"The symbolic integration and differential equation solving capabilities of Mathematica have saved me a lot of time."

• Numeric—integration
• Symbolic—differentiation and integration, simplification of large algebraic expressions, solution of differential equations

Supporting the furious quest for a cure for AIDS is a massive effort to understand as much as we can about this puzzling disease. Among the many unknowns lies the question: how much time typically passes between the time a person is infected and the time the person is diagnosed with the disease? In their search for answers, researchers at the U.S. Department of Health and Human Services' Centers for Disease Control and Prevention use Mathematica for vital computations.

Mathematical statistician Bob Byers discovered that the Weibull distribution--the probability distribution most widely used to estimate the incubation period--does not exhibit some important characteristics of data gathered in AIDS research so far. "While data shows that the probability of being diagnosed with AIDS reaches a plateau at around seven years, the Weibull's 'hazard function' does not," Byers explains.

According to Byers, being able to more accurately estimate this incubation period will benefit both patients and their physicians by helping them determine when to begin more aggressive treatment, and sometimes helping them to reconstruct the incidence of infection. Knowing the time to AIDS will also aid health-care analysts and economists predict the effect of AIDS cases on the health-delivery system.

"I used Mathematica to solve a differential equation which allows the 'hazard function' to follow a logistic distribution," explains Byers. "This new distribution fits the observed data significantly better than the Weibull." Without Mathematica, Byers says he would have been faced with the tedious and time-consuming task of solving the equation by hand.

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