We constantly hear at the UK daily Covid press conferences that we must keep R <1 to avoid a second wave of infections. Prof. Chris Witty warns us that if R goes above 1 then we will see an “exponential” increase in cases. However any second wave will depend critically on the exact value of R, because R defines both the speed of the outbreak and the total number of infections needed before it naturally ends. This is because it runs out of (an R -dependent) pool of susceptible (i.e. non-infected) people. This finally results in so-called “herd immunity”. Figure 1 shows some examples of how the rate of infections develop in time and severity for different values of R.

Fig 1. Infections for different values of R0. (cases for UK)

The initial rise is not really “exponential” but a power series where is the number of infection cycles. This strong dependence on R suggests that one possible strategy would be to restart the UK economy after lockdown but just try to keep R slightly above 1 – say 1.1. This would minimise total deaths in the long run and eventually achieve herd immunity, unless of course a vaccine can be found to achieve this sooner (see fig. 2). The value of R becomes a pay off between the length of an outbreak and its severity. Measuring R would depend on widespread public testing.

Fig 2. An epidemic with R=1.1 moves very slowly peaking only after 1.5 years but would minimise UK deaths (this assumes IFR = 0.5%)

This strategy appears to be exactly that being adopted in Sweden, as explained by Professor Johan Giesecke in his lockdown TV interview. Sweden has adopted a limited social distancing policy but kept primary schools, bars and restaurants open. They are planning for herd immunity while at the same time minimising deaths.

These are incredibly difficult decisions for any government to take. Apparently successful countries like Taiwan, New Zealand, Hong Kong and Iceland may succeed to eradicate Covid-19 entirely. However afterwards they would still need to isolate themselves from the rest of the world indefinitely in order to stop any possible reinfection. This is unsustainable long term. The global economy depends on trade and international travel. Any resultant collapse in global living standards would likely then cause more deaths than the virus itself.

Let’s hope a vaccine appears as soon as possible!

*Related*

You assume that immunity lasts. It is a corona virus, so it’s possible immunity will not last long: https://www.technologyreview.com/2020/04/27/1000569/how-long-are-people-immune-to-covid-19/

A minor point. It is indeed a power series, but it is the power series expansion of the exponential function.

Maybe true initially proportional

but as susceptible cases begin to decrease so the curve reduces away from any power series.

http://clivebest.com/blog/?p=9374

see

Good point about the R>1 scenario. If the virus can move through the not-vulnerable part of the population with an R>1 while keeping the vulnerable shielded, then the country that does it will achieve a worthwhile herd immunity effect when the vulnerable are allowed out again. It is a pity that that the UK has bottled it, and gone for a kind of reverse strategy of keeping R below 1 for the not-vulnerable while keeping it above 1 for the vulnerable by permitting exchanges of patients between care homes and hospitals and back again for those who recover.

The worst possible outcome of this is a mutation which preferentially culls the working age or young ( it’s happened before with viruses that previously culled the old ). The closure of primary schools has reduced the chances of younger people having the tool-set to fight the next mutation of this thing.

Giesecke does not decide the swedish strategy. According to this the authorities in Sweden do not plan for herd immunity:

https://edition.cnn.com/2020/04/28/europe/sweden-coronavirus-lockdown-strategy-intl/index.html

Their primary goal is to keep the health system afloat.

A cogent analysis. The key calculus is for each society to determine the tolerable rate at which unproductive people die. No jurisdiction – not even North Korea – can hermetically seal itself, and the longer an unexposed population exists the more likely it becomes to suffer intolerable outbreaks, akin to flu being introduced to isolated tribes, or smallpox to the Americas. Any disabled economy has a disabked healthcare system. The harsh realities are clear, irrespective of the lethality of the infection.

I was going to try these curves myself but you are much better at it. I am very interested in the sort of inverse function how herd immunity responds. Have heard herd immunity sets in at 70%?

That’s an interesting question. The fraction of the population who need to be infected to reach herd immunity depends critically on R0 (R at time zero). If you think about it if R0=2 then at the beginning with 65 million people susceptible the cases increase by 2^n every 5 days. Pretty soon though the number of infected cases reduces the available pool of susceptible people, which causes R to drop quickly until it reaches R = 1 at the peak. Thereafter R quickly drops rapidly to zero ending the epidemic. There are always a % of the population who escape the disease. If R0 = just 1.1 then only about 7 million people get infected. This means herd immunity is reached at just 11%.

Maybe I’ll look at this in detail in a new post.

Why do different values for R0 lead to less area under the curve ? I thought that the same number of people died regardless, presumably it’s a change in herd immunity percentage with different R0, in which case some curve flattening measures may be useful but certainly not the draconian measures we are currently experiencing.

But with the vulnerable safely tucked up in isolation, the best scenario then would be to remove all restrictions and let the virus run through the fit healthy, no comorbidities, not retired population, hopefully a large enough chunk of the population to provide herd immunity.

Yes it is because of different pools of susceptible and recovered persons when herd immunity is reached. Basically far less people get infected if R is just above 1, so as a consequence far less people die.

In the long run I fear it it is either a vaccine or herd immunity, unless we are incredibly lucky and the virus mutates to a mild form. There may even be evolutionary pressure for this to happen.

The ratio of the number of people who has been infected to the total number of people when the epidemic is over (I=0 for good ) is r=1+W(-z*exp(-z))/z where W(x) is the Lambert-w function and z is the I-weighted mean value of R0 for the whole period(Including all waves). The total area under the curve is r/g. For z=1 this is zero and goes to 1/g when z goes to infinity.

All restrictions can be stopped at the time R0*S-1=0, giving S=1/R0. This means r=1-S=1-1/R0. This is the herd immunity value. The pandemic is not stopped but will decrease to zero even with no restrictions. For R0=2.5 this is 60%.

For minimum number of deaths 1+W(-z*exp(-z))/z=1-/R0 with the solution z=R0/(R0-1)*ln(R0). If the mean value of R is less than this, the epidemic is not stopped, even if it lookes like it.

For R0=2,5 this magic number is 1,527.So if the restrictions are too intense you are bound to have new waves unless you have restrictions forever or have a vaccine.

Where do you estimate Sweden is now ?

R0 for Sweden with its low population density is lower than 2,5 .It is about 2,0 or even lower.This gives a critical number of 1,38. Swedens z is at the moment about 1.1, which probably means there will be another wave in the end of the year.After this wave herd immunity will probably be achieved.

Countries with lesser z then this have a greater risk of having 3 waves or even 4.