COVID-19 Dispersion Modeling of Data with Minimax Measurement

When a disease spreads across a landscape, it does not move like a single wave crashing on a shore. Instead, it appears as a scattered collection of outbreaks, some small and hidden, others large and visible, all shifting over time. To understand where a virus is going, scientists often try to draw a shape around these scattered points, looking for a center and a boundary that captures the whole picture. This is a classic challenge in mathematics: finding a circle that minimizes the weighted maximum distance to a group of points, a problem that dates back centuries to the work of mathematicians like Pierre de Fermat. In the context of a pandemic, solving this puzzle helps researchers see not just where cases are, but how the infection is expanding, contracting, or stabilizing. By treating the spread of illness as a geometric problem, scientists can track the invisible forces pushing the virus from one town to another, turning raw numbers into a clear map of movement.

In a study published in August 2026, researchers Dongyung Kim and Grace Kim applied this geometric thinking to the early days of the COVID-19 pandemic. They focused on a specific week in April 2020, a critical period when the virus was accelerating through the southwestern United States. Using data from Arizona, Colorado, and New Mexico, the team built a model that treated every confirmed case cluster as a point on a map. Their goal was to find a single, moving center point that best represented the spread of the disease, constrained by two fixed reference points that acted as anchors for the calculation. This approach allowed them to draw a "minimax circle"—a boundary optimized to minimize the weighted maximum distance to all known cases, adjusted for the weight or importance of each location. By watching how the size of this circle and its center changed from day to day, they could measure the strength and direction of the virus's dispersion.

The researchers analyzed daily data from April 1 to April 7, 2020, mapping the locations of confirmed cases in each state. They converted the real-world latitude and longitude of these clusters into a flat, two-dimensional grid to make the geometry work. For each day, they calculated the optimal center of the circle and its radius, which represents the reach of the outbreak. They also measured a "dispersion strength," a value that tells us how tightly packed the cases are within that circle. If the circle grows larger while the number of cases stays the same, the strength drops, meaning the virus is spreading out over a wider area. If the circle shrinks or stays the same size while cases increase, the strength rises, indicating a tight, concentrated outbreak.

The results revealed distinct patterns for each state. In Arizona, the model showed a clear shift over the week. During the first four days, the radius of the circle grew steadily, meaning the virus was jumping to new, distant locations. At the same time, the dispersion strength fell sharply, confirming that the outbreak was becoming more scattered. However, by the fifth day, the growth stopped. The circle's size stabilized, and the strength settled into a steady pattern. This suggested that the virus had reached a limit in how far it could spread within that specific week, hitting a saturation point where new cases were appearing within the existing boundary rather than pushing it outward.

Colorado told a different story. Throughout the entire week, the state maintained the highest dispersion strength of the three, meaning its cases remained tightly clustered. The model showed that the virus was concentrated heavily around the Denver metropolitan area, with the circle staying relatively small. This indicated that the outbreak was intense but geographically compact, with new infections appearing close to the established centers rather than spreading far into the countryside.

New Mexico presented the opposite extreme. Here, the dispersion strength continuously declined without ever stabilizing. The circle kept getting larger every day, reflecting the state's vast territory and scattered population. The virus was spreading slowly but steadily across a wide landscape, with new clusters appearing far from the original epicenters. Unlike Arizona, which found a temporary balance, New Mexico's outbreak was in a constant state of expansion, with no sign of reaching a saturation limit during that week.

The study confirms that this geometric method is a reliable tool for tracking disease movement, provided there is enough data. Sensitivity analysis indicates that with small datasets (N = 20), the results are unstable and sensitive to minor changes. However, as empirical data scales up to N = 200 and N = 800, the model demonstrates robust convergence behavior, validating its computational stability for larger-scale applications. The team found that the mathematical approach could successfully predict the trend of the spread, distinguishing between a virus that is exploding outward and one that is settling into a pattern.

By focusing on the shape and movement of the outbreak rather than just the total number of cases, this research offers a new way to visualize the pandemic. It shows that the spread of a virus is not just a matter of counting sick people, but of understanding the geometry of their distribution. The model suggests that different regions can react to the same virus in fundamentally different ways, driven by their geography and population density. For public health officials, this means that a single strategy might not work everywhere; what looks like a contained outbreak in one state might be a rapidly expanding frontier in another. The work does not claim to solve the pandemic or predict the future with absolute certainty, but it provides a clear, mathematical lens through which to watch the virus move, turning a chaotic spread of data into a readable story of expansion and containment.