Mathematical Model for the Epidemiology of Tuberculosis, with Estimates of the Reproductive Number and Infection-Delay Function

The authors used epidemiologic data on tuberculosis to construct a model for the time delay from initial latent infection to active disease, when infection transmission occurs. They used case rate tables in the United States to calculate the fractional rate of change per annum (A) in the incidence of active tuberculosis. They then derived estimates for the effective reproductive number (R) and the cumulative transmission, defined as the number of people whom one infected person will infect in his or her lifetime and over many multiple successive transmissions, respectively. For A of -4 percent per year, the average US condition from 1930 to 1995, they estimate the reproductive number to be about 0.55 and the cumulative transmission to be about 1.2. The estimated rate of the new latent infections in the United States is 80,000 per year, the estimated prevalence of latent infections is 5 percent, and the number of transmissions of infection per active case is 3.5. From the model, the authors predicted active case rates in various age groups and compared them with published tables. The comparison suggests that the risk of activation decreases rapidly, then gradually, for the first 10 years after initial infection; the risk is relatively constant from 10 to 40 years and may decrease again after 40 years. The authors also discuss how this model can be used to help make decisions about tuberculosis control measures in the population.