“Few scientific fields have a single metric that both insiders and outsiders obsess over as much as infectious disease epidemiology’s basic reproductive number, .”

Professor Alison Hill — Institute for Computational Medicine and the infectious disease dynamics group at Johns Hopkins University

November 12 2020 — All over the world, major intelligence agencies are involved in the fight to end the current pandemic. I intend to discuss these efforts in the future posts but first, I must define a few concepts. Follow us on Twitter: @INTEL_TODAY

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The 2011 film Contagion is certainly entertaining but it is not accurate when explaining the math behind an epidemic.

Jude Law’s Alan Krumwiede is a British blogger based in San Francisco who launches a crusade to try and expose what he believes is a conspiracy by both the government and the pharmaceutical companies to hide the truth about the deadly virus – and make money on a possible cure. [1]

“On day one, there were two people. And then, four and then, sixteen. In three months, it’s a billion. That’s where we’re heading.”

Alan Krumwiede — Contagion

In a tense scene, Alan Krumwiede describes the spread of the pandemic as being a problem you can do on a napkin.

Using a value of 2 for , he explains that the number of infected people grows daily from 2 to 4 to 16, and so on until a billion of people are infected 3 months later.

That is utter nonsense. For starter, is not a rate, but a dimensionless quantity.

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Moreover, one should always keep in mind that it is never easy to estimate and the result is always uncertain.

“What is the current level of Covid-19 transmission in the UK? This might sound like a straightforward question, but take a moment to think about what it’s actually asking. Are we interested in how many people will catch the virus from those currently infected? If so, this transmission hasn’t finished yet, which means we’d have to try to predict it somehow.

Or maybe the question implies we look backwards, and work out how much transmission there was when people who are currently infected originally picked up the virus. If so, what data should we use to do this? And how confident can we be in the results? These are the sorts of questions that epidemiologists around the world have been grappling with since January, against a background of changing transmission and uncertain knowledge.”

Adam Kucharski — Professor in infectious disease epidemiology at the London School of Hygiene & Tropical Medicine and author of The Rules of Contagion: Why Things Spread – and Why They Stop

What is ?

is defined as the average number of new cases, or secondary infections, caused by a typical infected individual in a susceptible population.

Easy to define but hard to estimate

Roughly speaking, R0 depends on the product of three factors:

— the contact rate, or the number of people an infected individual interacts with each day

— the transmissibility, or the probability per unit time that any given contact results in transmission

— the infection duration

The goal of most infectious disease control efforts is to reduce R0 by altering one or more of those components.

A Mathematical Model of

The number of secondary infections is often summarized by a negative binomial distribution:

is the mean of secondary infections,

k parameterizes the dispersion of secondary infections,

,

and is the gamma function.

Average and Variance

If all individuals have the same intrinsic infectiousness — that is, the variance is low (blue scenario on the right top pic) — then the number of secondary infections is expected to have a Poisson distribution (k→∞).

If the infectiousness is heterogeneous, the distribution is said to be over-dispersed and has a lower k, that is the variance is high. (yellow scenario on the right top pic)

Over-dispersion implies that a small number of individuals are responsible for a large percentage of secondary infections, whereas most others infect no one.

For COVID- 19 worldwide, a few studies have estimated k ≈ 0.5, albeit with high uncertainty.

From Mathematical Modelling to Policy Making

Let us assume for a moment that you are the boss. Would you order a general lockdown if the data show that a small fraction of the population is spreading the virus?

Or would you rather order a targeted/partial lockdown?

In Belgium, which has one of the highest infection rates in Europe at the moment, the virologist and government adviser Marc Van Ranst pointed to Germany’s partial lockdown in a tweet on 29 October, commenting: “We could and should have done this six weeks ago.”

Brussels, European capital of the COVID-19

Belgium is clearly the European center of the pandemic and no one is denying the truth anymore.

According to the European Centre for Disease Prevention and Control, the “14-day cumulative number of COVID-19 cases per 100 000” is 1498.7 for Belgium.  For comparison, the number for Germany is 168.4!

Considering the gravity of the situation, you would think that Belgian experts have a pretty good math modelling of the pandemic. So, what is the value of k in Belgium?

So far we do not know… At least, we have not been told. Why on earth not?

To begin with, there is no large-scale testing, no serious tracing and no analysis of waste-waters. 

In fact, not only ‘contact tracing’ is still not working, there is not even an attempt to use it efficiently to fight the pandemic.

In Belgium, the ‘contact tracing’ is merely forward looking, designed to locate people who have been exposed to an infected person.

What is needed is a ‘contact tracing’ that tracks backwards from that person to identify an outbreak’s source.

Why was it not done? Belgian politicians are mostly trying to satisfy as many of their voters as possible in order to stay in power.

A targeted lockdown could be very costly to a political party which gets a large number of votes from a super-spreading community.

If people knew that a lockdown is imposed on all of them because a small group is largely responsible for the transmission of the virus, there would be a revolution.

PS — Did you know… Some of today’s most prolific infectious disease modelers originally trained as physicists, including Neil Ferguson of Imperial College London, an adviser to the UK government on its COVID- 19 response, and Vittoria Colizza of Sorbonne University in Paris, a leader in network modeling of disease spread.

To the best of my knowledge, there is not a single physicist among the ‘experts’ advising the Belgian government. Pieter Bruegel’s “The Blind Leading the Blind” is of course one of the most famous Flemish Renaissance paintings. Not without reason…

How Infection Spread | Contagion

REFERENCES

[1] On Monday (Nov. 9 2020), Pfizer CEO Albert Bourla cashed in on his company’s coronavirus vaccine breakthrough. The executive sold more than $5.5 million worth of Pfizer stock

Contagion: How to Model It and What R-nought (R0) Actually Means — Dr Duncan Robertson

The Math Behind Epidemics — Alison H. Hill

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COVID-19 — From Mathematical Modelling to Policy Making