A Data Science's view of Herd Immunity, What does a simple epidemiological model tell us?

COVID-19 spreads like wildfire. this microorganism particle is rapidly changing the world. In less than four months, it has emerged from Wuhan (China) to almost every country in the world.

On the 13 March 2020, the prime minister of UK Boris Johnson gave the Nation the latest coronavirus briefing. At the time he mentioned that the UK would use scientific research to model the best strategy based on the current shreds of evidence that are available, this loose control strategy was later indicted by the UK’s chief scientific advisor Patrick Vallance as Herd Immunity.

So what exactly is the Herd Immunity?

Let’s discuss the topic today through the lens of Data Science.

I. A Simple Virus Transmission Model.

The virus transmission process is extremely complicated in its nature. There are various epidemiological models to explain or simulate the spread of infectious disease. But for a basic understanding, we start with a simple model.

In this model, we have two fundamental concepts of grounding (basic reproduction number and generation interval)

1.1 Basic Reproduction Number

The basic reproduction number (R0 or Rt), is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population.

It will help us to understand how quickly an infectious disease will spread across the population.

If the R0 is less than 1, for example, 0.5. Then one person will produce 0.5 new cases, the secondary 0.5 cases will pass the virus to the next 0.25 cases. Thus the infected population will be getting smaller. and die out without any chance to become a global epidemics.

If the R0 is close or equal to 1, meaning that one person will only pass the virus to another, then to the next one. We end up having a flat line of the infectious case. Hence we call it endemic (local spread), for example, chickenpox.

The last scenario is when the R0 is larger than 1, for example, 2, then the growth will be exponential, 2 to 4, 4 to 8 and eventually reach the tipping point that it cannot be contained in a local area. We now call it a pandemic.

Once a virus outbreak becomes a pandemic, there are normally two endings, the patient is either die (break the circulation) or develop an immune system against the virus. (Stop the spreading)

A more realistic calculation of R0

The WHO initially estimated the COVID-19 R0 to be 1.4–2.5 (average 1.95), however a recent review of 12 studies estimated the basic R0 to be 3.28 and the median R0 to be 2.79.

1.2 Generation Interval

It is the period of time separating sequential infections.

In a simple term, let’s assume a COVID-19 patient can pass the disease to 2 others, so how long does this transmission gonna take? 1 day, 2 days or a week. This is the other critical factor in the mathematical model, the shorter the interval between two-generation, the more contagious it was.

“A” is infected by the virus at the time that denoted as a circle. After a while, “A” shows some mild symptoms, denoted as a square.

The gap between the circle and the square is the incubation period. (the period between exposure to an infection and the appearance of the first symptoms) The incubation period for COVID-19 varies quite a lot, ranging from 3 days to 20 days. Let’s take a weighted average, 8 days.

To point out, patient “A” is contagious throughout the whole incubation period, thus can infect person B anytime between the 8 days. Then it’s gonna take another 8 days for B to show symptoms (marked as square as well). The time gap between the two squares is known as the generation interval. For COVID-19, it’s around 4 days.

" Take “Ro = 2.5” and “Generation Interval = 4”, how long will it take Covid-19 to spread to the entire Europe if there is no intervention?"

1.3 Estimated time of transmission

The above equation is deduct based on the summation of geometric progression, it calculates the accumulated predicted case of coronavirus in each generation (R0=2.5). The total population in Europe is 741 million then the result of t = 23 generations.

And when we apply 4 days per generation, the final result corresponds to 92 days.

Counting from 24/01/2020, by 25/04/2020 COVID-19 will reach every corner of the European continent without any non-pharmaceutical intervention (NPI).

Remember, this is an extremely simple model, the real-world situation is much more complicated and the magnitude of more variables need to be factor into the dynamic epidemiological model.

Here are a few Model that can be considered,

• The SI model — [Divide the population into susceptible, S(t), and the infectious, I(t). N be the total population size. Then we have S(t)+ I(t) = N.]
• The SIR model — [Apart from susceptible and infectious, this model assumes that people who recover from the infection become immune and cannot become infected a second time. S(t)+ I(t) +R(t)= N]
• The SIS model — [assume people who are recovered still susceptible to the disease]
• Bayesian hierarchical model (used by the Imperial College London? in its most recent report, this model is easy to explain and infer )
• etc. https://web.stanford.edu/~jhj1/teachingdocs/Jones-Epidemics050308.pdf

II. The Herd Immunity

Finally, we come to Herd Immunity.

Economist magazine summaries that the world governments are using three main methods to combat the COVID-19.

• Herd Immunity -> a situation that allowing the majority of the population to be infected to develop an immune system.
• Mitigation -> normally know as “flattening the curve” to make the pandemic less intense, via means including massive testing, contact tracking, isolating cases, quarantining.
• Suppression -> shutting down the city, closing public events, schools, universities to cut the spread.

As mentioned before, Britain is the first country that proposed the Herd Immunity approach to contain the disease( as shown form the graph above, the UK is the last country that enforces a lockdown).

Herd Immunity.is proposed a century ago (same as many Data Science classic models)

• First, let’s assume we have a group of people and amount them there is an outbreak of unknown viruses
• Most people will recover from the disease and build up an immune system against the virus. This rate is denoted as P, and the non-immune rate is (1-p)
• Take an R0 rate, the virus can only infect to the non-immune group. RO(1-P)
• If we manage to get the [R0 (1-P)] < 1, the disease will go away in a short period of time. To satisfy, we need to have P>1–1/R0 (simple calculation).
• Take the COVID-19 R0 = 2.5 in this equation, we will have a P> 60 %. This means 2/3 of the person in this group need to be infected and recover with immunity to stop the virus.

Let’s say if the British government go ahead and implement this approach, 66.4 M * 60 % ≈ 40 M people will be infected. Furthermore, the current case fatality rate (CFR) of COVID-19 is 4% and 8% in China and Italy prospectively, and let’s assume Britain has the best health care system in the world that could bring down this rate to just 1%, there will still be about 40 M * 1% = 400 K death.

" Similarly, Neil Ferguson, an Imperial College London professor who leads the research on COVID-19 non-pharmaceutical research [Paper], said his original estimate, which showed the coronavirus would kill 500,000 people (with limited NPI, Non-Parmarcutical Intervention )in the U.K., remains true, while a new model reading reflecting the influence of lockdown measures saw that estimate shrink to 20,000 or fewer. "

What is this number mean?

In 2018, the total death toll in the UK is just over 500K. And with the herd immunity strategy, COVID-19 will take almost the same amount of people’s lives in just a few months. And as a comparison, World War Two only cost over 450K lives in Britain including soldiers, nurses, citizens, etc. This implies COVID-19 will potentially have a larger impact in the UK than WWII.

The supporters of the Herd Immunity approach argue that without a vaccine, this epidemic would return within a few weeks of the restrictions being lifted. The government might need to suppress the disease each time it resurfaces. This on and off-cycle must be repeated until either the disease has worked through the population or there is a vaccine which would be 6 months away.

The Critics defend that there is no scientific evidence that supports the theory of a seasonal COVID-19 outbreak, and even if the public can build up the herd immunity it won’t be effective when the virus mutates.

III. Is Herd Immunity Ever Stop an Epidemic?

The best example to look at is Measles.

Premises:

• Measles has R0 of 12–18, extremely contagious.
• Human is the only host for Measles.
• A low case fatality rate (CFR), 0.3%.
• No mutation, humans can develop lifelong immunity for Measles.

For all these characters, Measles seems like the best candidate for herd immunity.

So did this disease die out soon?

Measles’ first appearance was in the 10th Century, it caused more than 2.6 million death each year till we discover vaccine in 1980.

Before the age of the MMR vaccine, 90% (or some experts say 99% )of the infants will get Measles. The typical symptoms include rash and pneumonia or even brain damage for small cases.

Even after we discovered the MMR. Measles is still hard to prevent and control.

Using the previous equation P>1–(1/R0) = 1–(1/15) = 93%. We need to let 93% of the population build up an immune system to contain the virus. (WHO suggest this number should be 95%).

But, due to some of the negative comments on the bad reaction after receiving the MMR vaccine, the actual vaccination rate is only 86%.

We already eliminated smallpox, the next in the line for WHO is Measles, Polio, and COVID-19.

Mitigation cost too many lives and suppression may be economically unsustainable, herd immunity? Are we able to put a price tag on people’s lives?

No one knows what we will face in the weeks ahead, but everyone knows enough to understand that COVID-19 will test our capacities to be kind and generous and to see beyond ourselves and our own interests. Our task now is to bring the best of who we are and what we do to a world that is more complex and more confused than any of us would like it to be. May we all proceed with wisdom and grace.

Reference

research paper

• https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf
• https://www.imperial.ac.uk/media/imperial-college/medicine/sph/ide/gida-fellowships/Imperial-College-COVID19-Global-Impact-26-03-2020.pdf
• https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data
• https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6
• https://corona.help/#countries-nav
• https://www.bloomberg.com/news/videos/2020-03-19/can-herd-immunity-help-stop-covid-19-video