Disease spread in compartmental spaces: Markov and semi-Markov population models

In the study of infectious diseases, scientists have long relied on mathematical models to predict how an illness moves through a population. Traditional approaches often assume that people mix together randomly, like drops of ink swirling in a glass of water, where every individual has an equal chance of meeting every other person. However, the real world is far more structured. People move through specific environments—offices, schools, hospitals, and shopping centers—spending different amounts of time in different rooms and interacting with only a subset of the people around them. This spatial organization, combined with the fact that people enter and leave these spaces constantly, creates a complex web of contact that simple models often miss. Understanding how disease spreads in these enclosed, compartmentalized spaces is crucial for designing effective public health strategies, especially when ventilation and shared air play a significant role in transmission.

A team of researchers from Mexico has developed a new way to model this complexity, moving beyond the assumption of random mixing to create a framework that accounts for the specific layout of a building and the movement of people within it. Their work, published as a research article, introduces two complementary mathematical approaches to track how an infection spreads through a network of connected rooms. The first approach treats the environment as a closed system, where a fixed number of people move between rooms but no one enters or leaves. The second, more flexible approach treats the environment as an open system, allowing people to arrive from the outside and depart at any time, while also accounting for the fact that people do not spend the same amount of time in every room. By combining these movement patterns with a specific mechanism for how the disease is caught, the researchers were able to calculate the expected number of healthy people, carriers, and infected individuals at any given moment.

The researchers divided the population into three distinct groups to better understand the flow of infection. The first group consists of susceptible individuals, who are healthy but can catch the disease. The second group comprises carriers, who already harbor the infectious agent and can pass it on to others. The third group includes those who have become infected during their time in the system. A key feature of this model is that it distinguishes between carriers and newly infected individuals. The researchers assumed that once a person becomes infected within the system, they do not immediately become contagious themselves; instead, they remain in the "infected" category without spreading the disease further during the time they are observed. This distinction allows the model to focus specifically on the moment of transmission from a carrier to a susceptible person, rather than a chain reaction of secondary infections.

To simulate how people move, the researchers used two different mathematical tools. In the first scenario, they modeled a closed building where the total number of people remains constant. People move from room to room based on fixed probabilities, much like a game where a player rolls a die to decide which room to enter next. In this closed system, the researchers found that they could accurately predict the average number of people in each room and how the infection would spread over time. They verified these predictions by running computer simulations with thousands of virtual people. The results showed that as the number of people in the simulation increased, the actual outcomes matched the mathematical predictions almost perfectly, confirming that their formulas for the average behavior of the population were correct.

The second scenario addressed the reality that most buildings are not closed systems. People walk in through doors and leave through exits, and the time they spend in a room varies widely. To capture this, the researchers used a semi-Markov model, which allows for more realistic movement patterns. In this framework, the time a person spends in a room is not fixed to a standard average; instead, it can follow any distribution, reflecting the fact that someone might linger in a waiting room for an hour but rush through a hallway in seconds. This model also includes "absorbing states," which represent the outside world. Once a person reaches an absorbing state, they have left the system and will not return. This setup allowed the researchers to calculate not just how many people are inside, but also how long they stay and how many leave while healthy, carrying the virus, or already infected.

The researchers applied their open model to a specific example involving a facility with two rooms and two entrances and exits. They simulated the arrival of healthy people and carriers using a standard statistical distribution for random arrivals. By running these simulations, they were able to determine the steady-state numbers of people leaving the building in each health category. Their findings revealed a clear relationship between the flow of people and the spread of the disease. As the number of carriers entering the system increased, the number of healthy people leaving the system decreased, indicating that more infections were occurring inside. Conversely, the number of people leaving with the infection increased as the rate of infected entries rose.

A particularly interesting finding emerged when the researchers compared a facility with a single entrance and exit to one with multiple entrances and exits. They found that the configuration of the building significantly influenced the outcome. In the model with a single entry and exit, the number of healthy people leaving the system was lower, and the number of infected people leaving was higher, compared to the model with multiple entry and exit points. This suggests that having more ways to enter and leave a space, which might seem counterintuitive, could actually reduce the overall number of infections by altering the movement patterns and reducing the time people spend in close contact with carriers.

The study concludes that by integrating the physical structure of a space with the movement of people and the mechanics of disease transmission, it is possible to create a much more accurate picture of how an outbreak evolves. The mathematical expressions derived by the researchers provide a way to calculate the expected number of susceptible, carrier, and infected individuals at any point in time. These tools can help public health officials and policymakers understand which parts of a building or which movement patterns pose the greatest risk. While the current work focuses on the theoretical framework and simulations, the authors suggest that these models could eventually be applied to real-world environments, using actual data on how people move to design better interventions and reduce the spread of infectious diseases in complex, structured spaces.