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,

by starting from a given  y₀,  and computing each rise as slope x run. That is,

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

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.

1. Use your helper application's differential equations solver, with the sample values of  b = 1/2  and  k = 1/3 in your worksheet, to generate graphical solutions of the SIR equations, starting from  s(0) = 1,  i(0) = 1.27 x 10^-6,  and r(0) = 0.
2. Now generate Euler's Method solutions for the three sectors of the population. Start with a relatively coarse step size of t = 10  days, and let  t  range up to 150 days. Superimpose these solutions on the "exact" solutions from Step 1. Do you think the Euler solutions closely track true solutions of the system? Why or why not? What characteristic of Euler's Method causes the approximate solutions to behave the way they do?
3. Change the step size to 1 day and replot the Euler solutions. Now do they closely track true solutions of the system? Why or why not?
4. Find a step size for which the Euler solutions appear to closely track true solutions of the system.

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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Copyright CCP and the author(s), 2000