Monday, 27 June, 2022
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These mathematical models are running our Covid world

From decisions on hospitals to restricting the movements of millions of people, predictions from these models are a matter of life and death amid the crisis.

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The coronavirus pandemic has thrust mathematical models into a leading role in human affairs. Their predictions are literally a matter of life and death, driving decisions on equipping hospitals, shutting down businesses and restricting the movements of millions of people.

Yet few understand the models’ inner workings. So here’s my data scientist’s take on them, with the aim of offering a sense of which ones merit trust.

Let’s start by defining what it means for a predictive model to work. Sometimes it’s solely a matter of accuracy: A good astronomy model, for example, will predict the return of Haley’s Comet with great precision. Other times it’s more complicated: The epidemiological models that feature in the Covid-19 crisis have a dual goal of prediction and persuasion. A good model should give people a realistic sense of what will happen if they don’t change their behavior, and also a persuasive picture of the outcomes they can achieve if they do.

There are, broadly, four types of models that have played different parts in the crisis so far. First comes the SIR, versions of which have been used in Washington and Pennsylvania. It projects the arc of a pandemic by focusing on three populations: the susceptible (which provides the opportunity to spread), the infectious (which informs the rate of spread) and the recovered (which, if immune, slows the spread).

The crucial parameter here is R(t), which is the average number of people each contagious person will infect (and has nothing to do with the R in SIR). If a virus is highly contagious and faces no obstacles, R(t) will be much greater than one and the infected population will grow exponentially — leading, in the case of the coronavirus, to an overwhelmed hospital system. If people manage to push it below one by staying at home, wearing masks and keeping their distance, cases will decline (as has happened in New York City). Also, R(t) inevitably changes with time: As more people are infected or immune, it gradually peters out.

The concept of R(t) makes SIR models very persuasive. It can be estimated from the data, and people can see how efforts to push it down will also flatten the curves of cases, hospitalizations and deaths. It’s something humans can control and track.

As regards predictions, though, SIR models aren’t so great. That’s because tiny changes in R(t) generate huge changes in the trajectory of a disease. Such is the nature of exponential growth. This helps explain, for example, why New York State thought it would need tens of thousands of ventilators but ended up with a surplus.

Next, there’s the agent-based or individual spread model. It can be populated with millions of virtual people, each assigned separate R(t)s for infecting family, colleagues, and people they meet in public, as well as other attributes such as propensity to move around. It’s great for modeling how infections spread over large areas quickly if people travel far and wide. It’s also highly persuasive, because it demonstrates how individuals can slow the spread by changing their behavior. One such model — the Imperial College London March 15th model, which originally predicted 2.2 million U.S. deaths — played a large role in convincing both U.S. and U.K. authorities to take more aggressive measures to contain the pandemic.

In terms of accuracy, it suffers from the same issues as the SIR models. Clearly, the U.S. is no longer likely to have anywhere near 2.2 million deaths, at least in this first round. But who knows? Such predictions might have been accurate had nothing changed, and may well happen if people refuse to stay at home in the future.

Then there’s the network model, which treat individuals as nodes in a vast lattice — sort of like friends in a social network such as Facebook. The idea is that certain people tend to be super-spreaders — because they interact with a lot of people (or a lot of super vulnerable people), because they do particularly risky things, or all of the above. Think nurses and other frontline workers, or politicians whose job it is to meet with people.

Network models can be valuable for specific purposes — such as deciding whom to vaccinate first, if a vaccine is in short supply. Inoculating the highly networked people will do most to bring down R(t), because they are the portion of the population with the highest propensity to infect others. The GLEAM group at Northeastern University uses network models to understand how things like airplane travel effects spread. Like SIR and agent-based models, the method is highly persuasive but marginally accurate.

Finally, there’s the curve-fitting model. It essentially takes the historical trend and projects it into the future. If, for example, infections have been doubling every five days, the model might forecast that they will keep doing so. This is rarely useful, as a cursory glance at different countries’ pandemic curves demonstrates. They all start with exponential growth, then take their own idiosyncratic paths depending on the containment measures adopted. There’s no pattern other than utterly predictable inaccuracy.

Curve-fitting models have generated some spectacularly stupid projections. Consider, for example, the one that White House advisor Kevin Hassett reportedly used to predict that Covid-19 deaths would decline to zero in 10 days. They’re also utterly unpersuasive, because they lack any parameters with which to project alternative scenarios – that is, they can’t tell you how people can change the future. By construction, they simply assume that the future will be like the past.

You might have noticed that the useful models have a common element: R(t). I’m a fan of the concept, so I think I’ll be writing about it more. – Bloomberg


Also read: A secret algorithm is deciding who will die in America


 

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1 COMMENT

  1. Mark Twain has said in a skeptical mood that there are three kinds of lies-“ Lies, damned lies and statistics “. While all mathematical models are useful and it is not possible for us to grasp reality without the help of numbers, common sense is the best weapon that we can have in such crisis situations. One may ask why countries like Taiwan, Germany and South Korea have been spared the brunt of the coronavirus? It is because they learnt quickly from the Chinese scenario and employed preventive measures at an early stage.
    As the learned author himself has conceded these models fail to predict the impact of a pandemic accurately. Imperial College, London’s March 15 th Model predicted deaths in USA to the tune of 2.2 million. Thus, it better to avoid over-reliance on these models. Yes, the Ferguson predictions were wildly exaggerating, but convinced Boris Johnson to discard the herd immunity model and stick to the more reliable lockdown strategy. Thus, these models, though predict in exaggerated ways, do provide some useful forewarnings to governments, so that they can adopt a suitable strategy.

    The ‘Curve-fitting’ model also has some utility. Firstly, layman can easily understand its logic and secondly the reality is furnished with a proper visual perspective. The fundamental principle that there is a pattern to be understood from the pandemic data is common to all models. If the reality is without any pattern or conceived as random and irrational, no mathematical method can interpret it. Thus, the underlying principle is that there is a pattern. If you grant this, the ‘Curve-fitting’ model is also useful and cannot be discarded.
    The highly useful method, in my view, is to calculate growth rates per day on the weekly average basis by using the compounding formula ( easily available on internet). Secondly, we should discard the cumulative infection numbers as the basis for knowing the reality. The cumulative numbers have only a historic value. What matters the most is the situation obtaining as of now. This is represented best by the Active Cases. Thus, predictions, if any, should be related to active cases to know the peak level and be prepared to meet the challenge of the peak level active cases. The trend of weekly growth rates can help us to roughly predict when the peak could be expected. For example, many countries have crossed the peak of active cases and are displaying ac downward trend, or negative growth rate. This is the most desirable situation. Here is the list in the following order : Name of the Country/ Peak Level Active Cases/ Date of attaining peak level/ current level of active cases ( This website doesn’t support tabular formats. So please bear with the data format).
    (1) Italy/ 108257/April 19/ 76440
    (2) France/ 93365/ April 28/ 91840
    (3) Spain/ 100106/ April 23/ 58845
    (4) Germany/ 72865/ April 06/ 16647
    (5) South Korea/ 7362/ Mar 11/ 969
    (6) China/ 57805/ Feb 18/ 101
    (7) Taiwan/ 311/ April 06/ 50
    (8) Singapore/ 20799/ May 12/ 20104
    There are, however, cases where reversal in the declining trend in Active Cases is noted : Iran and Sweden.
    For India, the peak seems to be far away. India’s compounded growth rates per day, as worked out by me , are as under : Week ended/ growth rate (%)
    (1) 19 Mar / 13.74
    (2) 26 Mar/ 21.43
    (3) 02 Apr/ 19.32
    (4) 09 Apr/ 14.44
    (5) 16 Apr/ 9.7
    (6) 23 Apr/ 6.39
    (7) 30 Apr/ 5.17
    (8) 7 May / 6.25
    (9) 14 May/ 4.52

    This does establish a declining pattern, except for week ended 7th May. You will find a declining pattern in growth rates, though not of the same values, for all nations. Based on this data, I may roughly predict that for India, peak level in active cases would be somewhere in mid-June at around 90000 to 1 lakh active cases. Please note that around 80-85 per cent of these would be asymptomatic patients or those with mild symptoms and critical patients would be around 3 to 4 per cent. The challenge is not insurmountable.

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