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Coronavirus Covid-19 Michael Levitt

Mechanistic and curve-fitting UK modelling comparison

Introduction

In my most recent post, I summarised the various methods of Coronavirus modelling, ranging from phenomenological “curve-fitting” and statistical methods, to the SIR-type models which are developed from differential equations representing postulated incubation, infectivity, transmissibility, duration and immunity characteristics of the SARS-Cov-2 virus pandemic.

The phenomenological methods don’t delve into those postulated causations and transitions of people between Susceptible, Infected, Recovered and any other “compartments” of people for which a mechanistic model simulates the mechanisms of transfers (hence “mechanistic”).

Types of mechanistic SIR models

Some SIR-type mechanistic models can include temporary immunity (or no immunity) (SIRS) models, where the recovered person may return to the susceptible compartment after a period (or no period) of immunity.

SEIRS models allow for an Exposed compartment, for people who have been exposed to the virus, but whose infection is latent for a period, and so who are not infective yet. I discussed some options in my late March post on modelling work reported by the BBC.

My model, based on Alex de Visscher’s code, with my adaptations for the UK, has seven compartments – Uninfected, Infected, Sick, Seriously Sick, Better, Recovered and Deceased. There are many variations on this kind of model, which is described in my April 14th post on modelling progress.

Phenomenological curve-fitting

I have been focusing, in my review of modelling methods, on Prof. Michael Levitt’s curve-fitting approach, which seems to be a well-known example of such modelling, as reported in his recent paper. His small team have documented Covid-19 case and death statistics from many countries worldwide, and use a similar curve-fitting approach to fit current data, and then to forecast how the epidemics might progress, in all of those countries.

Because of the scale of such work, a time-efficient predictive curve-fitting algorithm is attractive, and they have found that a Gompertz function, with appropriately set parameters (three of them) can not only fit the published data in many cases, but also, via a mathematically derived display method for the curves, postulate a straight line predictor (on such “log” charts), facilitating rapid and accurate fitting and forecasting.

Such an approach makes no attempt to explain the way the virus works (not many models do) or to calibrate the rates of transition between the various compartments, which is attempted by the SIR-type models (although requiring tuning of the differential equation parameters for infection rates etc).

In response to the forecasts from these models, then, we see many questions being asked about why the infection rates, death rates and other measured statistics are as they are, differing quite widely from country to country.

There is so much unknown about how SARS-Cov-2 infects humans, and how Covid-19 infections progress; such data models inform the debate, and in calibrating the trajectory of the epidemic data, contribute to planning and policy as part of a family of forecasts.

The problem with data

I am going to make no attempt in this paper, or in my work generally, to model more widely than the UK.

What I have learned from my work so far, in the UK, is that published numbers for cases (particularly) and even, to some extent, for deaths can be unreliable (at worst), untimely and incomplete (often) and are also adjusted historically from time to time as duplication, omission and errors have come to light.

Every week, in the UK, there is a drop in numbers at weekends, recovered by increases in reported numbers on weekdays to catch up. In the UK, the four home countries (and even regions within them) collate and report data in different ways; as recently as July 17th, the Northern Ireland government have said that the won’t be reporting numbers at weekends.

Across the world, I would say it is impossible to compare statistics on a like-for-like basis with any confidence, especially given the differing cultural, demographic and geographical aspects; government policies, health service capabilities and capacities; and other characteristics across countries.

The extent of the (un)reliability in the reported numbers across nations worldwide (just like the variations in the four home UK countries, and in the regions), means that trying to forecast at a high level for all countries is very difficult. We also read of significant variations in the 50 states of the USA in such matters.

Hence my reluctance to be drawn into anything wider than monitoring and trying to predict UK numbers.

Curve fitting my UK model forecast

I thought it would be useful, at least for my understanding, to apply a phenomenological curve fitting approach to some of the UK reported data, and also to my SIR-style model forecast, based on that data.

I find the UK case numbers VERY inadequate for that purpose. There is a fair expectation that we are only seeing a minority fraction (as low as 8% in the early stages, in Italy for example) of the actual infections (cases) in the UK (and elsewhere).

The very definition of what comprises a case is somewhat variable; in the UK we talk about confirmed cases (by test), but the vast majority of people are never tested (owing to a lack of symptoms, and/or not being in hospital) although millions (9 million to date in the UK) of tests have either been done or requested (but not necessarily returned in all cases).

Reported numbers of tests might involve duplication since some people are (rightly) tested multiple times to monitor their condition. It must be almost impossible to make such interpretations consistently across large numbers of countries.

Even the officially reported UK deaths data is undeniably incomplete, since the “all settings” figures the UK Government reports (and at the outset even this had only been hospital deaths, with care homes added (and then retrospectively edited in later on) are not the “excess” deaths that the UK Office for National Statistics (ONS) also track, and that many commentators follow. For consistency I have continued to use the Government reported numbers, their having been updated historically on the same basis.

Rather than using case numbers, then, I will simply make the curve-fitting vs. mechanistic modelling comparison on both the UK reported deaths and the forecasted deaths in my model, which has tracked the reporting fairly well, with some recent adjustments (made necessary by the process of gradual and partial lockdown relaxation during June, I believe).

I had reduced the lockdown intervention effectiveness in my model by 0.5% at the end of June from 83.5% to 83%, because during the relaxations (both informal and formal) since the end of May, my modelled deaths had begun to lag the reported deaths during the month of June.

This isn’t surprising, and is an indicator to me, at least, that lockdown relaxation has somewhat reduced the rate of decline in cases, and subsequently deaths, in the UK.

My current forecast data

Firstly, I present my usual two charts summarising my model’s fit to reported UK data up to and including 16th July.

On the left we see the the typical form of the S-curve that epidemic cumulative data takes, and on the right, the scatter (the orange dots) in the reported daily data, mainly owing to regular incompleteness in weekend reporting, recovered during the following week, every week. I emphasise that the blue and grey curves are my model forecast, with appropriate parameters set for its differential equations (e.g. the 83% intervention effectiveness starting on March 23rd), and are not best fit analytical curves retro-applied to the data.

Next see my model forecast, further out to September 30th, by when forecast daily deaths have dropped to less than one per day, which I will also use to compare with the curve fitting approach. The cumulative deaths plateau, long term, is for 46,421 deaths in this forecast.

UK deaths, reported vs. model, 83%, cumulative and daily, to 30th September

The curve-fitting Gompertz function

I have simplified the calculation of the Gompertz function, since I merely want to illustrate its relationship to my UK forecast – not to use it in anger as my main process, or to develop multiple variations for different countries. Firstly my own basic charts of reported and modelled deaths.

On the left we see the reported data, with the weekly variations I mentioned before (hence the 7-day average to make the trend clearer) and on the right, the modelled version, showing how close the fit is, up to 16th July.

On any given day, the 7-day average lags the barchart numbers when the numbers are growing, and exceeds the numbers when they are declining, as it is taking 7 numbers prior to and up to the reporting day, and averaging them. You can see this more clearly on the right for the smoother modelled numbers (where the averaging isn’t really necessary, of course).

It’s also worth mentioning that the Gompertz function fitting allows its analytical statistical function curve to fit the observed varying growth rate of this SARS-Cov-2 pandemic, with its asymmetry of a slower decline than the steeper ramp-up (sub-exponential though it is) as seen in the charts above.

I now add, to the reported data chart, a graphical version including a derivation of the Gompertz function (the green line) for which I show its straight line trend (the red line). The jagged appearance of the green Gompertz curve on the right is caused by the weekend variation in the reported data, mentioned before.

Those working in the field would use smoothed reported data to reduce this unnecessary clutter, but this adds a layer of complexity to the process, requiring its own justifications, whose detail (and different smoothing options) are out of proportion with this summary.

But for my model forecast, we will see a smoother rendition of the data going into this process. See Michael Levitt’s paper for a discussion of the smoothing options his team uses for data from the many countries the scope of his work includes.

Of course, there are no reported numbers beyond today’s date (16th July) so my next charts, again with the Gompertz equation lines added (in green), compare the fit of the Gompertz version of my model forecast up to July 16th (on the right) with the reported data version (on the left) from above – part of the comparison purpose of this exercise.

The next charts, with the Gompertz equation lines added (in green), compare the fit of my model forecast only (i.e. not the reported data) up to July 16th on the left, with the forecast out to September 30th on the right.

What is notable about the charts is the nearly straight line appearance of the Gompertz version of the data. The wiggles approaching late September on the right are caused by some gaps in the data, as some of the predicted model numbers for daily deaths are zero at that point; the ratios (c(t)/c(t-1)) and logarithmic calculation Ln(c(t)/c(t-1)) have some necessary gaps on some days (division by 0, and ln(0) being undefined).

Discussion

The Gompertz method potentially allows a straight line extrapolation of the reported data in this form, instead of developing SIR-style non-linear differential equations for every country. This means much less scientific and computer time to develop and process, so that Michael Levitt’s team can process many country datasets quickly, via the Gompertz functional representation of reported data, to create the required forecasts.

As stated before, this method doesn’t address the underlying mechanisms of the spread of the epidemic, but policy makers might sometimes simply need the “what” of the outlook, and not the “how” and “why”. The assessment of the infectivity and other disease characteristics, and the related estimation of their representation by coefficients in the differential equations for mechanistic models, might not be reliably and quickly done for this novel virus in so many different countries.

When policy makers need to know the potential impact of their interventions and actions, then mechanistic models can and do help with those dependencies, under appropriate assumptions.

As mentioned in my recent post on modelling methods, such mechanistic models might use mobility and demographic data to predict contact rates, and will, at some level of detail, model interventions such as social distancing, hygiene improvements and the use of masks, as well as self-isolation (or quarantine) for suspected cases, and for people in high risk groups (called shielding in the UK) such as the elderly or those with underlying health conditions.

Michael Levitt’s (and other) phenomenological methods don’t do this, since they are fitting chosen analytical functions to the (cleaned and smoothed) cases or deaths data, looking for patterns in the “output” data for the epidemic in a country, rather than for the causations for, and implications of the “input” data.

In Michael’s case, an important variable that is used is the ratio of successive days’ cases data, which means that the impact of national idiosyncrasies in data collection are minimised, since the same method is in use on successive days for the given country.

In reality, the parameters that define the shape (growth rate, inflection point and decline rate) of the specific Gompertz function used would also have to be estimated or calculated, with some advance idea of the plateau figure (what is called the “carrying capacity” of the related Generalised Logistics Functions (GLFs) of which the Gompertz functions comprise a subset).

I have taken some liberties here with the process, since my aim was simply to illustrate the technique using a forecast I already have.

Closing remarks

I have some corrective and clarification work to do on this methodology, but my intention has merely been to compare and contrast two methods of producing Covid-19 forecasts – phenomenological curve-fitting vs. SIR modelling.

These is much that the professionals in this field have yet to do. Many countries are struggling to move from blanket lockdown, through to a more targeted approach, using modelling to calibrate the changing effect of the various sub-measures in the lockdown package. I covered some of those differential effects of intervention options in my post on June 28th, including the consideration of any resulting “herd immunity” as a future impact of the relative efficacy of current intervention methods.

From a planning and policy perspective, Governments have to consider the collateral health impact of such interventions, which is why the excess deaths outlook is important, taking into account the indirect effect of both Covid-19 infections, and also the cumulative health impacts of the methods (such as quarantining and social distancing) used to contain the virus.

One of these negative impacts is on the take-up of diagnosis and treatment of other serious conditions which might well cause many further excess deaths next year, to which I referred in my modelling update post of July 6th, referencing a report by Health Data Research UK, quoting Data-Can.org.uk about the resulting cancer care issues in the UK.

Politicians also have to cope with the economic impact, which also feeds back into the nation’s health.

Hence the narrow numbers modelling I have been doing is only a partial perspective on a very much bigger set of problems.

Categories
Coronavirus Covid-19 Michael Levitt Reproductive Number Uncategorized

Current Coronavirus model forecast, and next steps

Introduction

This post covers the current status of my UK Coronavirus (SARS-CoV-2) model, stating the June 2nd position, and comparing with an update on June 3rd, reworking my UK SARS-CoV-2 model with 83.5% intervention effectiveness (down from 84%), which reduces the transmission rate to 16.5% of its pre-intervention value (instead of 16%), prior to the 23rd March lockdown.

This may not seem a big change, but as I have said before, small changes early on have quite large effects later. I did this because I see some signs of growth in the reported numbers, over the last few days, which, if it continues, would be a little concerning.

I sensed some urgency in the June 3rd Government update, on the part of the CMO, Chris Whitty (who spoke at much greater length than usual) and the CSA, Sir Patrick Vallance, to highlight the continuing risk, even though the UK Government is seeking to relax some parts of the lockdown.

They also mentioned more than once that the significant “R” reproductive number, although less than 1, was close to 1, and again I thought they were keen to emphasise this. The scientific and medical concern and emphasis was pretty clear.

These changes are in the context of quite a bit of debate around the science between key protagonists, and I begin with the background to the modelling and data analysis approaches.

Curve fitting and forecasting approaches

Curve-fitting approach

I have been doing more homework on Prof. Michael Levitt’s Twitter feed, where he publishes much of his latest work on Coronavirus. There’s a lot to digest (some of which I have already reported, such as his EuroMOMO work) and I see more methodology to explore, and also lots of third party input to the stream, including Twitter posts from Prof. Sir David Spiegelhalter, who also publishes on Medium.

I DO use Twitter, although a lot less nowadays than I used to (8.5k tweets over a few years, but not at such high rate lately); much less is social nowadays, and more is highlighting of my https://www.briansutton.uk/ blog entries.

Core to that work are Michael’s curve fitting methods, in particular regarding the Gompertz cumulative distribution function and the Change Ratio / Sigmoid curve references that Michael describes. Other functions are also available(!), such as The Richard’s function.

This curve-fitting work looks at an entity’s published data regarding cases and deaths (China, the Rest of the World and other individual countries were some important entities that Michael has analysed) and attempts to fit a postulated mathematical function to the data, first to enable a good fit, and then for projections into the future to be made.

This has worked well, most notably in Michael’s work in forecasting, in early February, the situation in China at the end of March. I reported this on March 24th when the remarkable accuracy of that forecast was reported in the press:

The Times coverage on March 24th of Michael Levitt's accurate forecast for China
The Times coverage on March 24th of Michael Levitt’s accurate forecast for China

Forecasting approach

Approaching the problem from a slightly different perspective, my model (based on a model developed by Prof. Alex de Visscher at Concordia University) is a forecasting model, with my own parameters and settings, and UK data, and is currently matching death rate data for the UK, on the basis of Government reported “all settings” deaths.

The model is calibrated to fit known data as closely as possible (using key parameters such as those describing virus transmission rate and incubation period, and then solves the Differential Equations, describing the behaviour of the virus, to arrive at a predictive model for the future. No mathematical equation is assumed for the charts and curve shapes; their behaviour is constructed bottom-up from the known data, postulated parameters, starting conditions and differential equations.

The model solves the differential equations that represent an assumed relationship between “compartments” of people, including, but not necessarily limited to Susceptible (so far unaffected), Infected and Recovered people in the overall population.

I had previously explored such a generic SIR model, (with just three such compartments) using a code based on the Galbraith solution to the relevant Differential Equations. My following post article on the Reproductive number R0 was set in the context of the SIR (Susceptible-Infected-Recovered) model, but my current model is based on Alex’s 7 Compartment model, allowing for graduations of sickness and multiple compartment transition routes (although NOT with reinfection).

SEIR models allow for an Exposed but not Infected phase, and SEIRS models add a loss of immunity to Recovered people, returning them eventually to the Susceptible compartment. There are many such options – I discussed some in one of my first articles on SIR modelling, and then later on in the derivation of the SIR model, mentioning a reference to learn more.

Although, as Michael has said, the slowing of growth of SARS-CoV-2 might be because it finds it hard to locate further victims, I should have thought that this was already described in the Differential Equations for SIR related models, and that the compartment links in the model (should) take into account the effect of, for example, social distancing (via the effectiveness % parameter in my model). I will look at this further.

The June 2nd UK reported and modelled data

Here are my model output charts exactly up to, June 2nd, as of the UK Government briefing that day, and they show (apart from the last few days over the weekend) a very close fit to reported death data**. The charts are presented as a sequence of slides:

These charts all represent the same UK deaths data, but presented in slightly different ways – linear and log y-axes; cumulative and daily numbers; and to date, as well as the long term outlook. The current long term outlook of 42,550 deaths in the UK is within error limits of the the Worldometers linked forecast of 44,389, presented at https://covid19.healthdata.org/united-kingdom, but is not modelled on it.

**I suspected that my 84% effectiveness of intervention would need to be reduced a few points (c. 83.5%) to reflect a little uptick in the UK reported numbers in these charts, but I waited until midweek, to let the weekend under-reporting work through. See the update below**.

I will also be interested to see if that slight uptick we are seeing on the death rate in the linear axis charts is a consequence of an earlier increase in cases. I don’t think it will be because of the very recent and partial lockdown relaxations, as the incubation period of the SARS-CoV-2 virus means that we would not see the effects in the deaths number for a couple of weeks at the earliest.

I suppose, anecdotally, we may feel that UK public response to lockdown might itself have relaxed a little over the last two or three weeks, and might well have had an effect.

The periodic scatter of the reported daily death numbers around the model numbers is because of the reguar weekend drop in numbers. Reporting is always delayed over weekends, with the ground caught up over the Monday and Tuesday, typically – just as for 1st and 2nd June here.

A few numbers are often reported for previous days at other times too, when the data wasn’t available at the time, and so the specific daily totals are typically not precisely and only deaths on that particular day.

The cumulative charts tend to mask these daily variations as the cumulative numbers dominate small daily differences. This applies to the following updated charts too.

**June 3rd update for 83.5% intervention effectiveness

I have reworked the model for 83.5% intervention effectiveness, which reduces the transmission rate to 16.5% of its starting value, prior to 23rd March lockdown. Here is the equivalent slide set, as of 3rd June, one day later, and included in this post to make comparisons easier:

These charts reflect the June 3rd reported deaths at 39,728 and daily deaths on 3rd June of 359. The model long-term prediction is 44,397 deaths in this scenario, almost exactly the Worldometer forecast illustrated above.

We also see the June 3rd reported and modelled cumulative numbers matching, but we will have to watch the growth rate.

Concluding remarks

I’m not as concerned to model cases data as accurately, because the reported numbers are somewhat uncertain, collected as they are in different ways by four Home Countries, and by many different regions and entities in the UK, with somewhat different definitions.

My next steps, as I said, are to look at the Sigmoid and data fitting charts Michael uses, and compare the same method to my model generated charts.

*NB The UK Office for National Statistics (ONS) has been working on the Excess Deaths measure, amongst other data, including deaths where Covid-19 is mentioned on the death certificate, not requiring a positive Covid-19 test as the Government numbers do.

As of 2nd June, the Government announced 39369 deaths in its standard “all settings” – Hospitals, Community AND Care homes (with a Covid-19 test diagnosis) but the ONS are mentioning 62,000 Excess Deaths today. A little while ago, on the 19th May, the ONS figure was 55,000 Excess Deaths, compared with 35,341 for the “all settings” UK Government number. I reported that in my blog post https://www.briansutton.uk/?p=2302 in my EuroMOMO data analysis post.

But none of the ways of counting deaths is without its issues. As the King’s Fund says on their website, “In addition to its direct impact on overall mortality, there are concerns that the Covid-19 pandemic may have had other adverse consequences, causing an increase in deaths from other serious conditions such as heart disease and cancer.

“This is because the number of excess deaths when compared with previous years is greater than the number of deaths attributed to Covid-19. The concerns stem, in part, from the fall in numbers of people seeking health care from GPs, accident and emergency and other health care services for other conditions.

“Some of the unexplained excess could also reflect under-recording of Covid-19 in official statistics, for example, if doctors record other causes of death such as major chronic diseases, and not Covid-19. The full impact on overall and excess mortality of Covid-19 deaths, and the wider impact of the pandemic on deaths from other conditions, will only become clearer when a longer time series of data is available.”

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Cambridge Conversations Coronavirus Covid-19 Michael Levitt

Cambridge Conversation 14th May 2020, and Michael Levitt’s analysis of Euro data

I covered the May 14th Cambridge Conversation in my blog post last week, and promised to make available the YouTube link for it when uploaded. It is now on the University of Cambridge channel at:

Cambridge Conversation – COVID-19 behind the numbers – statistics, models and decision-making

In my following, and most recent post, I also summarised Prof. Michael Levitt’s interview with UnHerd at my post Another perspective on Coronavirus – Prof. Michael Levitt which presents a perspective on the Coronavirus crisis which is at odds with earlier forecasts and commentaries by Prof. Neil Ferguson and Prof. Sir David Spiegelhalter respectively.

Michael Levitt has very good and consistent track record in predicting the direction of travel and extent of what I might call the Coronavirus “China Crisis”, from quite early on, and contrary to the then current thinking about the rate of growth of Coronavirus there. Michael’s interview is at:

Michael Levitt’s interview with UnHerd

and I think it’s good to see these two perspectives together.

I will cover shortly some of Michael’s latest work on analysing comparisons presented at the website https://www.euromomo.eu/graphs-and-maps, looking at excess mortality across several years in Europe. Michael’s conclusions (which I have his permission to reproduce) are included in the document here:

where as can be seen from the title, the Covid-19 growth profile doesn’t look very dissimilar from recent previous years’ influenza data. More on this in my next article.

As for my own modest efforts in this area, my model (based on a 7 compartment code by Prof. Alex de Visscher in Canada, with my settings and UK data) is still tracking UK data quite well, necessitating no updates at the moment. But the UK Government is under increasing pressure to include all age related excess deaths in their daily (or weekly) updates, and this measure is mentioned in both videos above.

So I expect some changes to reported data soon: just as the UK Government has had to move to include “deaths in all settings” by including Care Home deaths in their figures, it is likely they should have to move to including the Office for National Statistics numbers too, which they have started to mention. Currently, instead of c. 35,000 deaths, these numbers show c. 55,000, although, as mentioned, the basis for inclusion is different.

These would be numbers based on a mention of Covid-19 on death certificates, not requiring a positive Covid-19 test as currently required for inclusion in UK Government numbers.

Categories
Coronavirus Covid-19 Reproductive Number

Another perspective on Coronavirus – Prof. Michael Levitt

Owing to the serendipity of a contemporary and friend of mine at King’s College London, Andrew Ennis, wishing one of HIS contemporaries in Physics, Michael Levitt, a happy birthday on 9th May, and mentioning me and my Coronavirus modelling attempts in passing, I am benefiting from another perspective on Coronavirus from Michael Levitt.

The difference is that Prof. Michael Levitt is a Nobel laureate in 2013 in computational biosciences…and I’m not! I’m not a Fields Medal winner either (there is no Nobel Prize for Mathematics, the Fields Medal being an equivalently prestigious accolade for mathematicians). Michael is Professor of Structural Biology at the Stanford School of Medicine.

I did win the Drew Medal for Mathematics in my day, but that’s another (lesser) story!

Michael has turned his attention, since the beginning of 2020, to the Coronavirus pandemic, and had kindly sent me a number of references to his work, and to his other recent work in the field.

I had already referred to Michael in an earlier blog post of mine, following a Times report of his amazingly accurate forecast of the limits to the epidemic in China (in which he was taking a particular interest).

Report of Michael Levitt’s forecast for the China outbreak

I felt it would be useful to report on the most recent of the links Michael sent me regarding his work, the interview given to Freddie Sayers of UnHerd at https://unherd.com/thepost/nobel-prize-winning-scientist-the-covid-19-epidemic-was-never-exponential/ reported on May 2nd. I have added some extracts from UnHerd’s coverage of this interview, but it’s better to watch the interview.

Michael’s interview with UnHerd

As UnHerd’s report says, “With a purely statistical perspective, he has been playing close attention to the Covid-19 pandemic since January, when most of us were not even aware of it. He first spoke out in early February, when through analysing the numbers of cases and deaths in Hubei province he predicted with remarkable accuracy that the epidemic in that province would top out at around 3,250 deaths.

“His observation is a simple one: that in outbreak after outbreak of this disease, a similar mathematical pattern is observable regardless of government interventions. After around a two week exponential growth of cases (and, subsequently, deaths) some kind of break kicks in, and growth starts slowing down. The curve quickly becomes ‘sub-exponential’.

UnHerd reports that he takes specific issue with the Neil Ferguson paper, that along with some others, was of huge influence with the UK Government (amongst others) in taking drastic action, moving away from a ‘herd immunity” approach to a lockdown approach to suppress infection transmission.

“In a footnote to a table it said, assuming exponential growth of 15% for six days. Now I had looked at China and had never seen exponential growth that wasn’t decaying rapidly.

“The explanation for this flattening that we are used to is that social distancing and lockdowns have slowed the curve, but he is unconvinced. As he put it to me, in the subsequent examples to China of South Korea, Iran and Italy, ‘the beginning of the epidemics showed a slowing down and it was very hard for me to believe that those three countries could practise social distancing as well as China.’ He believes that both some degree of prior immunity and large numbers of asymptomatic cases are important factors.

“He disagrees with Sir David Spiegelhalter’s calculations that the totem is around one additional year of excess deaths, while (by adjusting to match the effects seen on the quarantined Diamond Princess cruise ship, and also in Wuhan, China) he calculates that it is more like one month of excess death that is need before the virus peters out.

“He believes the much-discussed R0 is a faulty number, as it is meaningless without the time infectious alongside.” I discussed R0 and its derivation in my article about the SIR model and R0.

Interestingly, Prof Alex Visscher, whose original model I have been adapting for the UK, also calibrated his thinking, in part, by considering the effect of the Coronavirus on the captive, closed community on the Diamond Princess, as I reported in my Model Update on Coronavirus on May 8th.

The UnHerd article finishes with this quote: “I think this is another foul-up on the part of the baby boomers. I am a real baby boomer — I was born in 1947, I am almost 73 years old — but I think we’ve really screwed up. We’ve caused pollution, we’ve allowed the world’s population to increase threefold in my lifetime, we’ve caused the problems of global warming and now we’ve left your generation with a real mess in order to save a relatively small number of very old people.”

I suppose, as a direct contemporary, that I should apologise too.

There’s a lot more at the UnHerd site, but better to hear it directly from Michael in the video.