Random results are different each time the simulation is run. In this simulation, the disease doesn't travel to men until one promiscuous woman is infected. This will only happen fairly rarely, so it may be that many liaisons must occur before the population of infected people grows. You can adjust this by changing the rate of transmission.

Another way to remove the variation in the simulation is to increase the size of the population. More people means that the law of averages smooths out the differences.

Back to Top

Why start with five infected men?
This is somewhat arbitrary because no one is sure about the actual rate of infection in the population. One popular estimate is 1 million people in the United States carry the virus. That's approximately 0.5 percent.

But the rate of infection varies significantly according to age. Younger people are more likely to carry the disease as well as to be promiscuous. Choosing five initial carriers is arbitrary, but it does very roughly approximate the rate in the population at risk.

Another reason to start with five men is that starting with less can make the simulation run much longer. The slowest part of the epidemic is at the beginning when only a few people are infected. Starting with five eliminates the need to wait for the first few infections to take place.

Back to Top

How do you set transmission risks?
Estimating the odds that the virus will be transmitted is complicated. Researchers look at the spouses or partners of people who carry HIV. Ideally, they concentrate on spouses who are monogamous with their infected partner and who do not use drugs, work in a medical arena or expose themselves to the virus in any other way. Then the researchers count the number of these people who are infected by their partners.

The amount of infection varies significantly. One study conducted by Nancy Padian, Steve Shiboski and Nicholas Jewell at the University of California at San Fransisco in 1991 examined 72 couples in which the female partner was infected. Only one man in the group tested positive, presumably having been infected by his partner. On the other hand, 61 women in a test group of 307 couples were infected by HIV-positive men.

But there are often great variations in the test populations. Another study, based in Europe, by Isabelle de Vincenzi in 1992 found a much greater rate of transmission from female-to-male. In a group of 159 couples with HIV-positive women, 19 men ended up with the virus; in a group of 404 couples with HIV-positive men, 82 women became infected.

A recent paper by Timothy Mastro and Isabelle de Vincenzi analyzed 10 recent studies from the United States, Europe and Thailand and found that 449 women in 1,919 couples (23.4 percent) had been infected, presumably by the male partner.

The paper also analyzed eight studies that focused on cases where the female partner had been infected first. In these samples, which were less numerous, 63 men out of 522 couples (12.1 percent) became infected, presumably by the female partner.

The researchers convert these values into basic odds by asking the couples to estimate the number of contacts they've had with each other. This process is also fraught with error. Memories fail, and people often have personal reasons for distorting the number of times they have had sex. Also, there is often no easy way to determine exactly when one partner was first infected.

Stephen Shiboski, an assistant adjunct professor of biostatistics at the University of California at San Fransisco says that the basic odds estimates from the different studies vary significantly. The odds that an infected man will transmit the virus to an uninfected woman can be as low as 5 out of 10,000 times or as high as 1 out of 10 times. The odds that an infected woman will transmit HIV to her uninfected male partner range from 3 out of 10,000 at the low end to a high of 3 out of 100.

Generally, Shiboski says, most studies produce estimates of transmission in about 1 out of 1,000 sexual encounters between infected men and uninfected women. The rates of transmission between infected women and uninfected men is about half of that, or 1 in 2,000.

But these estimates may rely on a simple model that doesn't accurately reflect how the virus actually spreads. Simple models assume that each liaison is equally likely to result in the virus moving from one partner to another. But in practice a wide variety of factors like the presence of genital warts or even unknown immunities seem to affect the transmission.

In addition, people seem more likely to catch the virus at the beginning of a relationship. In one study, eight uninfected women in a test group of 50 couples became infected by their partner after reporting fewer than 50 contacts, a rate of about 18 percent. But, the rates for couples that reported having had sexual intercourse more than 800 times were not much higher. In the same study, four uninfected women in a group of 12 couples caught the virus -- a rate of about 33 percent. If the rate of transmission was really equal, then a much smaller proportion of the low-contact group should have caught the virus.

Back to Top

What about condoms?
Condoms may dramatically slow the transmission of the disease, but there are no solid numbers on which to base an estimate.

Norman Hearst and Stephen Hulley of the University of California at San Fransisco offer the estimate that strict condom use cuts the risk of transmission by a factor of 10. In other words, if there is a 1 in 500 chance of the virus traveling between two heterosexual partners, they suggest that using a condom cuts the odds to 1 in 5,000.

Back to Top

How Long is a Time Unit?
This is a difficult question. Ideally, the time unit and the transmission rates should match. That is, a time unit is merely the amount of time that will allow the virus to be spread at the rate specified. So if 80 out of 1,000 liaisons between an infected man and an uninfected woman result in transmission, then a time unit should include enough time for couples to trade the virus that often.

The University of Chicago study reports that 43 percent of married men report having sex "a few times per month" and 36 percent say "2 to 3 times per week." A figure of 100 liaisons per year for a couple seems reasonable for this simulation.

If the odds are about 1 in 1,000, then a rate of 100 out of 1,000 might be appropriate when you think of a time unit as a year, and 5 to 10 out of 1,000 might be better for a month. But all of these are assume the study produced realistic averages.

What's more, the averages can be misleading. The University of Chicago study also reported that men and women who were not cohabitating reported fewer liaisons than married people reported. But these could include some who are simply not dating anyone or who are attempting to date someone. If you wanted to focus on a subgroup of the population who actively troll the world for conquests, then the rates can go up. Of course, the relationship between infection and promiscuity may not be directly linear.

Back to Top

Dealing with death

Setting the lifespan after infection is important to the overall performance of the simulation. Clearly, this model fails to even approximate reality because in the model everyone who is infected dies a fixed amount of time after infection.

In the real world, the time between infection and death is crucial in any study of an epidemic, because it ends the infected person's ability to infect others

The death of people in this simulation is a poor approximation of when they stop contributing to the epidemic. Many studies show that the level of contagion changes as HIV infection progresses to various stages of AIDS, and the flat approximation here is obviously not accurate.

Back to Top

What about births?
This model doesn't include new births. If the mother is not HIV-positive, new children are also uninfected. An epidemic can die out if the people who are infected die without passing on the disease.

Kremer's models include new births, which can reduce the percentage of infected people in the population, but this computer simulation does not. You can watch the epidemic end when the the percentage of those infected stops growing. This often happens when either all of the promiscuous men or all of the promiscuous women die.

Back to Top

What about transmission in marriage?
In each time unit, this model creates one liaison between each couple and also sends the male out looking for additional liaisons. It is entirely possible that a man could have 10 times as many sexual encounters outside of the marriage as he has with his spouse. Each pair may have the same chance of transmitting the disease. Reality, however, is never so exact, and people who meet more often may be more likely to exchange the virus. The model clearly only approximates what is going on and there is room for more attention to detail in the structure of the model.

Back to Top

Estimating promiscuity
Estimating the amount of promiscuity in the United States is also prone to error. Many surveys find that men as a group report more sexual liaisons than women report, despite the fact that these numbers should be exactly equal in a strictly heterosexual population. The latest and most ambitious surveys, like the 1994 study by the University of Chicago, attempt to correct for this by asking extra questions that can be used to corroborate the data, but they have not completely extinguished the effect.

Still, the numbers from the University of Chicago survey suggest that Americans as a whole are not very promiscuous. About 77 percent of the men and 89 percent of the women reported zero or one partner in the previous 12 months.

If you're experimenting with the simulator, you can approach simulating these effects in two ways. The first is simply to select this percentage for the simulator. Many couples will be made up of two nonpromiscuous people (approximately 68 percent) and neither partner will become infected with HIV during the entire simulation.

This may be accurate, but it is also a waste of computer resources. There is no reason to test a nonpromiscuous couple again and again. They will not catch the disease. You can also choose a higher number (like 100 percent of the men), and let the simulation concentrate on the promiscuous subculture.

Back to Top

Liaison Models
Is the "Relentless Male" or "Ten-Timing Male" the correct model? Obviously, this simulation excludes the possibility of female initiation just to simplify the process of testing whether more sex can slow the course of the AIDS virus. Other models may do a better job of illustrating this if they're able to better capture the sexual behavior of everyone.

But the question of how to model sexual behavior is a difficult one. Kremer calls his version of the "Relentless Male" setting, his "bar model." Anyone who wants a partner finds one. In one paper, Kremer devotes some attention to arguing that his model guaranteeing a liaison for everyone is correct, at least for people of high sexual activity. He writes, "This group is not likely to continue dating without sex, but instead to seek other sexual partners."

The opposite of this is his "dating model," where each person must spend time and energy ("search costs") to discover a willing partner. He concedes that this may be the best model for people of low sexual activity and suggests that future research might embrace a model that mixes both strategies.

Of course 10 potential conquests per "time unit" is clearly just an approximation. You are welcome to create your own version of the model. Both Landsburg and Kremer are economists, and they are naturally curious to examine how marketplaces can affect behavior. Landsburg suggests that the marketplace would benefit from objective knowledge about the sexual history of a potential partner. Just as a car odometer measures how many times the car has been around the block, a perfect sexual economy would make this information available. A better model of simulation might offer this information.

Back to Top

What's been left out?
The sexual marketplace is clearly a complicated place. This simulation has focused on one major part, the realm of heterosexual couples, but it has ignored many other important details. Potential partners are chosen at random, and they mate as if they are both promiscuous. Clearly, in the real world, other factors, including age, race and class, affect how people make sexual decisions. The lack of comingling by different groups can lead to a segregation of the disease. It may rage in one small subculture and not affect others who do not interact with that subculture. You're invited to experiment with building a more complicated model.

Back to Top

Differential equations versus computer simulations
Michael Kremer, like many academic economists, uses differential equations to build his model of how HIV spreads through a population. These equations, which are an extension of calculus, provide a way to take a few basic assumptions about the rates of change and convert them into equations that describe the flow of an epidemic. The process is based on paper-and-pencil mathematics developed long before the digital computer was invented.

This applet offers a computer simulation. There are no equations, and there is no reliance on symbolic mathematics. The computer merely creates a number of simulated humans and puts them through simulated sexual encounters. The virus is transmitted by a digital role of the dice.

These two different approaches can offer different types of solutions. For instance, it is easy to use Kremer's model to find the "steady-state prevalance" or the number of people infected when the world finally achieves a balance. You can also calcuate an absolute answer for what is the best strategy for society.

The computer simulation has trouble answering questions like that. You need to run it repeatedly to find optimal solutions because all you can do is put in some initial conditions and watch what comes out.

But the computer simulation has some distinct advantages. Only some differential equations can be solved; most cannot. So scientists and economists often rely upon models that work instead of those that might be the best at approximating reality. For instance, the same differntial equations are used to model the flow of heat through a radiator, the dispersion of poison gas and the value of stock options on Wall Street. All have a simple solution that was identified years ago. So it isn't surprising that many build their models around these easy-to-solve equations.

Computer simulations are not limited in this way. If you want to have one man approach 10 women then the computer simulation can do it. Any simple and direct recipe can be simulated. Of course, concepts like "attractiveness" are ephemeral notions that escape the rule of logic and thus can't be simulated.

Back to Top

Is more sex really safer sex?
The Slate article may suggest this, but the results are far from clear. If you believe that the model of a ten-timing man is accurate, then the answer is no. More promiscuity leads to more HIV.

Even if you completely believe that the model of a relentless male is correct, there may be no practical way of implementing the behavior. Kremer refuses to endorse encouraging people to have more partners. "Whatever the impact on the epidemic as a whole, people who have more partners put themselves at increased risk of HIV infection," he says. "So even though an uninfected person who goes out to the bars may be making life a little bit safer for another uninfected person out there, they are also risking becoming infected."

The details are even more murky. Kremer's models suggest that each person can choose an optimal behavior to do the most to benefit society. He suggests that this amount is a function that is about one half of the mean plus the variance divided by the mean. This implies that in populations with a high degree of variation in promiscuity levels, the sexual slackers have an obligation to be more promiscuous and the high-activity zealots have an obligation to moderate their activity.

But what happens if a population applies this rule repeatedly -- that is, if they calculate the target activity level that will minimize the spread of HIV in a world behaving according to Kremer's model? At the beginning, the low-activity people will add more sexual partners. Once everyone reaches the same target, however, the new target will be lower. Kremer's model suggests that it is one half of the current average when everyone has the same number of partners. After repeating this step several times, the target level drops to practically no promiscuity.

Back to Top

What were the sources for this report and simulator?
Timothy Mastro and Isabelle de Vincenzi, "Probabilities of sexual HIV-1 transmission" in Journal of AIDS, 1996, 10 (suppliment A) pp S:75-82.

Robert T. Michael, John H. Gagnon, Edward O. Laumann, and Gina Kolata, Sex in America, A Definitive Survey published by Little Brown (New York) in 1994. [The University of Chicago Study]

Norman Hearst and Stephen Hulley, "Preventing the Heterosexual Spread of AIDS: Are We Giving Our Patients the Best Advice?" in Journal of the American Medical Association, April 22/29, 1988, volume 259, number 16, pp 2428-2432.

R. Brookmeyer and HM Gail, AIDS Epidemiology: A Quantitative Approach published by Oxford University Press (New York) in 1994.

Michael Kremer's "Can Abstinence Increase Prevalence of AIDS?" in pre-print form. Similar work can be found at his web site.

Back to Top