Chapter 7. The AIDS epidemic and models of bank accounts

AIDS is an acronym that stands for acquired immumodeficiency syndrome and is an infectious disease in which the immune system collapses. This disease is caused by the human immunodeficiency virus (HIV). Most HIV infections, both in the U.S. and worldwide, are sexually transmitted; however, another cluster of infections has come from IV-drug users sharing needles. HIV infections are atypical of other viral infections in several ways: HIV typically kills its host around a decade after infection yet is virtually asymptomatic during the first several years; the viral infection is not, by itself, the cause of death; the virus specifically infects key cells of the immune system and thereby ultimately destroys the immune response; the virus can "hide-out" inside cells for years and then emerge long after any treatment has stopped. More government money is spent on AIDS than on all other infectious diseases put together, yet we are no closer to a cure now than we were when AIDS was first discovered nearly 20 years ago. Vaccines are in trials, but no one is holding their breath.

The occurrence of a long-term, untreatable, deadly infection raises all sorts of social and legal issues in our society. Dental and invasive medical treatments now use precautions (gloves, masks) to prevent acquisition and spread of HIV. People knowingly infected with HIV have been incarcerated for having unprotected sex with uninfected individuals. A pending attempted murder case in Louisiana argues that a physician deliberately injected his mistress with blood containing HIV; the woman is alive but is infected with the virus. The U.S. response to the AIDS epidemic has been to allow infected individuals the same rights as uninfected individuals, but laws still exist that allow quarantine of individuals infected with other diseases (for example, tuberculosis).

There seems to be a general rule of human nature that a social stigma is associated with deadly diseases. In Britain at least, deaths from TB were closely-guarded family secrets into the 20th century. AIDS, too, has been a disease to keep secret, perhaps in part because of its initial association with gay sexual practises. Celebrities infected with HIV have made bold forays into public attention by admitting their disease (Arthur Ashe and Magic Johnson were two of the early admissions). This intrinsic human shyness about deadly illness, coupled with HIV being sexually transmitted, has made it difficult to study this disease. People are not very open about their sexual practises, so it is difficult to get good data on who they had sex with, how often, and which orifices and appendages they used.

The epidemic

AIDS was an unknown disease in 1975. By the early 1980s, it had been identified in a cluster of gay individuals in San Francisco and then in a few other major cities. The number of cases grew rapidly in just a few years, and disease experts worried that a new, infectious disease was spreading rapidly. They were right, but what they didn't know at the time was that the people coming down with this disease in the mid 1980s had acquired it 5-10 years earlier. We don't know when HIV entered the human population for good. From blood samples and autopsy reports, the first case dates to the 1950s. But it was not until nearly 1980 that it became common enough to be noticed. Today, 30 million people are infected worldwide. A huge fraction of Africa's population in some countries is infected, the epidemic is spreading exponentially in Asia. (Five years ago, it was estimated that 10% of Thailand's military was infected.) Drugs are available that prolong the life of infected individuals, but the regimen is rigorous (daily doses of several drugs), and the virus has been able to evolve resistance to every drug tried. Furthermore, third world countries cannot afford the drugs for most of the people. Vaccines are being tested, but there is no word on whether they offer protection.

When is the virus transmitted?

The rapid spread of the epidemic suggests that the virus is easily transmitted. In the U.S., HIV infections have disproportionately occurred among gay men, but in Asia and Africa, the bulk of those infected are heterosexual. It is suspected that the greater prevalence of heterosexual transmission in other countries is due to a high prevalence of open sores on genitals from other sexually-transmitted diseases.

HIV is not as easily transmitted per sex act as one might think. A study several years ago looked at how often an infected person infected his HIV-free partner. Only a small number of couples was studied, because it was difficult to find "discordant" couples (one partner was infected and the other not) who were willing to provide details on the frequency of sex. The results showed the surprising result that the probability of infecting the HIV-free partner did NOT depend on how many times they had sex. In fact, the probability of getting infected during sex with an HIV-infected person can be high or almost zero, depending on the stage of the infection.

The profile of an HIV infection

Although it has been difficult to obtain data on the transmission of HIV, much has been learned about the infection. The virus infects different tissues, but if one looks at the amount of virus present in the blood, it follows a common pattern (figure 5.A.1). Within a few weeks of being infected, the virus multiplies to high levels and is easily detectable in blood (the "primary infection"). The immune system -- which is ultimately destroyed by the virus -- responds at this time and brings circulating viral levels down to the point that they are undetectable. This immune response includes antibodies against the virus, so at this point, the infected person will test positive by the standard tests (such a person is said to be HIV+ but does not yet have AIDS). The infection persists however, with low levels of virus in the blood (but high levels in lymph nodes). This "asymptomatic" phase lasts several years. Eventually (typically 8 years into the infection), the person develops AIDS, which is fatal unless successfully treated.

Profile of blood-borne concentration of HIV over the course of infection. The early peak (known as the "primary infection") occurs with a few weeks after acquiring the virus and does not have any obvious symptoms. The long asymptomatic phase is accompanied by viral proliferation in lymph nodes, but blood-borne virus is low, and it is thought that the infected person does not easily transmit the virus sexually. In the third phase, years, after infection.viral levels increase again, and the person develops AIDS.

Some mathematically-minded individuals studying this disease wondered if the probability of infecting another person might vary with the stage of infection. Some fancy modeling suggested that transmission during the asymptomatic phase was virtually zero. This only leaves the possibility of transmission during the primary infection and during the later stages of infection (leading up to and during AIDS). Additional modeling suggested that the rapid spread of this disease was due chiefly to transmission during the primary infection.

Epidemic growth and interest rates

How does this work? The growth of a disease epidemic is much like the growth of an investment. The value of your investment in one year depends on its value now plus a change due to the growth rate of the account. If the growth rate is high, then the value of your investment will double in a short time; if the growth rate is zero, then the value of your investment does not change. Furthermore, when the growth rate is positive, big investments make more money than small investments -- 1% of a million dollars is $10,000, whereas 1% of one hundred dollars is only $1.

Epidemics have the same properties. An epidemic is the spread of a disease through a population. We can talk about the rate of growth of the epidemic in just the same way we talk about the growth of an investment account. The number of people infected may be big or small, and the growth rate doesn't tell us how many are infected. However, even if the number of infected people starts out small, an epidemic with a high growth rate will soon be infecting many people -- the AIDS epidemic started from a handful of people in the 1970s (it wasn't detected until the early '80s), and 30 million people are now infected.

The growth rate of your account depends on two properties: the interest rate and the interval over which it is compounded. An interest rate of 1% per year is not much in today's economic climate, but an interest rate of 1% per day would give you astounding growth after one year. The following table shows the effect of different interest rates compounded over different intervals. The rates all multiply out to be about 50% per year (1% per week x 52 weeks per year, 4.2% x 12 months per year, ...), but because of the different intervals of compounding, the weekly compounding gives a much higher net return.

Which is better -- low or high interest rates?

How does all this bear on our understanding of the AIDS epidemic? If transmission occurs during the primary infection, it acts like interest being compounded every couple of months. If transmission occurs only during late stages of the infection, it acts like interest being compounded only every 5-10 years. Thus, relatively few new infections transmitted during the primary infection are enough to sustain the epidemic, but many, many infections must be transmitted from the AIDS stage if they are to maintain the epidemic. Some crude estimates suggest numbers approximately as follows:

if during the primary infection: 1.1 new new transmissions per infection

if during the AIDS stage: 70 new new transmissions per infection

Thus the worldwide spread of HIV can be explained if transmission occurs during the primary phase to infect slightly more than 1 new individual (on average). The alternative, transmission during the late stages of infection, requires that everyone infected must be infecting 50-100 people. The available data implicate early transmission.

Implications of early transmission

Early transmission has some sobering implications. Most obviously, it means that people are transmitting the disease even before the standard tests show that they are infected. If you and your HIV-free partners agree to be tested every 2 months for infection, by the time one of you shows up positive, the others may be infected. More generally, by the time people in a sexually-promiscuous group develop AIDS, the epidemic will likely be over, with everyone already infected (which is essentially what happened in San Francisco at the inception of AIDS).

A second implication is more subtle and becoming irrelevant as the worldwide epidemic reaches into all corners of civilization. It may be difficult to develop a vaccine that prevents HIV infection. But a vaccine may well help the immune system suppress viral levels early during the infection (lower the level of the virus during the primary infection). Such a vaccine, if widely administered, could reduce the rate of transmission enough to curtail the epidemic. But if the vaccine did not protect a person from getting the disease or prolong their life, would anyone take it? In today's market-driven medical environment, we'll never know, because there developing such a vaccine entails too high of an economic risk.