Immunity Passports and Unintended Consequences

A well-known anecdote relates how the British introduced bounties on cobras in an attempt to reduce the numbers of these dangerous snakes in Delhi. Locals responded by breeding cobras to be presented for the bounty. When the British learned that their program was resulting in the breeding of additional cobras, they discontinued the program ... upon which the locals set loose the cobras they'd bred, resulting in an increase in the cobra population.

Fast forward to 2020, and British authorities seem set on teaching the world another lesson about incentives and the laws of unintended consequences.

This time, British authorities are discussing the introduction of an immunity passport, perhaps to be re-branded as a 'health passport.' The idea is that a person who has recovered from COVID-19 is no longer able to become infected or to introduce the virus to anyone else, so the passport holder should be free to opt out of the measures that the rest of the population are undertaking to prevent the spread of the virus.

There are obvious problems with such a scheme, most notably the untested assumption that people who have recovered from infection with SARS-CoV-2 enjoy long-term immunity. And I won't even address the ethical issues involved in segmenting the population this way.

But the most obvious problem with this health passport scheme is that it creates an obvious perverse incentive -- namely, the incentive, particularly among young people, to become infected so as to claim the bounty -- er, passport.

How much might this passport be worth to a young person? A resonable guess might be based on the results from a paper titled, The Effects of Graduating from College During a Recession on Living Standards, published in the journal, Economic Inquiry, by Professors Daiji Kawaguchi and Ayako Kondo, of the University of Tokyo. Studying US data, they report:

"Those who graduate in a local labor market with a 1 percentage point higher unemployment rate experience about a 7%–9% reduction of hourly wage up to 18 years after graduation."

These results are similar to those published earlier by Lisa Kahn in a paper titled, "The Long-Term Labor Market Consequences of Graduating from College in a Bad Economy," published in Labour Economics. In particular, Kahn reported 

"...an initial wage loss of 6 to 8% for a 1 percentage point increase in the unemployment rate measure. This effect falls in magnitude by approximately a quarter of a percentage point each year after college graduation. However, even 15 years after college graduation, the wage loss is 2.5% and is still statistically signficant. Using state rates, the OLS results are insignificant but the IV estimates imply a 10% wage loss which persists, remaining statistically significant 15 years after college graduation."

Of course, the circumstances in these two papers are different from the circumstances surrounding health passports, but the broad principals appear applicable. Graduating when the labor market is depressed can result in a material decline in income for at least two decades.

Given these estimates, the present value of the magnitude of this effect might be on the order of GBP 100,000. For some students, the magnitude of the effect is likely to greater, and for other students the magnitude of the effect is likely to be more moderate. And it's worth noting that the estimates cited in these papers involved recessions that were far less severe than the current downturn -- another reason that this GBP 100,000 estimate doesn't strike me as unreasonable.

Let's imagine that every 20-something in the UK were offered 100,000 pounds to contract COVID-19. Of course, some would decline straightaway. But some significant percentage is likely to consider the offer. After all, the disease is reported to be moderate in young people. They may not even develop symptoms. And if they develop symptoms, the symptoms probably will be mild. The symptoms are unlikely to become sufficiently severe to require hospitalization. But even then, the chances of requiring an ICU bed or a ventilator are pretty remote. And the chance of death from COVID-19 is quite modest. So for some material segment of the population, getting a health passport in exchange for acquiring COVID-19 will appear to be a reasonable proposition.

Of course, those people intent on acquiring these health passports won't even know whether they're infected until 5-6 days after the infection has been acquired, during which time they'll be spreading the infection, including to people in high-risk groups or who otherwise have no interest in becoming infected.

Passport seekers might argue that they could be tested daily so as to avoid infecting anyone else during this period. But the notion of allocating scarce tests to people trying to acquire COVID-19 impresses me as a perverse outcome.

Overall, an immunity passport strikes me as a terrible idea that neglects the simple but fundamental notion that incentives matter. If we give people an incentive to acquire COVID-19, some people will respond by acquiring the infection. And their efforts to acquire the infection will result in the virus spreading more quickly within the population -- an outcome that is diametrically opposed to the explicit goals of the Government.

If the strategy of the Government were to develop herd immunity throughout the population, the use of immunity passports might be a reasonable tactic, since the incentive created by the passport would be consistent with the strategy in that case. But given that the UK Government is ostensibly pursuing a strategy in which people are being asked to avoid getting the virus, the incentive created by the passport appears perverse.

I urge policy makers to reject proposals to introduce an immunity passport in favor of the current plan: to reduce the number of infected persons to a level that would allow the introduction of intensive testing and contact tracing in an attempt to further reduce local infections, similar to the policy enacted in South Korea.

Comparing Sweden to Norway and Denmark

Sweden is often cited as an example of a country that has imposed relatively modest restrictions on movement and commerce in response to the ongoing COVID-19 pandemic. So as Western nations focus on easing COVID-related restrictions, I thought it might be useful to compare the experience of Sweden to date with those of its Nordic neighbors, Norway, and Denmark.

The first graph below shows the number of daily cases in three countries.

Source: European Centre for Disease Prevention and Control

This next chart shows the number of daily COVID-related deaths.

Source: European Centre for Disease Prevention and Control

Taken together, they support an initial impression that Sweden's experience has been more severe than those of Norway and Denmark. (Note that the number of daily deaths in Sweden exhibits a marked weekly seasonality. I presume this is a reporting effect rather than a genuine seasonality in actual deaths.)

We can gain some further perspective on this data by viewing cases and deaths on a cumulative basis, shown in the next two graphs.

Source: European Centre for Disease Prevention and Control

Source: European Centre for Disease Prevention and Control

Again, the impression created by viewing the data from this perspective is that the pandemic has spread more broadly and resulted in greater mortality in Sweden than it has in Norway and Denmark.

Of course, Sweden has a larger population than do Norway and Denmark. In particular, the population of Sweden is slightly greater than 10 million -- almost twice the populations of Norway and Denmark (5.4 million and 5.8 million). With this in mind, we also should consider infection and mortality on a per capita basis in each country, as shown in the next two graphs.

Source: European Centre for Disease Prevention and Control

Source: European Centre for Disease Prevention and Control

The cumulative numbers of cases as a percent of the population in each country are broadly similar, highlighting the importance of adjusting for population. On the other hand the cumulative numbers of deaths as a percent of the population in each country are quite different, with the death rate in Sweden significantly greater than the death rates in Norway and Denmark.

The population of Sweden is skewed slightly toward older age groups relative to Denmark and Norway, so we might expect a greater death rate per capita in Sweden than in Norway and Denmark simply due to Sweden's skew toward an older demographic profile.

The data in the next table helps us asses this situation. Of particular interest is the number of deaths in Sweden that would be expected if the mortality profile in Sweden as a function of age and sex were the same as the average profiles in Norway and Denmark.

Note that the number of reported deaths in Sweden is nearly four times larger than would be expected given the demographic adjustments derived from Norway and Denmark. So even after adjusting for population, sex, and age, the mortality figures in Sweden appear particularly bad.

Other demographic factors that have been mentioned are ethnicity and income. I don't currently have ethnicity data for these three countries, but data is available for immigration, which might be a rough proxy in these countries for ethnicity (and perhaps also for income). In particular, 20.0% of the Swedish population is foreign-born according to the UN Department of Economic And Social Affairs. The figures for Norway and Denmark are 16.1% and 12.5%.

Is the higher percentage of immigrants living in Sweden a likely cause of the higher rate of COVID-related deaths in Sweden? It seems doubtful. Even in the extreme scenario in which COVID-related deaths were concentrated entirely within immigrant populations in the three countries, the expected number of COVID-related deaths in Sweden would be only modestly greater than the figure in the previous table, assuming that the demographic profiles of the immigrant communities were roughly comparable to the demographic profiles of the overall population in each country. If anything, my sense is that the immigrant communities tend to skew toward younger age groups, making it even less likely that relative immigration percentages are responsible for the relatively large number of COVID-related deaths reported in Sweden.

Two Hypotheses

At this point, these data sets appear consistent with two hypotheses:

1. The Swedish health care system is less capable than those of Norway and Denmark in preventing COVID-related deaths
2. The relatively moderate response to the pandemic in Sweden has resulted in a greater infection rate in Sweden.

I'm not aware of any evidence suggesting the capabilities of the Swedish health care system are inferior to those in Norway and Denmark, but neither can I completely discount this possibility.

On the other hand, it seems entirely reasonable -- even expected --  that the relatively moderate response to the pandemic in Sweden would lead to a greater infection rate.

But if that were the case, why is the cumulative reported cases per capita in Sweden only modestly greater than those in Norway and Denmark?

As it happens, Sweden appears to have implemented a less intensive testing regime. For example, Norway is reported to have administered 28,614 tests per million people, while Denmark is reported to have administered 26,900 tests per million people. In contrast, the rate of test administration in Sweden is reported to be only 9,357 per million -- about one third the rates of Norway and Denmark. (Source: https://www.worldometers.info/coronavirus/#countries)

Taken together, my view is that the data suggests that the population in Sweden is suffering considerably greater COVID-related mortality than are the populations in Norway in Denmark because of the relatively modest restrictions placed on movement and commerce in Sweden. The fact that the reported infection rate in Sweden is only modestly greater than those in Norway and Denmark is due to the much lower level of testing administered in Sweden.

Implications

With restrictions being eased throughout much of Europe and the US, the experience of Sweden may well serve as a useful example. In particular, the Swedish experience suggests that infection rates and mortality rates are likely to increase in locations in which restrictions are eased.

Of course, there's nothing surprising about this conclusion. Infectious disease specialists have been warning us of the relation between restrictions and infection rates since the start of the pandemic. The point of this cursory analysis is simply to help dispel the notion that Sweden somehow demonstrates that there is no trade-off between the severity of restrictions and the rate of mortality. In fact, the comparison between Sweden, Norway, and Denmark provides considerable evidence for the existence of this trade-off.

Nothing about this analysis suggests that the choices made in Sweden are somehow inferior to those made in Norway and in Denmark. In particular, we've made no effort to address the relative economic costs borne by the populations in these countries. Neither have we addressed other, non-monetary costs, such as mortality due to other causes in these countries. And it's too early to assess whether Sweden's approach might lead to greater immunity and/or fewer deaths over the long run. 

But as societies weigh various trade-offs, it's useful to note that the data coming from these Nordic countries supports the notion that more moderate restrictions on movement and commerce can be expected to lead to an increase in rates of COVID-related infection and mortality, everything else equal.

Negative WTI Prices

Negative prices for May WTI futures have prompted a lot of confusing comments in the press. I thought I'd add a few thoughts.

The graph below shows spot prices for three crude benchmarks: WTI (Cushing, Oklahoma); Light Louisiana Sweet (St James); and Brent (North Sea).

LLS is coastal, and Brent is seaborne, so it's relatively straightforward to arrange transport for these benchmarks. WTI is inland, which means it needs to be transported via fixed pipelines, by rail, or by truck. For this reason, the difference between WTI and LLS typically reflects the marginal cost of transport between the two hubs.

When we observe the spread between LLS and WTI widen, it's typically an indication that the marginal cost of transport from Cushing to the Gulf Coast has increased. Note that current prices for LLS and Brent are still very much in line with one another, despite the thousands of miles separating the two reference points, as shipping is relatively straightforward to arrange.

The negative prices observed for WTI have been attributed to the fact that storage in Cushing is near capacity. And clearly, storing a barrel in Cushing for a month and delivering into the June WTI contract would be an option for people able to arrange storage. But it's worth remembering that an alternative to storing a barrel in Cushing is to transport the barrel elsewhere, where crude is selling for a higher price.

I've seen people suggesting that the higher price for June WTI futures suggests that oil market participants are expecting a rapid economic recovery in the US. Given the economics of storage and transport, I believe these comments are incorrect. Rather, the price difference reflects the fact that market participants believe they have time to contract for June storage and/or transport at reasonable rates. It's difficult to increase storage and pipeline capacity on short notice, but it is possible to arrange for additional rail cars and tanker trucks to travel to Cushing for June delivery. And in the meantime, traders now have plenty of time (and incentive) to close long positions well ahead of delivery.

One of the cheaper means of storing crude is to simply leave it in the ground. And of course that's the decision that the Saudis and Russians would like US producers to make. But some of these producers have hedged the prices they receive by selling the crude forward, including via the futures market. And in that case, they have every incentive to continue producing and to deliver their barrels at the agreed prices. But at some point, these hedges will roll off, in which case many of these producers will be facing spot and forward prices that are well below their marginal production costs.

Why isn't the Federal government taking advantage of these negative prices to further fill the strategic petroleum reserves? The SPR storage facilities are all along the Gulf Coast, so the managers of the SPR face the same transport bottlenecks as the rest of the market.

Are we likely to see negative prices for the June 2020 WTI contract, particularly going into delivery? Given the relatively inelastic nature of storage and pipeline transport capacity, it's a distinct possibility, particularly if producers have sold forward a lot of production using the June futures contract. In that case, I'd be concerned that the extra rail cars and tanker trucks that can brought into the region over the next month may be insufficient. My expectation is that one month is sufficient time for the market to make arrangements to avoid a fiasco of the sort seen this week, but I don't have data on actual production sold forward, so this is only a hunch.

Finally, it's worth noting that the debacle in Cushing this week ultimately stems from the fact that there isn't sufficient capacity to transport all the crude that the market would like to see transported to the Gulf Coast. And this is a common theme developing during this COVID-19 pandemic. In many places, there's an insufficient number of ICU beds, an insufficient number of ventilators, an insufficient number of masks and gloves, an insufficient number of test kits, and an insufficient number of trained medical personnel. There's insufficient capacity to get dairy products and produce into retail channels. Many places have insufficient broadband capacity as more people work from home.

Just-in-time practices have improved capital efficiency under normal circumstances, but of course they're not robust to disruptions. And I suspect we'll be seeing more investment in infrastructure that would be judged inefficient during normal circumstances but that would be greatly appreciated during crises. At least I hope so.

S&P 500 Relative to Trend

When considering the current valuation of the S&P 500 index, it's useful to consider the longer-term context.

The graph below shows the S&P since the beginning of 1928, with the red point representing Friday's ending value.

The time series appears as if it's growing exponentially over time, so the natural logarithm of the S&P is shown in the next graph.

This gives the impression of being subject to a long-term upward trend, so we show the series again in the next graph, this time with a trend line added.

This gives the impression that the log of the S&P 500 is reverting around a long-term upward trend. In fact, Robert Merton, in his well-known 1970 paper, Optimal Portfolio and Consumption Rules in A Continuous-Time Model, proposed just a process for the natural logarithm of stock prices. In particular, if yt is the natural logarithm of the stock price, he proposed the stochastic differential equation:

dyt = k (mxt + b - yt) dt + S dWt 

where

• xt is the index used to represent time (eg, could simply be t)
• mxt + b is the linear time trend, with constants m and b
• dt is an infinitesimal change in time, t
• dWt is the infinitesimal change in a standard Wiener process, {Wt}
• S is a constant that scales the Wiener process.

In fact, if we estimate the parameters of this process using the data shown in these graphs, it appears as if the log of the S&P 500 is reverting around a time trend that is increasing at a rate of 6.5% per year, with a half-life of four years.

The next graph shows the de-trended natural log of the S&P 500 -- ie, the difference between the blue line and the green line in the graph above.

The S&P 500 was as much as 34% above trend as recently as February 19 of this year. But as governments responded to the continuing spread of the SARS-CoV-2 virus, the S&P declined to the point that it was 11.9% below trend on March 23. In the latter half of last week, in the wake of fiscal and monetary support in a number of countries, the index regained a significant portion of its recent losses, increasing 13.6% -- essentially bringing it back to trend. (As of Friday's close, the S&P was 0.1% below trend.)

To help put this in perspective, the de-trended index is shown again in the graph below -- this time shown along with recessions, as determined by the National Bureau of Economic Analysis business cycle dating committee.

Note that most recessions involved the log of the S&P 500 index being well below trend at some point during the recession. In particular, the worst recessions saw the de-trended natural log of the S&P 500 down to -0.5. For example, during the great financial crisis, the de-trended log of the S&P 500 index reached -0.60 on March 9, 2009 -- 45% below trend.

In the event that the natural log of the S&P 500 index were to to decline to 0.6 below trend now, that would correspond to an index value of 1,395 -- ie, 45% below Friday's ending value of 2,541.47.

This exercise isn't intended to provide a precise valuation metric for US equities. Clearly, there are a number of important factors that would be expected to effect equity valuations in the present environment, including unusually low long-term risk-free interest rates, heightened volatility, and elevated risk premia.

On the the other hand, this exercise does put the current index value in to broader context. More specifically, it appears:

• the index was well above trend as recently as mid-February
• It declined to 11.9% below trend early last week
• It moved back to trend by the end of the week.

My sense is that the current recession is likely to be at least as severe as the one associated with the financial crisis of 2007-09. For that reason alone, I'd expect the S&P 500 to hit a level more than 11.9% below trend. 

But in addition, the mathematical models of virus transmission considered in previous posts all point in the same direction -- with the pandemic increasing considerably during the next few weeks. For example, between March 18 and March 29, reported deaths due to COVID-19 increased at an average daily rate of 13.4% in Italy, 24.9% in Spain, 29.4% in the UK, and 31.5% in the US.

As of March 29, deaths reported outside of China (where the virus currently appears contained) totaled 28,939. If this figure increased at a daily rate of 24.6% (the geometric mean of the death rates for Italy, Spain, the UK, and the US), we'd expect an additional 600,00 deaths to be reported between now and April 12. 

Over the same period, reported deaths globally ex China have increased at a daily rate of 15%. This is probably an underestimate of the death rate that will be experienced over the next few weeks, as the virus has only recently been reported in some regions, and death rates lag reported infection rates. But if reported deaths increased 'only' at a daily rate of 15%, this would still correspond to more than 175,000 additional deaths.

In the event that hundreds of thousands of additional deaths are reported in the next two weeks, I expect risk aversion levels among investors and traders would increase dramatically. In particular, I believe typical levels of risk aversion among investors and traders would easily exceed the levels experienced during the financial crisis of 2007-09.

Given my expectation that this recession is likely to be worse than the recession of 2007-09, and given my expectation that levels of risk aversion are likely to exceed those of 2007-09, it wouldn't be at all surprising for equity valuations to decline further below trend in this downturn than they did during the bear market of 2007-09.

None of this analysis is sufficient to provide a point forecast for this cycle's low in the S&P 500, but it seems unlikely to me that the S&P 500 is going to remain at or above trend as we go into this next, quite ugly phase of the current crisis in the next few weeks.

Oxford University's COVID-19 Analysis

Researchers at Oxford University just released a draft working paper titled, Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic. While the authors make claims with significant implications for the evolution of the epidemic in the UK, it's also been roundly criticized by other academics. With that in mind, I thought it might be useful to offer a few thoughts.

The ostensible purpose of the paper is to estimate the percentage of the UK population that already has been exposed to SARS-CoV-2 and that therefore reasonably might be expected to have developed some degree of immunity to the virus. If this proportion is sufficiently large, it may imply that the UK population already has developed a degree of  herd immunity, in which case public policy measures designed to slow the spread of the virus -- including those that are hobbling the economy -- may be relaxed.

The authors use the same sort of SIR model that I've discussed in previous posts. More specifically, they assume a range of plausible values for most of the parameters appearing in this model, and then they calibrate the model -- in particular, the day on which the virus first appeared in the population -- to a small subset of initial data. In particular, they use reported deaths for the first fifteen days after the first reported death in the population. They perform separate analyses for the UK and for Italy.

With this approach, the authors suggest that the virus was introduced into each population roughly one month before the first reported death in each population. In that case, the implication is that the virus would have spread widely among individuals in both populations by now. For the UK, they suggest that between 36% and 68% of the population would have been exposed to the virus by now. For Italy, the figure is estimated to be between 60% and 80%.

If true, this would mean that the policy of isolation is not necessary and that policy measures that have been crippling the UK and Italian economies can be relaxed. But does their analysis really support their conclusions?

One insurmountable problem with this analysis is the size of the data set: 15 days in each country. The authors choose this small subset in order to avoid using data from periods during which public policy interventions may have changed the course of the epidemic in each population.

Fifteen data points is far too small to reasonably calibrate the various parameters of their model, so the authors assume values that they believe are reasonable for many of these parameters, including

• the basic reproduction number, R0
• the duration of an infection
• the time between infection and death
• the probability of dying with severe disease
• the proportion of the population at risk of developing a severe infection.

But even if the authors had only one parameter to estimate -- the time of introduction of the virus into each population -- fifteen data points is almost surely far too few to estimate this time with any reasonable level of confidence. In fact, the authors don't report standard errors or confidence intervals for these estimates. Even with their many assumptions, I suspect these confidence intervals would be quite large. And if they were required to estimate all the parameters of their model using their fifteen data points, the results almost surely would be nonsensical, as there are simply too few data points per parameter.

A reasoned critique of the Oxford paper appeared today in a BMJ article titled, Covid-19: experts question analysis suggesting half UK population has been infected. It's useful to quote an excerpt from the BMJ critique. [Emphasis added.]

Neil Ferguson, director of the MRC Centre for Global Infectious Disease Analysis at Imperial College London, was asked about the study when he appeared before a parliamentary select committee hearing on 25 March. It was his analysis that showed that without physical distancing there would be 260 000 deaths in the UK from covid-19 and that led to change in government policy.

Ferguson said, “We’ve been analysing data from a number of Italian villages at the epicentre for the last few weeks where they did a viral swab on absolutely everybody in the village at different stages of the outbreak. And we can compare that with official case numbers, and those data all point to the fact that we are nowhere near the Gupta [the Oxford analysis] scenario in terms of the extent of the infection.”

Paul Hunter, professor in medicine at the University of East Anglia, said that the simple model “assumes complete mixing of the population,” which is “almost always wrong” at a country level. “We do not all have an equal random chance of meeting every other person in the UK.” He said that reproduction number was a “very clumsy” measure of how disease spreads, which is likely to change over time. He also criticised the researchers’ assumption that only a very small proportion of the population was at risk of being admitted to hospital because of the disease. “This is a big assumption and it is far too early in the epidemic to know what this value is,” he said.

Market Implications

If the Oxford results were true, we might reasonably expect significantly less economic disruption than many in the market have been expecting, in the UK, in Italy, and quite possibly across Europe and in the US as well. Given the historic reduction in economic activity that appears to be occurring, such a result would have significant implications for equity prices, bond yields, credit spreads, commodity prices, etc.

To be clear, I do believe the number of infections in the UK, the rest of Europe, and the US are very likely to be significantly higher than reported, particularly since there has been relatively little testing in these regions.

But the results of the comprehensive tests in Italian villages, mentioned by Dr Ferguson, suggest that the infection rates are likely significantly lower than suggested in the Oxford paper.

As I result, I continue to expect that the peak infection rates in the US and the UK are still weeks if not months ahead, with much higher infection rates becoming clear in coming weeks. As a consequence, I expect public policy measures will become more disruptive for the economy rather than less disruptive. And in that scenario, it seems quite unlikely that we've already seen the lows in the S&P 500 and in Treasury yields.

Empirical Infection Curves

In Time-Varying Parameters in Virus Transmission Models, I illustrated the effect that non-pharmaceutical interventions could have on the infection curve in a basic SIR virus transmission model. In particular, it's difficult to estimate the parameters of these models when there's reason to believe they're varying over time. And for that reason, it's useful to consider the empirical curves, when available.

For example, the graph below shows confirmed cases, recovered cases, infected cases, and deceased cases in recent months in China.

Source: John Hopkins

Note that none of the curves appear as simple as our basic SIR model would predict. Of course, there are numerous reasons for this. For example, the basic SIR may be too simple to capture dynamics in a real-world epidemic. In addition, Chinese authorities intentionally changed the basis on which some of the data was reported in mid-February. And important for our discussion: interventions on the part of authorities in China almost surely changed the dynamics of viral transmission within China. In particular, lowering the reproduction number, R0, is an important tactic in all efforts to restrict viral transmission.

Notwithstanding, the basic characteristics of viral transmission are clear in these reported figures. The 'infected' curve starts with near-exponential growth; it then reaches an inflection point followed by a peak, after which the number of people infected with the virus declines.

The next graph shows similar curves for South Korea.

Source: John Hopkins

Here again, note the initial growth, followed by an inflection point, followed by a peak, followed by a decline. South Korea appears to have rigorously followed WHO advice for people to isolate and for authorities to engage in intensive testing and contact tracing. And these curves give the impression that the South Koreans have made notable progress in recovering from the infection.

In contrast, the figures for Italy are shown in the next graph.

Source: John Hopkins

The 'infected' curve for Italy is no longer growing at a near-exponential rate, but neither has it reached an inflection point. As a result, it appears too early to forecast when Italians might experience the peak and the number of people who will be infected at the peak.

The same is true of Spain.

Source: John Hopkins

The reported transmission curves for the UK are shown in the next graph below.

Source: John Hopkins

Unfortunately, the number of reported infections in the UK is still exhibiting near-exponential growth. More specifically, simply fitting an exponential curve to the 'infected' curve in the UK still provides quite a good fit, as seen in the graph below.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the UK since these first topped 100, on 5-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the UK infection is still in its early stages of transmission. 

How about the US? The confirmed, infected, recovered, and deceased curves are shown in the graph below.

From this graph, it's difficult to tell whether the US infection is still at the initial near-exponential stage or whether it's beginning to bend toward an inflection point. The next graph below, with a fitted exponential curve, confirms that the US still appears to be at the near-exponential stage of growth. As with the UK, there's no basis in this data for predicting the timing or severity of peak infection in the US.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the US since these first topped 100, on 3-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the US infection is still in its early stages of transmission.

Even when circumstances are changing and the parameters governing the evolution of the infection are changing, it's useful to analyze the empirical data, when available -- particularly to look for

• the period of near-exponential growth in the infection curve
• the subsequent inflection point, when the rate of growth stops increasing
• the peak of the infection curve
• the subsequent decline in the incidence of infection.

Market implications

Of course, our interest isn't in epidemiology per se. We're ultimately interested in assessing the likely effect of the pandemic on economic growth, inflation, employment, and the financial markets.

With that in mind, what might we have learned from this exercise? A few things:

• China and South Korea appear to have turned the tide on their epidemics.
• The virus still appears to be spreading rapidly in Europe -- eg, in Italy and Spain, where the rate of infection is still increasing, though not at a near-exponential rate.
• In the US and the UK, the rate of infection still appears to be in the initial, near-exponential stage.

Equity markets have reacted favorably to news of fiscal stimulus in the US, the UK, and elsewhere -- and perhaps also to President Trump's insistence that US workers should return to work en masse by Easter (April 12). But given the still rapid rate at which the virus appears to be spreading in the US, it seems more likely that more American workers will be in isolation in mid-April than there are now. 

In fact, if the number of infections were to continue to grow at the current exponential rate between now and Easter, the total number of reported infections in the US would be on the order of 15 million.

Of course, it's quite possible that the infection curve will begin to move away from this near-exponential growth between now and then. It's even possible that the infection curve in the US will have reached an inflection point between now and Easter, in which case the number of reported infections may be well below 15 million.

But to put this in context, the peak number of active infections in China was reported to be about 58,000. Even if the number of infections in the US peaked at, say, 5 million, the number of infected would be roughly 86 times the peak number of infected people in China.

In Italy, 12% of all detected COVID-19 cases were admitted to the intensive care unit. If that percentage were observed in the US, and if the peak number of infections in the US were 'only' 5 million, that would still point to a simultaneous demand for 600,000 ICU beds in the US -- far outstripping the available supply, estimated to be on the order of 100,000.

I should stress that these figures are all just attempts to produce first-order approximations of the eventual numbers. But they're broadly consistent with figures produced elsewhere. For example, The Harvard Global Health Institute produced an online report, in which ICU beds in the US were estimated at just over 87,000. The number of patients requiring hospitalization for COVID-19 over time was predicted to be 10 million, and the number of patients requiring an ICU bed was predicted to be just over 2 million. These Harvard figures aren't for a single point in time, as were my attempts to predict peak demand for ICU beds. But these sorts of figures at least put the problem into broad perspective.

All in all, what are the conclusions from analyses of this sort?

1. The rate of infection in the US is still increasing at a near-exponential rate.
2. Peak infection is still well in the future in the US, with the peak infection rate many times greater than the infection rate currently reported.
3. The situation at Easter will almost surely be far worse than it is currently.
4. Under these circumstances, it is difficult to see droves of US employees returning to their workplaces, admonitions by the President notwithstanding. More likely, an even larger percentage of the US population will be under strict lock-down orders at Easter.

The rally in the S&P yesterday and today, from a low of 2174 (June S&P futures) at the start of the week to 2498 now (up by nearly 15%) is impressive. But my sense is that with peak infection well ahead of us, it's quite unlikely that we've seen the lows in the June S&P futures -- particularly if the number of peak infections is even close to the estimates consistent with the analysis discussed here.

Time-Varying Parameters in Virus Transmission Models

In Modeling the Incidence of COVID-19 Over Time, I mentioned some of the models I follow for the purpose of tracking the spread of the SARS-CoV-2 virus. These models are particularly useful in projecting the incidence of infection, recovery, death, etc., given a set of parameter values describing characteristics such a the average number of contacts per person per unit time, the probability of a contact resulting in an infection, and the typical duration of an infection.

In practice, most governments have introduced public policy measures designed to slow or reverse the spread of the virus, including social distancing, testing, and contact tracing. 

From the perspective of a transmission model, these non-pharmaceutical interventions are intended primarily to reduce the reproduction number, R0 of the pandemic. And to the extent these interventions are successful, we need to reflect that in our models.

For example, the graph below shows two infection curves for a hypothetical virus, in a population the size of the UK (67.9 million people). The blue curve assumes:

1. the average person has 20 contacts per day
2. the probability of a contact resulting in an infection is 0.01
3. the duration of the infection is 28 days.

With these three assumptions, the reproduction number, R0, is 5.6. The maximum number of infections under this scenario is 34.9 million people (51.4% of the poplulation), reached 114 days into the epidemic. Infections then decrease as individuals move from being infected to being removed (ie, immune or deceased).

The blue curve shows the trajectory of infections when the reproduction number, R0, is held constant at 5.6. The orange line shows the alternative trajectory in the event measures are introduced to reduce R0 to 0.7 -- in this case when the number of infected people first exceeds 10 million.

Now let's imagine that public policy interventions are introduced as the number of infections first reaches 10 million. In particular, let's assume these interventions reduce the average number of contacts per person per day from 20 to 5 and that the probability of a contact resulting in an infection declines from 0.01 to 0.005.

In this case, the reproduction number decreases from 5.6 to 0.7, with the result that that trajectory of the virus changes dramatically, from a rapid rate of increase to a moderate rate of decline, as illustrated by the orange line in the graph.

Now consider a country like the UK or the US, for which public policy measures have been changing nearly every day in recent weeks. These continual changes in policy measures create a situation in which R0 can't be assumed to be constant over the period of analysis one might otherwise use to estimate the parameters of the model.

As the experience of Imperial's MRC Centre illustrates, a model with inappropriate parameter values may actually be worse than useless in the event it supports policy measures that do more harm than good. So in that case, what good are these models -- particularly for financial analysts wishing to consider implications for the macroeconomy and financial markets?

From my perspective, these sorts of models offer two types of benefits. First, they help us understand the reasons that viral infection curves observed in practice are characterized by a period of near-exponential growth, followed by an inflection point, followed by a maximum, followed by a steady decline. Second, they provide a framework we can use to produce first-order approximations of viral transmission under various public policy initiatives. These projections needed to be treated with caution, due to the combination of model error, parameter estimation error, and data errors. (For example, who really knows the actual current number of infectious persons in the UK?)

And speaking of infection curves observed in practice, it's time to turn to some actual data...