ASU Professor Carlos Castillo-Chavez is seeking ways of stopping the spread of tuberculosis (TB), alcoholism and extreme ideologies, and he is doing it with mathematics.

When Castillo-Chavez models the spread of fanaticism, or SARS, or the use of ecstasy at raves, he starts with a question. “Then I try to make a model to address the question, and the simpler the model the better,” he said.

Castillo-Chavez asks questions that you wouldn't expect to hear from a professor of mathematics. He asks questions like, “What is the effect of long incubation time on the spread of sexually transmitted diseases,” or “How will smallpox spread through the New York City subway if there is a bioterroism attack?”

According to Fred Roberts, a mathematics professor at Rutgers University and colleague of Castillo-Chavez, mathematical modeling has provided new insights into issues like the development of drug resistance in bacteria, the rate of the spread of infection, and the effects of treatment and vaccination.

“Modeling is especially crucial in epidemiology since in most cases we cannot do experiments,” he wrote in the February issue of Science's Next Wave.

There is a common important feature in all of the questions and the models Castillo-Chavez develops.

He said that in each model he tries to capture the heterogeneity of the specific population he is examining, whether it be New York subway passengers or teenagers at a rave.
Some of the networks that Castillo-Chavez has modeled, like the movement of people between locations in Portland, resemble other networks that we are familiar with like the Internet and friendships.

These networks are called “small worlds,” because of the high degree of clustering between nodes, and the short average distance between them. For a friendship network, a high degree of clustering means that there is a good probability that the friends of one person are also friends of each other.

Capturing the heterogeneity of a particular group of people involves using a sociologist’s eye to pick out important characteristics in a dynamic social landscape. Castillo-Chavez said his research, “lies at the interface of the natural and social sciences.”
Modeling the spread of tuberculosis is a good example of research conducted at this interface.

Tuberculosis reemerged in Arizona in 2003 with more than triple the number of pediatric cases of active TB compared to the year before. The Centers for Disease Control and Prevention (CDC) is currently investigating these cases, according to the Arizona Department of Health Services.

The first models that were developed to describe the way tuberculosis spreads, divided people into three epidemiological classes, according to a recent paper by Castillo-Chavez in the September issue of Mathematical Biosciences and Engineering. People could be infected, not infected, or infected but not contagious (called latent TB).

Initial models assumed that the number of new cases of tuberculosis depended only on the number of people that were infectious. Simply speaking, the more people that were around to spread the disease, the quicker the disease would spread.

In the early development of models for the spread of tuberculosis, parameters were added to take into account the control strategies that were used in different countries. These control strategies involved finding and treating more cases of active TB, and vaccination. The newer models then incorporated vaccinated and a treated populations, which were groups with reduced susceptibility to tuberculosis.

A recent example of this type of modeling, but with a different disease, occurred during the 2003 SARS outbreak in Toronto. Castillo-Chavez modeled the outbreak, and found that the early diagnosis and isolation of infected individuals in Toronto was the greatest factor in keeping the number of new cases so low.

There are also factors specific to each disease that are taken into account that add another layer of complexity to current models. An important feature of tuberculosis that the probability of developing active TB is highest when a person is initially infected with the bacteria, according to Castillo-Chavez. The probability of developing active TB then falls significantly with the age of the infection.

It is estimated that only 5-10 percent of people who are infected with tuberculosis will become sick or infectious at some time in their life, according to the World Health Organization (WHO). Many of the new cases of tuberculosis can be attributed to the reactivation of old dormant infections.

Tuberculosis bacteria can remain dormant in a person's system for years because their thick waxy coat gives them protection against the immune system. So, many people infected with the bacterium may not become sick for years, or maybe not ever, according to the WHO.

Liz Williams, an epidemiologist at the ADHS, said that she was not aware of current tuberculosis models but said that monitoring its spread was a difficult endeavor. The long latency period of the disease makes it difficult to track down and control, according to Williams.

Williams said that there are also several risk factors associated with developing active TB. People who are homeless or use illegal drugs are more likely to develop active TB. People who have AIDS and are infected develop active TB in 100 percent of cases, according to Williams.

ASU Professor James Collins, who studies population dynamics in amphibians, said that mathematical models that incorporate variation among individuals in a population would be useful in his area of research.

“Mathematical models that include such variation are more realistic, and more likely to yield useful predictions,” Collins said. “In human populations especially, it is likely that social factors will alter an individuals susceptibility considerably.”

Refining models for the spread of disease is important because it helps to better evaluate intervention and control strategies, according to Castillo-Chavez. He said that he has noticed interesting trends in other systems that he has modeled, and that there are lessons to be learned that apply across many different fields.

Castillo-Chavez said that models of the spread of HIV have shown that the single most important factor that controls the transmission of the disease is not an epidemiological one. He said the most important factor was the network of social interactions.

Changing human behavior can be a very effective way of battling the spread of disease, according to Castillo-Chavez. In most cases, the best thing that people could do in the case of a disease outbreak in a densely populated city would be to stay home. Castillo-Chavez said that people are smart, and if they are informed properly they will do a great deal to prevent the spread of disease.

Much emphasis has been put on developing new vaccines to prevent the transmission of disease, but in Castillo-Chavez's models the most effective way of preventing new cases is by changing the number of possible transmission lines between people. Castillo-Chavez has used this realization to study the transmission of “diseases” like alcoholism, radical ideologies, or the use of ecstasy.

Studies have shown that most people who use ecstasy get the drug at crowded raves, according to Castillo-Chavez. He is currently examining the possibility that reducing crowd density at raves might be effective at reducing the number of people who are exposed to ecstasy.

Castillo-Chavez is a mathematician of a special ilk in his awareness of social issues. He was nationally recognized for his efforts to increase the number of minorities and women in science and mathematics when he received the Distinguished Scientist Award by the Society for Advancement of Chicanos and Native Americans in Science (SACNAS) in 2001.