HIV/AIDS and mathematics

Summary: Epidemiological models track and estimate human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS) while immunological studies generate probable values for use in immune dynamics models to evaluate possible treatments.

The human immunodeficiency virus (HIV) is a type of retrovirus that targets the immune system. Retroviruses replicate by encouraging host cells to make copies of their own ribonucleic acid (RNA) after invading them. The immune system is designed to fight viruses and infections, but HIV targets the immune system and progressively destroys the body’s ability to fight infections and certain kinds of cancer. People with HIV may get life-threatening diseases called “opportunistic infections,” and they can later develop what is known as acquired immunodeficiency syndrome (AIDS). Mathematical and statistical techniques are used to track the spread of disease, estimate the number of cases, define various parameters for describing incidence and prevalence, and evaluate the clinical tests that are used to identify HIV and AIDS.

Historically, AIDS has been defined as a syndrome of several different illnesses that occur when the immune system fails. AIDS was first clinically identified in the 1980s and called “gay-related immune deficiency” (GRID), because the illness initially appeared in men who had sex with men. The cause was not known, but there were many theories, including cytomegalovirus and certain drugs. Later, AIDS cases emerged in both males and females who had received blood transfusions, suggesting an infectious agent with bodily fluids as a transmission vector. As early as 1983, scientists isolated a virus later named HIV, which was ultimately correlated with AIDS. As with most diseases, cause is inferred though animal testing and by making comparisons between people with and without the proposed causal agent. In 1993, a more precise definition was adopted by the Centers for Disease Control and Prevention (CDC), which added the criterion of a person having less than 200 CD4+ T-lymphocytes/μL or less than 14% CD4+ T-lymphocytes. In 2009, evidence suggested the human form of HIV developed sometime between 1884 and 1924, far earlier than originally believed.

Prevalence

HIV prevalence refers to the overall percentage of a population that has HIV, while HIV incidence refers to the rate of new infections that occur in a given year. Historically, tracking HIV has been difficult because AIDS surveillance registries relied upon AIDS cases that were reported, and data were then extrapolated with statistical models to estimate the prevalence in the wider population. Other data collection alternatives are available. Some methods involve surveys of different groups of people, including high-risk groups. Others, like prospective cohort studies, track samples of people over time. Globally, at the start of the twenty-first century, heterosexual women were most at risk of acquiring HIV. In Western countries, men who have sex with men were considered a high-risk group. Prisoners are also at risk because inmates may engage is high-risk behaviors. HIV is considered a pandemic, and a typology was developed to classify geographic regions according to the type of epidemic. Generalized type means that HIV prevalence is greater than 1 percent in pregnant women. Concentrated type means the prevalence is greater than 5 percent in some subpopulation but less than 1 percent in pregnant women. The low-level type has a prevalence of less than 5 percent in any subpopulation and less than 1 percent in pregnant women.

Epidemiology

There are many types of epidemiological models that are used in HIV/AIDS tracking and estimation. They take into account different variables, including the size of the overall population; the proportion of people who are already infected; the size of sexually active or other at-risk subgroups; the number of people who leave the population for various reasons, including deaths from AIDS; and the number of new sexual partnerships that people may form. They may calculate quantities such as risk of transmission between individual members of the population or overall population rate of transmission. A statistic called the “basic reproductive number” is often used to quantify transmissibility. A value less than 1 for this measure implies that a disease will eventually die out, assuming that the values entered into the model do not change over time. A value greater than one implies that the infection will spread. Very large values imply an epidemic, which may be difficult to control. Treatments and other interventions can reduce the infectiousness of HIV, which affects values like the basic reproductive number.

Many models are simplified, assuming that all individuals in a hypothetical population have the same patterns of sexual behavior. Factoring in individual differences in sexual behavior may increase realism in statistical models of HIV risk and infection. Many parameters could change, depending on factors like age, particularly sexual behavior and infectiousness of the virus. Some other variables that could be considered are types of sexual activity, the number and type of sexual partners, condom use, and HIV testing. The role of treatment in mitigating transmission could also be considered. Researchers from the Amsterdam Cohort Study created a more complex mathematical model that described the spread of HIV in one high-risk group. The researchers took into account many individual behavioral variables and were able to show that the majority of new HIV infections were because of main partners, not casual partners as was previously assumed. This had important implications for targeting risk-reduction messages to men in long-term relationships.

Testing

In the twenty-first century, politicians and reporters have questioned whether mandatory HIV testing should be required. In 2003, the CDC initiated the Advancing HIV Prevention: New Strategies for a Changing Epidemic program and tested the possibility of rapid HIV tests in some emergency rooms. HIV tests have a high sensitivity rating, with some tests listed as 99.7 accurate, meaning only 0.3 percent of the people with HIV will falsely test negative. The remaining people with the disease will correctly test positive. However, even with this degree of accuracy, one concern about universal testing is the possible number of false positives (people who test positive but do not actually have the disease). The percentage of false positives with these tests is also small. Yet, in the United States, because most of the population is HIV negative, the small percentage would be multiplied by the very large number of people in the HIV-free population and would result in many false positives, perhaps more than true positives. Repeated testing or the development of more accurate tests can mitigate the impacts of false positive results.

In the United States, prevention is having some effect. In 2015 there were 39,513 new cases of HIV; the number of new cases decreased 9 percent between 2010 and 2014. In 2013 there was an estimated 1.2 million people in the United States living with HIV; of these, 13 percent did not know that they had the disease. There were 2.1 million new cases were diagnosed worldwide in 2015.

Immunology

Immunology is the study of the immune system’s response to a pathogen, and mathematical models of HIV immune dynamics can be constructed. Data from experiments can be used to find plausible values for the mathematical model, such as the expected life of CD4+ T-cells. These models are particularly useful when evaluating or predicting the success of treatments that interfere with the replication of retroviruses. Mathematical models of immune dynamics are very complex and require revisions as knowledge about HIV changes. Collaboration between mathematicians and clinicians is important so that models can be maximally effective in preventing HIV spread and improving health outcomes for people infected with the disease.

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