Monday, December 27, 2004

Math Problem II

According to information provided by proponents of abstinence-only education, half of the gay male teenagers in the United States have tested positive for the AIDS virus. Proponents of abstinence-only education also teach that condoms fail to prevent HIV transmission during heterosexual intercourse as often as 31% of the time. Teens make up about 10% of the U.S. population at present, and the U.S. population is roughly 295 million.

Assume the following:
1) Presently 50% of gay male teens harbor the HIV virus.
1) HIV-positive gay teens do not self-sort according to HIV status; i.e., intercourse between gay teens is random with respect to HIV infection.
2) Condoms fail during homosexual intercourse as often as during heterosexual intercourse (proponents of abstinence-only sex education probably believe that condoms fail more often during homosexual intercourse, but I haven't been able to find statistics to this effect).
3) Gay teens have intercourse only with other gay teens, and not with any opposite-sex partners, or straight or bisexual male partners, or adults.
4) A gay teen meets a new sexual partner about every four months.
5) A gay teen has sex with a given sexual partner, on average, ten times before moving on to the next.

Question 1: Given these assumptions, what percentage of this group will be HIV-positive in a) four months? b) Eight months? c) A year? SHOW YOUR WORK!

Question 2: Given your results, do these statistics and assumptions seem to be plausible? Why or why not? If the statistics and assumptions are inaccurate, what seems like the most reasonable correction? Defend your answer.

See comments for answers.


Jessi Guilford said...
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Jessi Guilford said...

Question 1: First we must calculate the condom failure rate. With a 31% chance of failure, "safe" sex is actually safe 69% of the time. Repeated over ten different occasions, this means that the odds that all ten occasions were safe is (69/100)^10 = about 2.44%.

There are four different combinations of HIV status for teen A and teen B:

I. A positive, B positive
II. A positive, B negative
III. A negative, B positive
IV. A negative, B negative

The ones which concern us for purposes of HIV transmission are cases II and III. This is because if both teens are of the same HIV status, there is no net transmission of HIV.

In the initial condition, at time zero, 50% of the teens are infected. Thus there is a 50% chance that teen A is infected, and a 50% chance that teen B is infected. The odds of each case (I, II, III, and IV) are thus all equal to 25%, and the total of case II and III is 50%.

The 25% of teens in case I are all infected, and remain infected at the end of four months.

The 25% of teens in case IV are all clean, and remain clean at the end of four months.

Half of the teens in cases II and III were infected at time zero, and by time = four months, they will remain infected. Of the other half, the half which was HIV- at t=0, only 2.44% of these will still be HIV- at t=4 mo.

Suppose we begin with 100 gay teens. At the end of the fourth month, 25 are in case I, and are infected.

25 are in case II or III and were HIV-positive to begin with. This brings us up to the original 50.

25 are in case II or III and were HIV-negative to begin with, and have a 97.56% chance of being infected, which means that there are 24 new HIV cases in this group. (.9756*25=24.39)

a) Thus the rate of infection at the end of four months is 74.39%.

At the end of eight months, the calculation is similar:

case I (+/+) 55.34% of sample (0.7439)^2 = .5534
case II (+/-) 19.05% of sample (0.7439)(0.2561) = 0.1905
case III (-/+) 19.05% of sample (0.2561)(0.7439) = 0.1905
case IV (-/-) 6.56% of sample (0.2561)^2 = 0.0656

b) 55.34 + (19.05*2)(.9756) = 92.51% at the end of eight months.

And by one year:

case I (+/+) (0.9251)^2 = 85.58% of sample
case II (+/-) (0.9251)(0.0749) = 6.93% of sample
case III (-/+) (0.0749)(0.9251) = 6.93% of sample
case IV (-/-) (0.0749)^2 = 0.56% of sample

c) 85.58 + (6.93*2)(.9756) = 99.10% at the end of a year.

Question 2: No. The assumptions are not plausible. Even if homosexual teens only make up 1-2 percent of the teen population (the number frequently put forward by anti-gay activists; a more realistic number is 7%), teens themselves are about 10% of the U.S. population (approximately 295 million) at the moment. Thus, if 50% of gay male teens are HIV-positive right now, this would be (295,000,000 U.S. citizens)(10 teens / 100 U.S. citizens)(1 male / 2 teens)(1.5 gay males / 100 teens)(1 infected gay teen male / 2 gay teen males) = 110,625 cases of pediatric HIV infection. If the assumptions are correct, and 99.10% would be infected within the next year, this would be the equivalent of adding another 100,000 or so cases of HIV infection in a single year, which does not seem to be happening.

Students' answers may vary, for the part of the question about correcting the assumptions. However, answers should not be given full credit if they neglect to acknowledge that the 50% initial infection premise is ridiculously high: 110,625 cases of pediatric AIDS would almost certainly be noticed by the media.

In actuality, all of the assumptions are probably flawed to some degree or another: teens who know their HIV status would be expected to self-sort. Condoms are far more effective in preventing HIV transmission than the abstinence-only data would suggest (upwards of 99% if used correctly). Some gay teens will have sex with both same-sex and opposite-sex partners while coming to terms with their sexuality, which would lead to numerous cases of HIV among teenage heterosexual women, inflating the numbers of HIV infection in the teen age group even more. Also the range of sexual behavior among gay teens is much more variable than the model above (one partner every four months, ten occasions of intercourse per partner): some may be abstinent throughout their teen years, while others may have several partners in a given month. The model above was constructed to provide for ease of calculation: in real life, a more nuanced model would be required.