Part 3: Euler's Method for Systems
In Part 2, we displayed solutions of an SIR model without any hint of solution formulas. This suggests the use of a numerical solution method, such as Euler's Method, which was discussed in Part 4 of An Introduction to Differential Equations. For a system of equations, the method is discussed in Systems of Differential Equations . We review the basic concepts here.
Recall the idea of Euler's Method: If we have a "slope formula," i.e., a way to calculate dy/dt at any point (t,y), then we can generate a sequence of y-values,
y0, y1, y2, y3, ...
by starting from a given y0, and computing each rise as slope x run. That is,
yn = yn-1 + slopen-1t
where t is a suitably small step size in the time domain.
It really doesn't matter in this calculation if the slope formula happens to depend not just on t and y but on other variables, say x and z -- as long as we know how x and z are related to t and y. If x and z happen to be other dependent variables in a system of differential equations, we can generate values of x and z in the same way.
Of course, for the SIR model, we want the dependent variable names to be s, i, and r. Thus we have three Euler formulas of the form
sn = sn-1 + s-slopen-1 t,
in = in-1 + i-slopen-1 t,
rn = rn-1 + r-slopen-1 t,
More specifically, given the SIR equations,
the Euler formulas become
Of course, to calculate something from these formulas, we must have explicit values for b, k, s(0), i(0), r(0), and t. In this part we explore the adequacy of these formulas for generating solutions of the SIR model. If your helper application has Euler's Method as an option, we will use that rather than construct the formulas from scratch.
As we noted inSystems of Differential Equations , Euler's Method is simple, but inefficient. Most of the more sophisticated methods (such as the one probably used by your computer algebra system) are similar in design. However, they use much more complicated formulas for the slopes at each step.
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