Covid-19 and the limitation of mathematical models

Every other day, somebody, somewhere in the world, presents a new mathematical model of how much worse the pandemic is going to get, or perhaps when the pandemic will end. Nobody can see into the future. So why should we believe these models any more than we believe astrological forecasts?

Herd immunity

Immunity results from vaccination or from getting infected and surviving. The Covid-19 pandemic will end when herd immunity is achieved; that is, when so many people have become immune to the disease that it’s hard for a newly infected person to infect anybody else.

The herd immunity threshold is estimated as (R0-1)/R0. In this formula, R0 is the average number of persons a newly infected patient transmits the disease to; so, on average, if each person infects 5 other persons, R0 is 5, and (5-1)/5 or 80% of the population must become immune for the pandemic to end.

Follow live updates on the coronavirus here

R0 is not easily estimated. Some persons infect many others because they are aerosol superspreaders; or merely because they live in crowded areas and can therefore infect many people. Disease spread is therefore very variable. It could vary because of differences in individual immune response, the presence of medical comorbidities, variations in population density, and variations in temperature and humidity. It could also vary depending on whether or not people wear masks and practice social distancing, and on how well partial or complete lockdown is enforced.

Mathematical models on how and when the pandemic will peak or end therefore need to take all these factors into consideration.

Unfortunately, we have poor data on all these factors. So models need to make assumptions about these factors. If the assumptions are wrong, the model will be wrong. Because most assumptions are based on educated guesses or on approximations from inaccurate data, models are inevitably approximations. This is why some models sensibly offer best and worst case scenarios.

Forecasting

A child weighs 3 kg at birth, 6 kg at 6 months, and 9 kg at 1 year, but not 63 kg at 10 years because weight gain is not uniform across time. Besides the nonlinear growth effect, gender, genes, nutrition, pregnancy, disease, and other factors can also affect weight. How would a mathematical model know all this in the case of something new, such as Covid-19? There are not only known unknowns, but unknown unknowns that can affect the Covid-19 mathematical model.

Herd immunity and the models depend on whether the vaccine or infection by the disease truly results in persistent immunity. If the immunity wears off early, then models that assume immunity will no longer be valid.

So whenever a new mathematical model is published, the reader must be told about what the assumptions were and what the error margin is. But, why have models, at all? Models are necessary to educate people about what might lie ahead, and to guide official policy-making. We must, however, be aware of the limitations of the models.

(The writer is Professor, Psychopharmacology at Nimhans, Bengaluru)