Math has been used to study disease for over 250 years – but for obvious reasons, in the past year or so, things have really got interesting. Although it can be hard to see a bright side to the ongoing coronavirus pandemic, the flourishing of math and science in an effort to tackle a global crisis is certainly one cause for optimism.
Even though it’s been the scientific breakthroughs that have made most of the headlines, the world of math has been quietly making some huge contributions throughout the pandemic. Let’s take a look at just a few of them.
The Math of a Pandemic
Epidemiology is, loosely put, the study of the spread of disease – and it’s a big deal in math. With the emergence of the COVID-19 pandemic, existing mathematical models of disease were put to the test like never before – and some were found wanting.
“The susceptible-exposed-infected-recovered model, SEIR, is a standard mathematical approach for forecasting the spread of an epidemic in a population,” explained Dr Rabih Ghostine of the Kuwait College of Science and Technology. “This model is based on several assumptions, such as homogeneous mixing of the population and the omission of migration, births or deaths from causes other than the epidemic.”
But such assumptions don’t work in the real world, so Ghostine and his team designed a new method, inspired by ocean modeling, to study the spread of COVID-19.
"We wanted to develop a robust mathematical model that takes into account such uncertainties and incorporates epidemic data in order to enhance forecasting accuracy," explained Ghostine.
In fact, the math of COVID-19 didn’t seem to work at all. Epidemics usually behave in a predictable way, with infection rates following a sort of S-shaped curve. The infection rates of COVID-19, in contrast, seemed to increase as expected initially, but then continue to grow linearly – much faster than “normal” epidemics.
“To explain this with standard epidemiological models would basically be impossible,” commented Peter Klimek, co-author of a 2020 paper explaining the phenomenon. And the reason turned out to be something surprising: not the super-spreader events that made the news so often over the past year, but normal socializing.
“Most people went to work, got infected and spread it to two or three people at home, and then those people went to work or school again,” explained co-author Stefan Thurner. “The infection was basically spreading from cluster to cluster.”
Over the past eighteen months, we’ve all learned certain habits. We wear masks, wash our hands while singing happy birthday, and we keep our distance from others. But are those really effective strategies for COVID-19 prevention? Or is it all just a placebo?
Well, according to the math, it works so long as we actually do it: a statistical analysis published in the journal Chaos found that it takes 60 percent of the population to stay responsible to prevent an outbreak.
"Neither social distancing nor mask wearing alone are likely sufficient to halt the spread of COVID-19, unless almost the entire population adheres to the single measure," study author Maurizio Porfiri said at the time. "But if a significant fraction of the population adheres to both measures, viral spreading can be prevented without mass vaccination."
In fact, math is seriously pro-mask. Another analysis, published in the Proceedings of the Royal Society A, used mathematical modeling to figure out the impact of various health measures. In every scenario the research team modeled, the R number was brought below 1.0 if at least 50 percent of the population stuck to wearing masks.
Social distancing and self-isolation proved important too. A study by the specially-formed University of Texas at Austin COVID-19 Modeling Consortium found that for every single day social distancing policies were delayed in a city, around 2.4 days were added to the length of the overall outbreak there. That same institution also helped overhaul care for homeless individuals in Austin, Texas, using mathematical models to determine how to shelter and isolate those infected.
"Our calculations indicated how many hotel rooms would be needed to isolate individuals when they developed symptoms or were exposed to the virus,” explained Consortium director Lauren Ancel Meyers. “[Lead author Tanvi Ingle] helped the City of Austin to protect some of its most vulnerable populations.”
Herd Immunity: a Viable Strategy?
Back at the beginning of the pandemic, a lot of us were hearing talk about herd immunity. The idea was that if enough people got exposed to COVID-19 and developed immunity as a result, the virus would stop being able to spread.
We now know that while herd immunity works fantastically well with vaccines, relying on it to take care of COVID-19 didn’t work. But could it have?
The short answer: no.
The long answer: according to mathematical models designed by scientists at the US Department of Energy's Brookhaven National Laboratory and the University of Illinois Urbana-Champaign, lasting herd immunity was unobtainable without vaccines. Instead, their models predicted many waves of what they termed “transient collective immunity”, followed by a wave of infections – and that’s exactly what the world is seeing.
Staying at Home
The early days of the pandemic saw some brave – or foolish – souls jetting off around the globe as vacation prices plummeted. But the fun couldn’t last, and travel restrictions soon shut down air traffic across the globe in an effort to slow the spread of the ever-mutating virus.
These measures weren’t uniformly supported, however, with even publications like the Lancet coming out against them. So what does the math say?
Just like public opinion, the results are mixed. Mathematical models using data from cities across China suggested that travel restrictions could be useful – but only if they are implemented quickly.
“Travel and mobility restrictions are the most useful right at the start, when local transmission has not yet become a factor,” explained study co-author Professor Samuel V. Scarpino. “After transmission is established, physical distancing and the quarantine of sick individuals will work, but it takes time.”
A major barrier to the effectiveness of international travel restrictions turned out to be the lack of cooperation between countries. Although researchers from Rensselaer Polytechnic Institute determined that travel restrictions in China may have saved six million lives globally, they also found that the vast majority of travel restrictions were basically pointless.
“According to the data we collected, about 63.2 percent of travel restrictions were ineffective," study co-author Lu Zhong said. “Because the travel restrictions were done in an uncoordinated way, they failed to contribute to the global good.”
As the pandemic rages on, the world has seen unprecedented pressure on hospitals and healthcare workers. This is another area where math has come to the rescue, with scientists creating sophisticated algorithms to optimize resource allocation and reduce demand.
In Austin, Texas, researchers used stochastic optimization methods to design a novel alert system that would monitor hospital admissions and trigger changes in local public health guidance. In a paper published in Nature Communications, the team demonstrated that this new system was markedly better than other methods at stopping hospitals from getting overwhelmed and preventing the need for lockdowns.
Meanwhile in the UK, math was helping the NHS keep their ICUs open. Researchers used a load-balancing algorithm to optimize patient allocation, providing intensive care beds for up to 1,000 extra patients and preventing hospitals in COVID-19 hotspots from getting overwhelmed.
There’s no doubt that the COVID-19 pandemic has been devastating, but what about the future? It turns out that math can help us there too: it can show the best way to open up businesses again, and help optimize vaccination strategies.
Perhaps most intriguing of all, the future of mathematical epidemiology might be in game theory. Parrondo’s paradox is a phenomenon where, by combining two losing strategies, a person can create a winning strategy. And according to one study, this paradox might be the key to minimizing the cost of future pandemics.
“Such novel strategies can be implemented to curb the spread of COVID-19 or future epidemics,” observed study lead Kang Hao Cheong. “[They] have the potential to alleviate suffering, preserve and promote health and well-being among the population."