R0 & The Economy
Today millions of businesses, from hairdressers to airlines are living a previously unthinkable reality. But as the initial outbreak of coronavirus is coming under control, eyes around the world are looking to the future. To a world in limbo, a world no longer threatened by an uncontrolled outbreak, but also a world without a vaccine, a cure or a clear path forward.
Do more deaths and infections now mean more movement, money and parties later? Will New York have enough immunity for life to go back to normal? Let's use a basic SIR model and find out.
To improve the economy you need to lessen the lockdown. With the relaxation of laws comes great risk. There is no rule that states that the once you’ve flattened curve once, it can’t shoot straight back up. — The Market Optimism Is Naive
In the most recent IHME model (a health organisation backed by the Bill & Melinda Gates Foundation) predicted between 9,000 and 24,000 deaths in the state of New York due to the current wave of coronavirus infections¹. What it doesn’t tell us is what happens when Donald Trump “opens up the country”.
Assuming 15,000 deaths and a 0.5% case fatality rate we can assume that 3 million or 15% of the state’s population would have been infected. A recent German anti-body study in the hard-hit district of Gangelt found only 15% immunity⁶. The region has remarkably similar statistics to New York City but at a much smaller scale.
With all these infections does that increases immunity allow things to return to normal? The SIR model holds the answers.
While it is important to note that this is a toy model which does not contain many of the complexities of the real world it is an incredibly powerful tool that allows us to conceptualise the impact of immunity on infection rates.
Rather than solving differential equations, the essence of the SIR model can be understood using only two values, R0 and Re.
The basic reproductive number (R0) as the number of people the average sick person will infect assuming no-body is immune. The effective reproductive number (Re) takes the number of susceptible people into account. Using this model it can be deduced that:
“It is important to note that given where τ is the transmissibility (i.e., probability of infection given contact between a susceptible and infected individual), ¯c is the average rate of contact between susceptible and infected individuals, and d is the duration of infectiousness.” — Department of Anthropological Sciences Stanford University
We can reduce R0 by reducing the time spent in contact with others (ie. isolation, working from home) and reducing the likelihood of the virus being transmitted when people are in contact with each other (ie. social distancing, wearing masks).
Here, 1 is the magic number. When Re < 1 there is a reduction in cases, when Re > 1, cases grow exponentially.
Once the number of cases drops to a stable and manageable level, the goal of policymakers across the world will be to slowly unwind restrictions on the economy and ensure that the effective reproductive value (Re) does not increase above 1.
Here, the world will encounter the trade-off between reducing R0 directly or reducing Re. This is the trade-off between increasing herd immunity & decreasing transmissibility, or saving lives or saving the economy.
Governments can either reduce R0 directly by implementing restrictive that limit interactions between people or Re can be reduced by simply increasing the number of infected and therefore immune people.
Method’s of reducing R0 can be broken into 3 categories:
1 — Impacts The Economy: Closing or restricting non-essential businesses
2 — Impacts Mental Health: Restricting or banning social events, social distancing, closing schools
3 — Impacts Nothing: Wear masks, widespread testing, forensic tracking of infections
Method’s of reducing RE can be broken into 1 category:
1 — Increase Total Immunity: Infect & kill more people
Obviously reducing R0 is prefered, but with mask shortages, testing limitations, overstretched medical resources, and an economy in free fall that might not be an option. J.P. Morgan suggests a 10% decline in peak to trough GDP this year². The temptation to get things back to normal before a vaccine arrives in 18 months is alluring. But how much does increasing the number of infected now decrease the spread later?
The R0 value of coronavirus suspected to be around 2.5 ³. Because herd immunity requires Re< 1, we determine that percentage of the population need to be infected for herd immunity to be reached is 60%.
Previously we conducted a back of an envelope calculation that approximately 15% of New Yorkers will be infected. This is far from the 60% required for things to get back to normal. Even with 15% of the population immune, the effective reproduction number is still 2.13. What about places with fewer infections like Rome, Berlin, London, Brussels or Amsterdam?
Using these equations we can trace just how much the spread will slow when more and more people become infected.
In New York, if things went back to normal, the increased immunity would likely make little difference. This is because virtually all the benefits of increasing immunity occur in the final stages when herd immunity is almost reached.
Without reaching close to that 60% mark, measures must be put in place to reduce the transmission of the disease. All countries with major outbreaks of coronavirus have been able to reduce the effective reproductive number to below 1 at the significant economic cost of nationwide lockdowns.
However, there are a number of countries contained the virus without significant economic interventions.
Australia: Prior to the first significant restrictions when bars, restaurant and gyms were closed indefinitely nationwide the Re was already below the critical threshold of 1.
Sweden: Sweden is currently pursuing a risky herd immunity strategy. They have very limited restrictions, lowering the of permitted gatherings from 500 to 50 on March 29. Despite this, behaviours have changed with ridership on public transport down 60 per cent and nearly all domestic flights cancelled⁴.
South Korea: Korea is the only one of these countries to suffer a serious outbreak with up to 900 cases a day being confirmed at its peak. The country has been widely praised in its ability to reverse the outbreak. The sustained pressure required to keep the virus at bay has come at a relatively small cost. Annual GDP is projected to decline by only 1% annually⁵.
Whether it be widespread testing, wearing masks or cultivating socially responsible culture, Australia, Sweden and South Korea have shown it possible to reduce the reproductive rate of the virus without risking millions of lives or decimating the economy. The trade-off between herd immunity and health is naive. Using the SIR model it is possible to show that medium level immunity as developed in hard-hit areas like New York will give limited protection to future outbreaks without additional measures. Where medium level immunity fails, effective herd immunity can be developed by reducing the transmissibility of the disease. These methods are far more effective than mid-level immunity that comes at the cost of thousands of lives and economically destructive lockdowns.