One of the biggest challenges for modern medical science will be dealing with diseases as the human population grows. On the one hand, we have lived in a golden age, with vaccines for the most common viral diseases and antibiotics for bacterial infections. However, bacteria have evolved to resist antibiotics, and recent estimates indicate that the most resistant forms of bacteria (tuberculosis and E. Coli variants) respond to only one antibiotic in the medical inventory. Perhaps even worse, as human populations increase humans come into contact with reservoirs of viral agents which we have no natural immunities to (HIV, Ebola, Marburg, Hantavirus...). With modern capitals all less than a day from one another in travel time, catastrophic amplification and spread of disease is one of the most realistic `doomsday' scenarios in the modern world.

The mathematical modelling of diseases began with Kermack and McKendrick studying the advent of plague in Bombay in 1927, and was put on firm theoretical setting by Anderson and May in the 70's. The goals for this exercise are to

- Build a game simulation which captures the basic dynamics of disease propagation.
- Understand the mathematics of logistic models.
- Explore ways to `fit' models to realistic data.

Divide into groups of three or four. Each group will get two transparent hex `playground's for the disease simulation. We will simulate children at school who interact on the play ground and infect one another. The population of children will be 50, of which one is initially infected. One team member will play the disease, one will play the susceptible, one will record data, and one will count new infections. Diseased and susceptible children will be placed separately on the play ground, and then new infections will be assessed according to interaction rules. The transparencies will be superposed and interactions evaluated by the judge. These new infections are added to the infected population for the next day.

Each group will simulate two different cases: a disease with and without a particular infectious window. Decide how infectious you want your disease to be (e.g. if disease requires direct transferral of body fluids, perhaps infected and susceptible persons need to overlap exactly. A `spray' infection might occur if infected and susceptible members are within one `hex' of one another. A really infectious disease might occur if infected members left of `trail' of sites, which infect any susceptible population passing through the site). Simulate also a case where population members are infectious for only a set number of days, at which point they are `removed' from being infectious (and also from being susceptible).

Each group must have two transparent hex sheets, transparency markers in at least two colors, paper towels and water to clean the transparencies, and paper and pencils to record data. Before your group begins a simulation you must decide the following:

- Decide on and record the rules for this simulation, including:
- How many `hexes' around and infected individual are also infectious. Can an infectious individual infect more than one new individual at a time?
- How long an infected individual remains infectious.
- Will infected individuals be `removed' from the population at some rate?
- Will there be a nontrivial environment (playground equipment which becomes infectious if touched, sources of disease like bad water....)?

- Decide who will play the infected population, the susceptible population, record data, and judge infections.
- Decide on how many infected and susceptible individuals to begin
with.

- Infected and susceptible players allocate their populations, and , one per hex on their transparency, in different colors.
- Overlay transparencies and determine which interactions lead to new infected individuals.
- Subtract newly infected individuals from susceptible population, and add in individuals who were infected but have recovered (if necessary). The resulting population is .
- Add newly infectious individuals to , subtracting any recovered individuals. The resulting number is .
- Record and .
- Continue iterating until the infected fraction becomes stable from
step to step.

Below is an outline for a group report on the Disease Game.

- Write down a logistic model, ( ), for the spread of disease. Include a term which accounts for the `infectious window,' and make sure to justify all the terms in your model and what the associated parameters mean.
- The general solution to the equation ( ), generated through separation of variables, partial fractions integration, and inversion.
- Choices of parameters which `best' fit the data generated in your simulations. What were your decision criteria?
- Plots of your simulation data and the solution curve which models it. Be sure to describe the rules you used to generate the data.
- What do you think about these approaches to disease modelling? Was the simulation experiment realistic? How good is the mathematical model?

Tue Feb 18 12:21:04 MST 1997