In the midst of the global COVID-19 pandemic, after the general public had become familiar with the reproduction number and exponential curves, some media outlets have was able to criticize mathematical models and their doomsday predictions, which policymakers would be wrong to believe.
And yet, policymakers need forecasts to make critical decisions, and only mathematical models can provide such predictions. This should not obscure the fact that the numerical predictions from epidemic models must be treated with great caution: we aim to explain what limits their reliability and how that reliability improves as we learn more about COVID-19.
Since when was the second wave predictable?
If we look at the data on the pandemic in France since early March, we can see that the pandemic went through four phases leading up to the start of the second lockdown, as illustrated in the figure below.
It had been clear since around September that the rise in the number of cases would lead to a new health crisis if it was not curbed. While, as we explain below, the exact scale of this crisis was difficult to predict, it would likely have been more severe than the one we experienced in the spring.
Read more: Understanding the Basics of Mathematical Models of Epidemics
In the spring, the epidemic’s reproduction number was brought down below 1 thanks to the lockdown. Also known as «effective R,» the reproduction number is an estimate of the average number of people infected by a single infected individual over the past 7 days. The relatively slow increase in the number of Covid-19-related hospitalizations and deaths observed since the first lockdown is likely the result of measures such as mask-wearing and social distancing, as well as the fact that, in the summer, people spend more time outdoors than indoors, etc.
This fall, the goal is to reduce the reproduction number from about 1.4 to less than 1—whereas in the spring, it had to be reduced from more than 3 to less than 1. This goal is, in principle, easier to achieve and explains why this second lockdown was effective even though it is less strict than the first one.
Should we be concerned about a third wave of the epidemic?
To prevent a resurgence of the epidemic—and thus a new wave—we need an effective and sustainable way to reduce the transmission of the disease. As long as a significant proportion of the population remains unimmunized (what is known as herd immunity), this requires reducing social contact and following preventive measures that limit the spread of the disease. According to several published studies, as of the 1ster September 2020, Only about 5 % of the French population had been infected with COVID-19, which is too low to significantly curb the epidemic. In fact, under these conditions, for the epidemic to subside, the R0 must remain below 1.05, compared to 1 if the population is not immune. Furthermore, even if the vaccines that will soon be available prove capable of providing lasting protection and preventing the spread of the disease—which is not yet certain, since they have only been tested over a short period—it will take many months to vaccinate a significant portion of the population.
In the absence of a strict policy for testing close contacts and isolating infectious individuals, there is concern that, starting in January 2021, the reproduction number will rise above 1. Authorities will need to respond quickly at the first signs of a resurgence of the epidemic if they want to avoid a third wave and a third lockdown: the later measures are implemented, the longer they will have to remain in place in order to bring the daily number of new cases back below a set threshold.
Read more: COVID-19: To Avoid Further Lockdowns, People Who Are Contagious Must Self-Isolate Sooner
What limits the reliability of mathematical model predictions?
Number of confirmed COVID-19 cases, by country, as of December 14, 2020 (from black to red, then pink and gray: more than 10,000,000; 1,000,000–9,999,999; 100,000–999,999; 10,000–99,999; 1,000–9,999; 100–999; 1–99; 0 or no data
How many deaths would there have been in France if the population had not been placed under lockdown in the spring? Several epidemiologists have attempted to answer this question using mathematical models, but no consensus seems to be emerging within the scientific community. The predictions by N. Fergusson of Imperial College London have at times been strongly criticized as overly alarmist, since they are based on assumptions too simplistic or unrealistic. How much weight should we give to the figures presented? What actually limits their reliability?
The primary source of uncertainty stems from a number of parameters that the modeler needs to know in order to make predictions. In this regard, one of the main unknowns at the start of the epidemic is the disease’s case fatality rate—that is, the proportion of infected individuals who die as a result of the infection. It is this parameter that allows us to estimate the number of infected individuals based on the number of deaths, and thus to estimate the proportion of the population that is immune. The figure below shows how the value of this parameter affects the models’ predictions of what might have happened without the second lockdown. Uncertainty regarding the disease’s case fatality rate is therefore a major source of variability in the predictions.
Furthermore, even the simplest mathematical models make an homogeneity assumption, which assumes that each infectious individual infects any susceptible individual with the same probability. When a sufficiently large proportion of the population is immune, the epidemic begins to subside, because a significant portion of the people an infectious individual comes into contact with are already immune, which limits the spread of the disease.
More realistic models take into account the spatial distribution of individuals, the differing behaviors of various age groups with respect to the disease, and the network of social relationships. But not all individuals are equal when it comes to the spread of the epidemic. A certain portion of the population has more social interactions than others. These individuals will tend to spread the epidemic more quickly than others, but they will also become infected earlier, on average. The epidemic therefore grows more rapidly at the outset and slows down once a significant proportion of these «super-active» individuals has been infected and immunized. This slowdown is then greater than what would be observed if the same number of individuals chosen at random from the population were immunized.
In short, the proportion of the population that must be immunized to achieve herd immunity is actually lower than what the homogeneous model predicts. Mathematicians who study epidemics are therefore striving to model the heterogeneity of individuals within the population as accurately as possible, but there is still a long way to go.
Although the details of the model used by Neil Ferguson are unknown, we can therefore assume that his predictions were pessimistic. But by exactly how much? It’s impossible to say. Without the first lockdown in France, our hospitals would have been overwhelmed in any case. Were the measures taken properly calibrated and appropriate for the severity of this disease? That’s another question.
Since the start of the health crisis, many scientists have dedicated themselves to understanding and combating this epidemic, leading to major advances in several fields. This is also true for the mathematical modeling of epidemics. Although scientific progress is slow, the work currently underway will lead to a better understanding in the future of the reliability of mathematical models and their predictions.
