Partial Differential Equations

• the spread of a contagious illness (the SIR model);
• vibrating strings and drums (the wave equation);
• heat diffusion (the heat and Laplace equations);
• bioassays (reaction-diffusion equation)

The spread of a contagious illness

• HTML version
• DVI version

To construct the model, let $$I(t) be the number of infected, $$S(t) the number of susceptibles, and $$R(t) the number of recovered (or otherwise "removed") in the population at time $$t. Then there are constants $$a and $$b for which

However, this model does not deal with the way the infection spreads though space. For this, partial differential equations are needed. The book by Jones and Sleeman extends the SIR model to account for variation over a 1-dimensional space, and the same ideas can be used to extend the model to a 2-dimensional space:

Here $$R and its differential equation are unchanged, but $$I(t, x, y) and $$S(t, x, y) are now functions of position $($x, y), and $$a is replaced by an expression involving two other constants related to the rate at which the infection diffuses through space.

We can construct approximate solutions to these equations using Euler's method. For example, with one space variable $$x we can write

To approximate the second derivative of $$I we then use

With a similar discrete version of the partial differential equation for $$I we can construct approximate numerical solutions. We have Mathematica notebooks that do this for several different initial conditions. In the first five notebooks, space is 1-dimensional and solutions are plotted in two different forms: as surfaces in $($t, x, z)-space and as curves in the $($x, z)-plane animated over time $$t. In the sixth, space is 2-dimensional and solutions are surfaces in $($x, y, z)-space animated over time $$t.

You can view these Mathematica notebooks from your Web browser if it is set up to launch Mathematica. Each notebook has two versions:

• pre-compiled: the 3D-graphics are drawn and the animations are ready to run;
• uncompiled: quick to load

• sir1.nb (2.7 MB): the susceptible population is uniformly high over a line interval (uncompiled version).
• sir1b.nb (3.1 MB): the susceptible population is uniformly low over a line interval. This time the infection does not take hold.
• sir1c.nb (4.5 MB): inputs here are one-tenth their values in part 1. Again the infection fails to take hold. This means that, even though the input is a scaled-down version of part 1, the output is not.
• sir2.nb (3.7 MB): the susceptible population is high on one third of an interval, low on another third, and is linearly interpolated in the middle third (uncompiled version)
• sir3.nb (5.5 MB): the susceptible population drops linearly from a high to a low value along an interval. This demonstrates how speed of propagation of the illness depends on the size of the susceptible population (uncompiled version)
• sir4.nb (6.2 MB): the susceptible population is uniformly high except for a short interval in the middle where drops discontinuously to a low value. This demonstrates how the illness can "tunnel" through a region where the susceptible population is below the threshhold necessary to sustain an epidemic (uncompiled version)
• sir5.nb (0.6 MB): the susceptible population drops discontinously from a high to a low level (uncompiled version)
• sir6.nb (35 MB): like SIR5 except on a 2-dimensional space (uncompiled version)

Vibrating drumhead

Here $$u(t, x, y) is the displacement from rest at time $$t and position $($x, y) in rectangular coordinates, and $$U is the same in polar coordinates. If we use the polar form for a circular drumhead, we can find basic solutions by separating the variables; then the ordinary differential equation for the radial variable $$r is the Bessel equation.

In the resulting basic solution $$S_n,k, $$b_n,k is the $$k-th positive root of the Bessel function $$J_n. For any $$k, the solution $$S_n,k has $$n nodal diameters--that is, $$n equally-spaced diameters that remain motionless during the oscillation of the drum head.

The following are 3-D animations of a circular drumhead with 0, 1, 2, or 3 nodal diameters. The first-mentioned version in each case is compiled and ready-to-run, but large.

• drum0.nb (1.9 MB), solutions with 0 and 1 nodal diameters. (uncompiled version.)
• drum2.nb (4.9 MB), solutions with 2 and 3 nodal diameters. (uncompiled version.)

Vibrating string

• String1.nb (4.9 MB) demonstrates the superposition of harmonics. It also has animations of a string plucked on-center and off-center.
• String3.nb (4.9 MB) shows how an initial spike at the center of a string propagates as a pair of spikes of half height that travel to the left and right and become inverted when they reflect off the end-points. (String2.nb (4.9 MB) is similar, but approximates the shapes with Fourier sums.

Heat diffusion

The heat equation is a diffusion equation like the SIR model, and we can get approximate numerical solutions the same way, using Euler's method.

Solutions $$u(x, y)to Laplace's equation (harmonic functions) satisfy the average value property: $$u(a, b) equals the average value of $$u on any circle centered at $($a, b). Therefore, in a square grid in the $($x, y)-plane, the value of $$u at a point is approximately equal to its average value at the four immediately neighboring points on the grid.

Therefore, using the given boundary values of $$u and assuming $$u = 0 at every interior point, a new value can be obtained for $$u at each grid point by averaging the current values at the four neighboring points. By repeating these calculations until they stabilize, we obtain an approximate solution to Laplace's equation on the grid points.

The following Mathematica notebooks construct approximate solutions to the 2-dimensional heat and Laplace equations. In each case the domain is a square, while the boundary conditions vary.

• The heat equation
  • heat1.nb One side is held at a high temperature, the other three sides at a low temperature.
  • heat2.nb Opposite sides are held at different temperatures, the other two sides are insulated.
  • heat3.nb One side is held at a high temperature, the other three sides are insulated.
  • heat4.nb One side is held at temperature that peaks in the middle, the other three sides are insulated.
  • heat5.nb Two parts of one side are held at different temperatures (a "window" and a "radiator"), the other three sides are insulated. (The same with a time interval twice as long: heat6.nb)
• The Laplace equation
  • laplace1.nb One side is held at a high temperature, the other three sides at a low temperature.
  • laplace2.nb Opposite sides are held at different temperatures, the other two sides are insulated.
  • laplace3.nb One side is held at a high temperature, the other three sides are insulated.
  • laplace4.nb One side is held at temperature that peaks in the middle, the other three sides are insulated.

Mutagen bioassays

• Awerbuch, T. E. et al., A Quantitative Method of Diffusion Bioassays, J. Theor. Biol. (1979), 79, 333-340.
• Edelstein-Keshet, Leah, Mathematical models in biology, New York, Random House, 1988, pp. 416-419.

Awerbuch, et al. modelled the diffusion of the substance by a partial differential equation:

They go on to derive an approximate solution. In assay1.nb we solve a discrete version of the diffusion equation and compare it to the Awerbuch approximation.