A Tuberculosis Model That Works in Both Continuous and Discrete Time

Tuberculosis remains one of the deadliest infectious diseases in the world. It is also hard to model, because infected people can carry the bacteria for years before they become sick. A new paper in Mathematics and Computers in Simulation offers a model that handles this long latency and can run on either continuous time or fixed reporting intervals.

The McCoy College of Business author is Dincer Konur, Associate Professor in the Department of Information Systems and Analytics. His co-authors are Elvan Akın of Missouri University of Science and Technology, Gülşah Yeni of Missouri University of Science and Technology and Pennsylvania State University, Mahmut R. Işık of the Dicle University Medical School in Turkey, and S. R. Işık, whose affiliation is not listed in the indexing records.

The study

The model uses the SEIR framework. That means the population is split into four groups: susceptible, exposed (infected but not yet infectious), infected, and recovered. People flow from one group to the next over time. Most SEIR models are written in one of two ways. Continuous models treat time as a smooth line. Discrete models treat time as a series of steps, such as weekly or monthly case counts.

The authors use a mathematical approach called time scales. It lets them write the model once, in a unified form, and then choose the time domain later. On the real number line the result is a standard continuous model. On a grid of numbers spaced by a step size h, the result is a discrete model. The two are not the same model with different clocks. The discrete version has a saturated incidence rate, which means the rate of new infections levels off as cases rise, while the continuous version does not.

What the researchers found

For both versions, the authors prove when the disease-free state and the endemic state are stable. The deciding quantity is the basic reproduction number, the average number of people one infected person infects. Above a threshold the outbreak takes hold. Below it the disease dies out. This holds in the continuous model and in the discrete one.

The paper then runs numerical experiments using parameters estimated in earlier studies of tuberculosis in the Philippines and South Korea. These experiments show that the step size h is not a technical detail. Changing it changes the predicted dynamics in ways that have biological meaning. In other words, the interval at which data are collected is part of the model, not just a setting.

The broader point is that time scales can generate new continuous and discrete models that are not simple approximations of each other. The new parameters that appear, such as h, can stand for real features of how disease data are gathered.

What it means for health system managers and analysts

Managers who rely on forecasts of disease spread should ask what time step their model assumes. A model calibrated to daily data may not behave like the same model run on quarterly reports. The choice should reflect how case data actually arrive.

The basic reproduction number remains the number to watch. This study confirms that it governs whether an outbreak grows or fades in both model versions, so it is a reliable signal for planning capacity and interventions.

Finally, the paper shows the value of building models that can switch between time domains. Health systems that collect data at different intervals in different regions could use one model rather than several.

This summary is based on the paper’s abstract. The full article reports the data, methods, and detailed results.

What it means for managers

• Models built on continuous time and models built on discrete reporting periods are not interchangeable. The time step itself changes the predicted course of the disease.
• The basic reproduction number still decides whether an outbreak grows or fades, in both versions of the model. It remains the key number to track.
• When choosing or commissioning a forecasting model, match its time step to how your data are actually collected, whether daily, weekly, or quarterly.

Akın, E., Yeni, G., Konur, D., Işık, S. R., & Işık, M. R. (2026). An SEIR model on time scales with discrete applications to tuberculosis. Mathematics and Computers in Simulation, 241(A), 72–103. 10.1016/j.matcom.2025.08.004