# Mathematics and HIV

In anticipation of the 19th International AIDS Conference — being held in Washington, D.C., July 22-27 — Work in Progress is turning its attention to the upcoming collection of articles on HIV Treatment as Prevention published this week. I’ll have several posts here about the collection, looking at some of the concepts behind the work being presented, and then delving into specific articles in the collection. (See a Los Angeles Times article on the collection here.)

Woven throughout the collection is the use of mathematical modeling as a way to evaluate the impact of potential interventions that could diminish the spread of HIV/AIDS. Last year, the landmark HPTN-052 study showed that early antiretroviral therapy (ART) for people infected with HIV could reduce the risk of the sexual transmission of the virus to an uninfected partner by 96 percent. (The complete report of this breakthrough study, published in the New England Journal of Medicine and named “Breakthrough of the Year” by Science, is available online here.)

In the wake of that finding — 96 percent! — the World Health Organization and other agencies are trying to figure out the best way to implement ART as prevention (the concept of treatment as prevention will be the focus of the next blog post on this collection). And the best way to figure out that best way is through mathematical modeling.

Thirty-three million people were living with HIV in 2009, according to UNAIDS. Considering the scope of the current HIV epidemic, the mathematical modeling of ART treatment as prevention could spur the policy-making powers that be into actions that could make a dent.

So, onto the important questions: What is a mathematical model? How are they being used to study HIV treatment as prevention? What are the potential problems with these models in this arena?

What is a mathematical model?

Like any equation, a model is like a system represented by numbers and mathematical functions. By applying numbers to all the variables within a system — such as, the number of people infected or uninfected, the rate at which new infections may occur in the presence versus absence of treatment — it is possible to generate a numerical value that reflects the real-world value of whatever is being studied (in this case, the role of ART in preventing new HIV infections).

The usefulness of models depends on how well defined any particular factor is. Sexually transmitted diseases are particularly well-suited for modeling because the route of contact is clearly defined: it’s sex. In addition, there are so many questions at play when it comes to addressing the spread of HIV that modeling is really the only practical approach. The broad range of the epidemic also means that modeling is much more feasible than studying interventions in people over several years. And because the outcome of an HIV study can mean life or death for many, many people, you want to get those studies done as quickly as possible. Although the models being applied to studying ART treatment as prevention do have holes, the insights provided are much better than waiting five years for more study results. Plus, modeling is a lot less costly.

How do mathematical models work?

Mathematical models come in several varieties. They can be compartmental or distributional, where the former groups together all people with an infection and the latter describes a gradation of symptoms. They can be discrete or continuous, depending on whether you want to examine theoretical changes in a population as a smooth, continuous process or in chunks of discrete steps. Deterministic models are not subject to chance, whereas stochastic models incorporate chance into the equation.

Then there are more factors to consider. Do you want to study a population average or do you want to simulate how life is for an individual? Do you want a linear model or a nonlinear model? A linear model often has quite predictable results because the variables involved are linked tightly together: effectively treating one individual reduces the number of cases of a disease by exactly one (as would be the case with, say, breast cancer). HIV and other infectious diseases are modeled as nonlinear. Do you want an analytical or a numerical solution? Analytical solutions can show exactly how a given intervention will impact a community. But with complex problems like the spread of HIV, where sex, age, and sexual activity must all be factored into the equation, the solution must be numerical.

These are a few of the considerations that go into deciding on what kind of mathematical model to use. For more about these points, this article is a good starting point, as is this one.

For more about the application of mathematical models in addressing HIV, the HIV Modeling Consortium is a good resource. (Much of the work published in the PLoS Collection was done in collaboration with this consortium.)

What have mathematical models shown about ART treatment as prevention?

The PLoS collection on HIV Treatment as Prevention includes a comparison of 12 different models evaluating the impact of ART treatment on preventing further infections. That comparison found that the incidence rate would be reduced by 35% to 54% if ART were given to 80% of individuals with HIV treated after their CD4 cell count reached 350 cells/microliter. The graph below, from the introduction to the PLoS collection, shows the cumulative distribution of new infections generated by a single HIV-infected individual over the course of their life since being infected, in the absence of treatment (the red line denotes the start of treatment).

CD4 cells are white blood cells that fight infection. Their quantity decreases as HIV progresses. As Hallett et al write, current ART treatment tends to be initiated when CD4 cell counts are well below 200 cells per microliter, diminishing the potential for ART treatment to prevent new infections. Most transmissions would have occurred before that cell count is reached. According to mathematical models, the number of new transmissions from an infected person could drop dramatically if treatment is started closer when counts are closer to 350 microliters.

This graph, from the evaluation of 12 models by Eaton and colleagues that is part of the PLoS collection, shows the impact of treatment when ART is initiated at 350 CD4 cells per microliter, with 80% access and 85% adherence to treatment:

And this, from the same article, shows the proportion of HIV reduction by the year 2020, according to each of the 12 models included in the analysis:

As Timothy Hallett and others write in the introduction to the collection, “If the average number of new infections arising from an infected person in a susceptible population exceeds one before treatment could be feasibly initiated, then treatment could not eliminate the HIV epidemic.” The models examine the initiation of ART therapy at different CD4 cell count starting points, and then factor in other preventive measures, such as circumcision, or a decrease in the number of new sexual partners. Through those calculations, policymakers can obtain a view on what policies make the most sense for preventing the continued spread of HIV. The current World Health Organization guidance stipulates that HIV-infected people begin ART when CD4 cell counts reach less than 350 cells per microliter (though patients with advanced disease or with advanced tuberculosis should receive ART upon diagnosis, regardless of cell counts).

Problems with HIV treatment as prevention modeling

As Hallet et al write, the HIV testing rate used in the models evaluated in the PLoS collection is much higher than the 52% reported in the South African National HIV Prevalence, Incidence, Behavior, and Communication Survey (available as a PDF). And, the models assume the link between testing and ART uptake to be 100%, whereas that figure is probably closer to 33%. The dropout rate from treatment programs is assumed to be about 1.7% in “the most optimistic model simulations,” according to the article by Hallett and colleagues. The real-world dropout rate is probably closer to 10%.

Another huge hurdle is cost. Hallett and colleagues write, “After years of rapid growth, funding commitments and disbursements have stabilized or been reduced, and only a few countries in sub-Saharan Africa are currently able to achieve the high levels of treatment coverage for those eligible recommended by current international guidelines.” Programs with high costs in the short-term might be impossible to implement, regardless of how convincing a case is made by a mathematical model.

To that end, the PLoS collection includes an article by Meyer-Rath and Over outlining what economic considerations should guide discussions about programs that may be spurred by what mathematical models reveal. The cold reality is that “to a decision-maker, savings that are accrued in the future may be worth less than those made today.” In other words, it’s hard for many agencies to implement expensive programs because they will save money in the long-run. In that case, “…potential future payoffs may be less attractive, and investment in programmes for other, more immediate causes of mortality would be a rational, if not necessarily an inspiring or ethical, response.”

What next?

Hallett and colleagues note that a gradual expansion of access to treatment may be the best middle-ground solution. Then, policymakers can also look at which groups to focus on in order of priority. Pregnant women who are already under care may be easier to reach for ART initiation than other groups of people infected with HIV. Stable couples might adhere better to treatment than other groups. Knowing that part of the goal here is to prevent future infections, should “serodiscordant” couples (that is, one infected and the other not) be placed as the top priority?

In concluding their introduction, Hallett and colleagues note that mathematical modeling alone is not enough to guide decisions on HIV treatment as prevention. Epidemiology, economics, demography, statistics, and biology must also weigh heavily into all discussions. But in light of the HPTN 052 findings, mathematical models are providing crucial insights about the potential impact that earlier treatment could have on slowing the spread of HIV — or perhaps stopping it altogether.

Here is an interview with Timothy Hallett on the PLoS Blog, Speaking of Medicine, providing further insights about the newly published collection.

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All images courtesy of PLoS Medicine