back to the main discussion.

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