Since the first news of COVID-19 emerged, mathematical modeling of its spread has been crucial to formulating government and public health authorities’ responses across the world.
The most popular model of disease spread is the SIR model. The SIR model, which stands for susceptible-infected-recovered, uses ordinary differential equations (ODEs) to estimate disease dynamics. However, there are certain drawbacks to the SIR model which new research seeks to address with a new type of epidemiological model.
At India’s National Institute of Technology and Indian Institute of Technology, researchers developed a model of COVID-19 dynamics using cellular automata. This approach allows modelers to study disease dynamics, like infection rates and mortality, similar to SIR models, but to also consider additional factors, like how geography and individual behavior affects these dynamics.
Cellular automata are discrete models that despite being based on very simple rules exhibit extraordinarily complex behavior. Sounds perfect for modeling epidemic spread, right? You may have come across cellular automata before in the form of mathematician John Conway’s Game of Life.
In their simplest form we can think of cellular automata as a grid of black and white cells in conjunction with a set of rules that determine how the color of cells may change. In Conway’s Game of Life, which is just one example of a cellular automata, the ‘player’ determines the initial black cells and the grid evolves on its own based on the rules of the game after that. The Game of Life’s rules can be summarized as follows:
- Any live cell (black) two or three live neighbors survives
- Any dead cell (white) with three live neighbors becomes a live cell
- All other live cells die in the next generation
The behavior of cellular automata is very complex and quantifiably unpredictable (they have been used as random number generators). The ability to model complexity with simple rules is part of what makes cellular automata so attractive to epidemiological modeling.
In this new research, countries are modeled as a grid and each cell in the grid represents a group of people and their current condition. Cells can be in any one of the following states: susceptible, exposed, infected, quarantined or removed (recovered or dead). The model evolves using probabilistic rules based on factors that affect disease transmission, such as proximity to infected individuals. By looking at changes in the conditions of cells we can observe how COVID-19 spreads and other related dynamics.
Saumik Bhattacharya, one of the researchers involved in the project, said “One particular thing we were interested in was individual interactions — predicting how individuals are behaving. ”
Cellular automata are well suited to do that. With the traditional SIR model we have to assume a countries’ population is totally homogenous, which is a clear oversimplification of the real world. However, with cellular automata we don’t have to assume everyone across the country behaves (quarantining when presenting symptoms, for example) in the same way or that the countries’ population is evenly distributed.
COVID-19 has affected the world unevenly in many respects, including geographically. A cellular automata model allows public health authorities to account for the differences in population density when predicting the spread of COVID-19 through, say, a major metropolitan area and a rural county.
Sayantari Ghosh and her co-author Bhattacharya were “already working on modeling viral marketing, which is kind of an epidemic spread” prior to the pandemic. When the pandemic hit they pivoted and used the cellular automata they were building to model viral marketing campaigns to study COVID-19 dynamics.
The team built their model using cutting edge genetic algorithms, and got promising results. Their model was able to closely fit data they collected from a diverse group of forty countries, as well as predict macro-dynamics, such as mortality rates.
Edward Robinson, a Professor of Mathematics at George Washington University, said using cellular automata for epidemiological models “have the advantage that they are really easy to understand,” but that “we don’t know to what extent they have proven themselves yet.”
The authors of the paper agree that more work has to be done on the topic and they are actively encouraging other researchers to expand on their model. The stakes for accurate modeling are very high, as we have seen over the past few months.
Despite their potential, cellular automata models are unlikely to replace ODE models. Bhattacharya says “If you want quick estimation of the nature of the spread, ODE models are necessary. We require both ODE and cellular automata models.” This suggests a likely scenario where cellular automata models are used in conjunction with ODE models. With cellular automata being used to answer questions better suited for their heterogenous approach, like what locations within a country are most likely to be run out of hospital beds.
