The coronavirus is spreading at an exponential rate in many countries. In the US, President Donald Trump has warned of a “painful two weeks” that will increase the death toll. Experience with epidemics indicates that the exponential spreading phase is an unfortunate part of infectious disease transmission. The Spanish flu of 1918-20 was the last major pandemic on a global scale, and infected about 500 million and killed 17-50 million people. This flu too had its exponential spreading phase. Compared to a century ago, we are in a much better position today to predict how the pandemic will evolve in the exponential spreading phase and beyond. This improved skill in predicting pandemics comes from the mathematical models of disease spreading. What are these magical mathematical models, and how do they work?
Mathematical models are abstractions of reality. They are mathematical equations that encapsulate our understanding of the science behind the process. By solving them, it is possible to predict future outcomes. In 1687, Isaac Newton published his work on gravitation, a model for how two material objects interact with one another. Newton’s model could predict the future position of planets and stars to the trajectory of missiles and cricket balls. As one of the most successful models in the history of science, it is elevated to the status of fundamental physical law. Not all models become law and achieve eternal stardom. For instance, the sheer complexity of geophysical processes makes it exceedingly difficult to construct accurate models for the prediction of earthquakes and weather. Yet, over the decades, notwithstanding many imponderables, advances in scientific understanding and better data collection have led to improved weather prediction.
Unlike the physical laws, the models for disease spreading have to contend with the vagaries of human behavioural patterns. After all, disease spreading depends on how people interact with each other. The first of such models was initiated in 1927 by the English scientists Anderson McKendrick and William Kermack. A simple version of their model classifies the entire population into three compartments, namely, those who are susceptible to contracting a disease, those who are infected and those who have recovered from the infection. Initially, a small number of people become infected. As the infected people interact with the susceptible people, infection begins to spread. Ultimately, after some reasonable time, infected people recover from the disease. One possible outcome under this simple set of rules is an exponential growth of the infected population during the initial phase of infection. In the battle against coronavirus, many countries are currently in this phase or staring at the prospect of entering it. On the other hand, China appears to have reached declining infection rates.
In the dreaded exponential growth phase, the number of infected patients will increase rapidly, and this can overwhelm the capacity of hospitals to treat them. Curiously, an old Indian folk tale can be a model for what it means to be exponentially increasing. The king of a small kingdom decides to honour the sage who had invented the game of chess. The wealthy king tells the inventor to ask for a gift. The humble sage requested some grains of rice. The amount was to be decided as follows. Two rice grains on the first square of the chessboard, four on the second square, eight on the third square and so on until all the 64 squares are filled with rice.
In this fable, the rice grains increase exponentially because each square has twice that in the previous square. The grains on the 64th square alone would exceed the net rice production in the world. The king gladly promised the gift and only later did he realise the exponentially large quantity. The wise sage had cooked up a model and foreseen its outcome. The disappointed king could not keep his promise. Foretelling the outcomes is the key. The predictive power of models, as opposed to purely data analysis, arises from their ability to provide a broader picture of the outcomes. For instance, improved models can estimate the effectiveness of lockdowns and social distancing in suppressing the infection spread.
It is often less appreciated that model building is the core of science. Scientific advances must be reflected in improvements to models that capture a better sense of reality. Today, reasonably complex models for various purposes are employed to guide decision-making. Nowhere is this complexity more evident than in areas of social and economic interactions dominated by human sentiments and prejudices. Economics is an area that abounds in models of varying complexity. At the other extreme is the Turing machine, an iconic mathematical model for computation, characterised by its simplicity. The versatile computers and cell phones are real versions of the abstract Turing machine.
In the ultimate analysis, as the statistician George Box said, ‘All models are wrong, but some are useful.” At least some of the infection spreading models are useful for predicting the future course of Covid-19. The good news is that the exponentially proliferating infection will ultimately die out. The bad news is that there is significant uncertainty over when this will happen. The models still need improvement.
M S Santhanam
Physicist and a professor at the Indian Institute of Science
Education and Research, Pune. Views expressed are personal
Email: santh@iiserpune.ac.in