Scientific Computing

This module illustrates the numerical solution of the
Kermack-McKendrick model for the course of an epidemic in a
population. Ordinary differential equations (ODEs) have applications
in a wide variety of contexts, including biology and ecology. One such
application is the modeling of the spread of an epidemic in a
population. The Kermack-McKendrick model of this phenomenon tracks the
population of three different groups: *y*_{1},*y*_{2},*y*_{3}*c**d*,

*y*_{1}′ = − *c* *y*_{1}
*y*_{2}

*y*_{2}′ = *c*
*y*_{1} *y*_{2} −
*d**y*_{2}

*y*_{3}′ =
*d* *y*_{2}

The user first selects the rates of infection and removal, the initial
populations of each population group, and how far forward in time to
compute the solution. The user also selects a numerical method and
step size to be used in computing the approximate solution. Clicking
*Solve* then calculates the approximate solution using the chosen
numerical method and step size, and displays the results graphically.
Each component of the solution is graphed as a function of time. The
degradation in the quality of the solution using a lower-order method
or coarser step size is clearly visible, particularly for higher rates
of infection. Indeed, the approximate solution may be so poor that it
does not fit within the displayed portion of the graph, in which case
using a higher-order method or smaller step size will yield a more
accurate solution.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Computer Problem 9.2 on pages 418-419.

**Developers:** Evan VanderZee and Michael Heath