The words "virtual reality" have captured the public imagination. NASA, the Virtual Environment Technology Lab at the University of Houston, and others are exploiting this technology and developing impressive virtual learning environments in which students can walk in space or explore electric fields at the subatomic level. Perhaps more importantly, anyone can buy a Nintendo 64 and Super Mario for a bit more than $200.00 and explore -- running, swimming, and even flying -- rich and wonderful but entirely fanciful worlds.

In the not-so-distant future students will be able to immerse themselves in virtual worlds and live realities limited only by the power of their computers and the imagination of the people writing software. Because these worlds are so realistic they will be powerful tools for learning BUT they are not REAL -- they are produced by humans, just like paintings and sculpture. In fact, virtual reality is not really new. We have been creating virtual realities using movies, television, stage performances, and books for many years. What is, perhaps, new is the extent to which a person becomes immersed in the newest versions of virtual reality, so that they feel surprizingly close to real. Virtual reality - based laboratories can be wonderfully effective and powerful learning tools but there is a danger, as they become more and more realistic we can be lead into the belief that they are real -- they aren't!!

This module is about three kinds of laboratories that we use to discover more about our world.

• "R" BL -- "Reality" Based-Laboratories -- based on real and direct experience.

• VRBL Virtual Reality-Based Laboratories -- based on computer and calculator simulations.

• GRBL -- Gedanken Reality-Based Laboratories -- based on our mental models of reality.

"Reality" - Based Laboratories

The most important thing about "reality" - based laboratories is the quotation marks around the word reality. Although these laboratories use real bowling balls, people, and chemicals, they are limited by practicality. For example, we may study a four foot model of an airplane wing in a wind tunnel rather than the real thing at 30,000 feet because we have more control over the environment in a wind tunnel and we can experiment more cheaply and more safely. Similarly, we can work with data collected by the United State Census Bureau or data that we collect ourselves from our classmates, neighbors or over the Internet but people don't always answer questions truthfully and it is difficult for the Census Bureau to contact everyone.

In this module we look at the way in which an illness, like the common cold, spreads. We are interested in how rapidly it spreads, how many people are affected and strategies that we might use to limit the spread. In particular, we are interested in whether your school should have some policies about cold-related absences -- should your school encourage students to stay home to limit the spread of the cold? -- or should your school encourage students to attend even with colds because students who are susceptible will catch it in any case?

The most direct way to gather information about these kinds of questions is by looking at real colds in your own school.

Suppose that you have been asked by your school board to collect some data that they can use to help make decisions about cold-related absences. Consider the following questions.

• How would you collect data about how many students have colds each day?

• You will need a definition of "have a cold." What definition would you use?

• What factors that are beyond your control might influence the spread of a cold and might affect your data?

• Do you think it is practical to actually try out alternative cold-fighting strategies like requiring all students with colds to stay home while they are contagious? Why or why not?

• How much time would be required to collect meaningful and useful data?

• Do some library research. Can you find "reality" - based research done by other people that might be useful to the school board? Discuss several studies or articles that you found and explain why you think each one would be useful or not useful.

Virtual Reality - Based Laboratories

We want to emphasize two very important points about virtual reality-based laboratories.

• There is really nothing new about this kind of modeling -- we have been doing it for years under the name simulation. The only difference is that as the tools we use for virtual reality get better the simulations will appear more real.

• Virtual reality and simulations are really just a way of presenting gedanken - reality -- our mental model or picture of what is going on. In fact, the three realities are very closely interrelated. Gedanken - reality is the structure behind virtual reality and "real" reality is an important component of virtual reality because our gedanken models are based on what we know about the real world and often involve quantities that must be determined on the basis of real world data.

The left side of this applet shows 100 students. They are colored a pale salmon color to indicate they are all susceptible to the cold that is about to arrive at their school. You can start the cold by clicking anyplace in the applet. When you do so, five students will be infected by the cold and will turn a sickly pale green. They don't have any symptoms when they are first infected but they are contagious. The students will come in contact with each other randomly and each time an infected person comes in contact with a susceptible one there is some possibility that the susceptible one will catch the cold and turn a sickly pale green. After one day each infected student develops symptoms and turns a dark green. After a few days each infected student recovers and becomes immune to a repeat infection. As the students on the left become infected, spread the cold, develop symptoms, and recover, the three graphs at the right keep track of how many are susceptible, how many are infected, and how many have recovered and are now immune. The simulation in this applet lasts 30 days.

If you haven't already done so, click on the applet to see this simulation. Click on it several times to see several different simulations. They will be somewhat different because there is a large element of chance involved in which students come in contact with each other and whether contacts result in the transmission of the cold.

Write down your observations about the course of this infection at the school.

Gedanken Reality - Based Laboratories

To get the most out of virtual reality based - laboratories like the one above you must understand the gedanken reality behind them and the "real" reality on which both are based. Ideally you should be able to build these virtual realities yourself or be able to modify one built by someone else. In this section we look at the gedanken realities behind virtual realities like the one above and we discuss how these gedanken models are based on real data and how we can work with them to explore, for example, the consequences of alternative ways of fighting the spread of a cold. You will work with simulations based on these gedanken models in your CAS window and explore different ways of trying to stop the spread of a cold.

We begin by thinking about the course of a cold. After a person catches a cold they are typically symptomless but contagious for some period. To simplify our model we will assume that this situation lasts one day. In practice the length of this period is extremely important and must be based on real data. AIDS has a particularly long period when it appears symptomless but is contagious. This is one reason it has been so difficult to control the spread of AIDS. After one day, our infected gedanken people develop symptoms and for a while are still contagious. The length of this contagious period is also important and must be based on real data. We will assume, for simplicity, that this number is the same for everyone. In fact, it may not be. Eventually, the gedanken antibodies in the infected gedanken people overwhelm the gedanken colds and the infected people recover. We will assume that everyone recovers after the same number of days. Once someone has recovered he or she is immune to this particular cold. Our model will follow the course of this cold for n days. You can change this number if necessary. Our cold epidemic will begin when five people become infected from some outside source.

As we fill in the details of this model we will try to make the model flexible for two reasons.

• By making the model flexible we can adapt it to many different situations and many different kinds of colds or other illnesses. In fact, as we learn more about the particular cold we are studying (one thing that makes colds difficult to study is that there are many different varieties of colds) we can make our model more realistic.

• We can use the flexibility of the model to experiment with different cold fighting strategies.

We begin with a sequence of numbers -- t₁, t₂, ... t_n. The number t_i is the probability that a person who is in the i-th day of a cold will transmit it to a susceptible person when they come in contact with each other. The table below shows one possible set of these transmission probabilities.

This particular model represents a cold that is equally contagious on days 1, 2, 3, and 4. On each of these days on the average one-fourth of the contacts with a susceptible person will result in transmission. After day 4 the cold is no longer contagious. The TI-92 screen below shows how n and this set of transmission probabilities can be entered on the TI-92. Your CAS window will show you how to do this with your CAS system.

Our model has three additional parameters.

• p -- the number of people in the school.

• k -- the number of contacts each person has with other people during the course of an average day that are close enough to transmit the cold.

• d -- the number of days before an infected person recovers.

The TI-92 screen below shows the result of one simulation using this model. Different simulations will have different results because there is a large element of chance involved. Because the TI-92 screen does not display color, we cannot mark the different curves using different colors as we did in the applet above. Thus, we've added some text that labels the curves. This text will not show on your TI-92 screen but you can tell which curve is which from their general shapes. The susceptible curve starts out high and steadily decreases. The recovered curve starts out at zero and after a pause steadily increases. The infected curve usually starts out small, increases, and then decreases eventually to zero.

Your CAS window has the programs you need to reproduce the simulation above and to explore various cold-fighting strategies using this basic model.

• Run several simulations like the one above. Notice that there is some variation because of the element of chance. In the rest of this set of questions run several simulations and observe the variation.

• One possible strategy the school board might adopt is persuading students with colds to stay home. This strategy cannot be perfect since some students may show few symptoms and others may come to school with colds because they don't want to miss some special event. Suppose that the school could reduce the transmission probabilities to the probabilities shown in the table below. Notice we are using the same transmission probability for day one as we did earlier because our cold is asymptomatic on day 1. Notice also that the transmission probability for day 4 is higher than for days 2 and 3 because students are likely to return to school after missing two days.

  t1	0.25
  t2	0.15
  t3	0.15
  t4	0.20
  t5	0.00
  t6	0.00
  t7	0.00

  Run several simulations using these transmission probabilities and describe the effects of this cold-fighting strategy.

• Try some other cold-fighting strategies of your own design. Describe the results.

• Some colds last longer and are contagious for a longer period than others. Try some experiments with colds that are contagious for longer. How effective is the cold-fighting strategy above for these colds?

• Suppose that people have more transmission capable contacts than in the cold we looked at -- that is, suppose k = 4. Try some experiments with this cold. How effective is the cold-fighting strategy above for this cold?