On the Meaning of Herd Immunity

09 Saturday May 2020

Posted by pnuetz in Pandemic, Public Health, Risk

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Antibody, Antigen, Carl T. Bergstrom, Christopher Moore, Covid-19, Herd Immunity, Heterogeneity, Household Infection, Immunity, Infection Mortality Risk, Initial Viral Load, John Cochrane, Lockdowns, Marc Lipsitch, Muge Cevik, Natalie Dean, Natural Immunity, Philippe Lemoine, R0, Santa Fe Institute, SARS-CoV-2, Social Distancing, Super-Spreaders, Zvi Mowshowitz

Immunity doesn’t mean you won’t catch the virus. It means you aren’t terribly susceptible to its effects if you do catch it. There is great variation in the population with respect to susceptibility. This simple point may help to sweep away confusion over the meaning of “herd immunity” and what share of the population must be infected to achieve it.

Philippe Lemoine discusses this point in his call for an “honest debate about herd immunity“. He reproduces the following chart, which appeared in this NY Times piece by Carl T. Bergstrom and Natalie Dean:

Herd immunity, as defined by Bergstrom and Dean, occurs when there are sufficiently few susceptible individuals remaining in the population to whom the actively-infected can pass the virus. The number of susceptible individuals shrinks over time as more individuals are infected. The chart indicates that new infections will continue after herd immunity is achieved, but the contagion recedes because fewer additional infections are possible.

We tend to think of the immune population as those having already been exposed to the virus, and who have recovered. Those individuals have antibodies specifically targeted at the antigens produced by the virus. But many others have a natural immunity. That is, their immune systems have a natural ability to adapt to the virus.

Heterogeneity

At any point in a pandemic, the uninfected population covers a spectrum of individuals ranging from the highly susceptible to the hardly and non-susceptible. Immunity, in that sense, is a matter of degree. The point is that the number of susceptible individuals doesn’t start at 100%, as most discussions of herd immunity imply, but something much smaller. If a relatively high share of the population has low susceptibility, the virus won’t have to infect such a large share of the population to achieve effective herd immunity.

The apparent differences in susceptibility across segments of the population may be the key to early herd immunity. We’ve known for a while that the elderly and those with pre-existing conditions are highly vulnerable. Otherwise, youth and good health are associated with low vulnerability.

Lemoine references a paper written by several epidemiologists showing that “variation in susceptibility” to Covid-19 “lowers the herd immunity threshold”:

“Although estimates vary, it is currently believed that herd immunity to SARS-CoV-2 requires 60-70% of the population to be immune. Here we show that variation in susceptibility or exposure to infection can reduce these estimates. Achieving accurate estimates of heterogeneity for SARS-CoV-2 is therefore of paramount importance in controlling the COVID-19 pandemic.”

The chart below is from that paper. It shows a measure of this variation on the horizontal axis. The colored, vertical lines show estimates of historical variation in susceptibility to historical viral episodes. The dashed line shows the required exposure for herd immunity as a function of this measure of heterogeneity.

Their models show that under reasonable assumptions about heterogeneity, the reduction in the herd immunity threshold (in terms of the percent infected) may be dramatic, to perhaps less than 20%.

Then there are these tweets from Marc Lipsitch, who links to this study:

“As an illustration we show that if R0=2.5 in an age-structured community with mixing rates fitted to social activity studies, and also categorizing individuals into three categories: low active, average active and high active, and where preventive measures affect all mixing rates proportionally, then the disease-induced herd immunity level is hD=43% rather than hC=1−1/2.5=60%.”

Even the celebrated Dr. Bergstrom now admits, somewhat grudgingly, that hereogeniety reduces the herd immunity threshold, though he doesn’t think the difference is large enough to change the policy conversation. Lipsitch also is cautious about the implications.

Augmented Heterogeneity

Theoretically, social distancing reduces the herd immunity threshold. That’s because infected but “distanced” people are less likely to come into close contact with the susceptible. However, that holds only so long as distancing lasts. John Cochrane discusses this at length here. Social distancing compounds the mitigating effect of heterogeneity, reducing the infected share of the population required for herd immunity.

Another compounding effect on heterogeneity arises from the variability of initial viral load on infection (IVL), basically the amount of the virus transmitted to a new host. Zvi Mowshowitz discusses its potential importance and what it might imply about distancing, lockdowns, and the course of the pandemic. In any particular case, a weak IVL can turn into a severe infection and vice versa. In large numbers, however, IVL is likely to bear a positive relationship to severity. Mowshowitz explains that a low IVL can give one’s immune system a head start on the virus. Nursing home infections, taking place in enclosed, relatively cold and dry environments, are likely to involve heavy IVLs. In fact, so-called household infections tend to involve heavier IVLs than infections contracted outside of households. And, of course, you are very unlikely to catch Covid outdoors at all.

Further Discussion

How close are we to herd immunity? Perhaps much closer than we thought, but maybe not close enough to let down our guard. Almost 80% of the population is less than 60 years of age. However, according to this analysis, about 45% of the adult population (excluding nursing home residents) have any of six conditions indicating elevated risk of susceptibility to Covid-19 relative to young individuals with no co-morbidities. The absolute level of risk might not be “high” in many of those cases, but it is elevated. Again, children have extremely low susceptibility based on what we’ve seen so far.

This is supported by the transmission dynamics discussed in this Twitter thread by Dr. Muge Cevik. She concludes:

“In summary: While the infectious inoculum required for infection is unknown, these studies indicate that close & prolonged contact is required for #COVID19 transmission. The risk is highest in enclosed environments; household, long-term care facilities and public transport. …

Although limited, these studies so far indicate that susceptibility to infection increases with age (highest >60y) and growing evidence suggests children are less susceptible, are infrequently responsible for household transmission, are not the main drivers of this epidemic.”

Targeted isolation of the highly susceptible in nursing homes, as well as various forms of public “distancing aid” to the independent elderly or those with co-morbidities, is likely to achieve large reductions in the effective herd immunity ratio at low cost relative to general lockdowns.

The existence of so-called super-spreaders is another source of heterogeneity, and one that lends itself to targeting with limitations or cancellations of public events and large gatherings. What’s amazing about this is how the super-spreader phenomenon can lead to the combustion of large “hot spots” in infections even when the average reproduction rate of the virus is low (R0 < 1). This is nicely illustrated by Christopher Moore of the Santa Fe Institute. Super-spreading also implies, however, that while herd immunity signals a reduction in new infections and declines in the actively infected population, “hot spots” may continue to flare up in a seemingly random fashion. The consequences will depend on how susceptible individuals are protected, or on how they choose to mitigate risks themselves.

Conclusion

I’ve heard too many casual references to herd immunity requiring something like 70% of the population to be infected. It’s not that high. Many individuals already have a sort of natural immunity. Recognition of this heterogeneity has driven a shift in the emphasis of policy discussions to the idea of targeted lockdowns, rather than the kind of indiscriminate “dumb” lockdowns we’ve seen. The economic consequences of shifting from broad to targeted lockdowns would be massive. And why not? The health care system has loads of excess capacity, and Covid infection fatality risk (IFR) is turning out to be much lower than the early, naive estimates we were told to expect, which were based on confirmed case fatality rates (CFRs).

Private Social Distancing, Private Reversal

04 Monday May 2020

Posted by pnuetz in Liberty, Pandemic, Uncategorized

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Andrew Cuomo, Anthony Fauci, Apple Mobility, Bill De Blasio, Centre for Economic Policy Research, Donald Trump, Externalities, Forbes, Foursquare, Heterogeneity, John Koetsier, Laissez Faire, Lockdowns, Nancy Pelosi, Points of Interest, Private Governance, Safegraph, Social Distancing, Social Welfare, Stay-at-Home Orders, Vitamin D, Wal Mart, WHO

My original post on the dominance of voluntary social distancing over the mandated variety appears below. That dominance is qualified by the greater difficulty of engaging in certain activities when they are outlawed by government, or when the natural locations of activities are declared off-limits. Nevertheless, as with almost all regulation, people make certain “adjustments” to suit themselves (sometimes involving kickbacks to authorities, because regulation does nothing so well as creating opportunities for graft). Those “adjustments” often lead to much less desirable outcomes than the original, unregulated state. In the case of a pandemic, however, it’s tempting to view such unavoidable actions as a matter of compromise.

I say this now because the voluntary social distancing preceding most government lockdown orders in March (discussed in the post below) is subject to a degree of self-reversal. Apple Mobility Data suggests that something like that was happening throughout much of April, as shown in the chart at the top of this post. Now, in early May, the trend is likely to continue as some of the government lockdown mandates are being lifted, or at least loosened.

An earlier version of the chart above appeared in a Forbes article entitled, “Apple Data Shows Shelter-In-Place Is Ending, Whether Governments Want It To Or Not“. The author, John Koetsier, noted the Apple data are taken from map searches, so they may not be reliable indicators of actual movement. But he also featured some charts from Foursquare, which showed actual visits to various kinds of destinations, and some of theoe demonstrate the upward trend in activity.

In the original post below, I used SafeGraph charts lifted from a paper I described there. The four charts below are available on the SafeGraph website, which offered the services of the friendly little robot in the lower right-hand corner, but I demurred. You’ll probably need to click on the image to read the detail. They show more granular information by industry, brand, region, and restaurant categories. The upward trends are evident in quite a few of the series.

I should qualify my interpretation of the charts above and those in my original post: First, nine states did not have stay-at-home orders, though a few of those had varying restrictions on individuals and on the operation of “non-essential” businesses. The five having no orders of any kind (that I can tell) are lightly-populated, very low-density states, so the vast majority of the U.S. population was subject to some sort of lockdown measure. Second, eight states began to ease or lift orders in the last few days of April, Georgia and Colorado being the largest. Therefore, at the tail end, a small part of the increase in activity could be related to those liberalizations. Then again, it might have happened anyway.

The authoritarian impulse to shut everything down was largely unnecessary, and it did not accomplish much that voluntary distancing hadn’t accomplished already (again, see below). Healthy people need to stop cowering and take action. That includes the non-elderly and those free of underlying health conditions. Sure, take precautions, keep your distance, but get out of your home if you can. Get some sunny Vitamin D.

Committing yourself to the existence of a shut-in is not healthy, not wise, and it might destroy whatever wealth you possess if you are a working person. The data above show that people are recognizing that fact. As much as the Left wishes it were so, government seldom “knows better”. It is least effective when it uses force to suppress voluntary behavior; it is most effective when it follows consensus, and especially when it protects the rights of individuals to make their own choices where no consensus exists.

Last week’s post follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

How much did state and local governments accomplish when they decided to issue stay-at-home orders? Perhaps not much. That’s the implication of data presented by the authors of “Internal and external effects of social distancing in a pandemic” (starts on page 22 in the linked PDF). Social distancing began in the U.S. in a series of voluntary, private actions. Government orders merely followed and, at best, reinforced those actions, but often in ham-handed ways.

The paper has a broader purpose than the finding that social distancing is often a matter of private initiative. I’ll say a bit more about it, but you can probably skip the rest of this paragraph without loss of continuity. The paper explores theoretical relationships between key parameters (including a social distancing construct) and the dynamics of a pandemic over time in a social welfare context. The authors study several alternatives: a baseline in which behavior doesn’t change in any way; a “laissez faire” path in which actions are all voluntary; and a “socially optimal” path imposed by a benevolent and all-knowing central authority (say what???). I’d offer more details, but I’ll await the coming extension promised by the authors to a world in which susceptible populations are heterogenous (e.g., like Covid-19, where children are virtually unaffected, healthy working age adults are roughly as at-risk as they are to the flu, and a population of the elderly and health-compromised individuals for which the virus is much more dangerous than the flu). In general, the paper seems to support a more liberalized approach to dealing with the pandemic, but that’s a matter of interpretation. Tyler Cowen, who deserves a hat-tip, believes that reading is correct “at the margin”.

Let’s look at some of the charts the authors present early in the paper. The data on social distancing behavior comes from Safegraph, a vendor of mobility data taken from cell phone location information. This data can be used to construct various proxies for aggregate social activity. The first chart below shows traffic at “points of interest” (POI) in the U.S. from March 8 to April 12, 2020. That’s the blue line. The red line is the percentage of the U.S. population subject to lockdown orders on each date. The authors explain the details in the notes below the chart:

Clearly POI visits were declining sharply before any governments imposed their own orders. The next two charts show similar declines in the percent of mobile devices that leave “home” each day (“home” being the device’s dominant location during nighttime hours) and the duration over which devices were away from “home”, on average.

So all of these measures of social activity began declining well ahead of the government orders. The authors say private social distancing preceded government action in all 50 states. POI traffic was down almost 40% by the time 10% of the U.S. population was subject to government orders, and those early declines accounted for the bulk of the total decline through April 12. The early drops in the two away-from-home measures were 15-20%, again accounting for well over half of the total decline.

The additional declines beyond that time, to the extent they can be discerned, could be either trends that would have continued even in the absence of government orders or reinforcing effects the orders themselves. This does not imply that lockdown orders have no effects on specific activities. Rather, it means that those orders have minor incremental effects on measures of aggregate social activity than the voluntary actions already taken. In other words, the government lockdowns are largely a matter of rearranging the deck chairs, or, that is to say, their distribution.

Many private individuals and institutions acted early in response to information about the virus, motivated by concerns about their own safety and the safety of family and friends. The public sector in the U.S. was not especially effective in providing information, with such politicos as President Donald Trump, Nancy Pelosi, Andrew Cuomo, Bill De Blasio, and the mayor of New Orleans minimizing the dangers into the month of March, and some among them encouraging people to get out and celebrate at public events. Even Anthony Fauci minimized the danger in late February (not to mention the World Health Organization). In fact, “the scientists” were as negligent in their guidance as anyone in the early stages of the pandemic.

When lockdown orders were issued, they were often arbitrary and nonsensical. Grocery stores, liquor stores, and Wal Mart were allowed to remain open, but department stores and gun shops were not. Beaches and parks were ordered closed, though there is little if any chance of infection outdoors. Lawn care services, another outdoor activity, were classified as non-essential in some jurisdictions and therefore prohibited. And certain personal services seem to be available to public officials, but not to private citizens. The lists of things one can and can’t buy truly defies logic.

In March, John W. Whitehead wrote:

“We’re talking about lockdown powers (at both the federal and state level): the ability to suspend the Constitution, indefinitely detain American citizens, bypass the courts, quarantine whole communities or segments of the population, override the First Amendment by outlawing religious gatherings and assemblies of more than a few people, shut down entire industries and manipulate the economy, muzzle dissidents, ‘stop and seize any plane, train or automobile to stymie the spread of contagious disease,’…”

That is fearsome indeed, and individuals can accomplish distancing without it. If you are extremely risk averse, you can distance yourself or take other precautions to remain protected. You can either take action to isolate yourself or you can decide to be in proximity to others. The more risk averse among us will internalize most of the cost of voluntary social distancing. The less risk averse will avoid that cost but face greater exposure to the virus. Of course, this raises questions of public support for vulnerable segments of the population for whom risk aversion will be quite rational. That would certainly be a more enlightened form of intervention than lockdowns, though support should be offered only to those highly at-risk individuals who can’t support themselves.

Christopher Phelan writes of three rationales for the lockdowns: buying time for development of a vaccine or treatments; reducing the number of infected individuals; and to avoid overwhelming the health care system. Phelan thinks all three are of questionable validity at this point. A vaccine might never arrive, and Phelan is pessimistic about treatments (I have more hope in that regard). Ultimately a large share of the population will be infected, lockdowns or not. And of course the health care system is not overwhelmed at this point. Yes, those caring for Covid patients are under a great stress, but the health care system as a whole, and patients with other maladies, are currently suffering from massive under-utilization.

If you wish to be socially distant, you are free to do so on your very own. Individuals are quite capable of voluntary risk mitigation without authoritarian fiat, as the charts above show. While private actors might not internalize all of the external costs of their activities, government is seldom capable of making the appropriate corrections. Coercion to enforce the kinds of crazy rules that have been imposed during this pandemic is the kind of abuse of power the nation’s founders intended to prevent. Reversing those orders can be difficult, and the precedent itself becomes a threat to future liberty. Nevertheless, we see mounting efforts to resist by those who are harmed by these orders, and by those who recognize the short-sighted nature of the orders. Private incentives for risk reduction, and private evaluation of the benefits of social and economic activity, offer superior governance to the draconian realities of lockdowns.

Social Distancing Largely a Private Matter

26 Sunday Apr 2020

Posted by pnuetz in Liberty, Pandemic, Uncategorized

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Andrew Cuomo, Anthony Fauci, Bill De Blasio, Centre for Economic Policy Research, Donald Trump, Externalities, Heterogeneity, Laissez Faire, Lockdowns, Nancy Pelosi, Points of Interest, Private Governance, Safegraph, Social Distancing, Social Welfare, Stay-at-Home Orders, Wal Mart, WHO

In March, John W. Whitehead wrote:

Spanish Flu: No Guide for Covid Lockdowns

25 Saturday Apr 2020

Posted by pnuetz in Pandemic

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Cost of Lost Output, Covid-19, Cytokine Storm, Economic Costs, Excess Mortality, Herd Immunity, Life-Years, Lockdown, Non-Prescription Measures, Novel Coronavirus, Pandemic, Quarantines, Reason.com, Serological Testing, Skilled Care, Social Distancing, South Korea, Spanish Flu, World War I

The coronavirus pandemic differs in a few important ways from the much deadlier Spanish flu pandemic of 1918-19. Estimates are that as much as 1/3rd of the world’s population was infected during that contagion, and the case fatality rate is estimated to have been 10-20%. The current pandemic, while very serious, will not approach that level of lethality.

Another important difference: the Spanish Flu was very deadly among young adults, whereas the Coronavirus is taking its greatest toll on the elderly and those with significant co-morbidities. Of course, the Spanish Flu infected a large number of soldiers and sailors, many returning from World War I in confined conditions aboard transport vessels. A major reason for its deadliness among young adults, however, is thought to be the “cytokine storm“, or severe inflammatory response, it induced in those with strong immune systems.

It’s difficult to make a perfect comparison between the pandemics, but the charts below roughly illustrate the contrast between the age distribution of case mortality for the Spanish Flu in 1918, shown in the first chart, and Covid-19 in the second. The first shows a measure of “excess mortality” for each age cohort as the vertical gap between the solid line (Spanish flu) and the dashed line (the average of the seven previous seasons for respiratory diseases). Excess mortality was especially high among those between the ages of 15 and 44.

The second chart is for South Korea, where the Covid-19 pandemic has “matured” and was reasonably well controlled. We don’t yet have a good measure of excess case mortality for Covid-19, but it’s clear that it is most deadly among the elderly population. Not to say that infected individuals in younger cohorts never suffer: they are a higher proportion of diagnosed cases, severe cases are of extended duration, and some of the infected might have to deal with lasting consequences.

One implication of these contrasting age distributions is that Covid-19 will inflict a loss of fewer “life years” per fatality. If the Spanish flu’s median victim was 25 years old, then perhaps about 49 life years were lost per fatality, based on life expectancies at that time. At today’s life expectancies, it might be more like 54 years. if Covid-19’s median victim is 70 years old, then perhaps 15 life-years are lost per fatality, or about 73% less. And that assumes the the median Covid victim is of average health, so the loss of life years is probably less. But what a grisly comparison! Any loss is tragic, but it is worth noting that the current pandemic will be far less severe in terms of fatalities, excess mortality (because the elderly always die at much higher rates), and in life-years lost.

Is that relevant to the policy discussion? It doesn’t mean we should throw all caution to the wind. Ideally, policy would save lives and conserve life-years. We’d always put children on the lifeboats first, after all! But in this case, younger cohorts are the least vulnerable.

The flu pandemic of 1918-19 is often held to support the logic of non-prescription public health measures such as school closures, bans on public gatherings, and quarantines. Does the difference in vulnerabilities noted above have any bearing on the “optimal” level of those measures in the present crisis? Some argue that while a so-called lockdown confers health benefits for a Spanish flu-type pandemic in which younger cohorts are highly vulnerable, that is not true of the coronavirus. The young are already on lifeboats having few leaks, as it were.

My view is that society should expend resources on protecting the most vulnerable, in this case the aged and those with significant co-morbidities. Health care workers and “first responders” should be on the list as well. If well-targeted and executed, a Covid-19 lockdown targeted at those groups can save lives, but it means supporting the aged and afflicted in a state of relative isolation, at least until effective treatments or a vaccine prove out. A lockdown might not change living conditions greatly for those confined to skilled care facilities, but much can be done to reduce exposure among those individuals, including a prohibition on staff working at multiple facilities.

Conversely, the benefits of a lockdown for younger cohorts at low risk of death are much less compelling for Covid-19 than might be suggested by the Spanish flu experience. In fact, it can be argued that a complete lockdown denies society of the lowest-hanging fruit of earlier herd immunity to Covid-19. Younger individuals who have more social and economic contacts can be exposed with relative safety, and thus self-immunized, as their true mortality rate (including undiagnosed cases in the denominator) is almost zero to begin with.

Then we have the economic costs of a lockdown. Idle producers are inherently costly due to lost output, but idle non-producers don’t impose that cost. For Covid-19, prohibiting the labor of healthy, working age adults has scant health benefits, and it carries the high economic costs of lost output. That cost is magnified by the mounting difficulty of bringing moribund activities back to life, many of which will be unsalvageable due to insolvency.

The lockdown question is not binary. There are ways to maintain at least modest levels of production in many industries while observing guidelines on safety and social distancing. In fact, producers are finding inventive ways of maximizing both production and safety. They should be relied upon to create these solutions. The excess mortality rates associated with this pandemic will continue to come into focus at lower levels with more widespread serological testing. That will reinforce the need for individual autonomy in weighing risks and benefits. Hazards are always out there: reckless or drunk drivers, innumerable occupational hazards, and the flu and other communicable diseases. Protect yourself in any way you see fit, but if you are healthy, please do so without agitating for public support from the rest of us, and without imposing arbitrary judgments on which activities carry acceptable risk for others.

Coronavirus: Framing the Next Few Weeks #3

05 Sunday Apr 2020

Posted by pnuetz in Pandemic, Risk Management

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80000 Hours, Chris Murray Model, Christopher Monkton, Christopher Murray, Co-morbidity, Confidence Interval, Coronavirus, Covid Tracking Project, Covid-19, Economic Restart, Indur M. Goklany, Institute for Health Metrics and Evaluation, Lockdowns, Pending Tests, Positive Test Ratio, Social Distancing, Stay-at-Home Orders, White House Coronavirus Task Force

There were a few encouraging signs of change over the past few days in the course of the coronavirus pandemic in the U.S. This is the third of my quaint efforts to provide perspective on the coronavirus pandemic with “tracking” or “framing” posts. The first two were: Coronavirus: Framing the Next Few Weeks, on March 22, and Framing Update on March 28. In both of those posts, I charted confirmed cases of Covid-19 in the U.S. along with optimistic and pessimistic scenarios. I speculated that over the course of a few weeks, social distancing would lead to a reduction in the daily number of new confirmed cases. Unfortunately, it’s not clear whether that curve has started to “bend” rightward, but new confirmed cases on Sunday, April 5 — the number of those testing positive — was down almost 30% from Saturday.

An updated version of the earlier chart appears below, accompanied by a table. The number of confirmed cases (red line) has mounted over the past week, as has the daily increase in confirmed cases, though yesterday’s number was better. The table below the chart shows that the growth rate of confirmed cases (the far right-hand column) has decelerated, but it had leveled off at about 14% over the few days before Sunday. If Sunday’s drop persists it would be encouraging. Unfortunately, even moderate growth rates are destructive when the base of confirmed cases is large. The faster the growth rate declines, the faster the curve will bend.

I did not make any changes to the original “Very Good” and “Pretty Bad” scenarios, deciding that it was better to keep them as a consistent benchmark. As of Sunday, the top of the “Very Good” curve would still be about 2.3x the North Korean experience as of Sunday night, normalized for population. The top of the “Pretty Bad” scenario (which is not visible in this “zoomed-in” version) would be about 1.4x the Italian experience thus far. South Korea’s curve flattened substantially several weeks ago, and now even Italy’s curve is showing a rightward bend. Let’s hope that continues.

The case count obviously depends on the volume of daily testing, which has been increasing rapidly. As I’ve noted before, there has been a backlog of test requests. In addition, every day there is an overhang of “pending” test results. Interestingly, the number of tests stabilized on Saturday and the number of pending test results plunged (see the next chart, which uses data from the Covid Tracking Project). We’ll see if those developments persist. It would represent a milestone because daily case counts will advance as long as tests do, and the effort to work through the backlog has been inflating the speed of the advance in confirmed cases.

Another interesting development coincident with the drop in pending tests has to do with the cumulative percentage of positive test results: it has stabilized after growing for several weeks. This might mean we’ve reached a point at which the most severe incoming cases are fewer, but we’ll have to see if the flattening persists or even declines, which would be wonderful.

I’ve been grappling with potential weaknesses of the data on confirmed cases: first, the U.S. got a late start on testing, so there was the backlog of patients requiring tests just discussed above. That was presumed to have exaggerated the acceleration in the daily totals for new cases. Second, it’s possible that a continuing transition to more rapid test results would exaggerate the daily counts of new cases. Third, It’s possible that testing criteria are being relaxed, which, despite reducing the positive test rate, would increase growth in the “official” confirmed case count. Suspected cases should be tested, of course, but the change in standards is another factor that distorts the shape of the curve.

Any published statistic has its shortcomings. All test results are subject to false positives and negatives. Hospitalizations of patients with a positive coronavirus diagnosis are subject to the measurement issues as well, though they might be driven more by the severity of symptoms and co-morbidities than a positive Covid diagnosis per se. And hospitalizations of Covid patients might be subject to inconsistencies in reporting, and so might ICU admittances. Coronavirus deaths are subject to vagaries: reporting a cause of death is dictated by various criteria when co-morbidities are involved, and those criteria differ from country to country, or perhaps even hospital to hospital and doctor to doctor! In fact, some go so far as to say that all deaths should be tracked for coronavirus plus its co-morbidities and then compared to an average of the past five to ten years to obtain an estimate of “excess deaths”, which could conceivably be negative or positive. Finally, recoveries are even more impacted by inconsistent reporting, especially because many recoveries occur at home.

I’ll be highlighting coronavirus deaths going forward, and I’ll continue to focus mainly on the U.S. and only lightly on other countries. After all, death is obviously the most negative outcome. Again, however, the count of coronavirus deaths does not account for deaths that would have occurred over the same time frame due to co-morbidities or the effect of deaths that would not have occurred absent co-morbidities.

The predictions in the chart below are from the Chris Murray Model, upon which the White House Coronavirus Task Force has focused more recently. This model was developed by Dr. Christopher Murray at the Institute for Health Metrics and Evaluation at the University of Washington. I’ll be using the forecast starting from April 2 as the basis for my “framing” of actual deaths over the next few weeks, keeping it “frozen” at that level as a new benchmark. My apologies for the absence of dates on the horizontal axis, but the origin is at March 14 and there is only a single line up through April 1. That line represents actual cumulative Covid deaths recorded in the U.S. The red line is the mean model prediction. Deaths are expected to ramp up over the next week or so, much as we’ve seen in the confirmed case count, as deaths lag behind diagnoses by anywhere from a few days up to 17 days. This model predicts an ultimate death toll of about 94,000 at the top of the mean curve (not visible on the zoomed-in chart). Above and below that line are upper (blue) and lower (green) bounds, respectively, of a “confidence interval”. It’s unlikely we’ll see cumulative deaths breach either of those bounds. The lower bound would place the ultimate death toll at about 40,000; the upper bound would place it at just under 180,000. At this point, as of Sunday, April 5, actual deaths (black) are slightly below the mean or central tendency, but that’s difficult to see in the current chart.

The ongoing lockdowns in the U.S. and around the world are exceedingly controversial. There is a very real tradeoff between the benefits of extending the length of these lockdowns and the benefits of allowing economic activity to “restart”. But do lockdowns work? Christopher Monkton offers aggregate evidence that they truly do reduce the spread of the virus in “Are Lockdowns Working?” That they would work is intuitive, and decisions to scale them back should be made cautiously for certain “high exposure” activities, and in conjunction with isolating and trace-tracking contacts of all infected individuals, as they have done successfully in countries such as Taiwan and Singapore.

There may be signs that a bend in the U.S. case curve is in the offing, perhaps over the next week or two. That timing would roughly comport with the notion that over the past three weeks, efforts at social distancing and stay-at-home orders have allowed the U.S. to limit the spread of coronavirus through the first major “round” of infections and a much more limited second round. Perhaps these efforts will largely stanch a third and subsequent rounds of infection. Ultimately, if the number of coronavirus deaths is in the neighborhood of the mean value predicted by the Murray Model, about 94,000, that would place the severity of the toll at less than two times the severity of a bad flu season, though limiting the Covid death toll will have been achieved at much higher economic cost.

There are signs elsewhere around the globe that the pandemic may be turning a corner toward more favorable trends. See “Good News About COVID-19” at 80,000hours.org for a good review.