Drug overdose deaths – another exceptional US epidemic

Earlier this week, CDC released provisional figures for drug overdose deaths in the USA in 2019. They estimate almost 72,000 deaths, the highest annual number yet seen. Initially driven by prescription opioid painkillers, users migrated first to heroin and then to fentanyl, which is cheaper (much of it illicitly made). Synthetic opioids accounted for an estimated 36,500 deaths.

The WHO Global Health Estimates  included estimates of direct deaths due to drug use disorders, including accidental poisoning by drugs and overdose deaths, as well as deaths coded to drug dependence (see Technical Note for methods and data).   Last year I updated these to year 2017 using the latest death registration data from the WHO Mortality Database, as well as recent reports for selected countries, and estimates of drug deaths from the IHME GBD 2017. It was apparent at the time that the USA was a massive outlier globally, responsible for a disproportionate proportion of global overdose deaths. I’ve increasingly noticed that the USA is an outlier from other developed countries for a number of health and social indicators, most recently the coronavirus pandemic, police killings, gun homicides and incarceration rates. I was interested to see how the USA compared with other countries for drug-related deaths, given its enormous investment in and the human cost of the “war on drugs”. I’ve done a quick update as described below, but I am fairly sure it gives a reasonably good general picture, not too different from the earlier 2017 update for WHO or the IHME GBD 2017 estimates.

I estimate that globally, drug use disorders caused close to 181,000 deaths in 2019, of which 40% were in the USA. The global average death rate for drug use disorders was 15 per million population, compared with 219 per million for the USA. In other words, the USA rate is 15 times higher than the global average.

I have projected recent trends forward to 2019 and updated drug death estimates for the USA, Canada, Australia and China based on recent reports. The analysis of death registration data is complicated by the need to distribute the accidental poisoning category for “other and unspecified chemicals and noxious substances” (X49) to the specific categories for alcohol and drug use disorders (opioids, cocaine, amphetamines, cannabis and “other drugs”) and to accidental poisoning (non-drug and non-alcohol). Additionally, there is a category F19 in the mental health chapter for “multiple drug use and unspecified drug use disorders” which is used to code deaths in some countries and also must be redistributed appropriately. So there is some uncertainty associated with the estimates for countries with death registration data, and even more for countries without good data. The GHE includes cannabis use disorders as a category, but I have not shown it in the graphs here as there are no overdose deaths.

In terms of the overall drug disorder death rate in 2019, the four leading countries were the USA (219 per million), Canada (121), Australia (71) and Russia (63). Opioids are responsible for many more deaths than other types of drugs.  The following graph shows the drug use death rates for the 15 leading countries in 2019, with the contributions of opioids, cocaine, amphetamines and other drugs.

The following graph shows time trends for opioid death rates in selected countries from 2000 to 2019.

The USA and Canada stand out as having substantially increasing opioid overdose death rates over the last decade. In fact, the opioid death rate for the USA has been growing exponentially since the 1980s until 2017, when there was a 5% drop for 2018, followed by a rise to slightly higher than the 2017 peak in 2019.  Its still too early to say whether the US drug death rates are stabilizing or will start to fall. A recent paper by Jalal et al (Science 2018) shows that the US drug overdose death rate fits an exponential curve from 1980 to 2016 resulting from multiple subepidemics with changing patterns of overdose deaths by age distribution (see graph below).

Source: Jalal et al. Changing dynamics of the drug overdose epidemic in the United States from 1979 through 2016. Science. 2018 

The following regional graphs highlight how different the situation in North America is to the rest of the world.

In this post, I have focused on direct deaths associated with drug use. However, drug use also raises risks for road injury deaths and suicide, and injecting drug users also face increased risks of death from HIV, hepatitis B and hepatitis C. Together, direct and indirect deaths associated with drug use account for close to half a million deaths annually in the world, and the great majority of these (74%) are associated with opioid use [conference-presentations #131].

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Covid-19: which countries are succeeding and not succeeding?

There is now sufficient data on social distancing, lockdown, and other measures that researchers are starting to assess the effectiveness of these measures. The July 4th edition of The Economist has an article summarizing what is becoming clear.  Extensive testing and tracing of contacts was carried out by the Asian countries earliest affected, China and South Korea. China also locked down Wuhan and largely succeeded to confining the epidemic there. Italy was the earliest European country affected, and it locked down Lombardy on February 22, and succeeded in largely confining the epidemic to the north of the country. Some European countries, Australia and New Zealand locked down hard and very early while there were still very few deaths.  These were the most successful. As were Switzerland, Germany and Austria which locked down while they still had fewer than 60 deaths. Other European countries that waited till they had hundreds of deaths before locking down, such as Britain, were much less successful and the virus spread throughout the country.

The Imperial College modelling group have estimated that Europe’s lockdown policies prevented 3 million deaths by May 4th. The first wave in the USA, which primarily hit New York and New Jersey, was fairly successfully contained as in the European countries, but other states which had not closed their borders or successfully reduced the spread of the virus, opened prematurely and the US is now facing a runaway growth in new cases and deaths (see my previous post).

The country plots below show that it only takes about 5-7 weeks of strong interventions to get rid of the majority of cases. Taking half-measures as the US has done does not work. Particularly when the President repeatedly contradicts the advice of the public health experts and refuses to provide national guidance.

A couple of months ago, I was producing country plots of time trends in cases and deaths to see how the epidemic was evolving across the world. Since then, a very nice site  https://www.endcoronavirus.org/countries has been doing these plots and updating them almost daily. The site groups countries into three groups: (1) those that are beating Covid-19, (2) those that are nearly there, and  (3) those that need to take action. The plots show new cases per day averaged over the last week and each graph has been normalized so all the peaks are the same height. This better shows the trend in the data and the degree to which the epidemic is being controlled, though not the relative magnitudes of the epidemics in different countries.

I’ve selected the plots for some countries of interest to me, to show differences in the impact of control measures in the three groups of countries, using data up to 4 July. The trends in some countries in the second group look quite similar to those in the first. The reason they are still in the second group is that the total number of new cases per day is higher that in the first group, and that can allow the epidemic to still escape control fairly easily.

Some countries that are beating Covid-19

Some countries that are nearly there

Countries that need to take more action

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An out-of-control pandemic in most world regions

I downloaded the Covid-19 data up to 5 July from the JHU Github today and plotted smoothed new cases per day (average over last 7 days) as rates per million population by world region. The dramatic difference in trajectories for Western Europe and the Americas is obvious. Rates in all other regions are rising except for East Asia and Pacific, which is dominated by China. Australia, New Zealand, Japan and South Korea are also included in that region.

I thought it would be interesting to take a closer look at the situation in the USA, as North America appears to have the most out-of-control epidemic.  I classified the US states into blue and red, according to the popular vote in presidential elections. A state was blue if a majority voted for the Democrat candidate in more than half of the last four elections and red if it voted for the Republican candidate in more than half.  For three states which voted two times for each party’s candidate, I used the five last elections to break the tie.

The first peak in April for the blue states is dominated by New York. However from early June with premature re-opening of blue states there has been dramatic growth in new cases at a similar rate to the early stage in late March, and all benefits of the lockdowns have been wasted. There is now an upwards trend in blue states as well starting later in June.

I’ve done some back of the envelope calculations of the worst case scenario where growth in cases is not brought under control and the infected population reaches the estimated herd immunity level of around 60%.  Recent data from Geneva confirms other studies that have estimated the overall infection mortality rate at 0.6%.  Assuming infection fatality rates do not decrease due to improving treatment regimes, that would mean there would be approximately 1 mllion further deaths in the USA. After I had done this calculation, I came across a post by an Australian infectious disease expert which also predicts a worst case death toll of 1 million deaths for the USA.

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Two different pandemics: the USA and Europe

The trajectories of COVID-19 have been dramatically different in the USA and Europe. Currently there are close to 30,000 new cases per day in the USA, compared with around 3,000 per day in the European Union (and around 20 per day in Switzerland which is not an EU member). Testing rates per million are similar in USA and EU, and the EU population is 446 million compared to 332 million in USA. Almost half of US states are reporting significant upward trend in new daily cases. In most cases, this is not a “second wave” but a restart of growth after social distancing was relaxed while there had not been any substantial downturn in new cases.

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Imperial study estimates 3.1 million deaths averted by the lockdown in Europe

I’ve stopped doing my own graphs to check trends for coronavirus. The 91-divoc.com website has added daily cases and daily deaths time series as well as moving 7-day average time trends to its visualizations, and has plots for countries for absolute numbers and rates per million population, and also for US States.  Looking at these yesterday, it is clear that apart from New York the US epidemic is not declining and for at least a dozen States, including Texas, North Carolina and California, the daily cases are trending upwards.  Likely as a result of the relaxation of social distancing with media reports of clusters of cases associated with various gatherings.

The Imperial College coronavirus modelling team published a new study on Monday which estimated that the lockdowns and other restrictions on daily life prevented around 3.1 million deaths in 11 European countries (Flaxman et al, https://www.nature.com/articles/s41586-020-2405-7). The study worked back from observed deaths (with all their limitations) up to 4 May to estimate transmission that occurred several weeks prior and hence the reproduction number R. They found that the various lockdowns and other restrictions on public life had reduced the reproduction number by an average 82%, bringing it below 1. They then ran the model to predict the number of deaths if no restrictions had been implemented to estimate the restrictions prevented 3.1 million deaths.

Overall, they estimated that between 12 and 15 million people had been infected in the period, or between 3.2 and 4 percent of the population of the 11 countries. The table below shows the estimated infection rate for each of the 11 countries.

I’d recently seen a plot of US states coronavirus rates against population density showing a strong correlation. I did a similar plot below for the 11 European countries, with linear trend line. Belgium and Sweden are outliers on the high side for infection rates, and Germany, Denmark, Austria and Switzerland on the low side. Of course, not too much should be read into this, as this analysis should really be done at subnational level, rural/urban at minimum or preferably, city, town, rural, which would require more detailed geographic breakdown within countries of the deaths.

Of course, the situation in Switzerland has been of particular interest to me as I live in Geneva, one of the hardest hit cantons. Regularly updated time series data are available for Swiss cantons at Covid-19 Information for Switzerland.

I plotted the total reported coronavirus deaths as at 8 June 2020 as rates per million population by canton in the graph below. I thought about doing a density plot, but as some cantons have populations confined to valleys between Alpine peaks, the density issue may not be straightforward. Instead, I highlighted the language differences. The yellow bar denotes the canton in which Italian is the dominant language, Ticino, and it has the highest death rates because it borders northern Italy where the epidemic hit early and hard, and the border was not closed until well into the epidemic. The blue denotes cantons where French is the dominant language, and red where German is the dominant language.  Clearly Romandy (the Swiss French population in the west of Switzerland) was hit much harder than Alemannic Switzerland (the part that speaks the Alemannic dialect of German known as Schweitzer Deutsche.

The Swiss Government has issued a statement explaining that the Swiss lock-down rules were applied equally in all cantons. The main reason for the high rates in French and Italian speaking cantons were because of the high rates in the neighbouring countries and high proportions of cross-border workers. Geneva has a very high population of frontaliers (people who live in neighbouring France and commute to Geneva to work or go to school) as well as a high level of international visitors associated with tourism, skiing, and business and NGO headquarters. Additionally, when the epidemic hit Europe there were large numbers of skiers heading through Geneva to the Alps, from countries including France and Britain. In contrast, infection rates in Germany and Austria, neighbouring the German-speaking parts of Switzerland, have been much lower than in France and Italy.

Social distancing has been progressively relaxing since April 27th and most children are back at school, and businesses open but with relatively unenforced limits on interpersonal distances. If rates start to go back up, as is happening in many US States now, I have no doubt the Swiss will efficiently and without polarizing debate reintroduce relevant restrictions.

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Machine learning projection model for Covid-19

Youyang Gu, an independent data scientist, has created a statistical projection model for Covid-19 (https://covid19-projections.com/) that uses machine learning techniques to fit a classical SEIR infectious disease model to the data for daily confirmed cases and deaths, taking into account the effects of social distancing and other factors. From the results I’ve looked at, it appears to be one of the better performing models around. The plots below show results for Switzerland, USA and United Kingdom based on data up to 31 May.

The second plot for each country shows R-t, the effective reproduction number at time t. When R is greater than 1, the epidemic is growing exponentially, and when R is less than 1 it is declining. The basic reproduction number in the absence of interventions to reduce transmission, R0, is typically around 2 for most countries, depending on factors such as population density and crowding. R0 was close to 4 for New York for example.

Looking across the country projections, it is interesting that R-t is currently slightly below 1 for countries such as Switzerland and UK, and marginally above 1 for the USA. It is more substantially below 1 for a few countries such as Norway and Australia, and above 1 for some countries, eg. Brazil, Russia and Nigeria.

A lot of people have now published strong criticisms of the IHME modelling, many identifying the major problem of fitting a mathematically symmetric curve to the epidemic which I noticed early on. Youyang Gu also compares IHME projections with his and shows severe under- and over-estimation issues with the IHME projections, which change wildly with model updates and iterations. See the plot below for a comparison.

Gu concludes:

“Models are going to make wrong predictions, but it’s important that we correct them as soon as new data shows otherwise. The problem with IHME is that they refused to recognize and update their wrong assumptions for many weeks. Throughout April, millions of Americans were falsely led to believe that the epidemic would be over by June because of IHME’s projections.

“On April 30, the director of the IHME, Dr. Chris Murray, appeared on CNN and continued to advocate their model’s 72,000 deaths projection by August. On that day, the US reported 63,000 deaths, with 13,000 deaths coming from the previous week alone. Four days later, IHME nearly doubled their projections to 135,000 deaths by August. One week after Dr. Murray’s CNN appearance, the US surpassed his 72,000 deaths by August estimate. It seems like an ill-advised decision to go on national television and proclaim 72,000 deaths by August only to double the projections a mere four days later.

“Unfortunately, by the time IHME revised their projections in May, millions of Americans have heard their 60,000-70,000 estimate. It may take a while to undo that misconception and undo the policies that were put in place as a result of this misleading estimate.”

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Police killings in the USA

The current protests in the USA made me curious to take a look at police killing rates compared to other countries. Wikipedia summarizes available statistics for most recent years, and I did some spot checks of the primary sources. The first graph shows a league table for high income countries with available data (this was all in the range 2017-2019).

For most recent year available, the US rate per 10 million population is 17 times higher than that for Australia and 57 times higher than that of United Kingdom. The police killed no-one in Switzerland in 2018 so the US ratio is infinite. The second graph breaks down US police killings by race (data for 2018 from Statistica.com).

The rate at which US blacks are killed by police is higher than the rate at which Iraqis are killed by police, and a similar magnitude to police killings in the Democratic Republic of Congo. But even the rates for US whites and US “Other” (mainly Asian) are also substantially higher than the rates of any of the other high income countries shown.

I haven’t done any systematic review of factors involved in the high US rates, but did come across an interesting article by a US Professor of Criminal Justice, which identified a number of factors that likely play key roles in the very high rates of US police killings.  These were:

  • Very high rates of gun ownership in the USA compared to other countries. In an earlier post, I found that there were 150 guns per 100 adults in the USA, compared to less than 9 per 100 for the rest of the world on average.
  • Racism (as evidenced by the high number of police shootings of unarmed African-Americans)
  • The localism of US policing with 15,500 separate municipal and county police forces (resulting in under-resourcing and staffing, inadequate disciplinary procedures and training)
  • Limited finances for local police often result in reliance on fines and asset seizure by police, resulting in more involuntary encounters with police.
  • Local policing results in high killing rates in small towns (25% of killings were in towns with fewer than 25,000 people)
  • Different standards for the use of deadly force. US police can use deadly force when they “reasonably” perceive imminent danger. In Europe, deadly force can only be used when “absolutely necessary” to achieve a lawful purpose and must be proportionate to the threat, with verbal warnings, warning shots, and shots at nonvital body parts where possible.
  • Compared to European countries where police training usually takes 2-3 years, the average recruit in the USA spends 19 weeks in training, with much more emphasis on training to use force than in conflict de-escalation.

The author also speculated that social and economic deprivation and injustice (and the lack of social safety nets), inadequate mental health care and intense desire to avoid harsh imprisonment may also result in higher levels of aggression in encounters by Americans with police.

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Country progress to 9 May in controlling COVID-19 epidemics

I downloaded the latest COVID-19 data for reported deaths and confirmed cases from Johns Hopkins this morning to see whether the data supports the relaxation of social isolation that is starting to happen in many countries. The USA now has 1.35 million confirmed cases (just under 1/3 of the global total) and 80,323 deaths (28% of the global total) and the trends are quite different for New York and the rest of the USA, as shown in the plot below. So I have done separate plots for New York and USA excluding New York in the following plots.

Daily deaths per million (dark blue line) and confirmed cases per 100,000 (red line).

The plots below show smoothed death rates (per million) and case rates (per 100,000) up to 9 May using simple moving averages. Inspired by some plots on www.endcoronavirus.org I’ve organized the countries into three groups:  those who have controlled the epidemic, those who are partway there (deaths are coming down, but not yet approaching zero) and those that need to take further action to turn things around. I excluded countries with population less than 1 million.

There are many more deaths and cases than recorded in this dataset. For countries with good data systems, excess deaths in the last two months is around 50 to 60% higher than confirmed coronavirus deaths, and the under-reporting is almost certainly much higher in most developing countries. However, these data probably provide a reasonably indication of the epidemic trends, at least in countries with reasonable testing levels and good data systems.

The final graph shows the top 20 countries in terms of deaths per million population, and their epidemic status. As to whether I’ve grouped the countries appropriately, there are a number where its debatable which group they should be in, and a few days more data may clarify what progress is occurring.

* Since almost all the deaths for China are in Hubei Province (Wuhan etc), I’ve used the population of Hubei rather than that of total China to calculate rates.

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COVID-19 projections and reality

On 27th April, I posted some short-run projections of COVID-19 cases and deaths. The plots below show how the daily new cases per million population and deaths per million population compare with reality (at least the confirmed case rates and death rates up to 4th May according to Johns Hopkins CSSEGIS Data.

Its a mixed bag. The projections match reasonably well for a few countries and are very different for others. I’ve revised the smoothing algorithm I used, and that may result in improved projections. But overall, I think I’m not doing much better than IHME, and should probably leave it to those with better models that use SEIR (susceptible-exposed-infected-recovered modeling) or computer simulations of case transmission.

Vox recently published an excellent article on the problems with the IHME modelling of COVID-19. The article also gives a link to a site which has been set up so that you can look at the US predictions made by old versions of the IHME model (and another model). The IHME models are frequently fairly far off. Here is the comparison for the USA as a whole (you can also examine State specific projections).

Its clear that the projection method takes the latest data point and plummets and essentially the same rate as the earlier rise. On second thoughts, I think my short-run projections are doing better than these for many countries.

 

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COVID-19 short-run projections

Purely statistical predictions of future trends in COVID-19 deaths or cases, even including predictive covariates, have been unable to make sensible forecasts that are not highly sensitive to slight additions of data. The only useful models have been the more traditional SIR and individual-based models of epidemic infectious disease spread in which scenarios allow some assessment of the impact of various social distancing measures.

It struck me there may be a way to make short-run projections of daily deaths, using information on recent trends in confirmed cases. I spent a couple of hours doing this yesterday, here is the result for Switzerland. I explain the simplistic modelling assumptions below, in what is intended only to be an exploratory analysis.

Reported daily new confirmed cases and deaths (dots) along with short-run projections.

A major issue is the quality of the available data on deaths and cases. Recent analysis of total mortality data for the pandemic period with similar data for previous years summarized in a recent New York Times article suggests that total deaths due to COVID-19 are around 60% higher than the reported COVID-19 deaths, which largely include only hospital deaths in many countries. These data are for countries with good death registration systems which cover 100% of deaths, albeit with some delays in registration and coding. For countries with poor or non-existent death registration, including most of Africa, the under-reporting will almost certainly be much higher. I’ve been looking at plots of daily deaths and noticed that there are occasional large increases in some countries for around 1 day in 7, and this may relate to timing of data compilation and reporting, or to “catch-up” when batches of deaths from outside the acute hospital sector are added. Additionally, the very strong age and sex dependence of fatality rates should mean that models need to take into account population age-sex structure and variation of case fatality rates and other factors by age and sex.

Confirmed case time series are also affected by the scope of testing and the extent to which the testing is restricted to specific risk groups or symptomatic people, or is extensive enough to approximate a population sampling. According to the data in Worldometers.info, country testing rates in developed countries, excluding those for some small populations under 1 million, range from around 10 per 1,000 in the UK up to around 35 per 1000 in some smaller European countries. For developing countries, rates can be much lower, less than 1 per 1,000 in many African and Asian countries. Trends in confirmed cases may be strongly influenced by trends in testing rates, as well as changes in the populations targeted for testing.

I downloaded latest data up to 25 April (CSSEGIS COVID-19 data) yesterday, and initially did some curve smoothing to look at trends in daily confirmed cases and deaths, mostly to see whether the epidemic does seem to have peaked in countries now discussing easing social distancing. It struck me that it may be possible to make use of these smoothed series to do some short-run projections of deaths out for around a week, without making strong assumptions about the shape of the curve, as was done by IHME in its recent modelling. Here is a plot for Switzerland of daily new confirmed cases and deaths.

My key insight was that trends in mortality should reflect trends in confirmed cases around 14 days earlier (based on a quick literature review). If testing rates are stable, this should allow projection of daily deaths out up to around 10-14 days based on the confirmed case time series. Furthermore, it should be possible to assess and project the trend in apparent case fatality rate (deaths divided by cases 14 days earlier) which should reflect the trend in testing rates and regime. So I spent a couple of hours yesterday having a go at doing this. This is not intended to be a serious attempt at prediction of short-run trends, as I’ve made some simplifying assumptions and picked a curve smoothing technique that was to hand, but not probably ideal. But I will compare reality in a week’s time with my projections, just for the heck of it.

In order to calculate the denominator (confirmed cases) for estimating apparent case fatality rates acfr, I assumed that the days d from diagnosis to death ranged from 8 to 21 days with a lognormal distribution with a mean of 14.1 days. In the limit where there was high levels of testing, I next assumed that the acfr should approach that observed for confirmed cases in Wuhan (2.2%) but adjusted for the age distribution of the country. So the long-term acfr, lacfr, will be higher for developed countries with older population distribution, such as Italy (4.3%) and lower for developing countries such as South Africa (1.2%).

I projected recent trend in ln(acfr-lacfr) using simple regression against time with an exponential weight, giving weight of 1 to the observation for the most recent day, and weights decreasing by factor 0.85 each day into the past. If the recent observed acfr was already lower than my estimated lacfr, I left it constant at its current value. For countries with recently declining acfr, the projection asymptotes at lacfr. For countries with increasing acfr, that increase is projected to continue. Only in a handful of countries does that projection result in dramatically increasing acfr. I’m not sure what that says about the data series, but there is clearly some issue with the data.

The following plots show two typical examples of the projection of apparent case fatality rates in which it starts very high (when deaths have started to occur but there is still very limited testing) and declines in a reasonably regular manner. The third example, for Sweden, is a country in which there is much more variability in the apparent case fatality rate, perhaps reflecting low numbers of cases and deaths, and also likely variations in data quality or scope.

Next, I did a similar short run projection of ln(smoothed daily confirmed cases) out 7 days into the future, and then calculated the denominators associated with smoothed daily deaths out 10 days into the future. This denominator projection actually uses only the first two days of the cases projection for the small fraction of the denominator associated with early deaths. I mostly did the projection of cases to see what a slightly longer projection of deaths looked like, but don’t present that here.

The plots below show examples of these projections for a few selected countries. I’ve added some comments in the captions.

Australia has a low case fatality rate, and appears to have indeed done extremely well in containing the epidemic, as is being claimed.

Switzerland shows clear evidence that the epidemic peaked in late March to early April and is in decline.

The US projections are for continuing increasing daily cases, and for slight decline in daily deaths, probably partly reflecting increasing levels of testing. These projections are probably not that meaningful, as the US has epidemics occurring with different timing in various States, and State level modelling would probably give more nuanced results.

Its unclear from this graph whether daily cases have plateaued, but a projected declining case fatality rate associated with increasing levels of testing has resulted in a projected decline in daily deaths. It will be interesting to see whether reality does better than this.

Clearly a continuing decline for Italy, though projected deaths are nearly flat. Again, reality will hopefully do better.

A rising apparent case fatality rate has resulted in a projected continuing increase in deaths. However, Germany has concluded that daily deaths are declining and it is time to relax social restrictions. Unclear from this data, but the death rate is much lower than for most other European countries.

France also show a peak followed by decline which is projected to continue.

The Netherlands has a death rate 40% higher than Switzerland, though the confirmed case rate is only 70% that of Switzerland. But both appear to be declining.

Larger variability in daily cases and deaths make it difficult to know whether these projected rising numbers are plausible. But there does not seem to be any good evidence the epidemic has peaked in Sweden.

These projections are simplistic, and really mainly to explore the data and the possibility of dealing with changing testing rates in doing projections.  A genuinely useful projection model of this type would not only need to have better evidence-based inputs but also ideally data disaggregated by age and sex, and for larger countries with epidemics in various population centres with different timing, to model at regional rather than national level. The likelihood of “later” epidemics starting in care home or other special population and either spreading into the community or causing later epidemic waves may also need to be taken into account. To a limited extent, it may be possible to treat the data on confirmed cases and hospital deaths as “indicators” of the epidemic and its dynamics, even though large numbers of cases and deaths are not included in such statistics. But large untested populations in institutions such as care homes or prisons could make a huge difference in some countries with relatively large institutionalized populations.

Still I conclude that there may be some value in using case data to make very short-run projections of deaths. Or perhaps to test the usefulness of such an approach using scenario results from one of the SIR or individual-based models.

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