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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 Imperial College Michael Levitt Reproductive Number

Phenomenology & Coronavirus – modelling and curve-fitting

Introduction

I have been wondering for a while how to characterise the difference in approaches to Coronavirus modelling of cases and deaths, between “curve-fitting” equations and the SIR differential equations approach I have been using (originally developed in Alex de Visscher’s paper this year, which included code and data for other countries such as Italy and Iran) which I have adapted for the UK.

Part of my uncertainty has its roots in being a very much lapsed mathematician, and part is because although I have used modelling tools before, and worked in some difficult area of mathematical physics, such as General Relativity and Cosmology, epidemiology is a new application area for me, with a wealth of practitioners and research history behind it.

Curve-fitting charts such as the Sigmoid and Gompertz curves, all members of a family of curves known as logistics or Richards functions, to the Coronavirus cases or deaths numbers as practised, notably, by Prof. Michael Levitt and his Stanford University team has had success in predicting the situation in China, and is being applied in other localities too.

Michael’s team have now worked out an efficient way of reducing the predictive aspect of the Gompertz function and its curves to a straight line predictor of reported data based on a version of the Gompertz function, a much more efficient use of computer time than some other approaches.

The SIR model approach, setting up an series of related differential equations (something I am more used to in other settings) that describe postulated mechanisms and rates of virus transmission in the human population (hence called “mechanistic” modelling), looks beneath the surface presentation of the epidemic cases and deaths numbers and time series charts, to model the growth (or otherwise) of the epidemic based on postulated characteristics of viral transmission and behaviour.

Research literature

In researching the literature, I have become familiar with some names that crop up or frequently in this area over the years.

Focusing on some familiar and frequently recurring names, rather than more recent practitioners, might lead me to fall into “The Trouble with Physics” trap (the tendency, highlighted by Lee Smolin in his book of that name, exhibited by some University professors to recruit research staff (“in their own image”) who are working in the mainstream, rather than outliers whose work might be seen as off-the-wall, and less worthy in some sense.)

In this regard, Michael Levitt‘s new work in the curve-fitting approach to the Coronavirus problem might be seen by others who have been working in the field for a long time as on the periphery (despite his Nobel Prize in Computational Biology and Stanford University position as Professor of Structural Biology).

His results (broadly forecasting, very early on, using his curve-fitting methods (he has used Sigmoid curves before, prior to the current Gompertz curves), a much lower incidence of the virus going forward, successfully so in the case of China) are in direct contrast to that of some some teams working as advisers to Governments, who have, in some cases, all around the world, applied fairly severe lockdowns for a period of several months in most cases.

In particular the work of the Imperial College Covid response team, and also the London School of Hygiene and Tropical Medicine have been at the forefront of advice to the UK Government.

Some Governments have taken a different approach (Sweden stands out in Europe in this regard, for several reasons).

I am keen to understand the differences, or otherwise, in such approaches.

Twitter and publishing

Michael chooses to publish his work on Twitter (owing to a glitch (at least for a time) with his Stanford University laboratory‘s own publishing process. There are many useful links there to his work.

My own succession of blog posts (all more narrowly focused on the UK) have been automatically published to Twitter (a setting I use in WordPress) and also, more actively, shared by me on my FaceBook page.

But I stopped using Twitter routinely a long while ago (after 8000+ posts) because, in my view, it is a limited communication medium (despite its reach), not allowing much room for nuanced posts. It attracts extremism at worst, conspiracy theorists to some extent, and, as with a lot of published media, many people who choose on a “confirmation bias” approach to read only what they think they might agree with.

One has only to look at the thread of responses to Michael’s Twitter links to his forecasting results and opinions to see examples of all kinds of Twitter users: some genuinely academic and/or thoughtful; some criticising the lack of published forecasting methods, despite frequent posts, although they have now appeared as a preprint here; many advising to watch out (often in extreme terms) for “big brother” government when governments ask or require their populations to take precautions of various kinds; and others simply handclapping, because they think that the message is that this all might go away without much action on their part, some of them actively calling for resistance even to some of the most trivial precautionary requests.

Preamble

One of the recent papers I have found useful in marshalling my thoughts on methodologies is this 2016 one by Gustavo Chowell, and it finally led me to calibrate the differences in principle between the SIR differential equation approach I have been using (but a 7-compartment model, not just three) and the curve-fitting approach.

I had been thinking of analogies to illustrate the differences (which I will come to later), but this 2016 Chowell paper, in particular, encapsulated the technical differences for me, and I summarise that below. The Sergio Alonso paper also covers this ground.

Categorization of modelling approaches

Gerard Chowell’s 2016 paper summarises modelling approaches as follows.

Phenomenological models

A dictionary definition – “Phenomenology is the philosophical study of observed unusual people or events as they appear without any further study or explanation.”

Chowell states that phenomenological approaches for modelling disease spread are particularly suitable when significant uncertainty clouds the epidemiology of an infectious disease, including the potential contribution of multiple transmission pathways.

In these situations, phenomenological models provide a starting point for generating early estimates of the transmission potential and generating short-term forecasts of epidemic trajectory and predictions of the final epidemic size.

Such methods include curve fitting, as used by Michael Levitt, where an equation (represented by a curve on a time-incidence graph (say) for the virus outbreak), with sufficient degrees of freedom, is used to replicate the shape of the observed data with the chosen equation and its parameters. Sigmoid and Gompertz functions (types of “logistics” or Richards functions) have been used for such fitting – they produce the familiar “S”-shaped curves we see for epidemics. The starting growth rate, the intermediate phase (with its inflection point) and the slowing down of the epidemic, all represented by that S-curve, can be fitted with the equation’s parametric choices (usually three or four).

This chart was put up by Michael Levitt on July 8th to illustrate curve fitting methodology using the Gompertz function. See https://twitter.com/MLevitt_NP2013/status/1280926862299082754
Chart by Michael Levitt illustrating his Gompertz function curve fitting methodology

A feature that some epidemic outbreaks share is that growth of the epidemic is not fully exponential, but is “sub-exponential” for a variety of reasons, and Chowell states that:

Previous work has shown that sub-exponential growth dynamics was a common phenomenon across a range of pathogens, as illustrated by empirical data on the first 3-5 generations of epidemics of influenza, Ebola, foot-and-mouth disease, HIV/AIDS, plague, measles and smallpox.”

Choices of appropriate parameters for the fitting function can allow such sub-exponential behaviour to be reflected in the chosen function’s fit to the reported data, and it turns out that the Gompertz function is more suitable for this than the Sigmoid function, as Michael Levitt states in his recent paper.

Once a curve-fit to reported data to date is achieved, the curve can be used to make forecasts about future case numbers.

Mechanistic and statistical models

Chowell states that “several mechanisms have been put forward to explain the sub-exponential epidemic growth patterns evidenced from infectious disease outbreak data. These include spatially constrained contact structures shaped by the epidemiological characteristics of the disease (i.e., airborne vs. close contact transmission model), the rapid onset of population behavior changes, and the potential role of individual heterogeneity in susceptibility and infectivity.

He goes on to say that “although attractive to provide a quantitative description of growth profiles, the generalized growth model (described earlier) is a phenomenological approach, and hence cannot be used to evaluate which of the proposed mechanisms might be responsible for the empirical patterns.

Explicit mechanisms can be incorporated into mathematical models for infectious disease transmission, however, and tested in a formal way. Identification and analysis of the impacts of these factors can lead ultimately to the development of more effective and targeted control strategies. Thus, although the phenomenological approaches above can tell us a lot about the nature of epidemic patterns early in an outbreak, when used in conjunction with well-posed mechanistic models, researchers can learn not only what the patterns are, but why they might be occurring.

On the Imperial College team’s planning website, they state that their forecasting models (they have several for different purposes, for just these reasons I guess) fall variously into the “Mechanistic” and “Statistical” categories, as follows.

COVID-19 planning tools
Imperial College models use a combination of mechanistic and statistical approaches.

Mechanistic model: Explicitly accounts for the underlying mechanisms of diseases transmission and attempt to identify the drivers of transmissibility. Rely on more assumptions about the disease dynamics.

Statistical model: Do not explicitly model the mechanism of transmission. Infer trends in either transmissibility or deaths from patterns in the data. Rely on fewer assumptions about the disease dynamics.

Mechanistic models can provide nuanced insights into severity and transmission but require specification of parameters – all of which have underlying uncertainty. Statistical models typically have fewer parameters. Uncertainty is therefore easier to propagate in these models. However, they cannot then inform questions about underlying mechanisms of spread and severity.

So Imperial College’s “statistical” description matches more to Chowell’s description of a phenomenological approach, although may not involve curve-fitting per se.

The SIR modelling framework, employing differential equations to represent postulated relationships and transitions between Susceptible, Infected and Recovered parts of the population (at its most simple) falls into this Mechanistic model category.

Chowell makes the following useful remarks about SIR style models.

The SIR model and derivatives is the framework of choice to capture population-level processes. The basic SIR model, like many other epidemiological models, begins with an assumption that individuals form a single large population and that they all mix randomly with one another. This assumption leads to early exponential growth dynamics in the absence of control interventions and susceptible depletion and greatly simplifies mathematical analysis (note, though, that other assumptions and models can also result in exponential growth).

The SIR model is often not a realistic representation of the human behavior driving an epidemic, however. Even in very large populations, individuals do not mix randomly with one another—they have more interactions with family members, friends, and coworkers than with people they do not know.

This issue becomes especially important when considering the spread of infectious diseases across a geographic space, because geographic separation inherently results in nonrandom interactions, with more frequent contact between individuals who are located near each other than between those who are further apart.

It is important to realize, however, that there are many other dimensions besides geographic space that lead to nonrandom interactions among individuals. For example, populations can be structured into age, ethnic, religious, kin, or risk groups. These dimensions are, however, aspects of some sort of space (e.g., behavioral, demographic, or social space), and they can almost always be modeled in similar fashion to geographic space“.

Here we begin to see the difference I was trying to identify between the curve-fitting approach and my forecasting method. At one level, one could argue that curve-fitting and SIR-type modelling amount to the same thing – choosing parameters that make the theorised data model fit the reported data.

But, whether it produces better or worse results, or with more work rather than less, SIR modelling seeks to understand and represent the underlying virus incubation period, infectivity, transmissibility, duration and related characteristics such as recovery and immunity (for how long, or not at all) – the why and how, not just the what.

The (nonlinear) differential equations are then solved numerically (rather than analytically with exact functions) and there does have to be some fitting to the initial known data for the outbreak (i.e. the history up to the point the forecast is being done) to calibrate the model with relevant infection rates, disease duration and recovery timescales (and death rates).

This makes it look similar in some ways to choosing appropriate parameters for any function (Sigmoid, Gompertz or General Logistics function (often three or four parameters)).

But the curve-fitting approach is reproducing an observed growth pattern (one might say top-down, or focused on outputs), whereas the SIR approach is setting virological and other behavioural parameters to seek to explain the way the epidemic behaves (bottom-up, or focused on inputs).

Metapopulation spatial models

Chowell makes reference to population-level models, formulations that are used for the vast majority of population based models that consider the spatial spread of human infectious diseases and that address important public health concerns rather than theoretical model behaviour. These are beyond my scope, but could potentially address concerns about indirect impacts of the Covid-19 pandemic.

a) Cross-coupled metapopulation models

These models, which have been used since the 1940s, do not model the process that brings individuals from different groups into contact with one another; rather, they incorporate a contact matrix that represents the strength or sum total of those contacts between groups only. This contact matrix is sometimes referred to as the WAIFW, or “who acquires infection from whom” matrix.

In the simplest cross-coupled models, the elements of this matrix represent both the influence of interactions between any two sub-populations and the risk of transmission as a consequence of those interactions; often, however, the transmission parameter is considered separately. An SIR style set of differential equations is used to model the nature, extent and rates of the interactions between sub-populations.

b) Mobility metapopulation models

These models incorporate into their structure a matrix to represent the interaction between different groups, but they are mechanistically oriented and do this by considering the actual process by which such interactions occur. Transmission of the pathogen occurs within sub-populations, but the composition of those sub-populations explicitly includes not only residents of the sub-population, but visitors from other groups.

One type of model uses a “gravity” approach for inter-population interactions, where contact rates are proportional to group size and inversely proportional to the distance between them.

Another type described by Chowell uses a “radiation” approach, which uses population data relating to home locations, and to job locations and characteristics, to theorise “travel to work” patterns, calculated using attractors that such job locations offer, influencing workers’ choices and resulting travel and contact patterns.

Transportation and mobile phone data can be used to populate such spatially oriented models. Again SIR-style differential equations are used to represent the assumptions in the model about between whom, and how the pandemic spreads.

Summary of model types

We see that there is a range of modelling methods, successively requiring more detailed data, but which seek increasingly to represent the mechanisms (hence “mechanistic” modelling) by which the virus might spread.

We can see the key difference between curve-fitting (what I called a surface level technique earlier) and the successively more complex models that seek to work from assumed underlying causations of infection spread.

An analogy (picking up on the word “surface” I have used here) might refer to explaining how waves in the sea behave. We are all aware that out at sea, wave behaviour is perceived more as a “swell”, somewhat long wavelength(!) waves, sometimes of great height, compared with shorter, choppier wave behaviour closer to shore.

I’m not here talking about breaking waves – a whole separate theory is needed for those – René Thom‘s Catastrophe Theory – but continuous waves.

A curve fitting approach might well find a very good fit using trigonometric sine waves to represent the wavelength and height of the surface waves, even recognising that they can be encoded by depth of the ocean, but it would need an understanding of hydrodynamics, as described, for example, by Bernoulli’s Equation, to represent how and why the wavelength and wave height (and speed*) changes depending on the depth of the water (and some other characteristics).

(*PS remember that the water moves, pretty much, up and down, not in the direction of travel of the observed (transverse) wave. The horizontal motion and speed of the wave is, in a sense, an illusion.)

Concluding comments

There is a range of modelling methods, successively requiring more detailed data, from phenomenological (statistical and curve-fitting) methods, to those which seek increasingly to represent the mechanisms (hence “mechanistic”) by which the virus might spread.

We see the difference between curve-fitting and the successively more complex models that build a model from assumed underlying interactions, and causations of infection spread between parts of the population.

I do intend to cover the mathematics of curve fitting, but wanted first to be sure that the context is clear, and how it relates to what I have done already.

Models requiring detailed data about travel patterns are beyond my scope, but it is as well to set into context what IS feasible.

Setting an understanding of curve-fitting into the context of my own modelling was a necessary first step. More will follow.

References

I have found several papers very helpful on comparing modelling methods, embracing the Gompertz (and other) curve-fitting approaches, including Michaels Levitt’s own recent June 30th one, which explains his methods quite clearly.

Gerard Chowell’s 2016 paper on Mathematical model types September 2016

The Coronavirus Chronologies – Michael Levitt, 13th March 2020

COVID-19 Virus Epidemiological Model Alex de Visscher, Concordia University, Quebec, 22nd March 2020

Empiric model for short-time prediction of Covid-19 spreading , Sergio Alonso et al, Spain, 19th May 2020

Universality in Covid-19 spread in view of the Gompertz function Akira Ohnishi et al, Kyoto University) 22nd June 2020

Predicting the trajectory of any Covid-19 epidemic from the best straight line – Michael Levitt et al 30th June 2020

Categories
Coronavirus Covid-19 Michael Levitt

Coronavirus model tracking, lockdown and lessons

Introduction

This is just a brief update post to confirm that my Coronavirus model is still tracking the daily reported UK data well, and doesn’t currently need any parameter changes.

I go on to highlight some important aspects of emphasis in the Daily Downing St. Update on June 10th, as well as the response to Prof. Neil Ferguson’s comments to the Parliamentary Select Committee for Science and Technology about the impact of an earlier lockdown date, a scenario I have modelled and discussed before.

My model forecast

I show just one chart here that indicates both daily and cumulative figures for UK deaths, thankfully slowing down, and also the model forecast to the medium term, to September 30th, by when the modelled death rate is very low. The outlook in the model is still for 44,400 deaths, although no account is yet taken for reduced intervention effectiveness (from next week) as further, more substantial relaxations are made to the lockdown.

Note that the scatter of the reported daily deaths in the chart below is caused by some delays and resulting catch-up in the reporting, principally (but not only) at weekends. It doesn’t show in the cumulative curve, because the cumulative numbers are so much higher, and these daily variations are small by comparison (apart from when the cumulative numbers are lower, in late February to mid-March).

UK Daily & Cumulative deaths, model vs. Government “all settings” data

It isn’t yet clear whether the imminent lockdown easing (next week) might lead to a sequence of lockdown relaxation, infection rate increase, followed by (some) re-instituted lockdown measures, to be repeated cyclically as described by Neil Ferguson’s team in their 16th March COVID19-NPI-modelling paper, which was so influential on Government at the time (probably known to Government earlier than the paper publication date). If so, then simpler medium to long term forecasting models will have to change, my own included. For now, this is still in line with the Worldometers forecast, pictured here.

Worldometers UK Covid-19 forecast deaths for August 4th 2020

The ONS work

The Office for National Statistics (ONS) have begun to report regularly on deaths where Covid-19 is mentioned on the death certificate, and are also reporting on Excess Deaths, comparing current death rates with the seasonally expected number based on previous years. Both of these measures show larger numbers, as I covered in my June 2nd post, than the Government “all settings” numbers, that include only deaths with a positive Covid-19 test in Hospitals, the Community and Care Homes.

As I also mentioned in that post, no measures are completely without the need for interpretation. For consistency, for the time being, I remain with the Government “all settings” numbers in my model that show a similar rise and fall over the peak of the virus outbreak, but with somewhat lower daily numbers than the other measures, particularly at the peak.

The June 10th Government briefing

This briefing was given by the PM, Boris Johnson, flanked, as he was a week ago, by Sir Patrick Vallance (Chief Scientific Adviser (CSA)) and Prof. Chris Whitty (Chief Medical officer (CMO)), and again, as last week, the scientists offered much more than the politician.

In particular, the question of “regrets” came up from journalist questions, probing what the team might have done differently, in the light of the Prof. Ferguson comment earlier to the Parliamentary Science & Technology Select Committee that lives could have been saved had lockdown been a week earlier (I cover this in a section below).

At first, the shared approach of the CMO and CSA was not only that the scientific approach was always to learn the lessons from such experiences, but also that is was too early to do this, given, as the CMO emphasised very clearly last week, and again this week, that we are in the middle of this crisis, and there is a very long way to go (months, and even more, as he had said last week).

The PM latched onto this, repeating that it was too soon to take the lessons (not something I agree with); and indeed, Prof. Chris Whitty came back and offered that amongst several things he might have done differently, testing was top of the list, and that without it, everyone had been working in the dark.

My opinion is that if there is a long way to go, then we had better apply those lessons that we can learn as we go along, even if, as is probably the case, it is too early to come to conclusions about all aspects. There will no doubt be an inquiry at some point in the future, but that is a long way off, and adjusting our course as we continue to address the pandemic must surely be something we should do.

Parliamentary Science & Technology Select Committee

Earlier that day, on 10th June, Prof. Neil Ferguson of Imperial College had given evidence (or at least a submission) to the Select Committee for Science & Technology, stating that lives could have been saved if lockdown had been a week earlier. He was quoted here as saying “The epidemic was doubling every three to four days before lockdown interventions were introduced. So had we introduced lockdown measures a week earlier, we would have reduced the final death toll by at least a half.

Whilst I think the measures, given what we knew about this virus then, in terms of its transmission and its lethality, were warranted, I’m second guessing at this point, certainly had we introduced them earlier we would have seen many fewer deaths.”

In that respect, therefore, it isn’t merely interesting to look at the lockdown timing issue, but, as a matter of life and death, we should seek to understand how important timing is, as well as the effectiveness of possible interventions.

Surely one of the lessons from the pandemic (if we didn’t know it before) is that for epidemics that have an exponential growth rate (even if only for a while) matters down the track are highly (and non-linearly) dependent on initial conditions and early decisions.

With regard to that specific statement about the timing of the lockdown, I had already modelled the scenario for March 9th lockdown (two weeks earlier than the actual event on the 23rd March) and reported on that in my May 14th and May 25th posts at this blog. The precise quantum of the results is debatable, but, in my opinion, the principle isn’t.

I don’t need to rehearse all of those findings here, but it was clear, even given the limitations of my model (little data, for example, prior to March 9th upon which to calibrate the model, and the questionable % effectiveness of a postulated lockdown at that time, in terms of the public response) that my model forecast was for far fewer cases and deaths – the model said one tenth of those reported (for two weeks earlier lockdown). That is surely too small a fraction, but even part of that saving would be a big difference numerically.

This was also the nature of the findings of an Edinburgh University team, under Prof. Rowland Kao, who worked on the possible numbers for Scotland at that time, as reported by the BBC, which talked of a saving of 80% of the lives lost. Prof Kao had run simulations to see what would have happened to the spread of the virus if Scotland had locked down on 9 March, two weeks earlier.

A report of the June 10th Select Committee discussions mentioned that Prof. Kao supported Prof. Ferguson’s comments (unsurprisingly), finding the Ferguson comments “robust“, given his own team’s experience and work in the area.

Prof Simon Wood, Professor of Statistical Science at the University of Bristol, was reported as saying “I think it is too early to talk about the final death toll, particularly if we include the substantial non-COVID loss of life that has been and will be caused by the effects of lockdown. If the science behind the lockdown is correct, then the epidemic and the counter measures are not over.

Prof. Wood also made some comments relating to some observed pre-lockdown improvements in the death rate (possibly related to voluntary self-isolation which had been advised in some circumstances) which might have reduced the virus growth rate below the pure exponential rate which may have been assumed, and so he felt that “the basis for the ‘at least a half’ figure does not seem robust“.

Prof. James Naismith, Director of the Rosalind Franklin Institute, & Professor of Structural Biology, University of Oxford, was reported as saying “Professor Ferguson has been clear that his analysis is with the benefit of hindsight. His comments are a simple statement of the facts as we now understand them.

The lockdown timing debate

In the June 10th Government briefing, a few hours later, the PM mentioned in passing that Prof. Ferguson was on the SAGE Committee at that time, in early-mid March, as if to imply that this included him in the decision to lockdown later (March 23rd).

But, as I have also reported, in their May 23rd article, the Sunday Times Insight team produced a long investigative piece that indicated that some scientists (from both Imperial College and the London School of Hygiene and Tropical Medicine) had become worried about the lack of action, and proactively produced data and reports (as I mentioned above) that caused the Government to move towards a lockdown approach. The Government refuted this article here.

As we have heard many times, however, advisers advise, and politicians decide; in this case, it would seem that lockdown timing WAS a political decision (taking all aspects into account, including economic impact and the wider health issues) and I don’t have evidence to support Prof. Ferguson being party to the decision, (even if he was party to the advice, which is also dubious, given that his own scientific papers are very clear on the large scale of potential outcomes without NMIs (Non Pharmaceutical Interventions).

His forecasts would very much support a range of early and effective intervention measures to be considered, such as school and university closures, home isolation of cases, household quarantine, large-scale general population social distancing and social distancing of those over 70 years, as compared individually and in different combinations in the paper referenced above.

The forecasts in that paper, however, are regarded by Prof. Michael Levitt as in error (on the pessimistic side), basing forecasts, he says, on a wrong interpretation of the Wuhan data, causing an error by a factor of 10 or more in forecast death rates. Michael says “Thus, the Western World has been encouraged by their lack of responsibility coupled with uncontrolled media and academic errors to commit suicide for an excess burden of death of one month.”

But that Imperial College paper (and others) indicate what was in Neil Ferguson’s mind at that earlier stage. I don’t believe (but don’t know, of course) that his advice would have been to wait until a March 23rd lockdown.

Since SAGE (Scientific Advisory Group for Emergencies) proceedings are not published, it might be a long time before any of this history of the lockdown timing issue becomes clear.

Concluding comment

Now that relaxation of the lockdown is about to be enhanced, I am tracking the reported cases and deaths, and monitoring my Coronavirus model for any impact.

If there were any upwards movement in deaths and case rates, and reversal of any lockdown relaxations were to become necessary, the debate about lockdown timing will, no doubt, revive.

In that case, lessons learned from where we have been in that respect will need to be applied.

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.”

Categories
Coronavirus Covid-19 Michael Levitt

Michael Levitt’s analysis of European Covid-19 data

Introduction

I promised in an earlier blog post to present Prof. Michael Levitt’s analysis of Covid-19 data published on the EuroMOMO site for European health data over the last few years.

EuroMOMO

EuroMOMO is the European Mortality Monitoring Project. Based in Denmark, their website states that the overall objective of the original European Mortality Monitoring Project was to design a routine public health mortality monitoring system aimed at detecting and measuring, on a real-time basis, excess number of deaths related to influenza and other possible public health threats across participating European Countries. More is available here.

The Excess Deaths measure

We have heard a lot recently about using the measure of “excess deaths” (on an age related basis) as our own Office for National Statistics (ONS) work on establishing a more accurate measure of the impact of the Coronavirus (SARS-CoV-2) epidemic in the UK.

I think it is generally agreed that this is a better measure – a more complete one perhaps – than those currently used by the UK Government, and some others, because there is no argument about what and what isn’t a Covid-19 death. It’s just excess deaths over and above the seasonal, age related numbers for the geography, country or community concerned, attributing the excess to the novel Coronavirus SARS-CoV-2, the new kid on the block.

That attribution, though, might have its own different issues, such as the inclusion (or not) of deaths related to people’s reluctance to seek hospital help for other ailments, and other deaths arising from the indirect consequences of lockdown related interventions.

There is no disputing, however, that the UK Government figures for deaths have been incomplete from the beginning; they were updated a few weeks ago to include Care Homes on a retrospective and continuing basis (what they called “all settings”) but some reporting of the ONS figures has indicated that when the Government “all settings” figure was 35,341, as of 19th May, the overall “excess deaths” figure might have been as high as 55,000. Look here for more detail and updates direct from the ONS.

The UK background during March 2020

The four policy stages the UK Government initially announced in early March were: Containment, Delay, Research and Mitigate, as reported here. It fairly soon became clear (after the outbreak was declared a pandemic on March 11th by the WHO) that the novel Coronavirus SARS-CoV-2 could not be contained (seeing what was happening in Italy, and case numbers growing in the UK, with deaths starting to be recorded on 10th March (at that time only recorded as caused by Covid-19 with a positive test (in hospital)).

The UK Government have since denied that “herd immunity” had been a policy, but it was mentioned several times in early March, pre-lockdown (which was March 23rd) by Government advisers Sir Patrick Vallance (Chief Scientific Adviser, CSA) and Prof. Chris Whitty (Chief Medical Officer, CMO), in the UK Government daily briefings, with even a mention of 60% population infection proportion to achieve it (at the same time as saying that 80% might be loose talk (my paraphrase)).

If herd immunity wasn’t a policy, it’s hard to understand why it was proactively mentioned by the CSA and CMO, at the same time as the repeated slogan Stay Home, Protect the NHS, Save Lives. This latter advice was intended to keep the outbreak within bounds that the NHS could continue to handle.

The deliberations of the SAGE Committee (Scientific Advisory Group for Emergencies) are not published, but senior advisers (including the CSA and CMO) sit on it, amongst many others (50 or so, not all scientists or medics). Given the references to herd immunity in the daily Government updates at that time, it’s hard to believe that herd immunity wasn’t at least regarded as a beneficial(?!) by-product of not requiring full lockdown at that time.

Full UK lockdown was announced on March 23rd; according to reports this was 9 days after it being accepted by the UK Government as inevitable (as a result of the 16th March Imperial College paper).

The Sunday Times newspaper (ST) published on 24th May 2020 dealt with their story of how the forecasters took charge at that time in mid-March as the UK Government allegedly dithered. The ST’s Insight team editor’s Tweet (Jonathan Calvert) and those of his deputy editor George Arbuthnott refer, as does the related Apple podcast.

Prof. Michael Levitt

Michael (a Nobel Laureate in Computational Biology in 2013) correctly forecast in February the potential extent of the Chinese outbreak (Wuhan in the Hubei province) at the end of March. I first reported this at my blog post on 24th March, as his work on China, and his amazingly accurate forecast, were reported that day here in the UK, which I saw in The Times newspaper.

On May 18th I reported in my blog further aspects of Michael’s outlook on the modelling by Imperial College, the London School of Hygiene and Tropical Medicine (and others) which he says, and I paraphrase his words, caused western countries to trash their economies through the blanket measures they have taken, frightened into alternative action (away from what seems to have been, at least in part, a “herd-immunity” policy) by the forecasts from their advisers’ models, reported as between 200,000 and 500,000 deaths in some publications.

Michael and I have been in direct touch since early May, when a mutual friend, Andrew Ennis, mentioned my Coronavirus modelling to him in his birthday wishes! We were all contemporaries at King’s College, London in 1964-67; they in Physics, and I in Mathematics.

I mentioned Michael’s work in a further, recent blog post on May 20th, when I mentioned his findings on the data at EuroMOMO, contrasting it with the Cambridge Conversation of 14th May, and that is when I said that I would post a blog article purely on his EurtoMOMO work, and this post is the delivery of that promise.

I have Michael’s permission (as do others who have received his papers) to publicise his recent EuroMOMO findings (his earlier work having been focused on China, as I have said, and then on the rest of the world).

He is senior Professor in Structural Biology at Stanford University School of Medicine, CA.

I’m reporting, and explaining a little (where possible!) Michael’s findings just now, rather than deeply analysing – I’m aware that he is a Nobel prize-winning data scientist, and I’m not (yet!) 😀

This blog post is therefore pretty much a recapitulation of his work, with some occasional explanatory commentary.

Michael’s EuroMOMO analysis

What follows is the content of several tweets published by Michael, at his account @MLevitt_NP2013, showing that in Europe, COVID19 is somewhat similar to the 2017/18 European Influenza epidemics, both in total number of excess deaths, and age ranges of these deaths.

Several other academics have also presented data that, whatever the absolute numbers, indicate that there is a VERY marked (“startling” was Prof. Sir David Spiegelhalter’s word) age dependency in the risk factors of dying from Covid-19. I return to that theme at the end of the post.

The EuroMOMO charts and Michael’s analysis

In summary, COVID19 Excess Deaths plateau at 153,006, 15% more than the 2017/18 Flu with similar age range counts. The following charts indicate the support for his view, including the correction of a large error Michael has spotted in one of the supporting EuroMOMO charts.

Firstly, here are the summary Excess Death Charts for all ages in 2018-20.

FIGURE 1. EuroMOMO excess death counts for calendar years 2018, 2019 & 2020

The excess deaths number for COVID19 is easily read as the difference between Week 19 (12 May ’20) and Week 8 (27 Feb ’20). The same is true of the 2018 part of the 2017/18 Influenza season. Getting the 2017 part of that season is harder. These notes are added to aid those interested in following the calculation, and hopefully help them in pointing out any errors.

The following EuroMOMO chart defines how excess deaths are measured.

FIGURE 2. EuroMOMO’s total and other categories of deaths

This is EuroMOMO’s Total (the solid blue line), Baseline (dashed grey line) and ‘Substantial increase’ (dashed red line) for years 2016 to the present. Green circles mark 2017/18 Flu and 2020 COVID-19. The difference between Total Deaths and Baseline Deaths is Excess Deaths.

Next, then, we see Michael’s own summary of the figures found from these earlier charts:

Table 3. Summary for 2020 COVID19 Season and 2017/18 Influenza Season.

Owing to baseline issues, we cannot estimate Age Range Mortality for the 2017 part of the Influenza season, so we base our analysis on the 2018 part, where data is available from EuroMOMO.

We see also the steep age dependency in deaths from under 65s to over 85s. I’ll present at the end of this post some new data on that aspect (it’s of personal interest too!)

Below we see EuroMOMO Excess Deaths from 2020 Week 8, now (on the 14th May) matching reported COVID Deaths @JHUSystems (Johns Hopkins University) perfectly (better than 2%). In earlier weeks the reported deaths were lower, but Michael isn’t sure why. But it allows him to do this in-depth analysis & comparison with EuroMOMO influenza data.

FIGURE 4. The weekly EuroMOMO Excess Deaths are read off their graphs by mouse-over.

The weekly reported COVID19 deaths are taken from the Johns Hopkins University Github repository. The good agreement is an encouraging sign of reliable data but there is a unexplained delay in EuroMOMO numbers.

Analysis of Europe’s Excess Deaths is hard: EuroMOMO provides beautiful plots, but extracting data requires hand-recorded mouse-overs on-screen*. COVID19 2020 – weeks 8-19, & Influenza 2018 – weeks 01-16 are relatively easy for all age ranges (totals 153,006 & 111,226). Getting the Dec. 2017 Influenza peak is very tricky.

(*My son, Dr Tom Sutton, has been extracting UK data from the Worldometers site for me, using a small but effective Python “scraping” script he developed. It is feasible, but much more difficult, to do this on the EuroMOMO site, owing to the vector coordinate definitions of the graphics, and Document Object Model they use for their charts.)

Figure 5. Deaths graphs from EurMoMo allow the calculation of Excess deaths

FIGURE 5. The Excess deaths for COVID19 in 2020 and for Influenza in 2018 are easily read off the EuroMOMO graphs by hand recording four mouse-overs.

The same is done for all different age ranges allowing accurate determination of the age range mortalities. For COVID19, there are 174,801 minus 21,795 = 153,006 Excess Deaths. For 2018 Influenza, the difference is 111,226 minus zero = 111,226 Excess Deaths.

Michael exposes an error in the EuroMOMO charts

In the following chart, it should be easy to calculate again, as mouse-over of the charts on the live EuroMOMO site gives two values a week: Actual death count & Baseline value.

Tests on the COVID19 peak gave a total of 127,062 deaths & not 153,006. Plotting a table & superimposing the real plot showed why. Baseline values are actually ‘Substantial increase’ values!! Wrong labelling?

Figure 6. Actual death count & Baseline value

In Figure 6, Excess Deaths can also be determined from the plots of Total and Baseline Deaths with week number. Many more numbers need to be recorded but the result would be the same.

TABLE 7. The pairs of numbers recorded from EuroMOMO between weeks 08 and 19

TABLE 7. The pairs of numbers recorded from EuroMOMO between weeks 08 and 19 of 2020 allow the Excess Deaths to be determined in a different way than from FIG. 5. The total Excess Deaths (127,062) should be the same as before (153,006) but it is not. Why? (Mislabelling of the EuroMOMO graph? What is “Substantial increase” anyway and why is it there? – BRS).

FIGURE 8. Analysing what is wrong with the EuroMOMO Excess Deaths count

FIGURE 8. The lower number in TABLE 7 is in fact not the Baseline Death value (grey dashed line) but the ‘Substantial increase’ value (red dashed line). Thus the numbers in the table are not Excess Deaths (Total minus Baseline level) but Total minus ‘Substantial increase’ level. The difference is found by adding 12×1981** to 127,062 to get 153,006. This means that the baseline is about 2000 deaths a week below the red line. This cannot be intended and is a serious error in EuroMOMO. Michael has been looking for someone to help him contact them? (**(153,006 – 127062)/12 = 25944/12 = 2162. So shouldn’t we be adding 12×2162, Michael? – BRS)

Reconciling the numbers, and age range data

Requiring the two COVID19 death counts to match means reducing the Baseline value by 23,774/12 = 1,981**. Mouse-over 2017 weeks 46 to 52 gave the table below. Negative Excess Deaths meant 2017 Influenza began Week 49 not 46. Michael tried to get Age Range data for 2017 but the table just uses 2018 Influenza data. (**see above also – same issue. Should be 25944/12 = 2162? – BRS)

TABLE 9. Estimating the Excess Deaths for the 2017 part of the 2017/18 influenza season

In TABLE 9, Michael tries to estimate the Excess Deaths for the 2017 part of the 2017/18 Influenza season by recording pairs of mouse-overs for seven weeks (46 to 52) and four age ranges. Because the Total Deaths are not always higher than the ‘Substantial increase’ base level, he uses differences as a sanity check. The red numbers for weeks 46 to 48 show that the Excess Deaths are negative and that the Influenza season did not start until week 49 of 2017.

TABLE 10. We try to combine the two parts of the 2017/18 Influenza season

TABLE 10 commentary. We try to combine the two parts of the 2017/18 Influenza season. The values for 2018 are straightforward as they are determined as shown in Fig. 5. For 2017, we need to use the values in Table 9 and add the baseline correction because the EuroMOMO mouse-overs are wrong, giving as they do the ‘Substantial increase’ value instead of the ‘Baseline’ value. We can use the same correction of 1981**(see my prior comments on this number – BRS) deaths per week as determined for all COVID19 data but we do not know what the correction is for other age ranges. An attempt to assume that the correction is proportional to the 2017 number of deaths in each age range gives strange age range mortalities.
Thus, we choose to use the total for 2017 (21,972) but give the age range mortalities just from the deaths in 2018, as the 2017 data is arcane, unreliable or flawed.

Michael’s concluding statement

COVID19 is similar to Influenza only in total and in age range excess mortality. Flu is a different virus, has a safe vaccine & is much less a threat to heroic medical professionals.

Additional note on the age dependency of Covid-19 risk

In my earlier blog post, reporting the second Cambridge Conversation webinar I attended, the following slide from Prof. Sir David Spiegelhalter was one that drew the sharp distinction between the risk to people in different age ranges:

Age related increase in Covid-19 death rates

Prof. Spiegelhalter’s own Twitter account is also quite busy, and this particular chart was mentioned there, and also on his blog.

This week I was sent this NHS pre-print paper (pending peer review, as many Coronavirus research papers are) to look at the various Covid-19 risk factors and their dependencies, and to explain them. The focus of the 20-page paper is the potential for enhanced risk for people with Type-1 or Type-2 Diabetes, but the Figure 2 towards the end of that paper shows the relative risk ratios for a number of other parameters too, including age range, gender, deprivation and ethnic group.

Risk ratios for different population characteristics

This chart extract, from the paper by corresponding author Prof. Jonathan Valabhji (Imperial College, London & NHS) and his colleagues, indicates a very high age-related dependency for Covid-19 risk, based on the age of the individual. The risk ratio for a white woman under 40, with no deprivation factors, and no diabetes, compared with a control person (a 60-69 year old white woman, with no deprivation factors, and no diabetes) is 1% of the risk. A white male under 40 with otherwise similar characteristics would have a risk of 1.94% of the control person.

Other reduction factors apply in the two 10-year age bands between 40-49 and 50-59, for a white woman (no deprivations or diabetes) in those age ranges of 11% and 36% of the risk respectively.

At 70-79, and above 80, the risk enhancement factors owing to age are x 2.63 and x 9.14 respectively.

So there is some agreement (at least on the principle of age dependency of risk, as represented by the data, if not the quantum), between EuroMOMO, Prof. Michael Levitt, Prof. Sir David Spiegelhalter and the Prof. Jonathan Valabhji et al. paper; that increasing age beyond middle age is a significant indicator of enhanced risk to Covid-19.

In some other respects, Michael is at odds with forecasts made by Prof. Neil Ferguson’s Imperial College group (and, by inference, also with the London School of Hygiene and Tropical Medicine) and with the analysis of the Imperial College paper by Prof. Spiegelhalter.

I reported this in my recent blog post on May 18th concerning the Cambridge Conversation of 14th May, highlighting the contrast with Michael’s interview with Freddie Sayers of UnHerd, which is available directly on YouTube at https://youtu.be/bl-sZdfLcEk.

I recommend going to the primary evidence and watching the videos in those posts.

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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.