Tag: Coronavirus

Adam Kucharski Alex de Visscher Coronavirus Covid-19 David Spiegelhalter Superspreader Worldometers

Model updates for UK lockdown easing points

Post author By Brian
Post date August 4, 2020
1 Comment on Model updates for UK lockdown easing points

Charts 4 and 13 for the comparison of cumulative & daily reported & modelled deaths, plus trend line, on the basis of 83.5% effectiveness, modified in 4 steps by -1%, -5% -10% and +2% successively — Charts 4 and 13 for the comparison of cumulative & daily reported & modelled deaths, plus trend line

Introduction

As I reported in my previous post on 31st July, the model I use, originally authored by Prof. Alex de Visscher at Concordia University in Montreal, and described here, required an update to handle several phases of lockdown easing, and I’m glad to say that is now done.

Alex has been kind enough to send me an updated model code, adopting a method I had been considering myself, introducing an array of dates and intervention effectiveness parameters.

I have been able to add the recent UK Government relaxation dates, and the estimated effectiveness of each into the new model code. I ran some sensitivities which are also reported.

Updated interventions effectiveness and dates

Now that the model can reflect the timing and impact of various interventions and relaxations, I can use the epidemic history to date to calibrate the model against both the initial lockdown on March 23rd, and the relaxations we have seen so far.

Both the UK population (informally) and the Government (formally) are making adjustments to their actions in the light of the threats, actual and perceived, and so the intervention effects will vary.

Model adjustments

At present I am experimenting with different effectiveness percentages relating to four principal lockdown relaxation dates, as described at the Institute for Government website.

In the model, the variable k11 is the starting % infection rate per day per person for SARS-Cov-2, at 0.39, which corresponds to 1 infection every 1/0.39 days ~= 2.5 days per infection.

Successive resetting of the interv_success % model variable allows the lockdown, and its successive easings to be defined in the model. 83.5% (on March 23rd for the initial lockdown) corresponds to k11 x 16.5% as the new infection rate under the initial lockdown, for example.

In the table below, I have also added alternative lockdown easing adjustments in red to show, by comparison, the effect of the forecast, and hence how difficult it will be for a while to assess the impact of the lockdown easings, volatile and variable as they seem to be.

Date	Day	Steps and example measures	Changes to % effectiveness
23rd March	52	Lockdown starts	+83.5%
13th May	105	Step 1 – Partial return to work Those who can work from home should do so, but those who cannot should return to work with social distancing guidance in place. Some sports facilities open.	-1% = 82.5% -4% = 79.5%
1st June	122	Step 2 – Some Reception, Year 1 and Year 6 returned to school. People can leave the house for any reason (not overnight). Outdoor markets and car showrooms opened.	-5% = 77.5% -8% = 71.5%
15th June	136	Step 2 additional – Secondary schools partially reopened for years 10 and 12. All other retail are permitted to re-open with social distancing measures in place.	-10% = 67.5% +10% = 81.5%
4th July	155	Step 3 – Food service providers, pubs and some leisure facilities are permitted to open, as long as they are able to enact social distancing.	+20% = 87.5% -6% = 75.5%
1st August	186	Step 3 additional – Shielding for 2m vulnerable people in the UK ceases	0% = 87.5%

Dates, examples of measures, and % lockdown effectiveness changes on k11

After the first of these interventions, the 83.5% effectiveness for the original March 23rd lockdown, my model presented a good forecast up until lockdown easing began to happen (both informally and also though Government measures) on May 13th, when Step 1 started, as shown above and in more detail at the Institute for Government website.

Within each easing step, there were several intervention relaxations across different areas of people’s working and personal lives, and I have shown two of the Step 2 components on June 1st and June 15th above.

I applied a further easing % for June 15th (when more Step 2 adjustments were made), and, following Step 3 on July 4th, and the end (for the time being) of shielding for 2m vulnerable people on August 1st, I am expecting another change in mid-August.

I have managed to match the reported data so far with the settings above, noting that even though the July 4th Step 3 was a relaxation, the model matches reported data better when the overall lockdown effectiveness at that time is increased. I expect to adjust this soon to a more realistic assessment of what we are seeing across the UK.

With the % settings in red, the outlook is a little different, and I will show the charts for these a little later in the post

The settings have to reflect not only the various Step relaxation measures themselves, but also Government guidelines, the cumulative changes in public behaviour, and the virus response at any given time.

For example, the wearing of face coverings has become much more common, as mandated in some circumstances but done voluntarily elsewhere by many.

Comparative model charts

The following charts show the resulting changes for the initial easing settings. The first two show the new period of calibration I have used from early March to the present day, August 4th.

On chart 4, on the left, you can see the “uptick” beginning to start, and the model line isn’t far from the 7-day trend line of reported numbers at present (although as of early August possibly falling behind the reported trend a little).

Chart 4 for the comparison of cumulative & daily reported & modelled deaths, plus trend line, on the basis of 83.5% effectiveness, modified in 4 steps by -1%, -5% -10% and +2% successively — **Charts 4 (with trend) and 13 for the comparison of cumulative & daily reported & modelled deaths**

Chart 13 for the comparison of cumulative reported and modelled deaths, on the basis of 83.5% effectiveness, modified in 4 steps by -1%, -5% -10% and -2% successively — **Charts 4 (with trend) and 13 for the comparison of cumulative & daily reported & modelled deaths**

On the linear axis chart 13, on the right, the reported and model curves are far closer than in the version in my most recent post on July 31st, when I showed the effects of lockdown easing on the previous forecasts, and I highlighted the difficulty of updating without a way of parametrising the lockdown easing steps (at that time).

Using the new model capabilities, I have now been able to calibrate the model to the present day, both achieving the good match I already had from March 23rd lockdown to mid-May, and then separately to adjust and calibrate the model behaviour since mid-May to the present day, by adjusting the lockdown effectiveness at May 15th, June 1st, June 15th and July 4th, as described earlier.

The orange dots (the daily deaths) on chart 4 tend to cluster in groups of 4 or 5 per week above the trend line (and also the model line), and 3 or 2 per week below. This is because of the poor accuracy of some reporting at weekends (consistently under-reporting at weekends and recovering by over-reporting early the following week).

The red 7-day trend line on chart 4 reflects the weekly average position.

Looking a little further ahead, to September 30th, this model, with the initial easing settings, predicts the following behaviour, prior to any further lockdown easing adjustments, expected in mid-August.

Chart 12 for the comparison of cumulative & daily reported & modelled deaths, on the basis of 83.5% effectiveness, modified in 4 steps by -1%, -5% -10% and +2% successively — **Chart 12 for the comparison of cumulative & daily reported & modelled deaths**

Finally, for comparison, the Worldometers UK site has a link to its own forecast site, which has several forecasts depending on assumptions made about mask-wearing, and/or continued mandated lockdown measures, with confidence limits. I have screenshot the forecast on October 1st, where it shows 48,268 deaths assuming current mandates continuing, and mask-wearing.

My own forecast shows 47,201 cumulative deaths at that date.

**Worldometers forecasts on the basis of mask wearing vs. no mandated measures, with confidence limits**

Alternative % settings in red

I now present a slideshow of the corresponding charts with the red % easing settings. The results here are for the same initial lockdown effectiveness, 83.5%, but with successive easings at -4%, -8%, +10% and -6%, where negative is relaxation, and positive is an increase in intervention effectiveness.

Chart 12 for the comparison of cumulative & daily reported & modelled deaths to 26th April 2021, adjusted by -4%, -8% +10% and -6%
Chart 12 for the comparison of cumulative & daily reported & modelled deaths to 30th Sep, adjusted by -4%, -8% +10% and -6%
Chart 4 comparison of cumulative & daily reported & modelled deaths, & reported trend line, for 83.5% effectiveness, adjusted by -4%, -8% +10% and -6%
Model forecast for the UK and reported deaths as at August 8th, for 83.5% lockdown effectiveness, adjusted by -4%, -8% +10% and -6%
Model forecast (linear axes) and reported UK deaths to August 8th, for 83.5% lockdown effectiveness, adjusted by -4%, -8% +10% and -6%

The model forecast here for September 30th is for 49, 549 deaths, and the outlook for the longer term, April 2021, is for 52,544.

Thus the next crucial few months, as the UK adjusts its methods for interventions to be more local and immediate, will be vital in its impact on outcomes. The modelling of how this will work is far more difficult, therefore, with fine-grained data required on virus characteristics, population movement, the comparative effect of different intervention measures, and individual responses and behaviour.

Hotspots and local lockdowns

At present, because the UK reported case number trend has flattened out and isn’t decreasing as fast, and because of some local hotspots of Covid-19 cases, the UK Government has been forced to take some local measures, for example in Leicester a month ago, and more recently in Manchester; the scope and scale of any lockdown adjustments is, therefore, a moving target.

I would expect this to be the pattern for the future, rather than national lockdowns. The work of Adam Kucharski, reported at the Wellcome Open Research website, highlighting the “k-number”, representing the variation, or dispersion in R, the reproduction number, as he says in his Twitter thread, will be an important area to understand.

The k-number might well be more indicative, at this local hotspot stage of the crisis, than just the R reproduction number alone; it has a relationship to the “superspreader” phenomenon discussed for SARS in this 2005 paper, that was also noticed very early on for SARS-Cov-2 in the 2020 pandemic, both in Italy and also in the UK. I will look at that in more detail in another posting.

Superspreading relates to individuals who are infected (probably asymptomatically or pre-symptomatically) who infect others in a closed social space (eg in a ski resort chalet as reported by the BBC on February 8th) without realising it.

The hotspots we are now seeing in many places might well be related to this type of dispersion. The modelling for this would be a further complication, potentially requiring a more detailed spatial model, which I briefly discussed in my blog post on modelling methods on July 14th.

Superspreading might also need to be understood in relation to the opening of schools, in August and September (across the four UK home countries). It might have been a factor in Israel’s experience of return to schools, as covered by the Irish Times on August 4th.

The excess deaths measure

There has been quite a debate on excess deaths (often a seasonal comparison of age-related deaths statistics compared with the previous 5 years) as a measure of the overall position at any time. As I said in a previous post on June 2nd, this measure does mitigate any arguments as to what is a Covid-19 fatality, and what isn’t.

The excess deaths measure, however, has its own issues with regard to the epidemic’s indirect influence on the death rates from other causes, both upwards and downwards.

Since there is less travel (on the roads, for example, with fewer accidents), and many people are taking more care in other ways in their daily lives, deaths from some other causes might tend to reduce.

On the other hand, people are feeling other pressures on their daily lives, affecting their mood and health (for example the weight gain issues reported by the COVID Symptom Study), and some are not seeking medical help as readily as they might have done in other circumstances, for fear of putting themselves at risk of contracting Covid-19. Both factors tend to increase illness and potentially death rates.

Even as excess deaths reduce, then, it may well be that Covid-19 deaths increase as others reduce. Possibly a crossover with seasonal influenza deaths, later on, might be masked by the overall excess deaths measure.

As I also mentioned in my post on July 6th, deaths in later years from other causes might increase because of this lack of timely diagnosis and treatment for other “dread” diseases, as, for example, for cancer, as stated by Data-can.org.uk.

So no measure of the epidemic’s effects is without its issues. Prof. Sir David Spiegelhalter covered this aspect in a newspaper article this week.

Discussion

The statistical interpretation and modelling of data related to the pandemic is a matter of much debate. Some commentators and modellers are proponents of quite different methods of data recording, analysis and forecasting, and I covered phenomenological methods compared with mechanistic (SIR) modelling in previous posts on July 14th and July 18th.

The current reduced rate of decline in cases and deaths in some countries and regions, with concomitant local outbreaks being handled by local intervention measures, including, in effect, local lockdowns, has complicated the predictions of some who think (and have predicted) that the SARS-Cov-2 crisis will soon be over (some possibly for political reasons, some of them scientists).

Even when excess deaths reduce to zero, this doesn’t mean that Covid-19 is over, because, as I mentioned above, illness and deaths from other causes might have reduced, with Covid-19 filling the gap.

There are also concerns that recovery from Covid-19 as a death threat can be followed by longer-lasting illness and symptoms, and some studies (for example this NHLBI one) are gathering evidence, such as that covered by this report in the Thailand Medical News.

This Discharge advice from the UK NHS makes continuing care requirements for discharged Covid-19 patients in the UK very clear.

It is by no means certain, either, that recovery from Covid-19 confers immunity to SARS-Cov-2, and, if it does, for how long.

Concluding comments

I remain of the view that in the absence of a vaccine, or a very effective pharmaceutical treatment therapy, we will be living with SARS-Cov-2 for a long time, and that we do have to continue to be cautious, even (or, rather, especially) as the UK Government (and many others) move to easing national lockdown, at the same time as being forced to enhance some local intervention measures.

The virus remains with us, and Government interventions are changing very fast. Face coverings, post-travel quarantining, office/home working and social distancing decisions, guidance and responses are all moving quite quickly, not necessarily just nationally, but regionally and locally too.

I will continue to sharpen the focus of my own model; I suspect that there will be many revisions and that any forecasts are now being made (including by me) against a moving target in a changing context.

Any forecast, in any country, that it will be all over bar the shouting this summer is at best a hostage to fortune, and, at worst, irresponsible. My own model still requires tuning; in any case, however, I would not be axiomatic about its outputs.

This is an opinion informed by my studies of others’ work, my own modelling, and considerations made while writing my 30 posts on this topic since late March.

Tags Coronavirus, Covid-19, forecasting, modeling, modelling, Superspreader

Coronavirus Covid-19 Office for National Statistics ONS PHE Public Health England Worldometers

The effect of lockdown easing in the UK

Post author By Brian
Post date July 31, 2020
2 Comments on The effect of lockdown easing in the UK

My forecast for the UK deaths as at July 30th, including trend line for reported deaths, for 83.5% lockdown effectiveness

Introduction

As reported in my previous post, there has been a gradual reduction in the rate of decline of cases and deaths in the UK relative to my model forecasts. This decline had already been noted, as I reported in my July 6th blog article, by The Office for National Statistics and their research partners, the University of Oxford, and reported on the ONS page here.

I had adjusted the original lockdown effectiveness in my model (from 23rd March) to reflect this emerging change, but as the model had been predicting correct behaviour up until mid-late May, I will present here the original model forecasts, compared with the current reported deaths trend, which highlights the changes we have experienced for the last couple of months.

Forecast comparisons

The ONS chart which highlighted this slowing down of the decline, and even a slight increase, is here:

Figure 6: The latest exploratory modelling shows incidence appears to have decreased between mid-May and early June

Public Health England had also reported on this tendency for deaths on 6th July:

The death rate trend can be seen in the daily and 7-day average trend charts, with data from Public Health England — The death rate trend can be seen in the daily and 7-day average trend charts

The Worldometers forecast for the UK has been refined recently, to take account of changes in mandated lockdown measures, such as possible mask wearing, and presents several forecasts on the same chart depending on what take-up would be going forward.

Worldometers forecast for the UK as at July 31st 2020

We see that, at worst, the Worldometers forecast could be for up to 60,000 deaths by November 1st, although, according to their modelling, if masks are “universal” then this is reduced to under 50,000.

Comparison of my forecast with reported data

My two charts that reveal most about the movement in the rate of decline of the UK death rate are here…

My forecast for the UK deaths as at July 30th, including trend line for reprted deaths — **My forecast for the UK deaths as at July 30th, including trend line for daily reported deaths, for 83.5% lockdown effectiveness**

My forecast for the UK deaths as at July 30th, compared with reported for 83.5% lockdown effectiveness — **My forecast for the UK deaths as at July 30th, including trend line for daily reported deaths, for 83.5% lockdown effectiveness**

On the left, the red trend line for reported daily deaths shows they are not falling as fast as they were in about mid-May, when I was forecasting a long term plateau for deaths at about 44,400, assuming that lockdown effectiveness would remain at 83.5%, i.e. that the virus transmission rate was reduced to 16.5% of what it would be if there were no reductions in social distancing, self isolation or any of the other measures the UK had been taking.

The right hand chart shows the divergence between the reported deaths (in orange) and my forecast (in blue), beginning around mid to late May, up to the end of July.

The forecast, made back in March/April, was tracking the reported situation quite well (if very slightly pessimistically), but around mid-late May we see the divergence begin, and now as I write this, the number of deaths cumulatively is about 2000 more than I was forecasting back in April.

Lockdown relaxations

This period of reduction in the rate of decline of cases, and subsequently deaths, roughly coincided with the start of the UK Govenment’s relaxation of some lockdown measures; we can see the relaxation schedule in detail at the Institute for Government website.

As examples of the successive stages of lockdown relaxation, in Step 1, on May 13th, restrictions were relaxed on outdoor sport facilities, including tennis and basketball courts, golf courses and bowling greens.

In Step 2, from June 1st, outdoor markets and car showrooms opened, and people could leave the house for any reason. They were not permitted to stay overnight away from their primary residence without a ‘reasonable excuse’.

In Step 3, from 4th July, two households could meet indoors or outdoors and stay overnight away from their home, but had to maintain social distancing unless they are part of the same support bubble. By law, gatherings of up to 30 people were permitted indoors and outdoors.

These steps and other detailed measures continued (with some timing variations and detailed changes in the devolved UK administrations), and I would guess that they were anticipated and accompanied by a degree of informal public relaxation, as we saw from crowded beaches and other examples reported in the press.

Model consequences

I did make a re-forecast, reported on July 6th in my blog article, using 83% lockdown effectiveness (from March 23rd).

Two issues remained, however, while bringing the current figures for July more into line.

One was that, as I only have one place in the model that I change the lockdown effectiveness, I had to change it from March 23rd (UK lockdown date), and that made the intervening period for the forecast diverge until it converged again recently and currently.

That can be seen in the right hand chart below, where the blue model curve is well above the orange reported data curve from early May until mid-July.

The long-term plateau in deaths for this model forecast is 46,400; this is somewhat lower than the model would show if I were to reduce the % lockdown effectiveness further, to reflect what is currently happening; but in order to achieve that, the history during May and June would show an even larger gap.

My forecast for the UK deaths as at July 30th, including trend line for reported deaths, for 83% lockdown effectiveness — **My forecast for the UK deaths as at July 30th, including trend line for daily reported deaths, for 83% lockdown effectiveness**

The second issue is that the rate of increase in reported deaths, as we can also see (the orange curve) on the right-hand chart, at July 30th, is clearly greater than the model’s rate (the blue curve), and so I foresee that reported numbers will begin to overshoot the model again.

In the chart on the left, we see the same red trend line for the daily reported deaths, flattening to become nearly horizontal at today’s date, July 31st, reflecting that the daily reported deaths (the orange dots) are becoming more clustered above the grey line of dots, representing modelled daily deaths.

As far as the model is concerned, all this will need to be dealt with by changing the lockdown effectiveness to a time-dependent variable in the model differential equations representing the behaviour of the virus, and the population’s response to it.

This would allow changes in public behaviour, and in public policy, to be reflected by a changed lockdown effectiveness % from time to time, rather than having retrospectively to apply the same (reduced) effectiveness % since the start of lockdown.

Then the forecast could reflect current reporting, while also maintaining the close fit between March 23rd and when mitigation interventions began to ease.

Lockdown, intervention effectiveness and herd immunity

In the interest of balance, in case it might be thought that I am a fan of lockdown(!), I should say that higher % intervention effectiveness does not necessarily lead to a better longer term outlook. It is a more nuanced matter than that.

In my June 28th blog article, I covered exactly this topic as part of my regular Coronavirus update. I referred to the pivotal March 16th Imperial College paper on Non-Pharmaceutical Interventions (NPIs), which included this (usefully colour-coded) table, where green is better and red is worse,

PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions)

which provoked me to re-confirm with the authors (and as covered in the paper) the reasons for the triple combination of CI_HQ_SD being worse than either of the double combinations of measures CI_HQ or CI_SD in terms of peak ICU bed demand.

The answer (my summary) was that lockdown can be too effective, given that it is a temporary state of affairs. When lockdown is partially eased or removed, the population can be left with less herd immunity (given that there is any herd immunity to be conferred by SARS-Cov-2 for any reasonable length of time, if at all) if the intervention effectiveness is too high.

Thus a lower level of lockdown effectiveness, below 100%, can be more effective in the long term.

I’m not seeking to speak to the ethics of sustaining more infections (and presumably deaths) in the short term in the interest of longer term benefits. Here, I am simply looking at the outputs from any postulated inputs to the modelled epidemic process.

I was as surprised as anyone when, in a UK Government briefing, in early March, before the UK lockdown on March 23rd, the Chief Scientific Adviser (CSA, Sir Patrick Vallance), supported by the Chief Medical Officer (CMO, Prof. Chris Whitty) talked about “herd immunity” for the first time, at 60% levels (stating that 80% needing to be infected to achieve it was “loose talk”). I mentioned this in my May 29th blog post.

The UK Government focus later in March (following the March 16th Imperial College paper) quickly turned to mitigating the effect of Covid-19 infections, as this chart sourced from that paper indicates, prior to the UK lockdown on March 23rd.

Projected effectiveness of Covid-19 mitigation strategies, in relation to the utilisation of critical care (ICU) bedsProjected effectiveness of Covid-19 mitigation strategies, in relation to the utilisation of critical care (ICU) beds — Projected effectiveness of Covid-19 mitigation strategies, in relation to the utilisation of critical care (ICU) beds

This is the imagery behind the “flattening the curve” phrase used to describe this phase of the UK (and others’) strategy.

Finally, that Imperial College March 16th paper presents this chart for a potentially cyclical outcome, until a Covid-19 vaccine or a significantly effective pharmaceutical treatment therapy arrives.

The potentially cyclical caseload from Covid-19, with interventions and relaxations applied as ICU bed demand changes

In this new phase of living with Covid-19, this is why I want to upgrade my model to allow periodic intervention effectiveness changes.

Conclusions

The sources I have referenced above support the conclusion in my model that there has been a reduction in the rate of decline of deaths (preceded by a reduction in the rate of decline in cases).

To make my model relevant to the new situation going forward, when lockdowns change, not only in scope and degree, but also in their targeting of localities or regions where there is perceived growth in infection rates, I will need to upgrade my model for variable lockdown effectiveness.

I wouldn’t say that the reduction of the rate of decline of cases and deaths is evidence of a “second wave”, but is rather the response of a very infective agent, which is still with us, to infect more people who are increasingly “available” to it, owing to easing of some of the lockdown measures we have been using (both informally by the public and formally by Government).

To me, it is evidence that until we have a vaccine, we will have to live with this virus among us, and take reasonable precautions within whatever envelope of freedoms the Government allow us.

We are all in each others’ hands in that respect.

Tags Coronavirus, Covid-19, forecasting, modeling, modelling

Coronavirus Covid-19 Michael Levitt

Mechanistic and curve-fitting UK modelling comparison

Post author By Brian
Post date July 18, 2020
2 Comments on Mechanistic and curve-fitting UK modelling comparison

Cumulative, daily and 7-day average model UK deaths, 83%, 6th March - 30th Sep, with the daily ratio of cumulative deaths, and the H(t) function, the natural logarithm of that ratio, and the negative natural log of H(t) — Model daily ratio of cumulative deaths, and the H(t) function, the natural log of that ratio

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.

UK deaths, reported vs. model, 83%, cumulative, to 16th July 2020

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.

Cumulative, daily and 7-day average reported UK deaths, 83%, 6th March - 16th July — Cumulative, daily & 7-day avge reported UK deaths, 83%, 6th March – 16th July

Cumulative, daily and 7-day average model UK deaths, 83%, 6th March - 16th July — Cumulative, daily & 7-day avge reported UK deaths, 83%, 6th March – 16th July

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.

Reported deaths data with the daily ratio of cumulative deaths, and the H(t) function, the natural logarithm of that ratio — Reported & daily ratio of cumulative deaths, c(t)/c(t-1), and H(t) = Ln(c(t)/c(t-1))

Cumulative, daily and 7-day average reported UK deaths, 83%, 6th March - 16th July, with the daily ratio of cumulative deaths, and the H(t) function, the natural logarithm of that ratio, and the negative natural log of H(t) with its trend line in red — Reported & daily ratio of cumulative deaths, c(t)/c(t-1), and H(t) = Ln(c(t)/c(t-1))

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.

Cumulative, daily and 7-day average model UK deaths, 83%, 6th March - 16th July, with the daily ratio of cumulative deaths, and the H(t) function, the natural logarithm of that ratio, and the negative natural log of H(t) — Adding the negative natural log of H(t) for reported data with its trend line in red

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.

Tags Coronavirus, Covid-19, Imperial College, mechanistic, Michael Levitt, modeling, modelling, phenomenology

Coronavirus Covid-19 Imperial College Michael Levitt Reproductive Number

Phenomenology & Coronavirus – modelling and curve-fitting

Post author By Brian
Post date July 14, 2020
3 Comments on Phenomenology & Coronavirus – modelling and curve-fitting

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

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).

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, in an elliptical path for any fluid “particle”, not in the direction of travel of the observed (largely 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

Tags Coronavirus, Covid-19, curve fitting, modeling, modelling, SARS-CoV-2, SIR

Coronavirus Covid-19 Worldometers

Coronavirus modelling update

Post author By Brian
Post date July 6, 2020
2 Comments on Coronavirus modelling update

Cumulative charts to date, with 83% vs. 83.5% intervention effectiveness since lockdown

Introduction

In my previous post on June 28th, I covered the USA vs. Europe Coronavirus pandemic situations; herd immunity, and the effects of various interventions on it, particularly as envisioned by the Imperial College Covid-19 response team; and the current forecasts for cases and deaths in the UK.

I have now updated the forecasts, as it was apparent that during the month of June, there had been a slight increase in the forecast for UK deaths. Worldometers’ forecast had increased, and also the reported UK numbers were now edging above the forecast in my own model, which had been tracking well as a forecast (if very slightly pessimistically) until the beginning of June.

This might be owing both to informal public relaxation of lockdown behaviour, and also to formal UK Government relaxations in some intervention measures since the end of May.

Re-forecast

I have now reforecast my model with a slightly lower intervention effectiveness (83% instead of 83.5% since lockdown on 23rd March), and, while still slightly below reported numbers, it is nearly on track (although with the reporting inaccuracy each weekend, it’s not practical to try to track every change).

My long term outlook for deaths is now for 46,421 instead of 44,397, still below the Worldometers number (which has increased to 47,924 from 43,962).

Here are the comparative charts – first, the reported deaths (the orange curve) vs. modelled deaths (the blue curve), linear axes, as of July 6th.

UK deaths, reported vs. model, 83%, cumulative, to 30th September 2020

UK deaths, reported vs. model, 83.5%, cumulative, to 30th September 2020UK deaths, reported vs. model, 83.5%, cumulative, to 30th September 2020 — UK deaths, reported vs. model, 83%, cumulative, to 30th September 2020

Comparing this pair of charts, we see that the .5% reduction in lockdown intervention effectiveness (from March 23rd) brings the forecast, the blue curve on the left chart, above the reported orange curve. On the right, the forecast, which had been tracking the reported numbers for a month or more, had started to lag the reported numbers since the beginning of June.

I present below both cumulative and daily numbers of deaths, reported vs. forecast, with log y-axis. The scatter in the daily reported numbers (orange dots) is because of inconsistencies in reporting at weekends, recovered during each following week.

UK deaths, reported vs. model, 83%, cumulative and daily, to 30th September 2020 — UK deaths, reported vs. model, 83%, cumulative/daily, to 30th September 2020

UK deaths, reported vs. model, 83.5%, cumulative and daily, to 30th September 2020 — UK deaths, reported vs. model, 83%, cumulative/daily, to 30th September 2020

In this second pair of charts, we can just see that the rate of decline in daily deaths, going forward, is slightly reduced in the 83% chart on the left, compared with the 83.5% on the right.

This means that the projected plateau in modelled deaths, as stated above, is at 46,421 instead of 44,397 in my modelled data from which these charts are drawn.

It also shows that the forecast reduction to single digit (<10) deaths per day is pushed out from 13th August to 20th August, and the forecast rate of fewer than one death per day is delayed from 21st September to 30th September.

ONS & PHE work on trends, and concluding comments

Since the beginning of lockdown relaxations, there has been sharpened scrutiny of the case and death numbers. This monitoring continues with the latest announcements by the UK Government, taking effect from early July (with any accompanying responses to follow from the three UK devolved administrations).

The Office for National Statistics has been monitoring cases and deaths rates, of course, and the flattening of the infections and deaths reductions has been reported in the press recently.

**July 3rd Times reporting ONS regarding trends in Covid-19 incidence rates and deaths**

As the article says, any movement would firstly be in the daily number of cases, with any potential change in the deaths rate following a couple of weeks after (owing to the Covid-19 disease duration).

Source data for the reported infection rate is on the following ONS chart (Figure 6 on their page), where the latest exploratory modelling, by ONS research partners at the University of Oxford, shows the incidence rate appears to have decreased between mid-May and early June, but has since levelled off.

The death rate trend can be seen in the daily and 7-day average trend charts, with data from Public Health England.

**Daily and 7-day average trends for deaths in the UK, The Times, July 3rd**

The ONS is also tracking Excess deaths, and it seems that the Excess deaths in 2020 in England & Wales have reduced to below the five-year average for the second consecutive week.

The figures can be seen in the spreadsheet here, downloaded from the ONS page. The following chart appears there as Figure 1, also showing that the number of deaths involving Covid-19 decreased for the 10th consecutive week.

**Number of deaths registered by week, England & Wales, Dec 2019 to 26th June 2020**

There are warnings, however, also reported by The Times, that there may be increased mortality from other diseases (such as cancer) into 2021 because worries about the pandemic haves led to changes in patterns of use of the NHS, including GPs, with fewer people risking trips to hospital for diagnosis and/or treatment. The report referred to below from Data-can.org.uk highlights this

**The Times report on July 6th concerning the impact on other diseases of the pandemic**

I will make any adjustments to the rate of change as we go forward, but thankfully daily numbers are just reducing at the moment in the UK, and I hope that this continues.

Tags Coronavirus, Covid-19, forecasting, modeling, modelling, Worldometers

Coronavirus Covid-19 Herd Immunity Imperial College Reproductive Number

Some thoughts on the current UK Coronavirus position

Post author By Brian
Post date June 28, 2020
2 Comments on Some thoughts on the current UK Coronavirus position

New confirmed cases of coronavirus, 7-day rolling average, USA vs Europe

Introduction

A couple of interesting articles on the Coronavirus pandemic came to my attention this week; a recent one in National Geographic on June 26th, highlighting a startling comparison, between the USA’s cases history, and recent spike in case numbers, with the equivalent European data, referring to an older National Geographic article, from March, by Cathleen O’Grady, referencing a specific chart based on work from the Imperial College Covid-19 Response team.

I noticed, and was interested in that reference following a recent interaction I had with that team, regarding their influential March 16th paper. It prompted more thought about “herd immunity” from Covid-19 in the UK.

Meanwhile, my own forecasting model is still tracking published data quite well, although over the last couple of weeks I think the published rate of deaths is slightly above other forecasts as well as my own.

The USA

The recent National Geographic article from June 26th, by Nsikan Akpan, is a review of the current situation in the USA with regard to the recent increased number of new confirmed Coronavirus cases. A remarkable chart at the start of that article immediately took my attention:

**7 day average cases from the US Census Bureau chart, NY Times / National Geographic**

The thrust of the article concerned recommendations on public attitudes, activities and behaviour in order to reduce the transmission of the virus. Even cases per 100,000 people, the case rate, is worse and growing in the USA.

**7 day average cases per 100,000 people from the US Census Bureau chart, NY Times / National Geographic**

A link between this dire situation and my discussion below about herd immunity is provided by a reported statement in The Times by Dr Anthony Fauci, Director of the National Institute of Allergy and Infectious Diseases, and one of the lead members of the Trump Administration’s White House Coronavirus Task Force, addressing the Covid-19 pandemic in the United States.

**Reported Dr Fauci quotation by the Times newspaper 30th June 2020**

If the take-up of the vaccine were 70%, and it were 70% effective, this would result in roughly 50% herd immunity (0.7 x 0.7 = 0.49).

If the innate characteristics of the the SARS-CoV-2 virus don’t change (with regard to infectivity and duration), and there is no other human-to-human infection resistance to the infection not yet understood that might limit its transmission (there has been some debate about this latter point, but this blog author is not a virologist) then 50% is unlikely to be a sufficient level of population immunity.

My remarks later about the relative safety of vaccination (eg MMR) compared with the relevant diseases themselves (Rubella, Mumps and Measles in that case) might not be supported by the anti-Vaxxers in the US (one of whose leading lights is the disgraced British doctor, Andrew Wakefield).

This is just one more complication the USA will have in dealing with the Coronavirus crisis. It is one, at least, that in the UK we won’t face to anything like the same degree when the time comes.

The UK, and implications of the Imperial College modelling

That article is an interesting read, but my point here isn’t really about the USA (worrying though that is), but about a reference the article makes to some work in the UK, at Imperial College, regarding the effectiveness of various interventions that have been or might be made, in different combinations, work reported in the National Geographic back on March 20th, a pivotal time in the UK’s battle against the virus, and in the UK’s decision making process.

This chart reminded me of some queries I had made about the much-referenced paper by Neil Ferguson and his team at Imperial College, published on March 16th, that seemed (with others, such as the London School of Hygiene and Infectious Diseases) to have persuaded the UK Government towards a new approach in dealing with the pandemic, in mid to late March.

**Possible intervention strategies in the fight against Coronavirus**

The thrust of this National Geographic article, by Cathleen O’Grady, was that we will need “herd immunity” at some stage, even if the Imperial College paper of March 16th (and other SAGE Committee advice, including from the Scientific Pandemic Influenza Group on Modelling (SPI-M)) had persuaded the Government to enforce several social distancing measures, and by March 23rd, a combination of measures known as UK “lockdown”, apparently abandoning the herd immunity approach.

The UK Government said that herd immunity had never been a strategy, even though it had been mentioned several times, in the Government daily public/press briefings, by Sir Patrick Vallance (UK Chief Scientific Adviser (CSA)) and Prof Chris Whitty (UK Chief Medical Officer (CMO)), the co-chairs of SAGE.

The particular part of the 16th March Imperial College paper I had queried with them a couple of weeks ago was this table, usefully colour coded (by them) to allow the relative effectiveness of the potential intervention measures in different combinations to be assessed visually.

Why was it, I wondered, that in this chart (on the very last page of the paper, and referenced within it) the effectiveness of the three measures “CI_HQ_SD” in combination (home isolation of cases, household quarantine & large-scale general population social distancing) taken together (orange and yellow colour coding), was LESS than the effectiveness of either CI_HQ or CI_SD taken as a pair of interventions (mainly yellow and green colour coding)?

The explanation for this was along the following lines.

It’s a dynamical phenomenon. Remember mitigation is a set of temporary measures. The best you can do, if measures are temporary, is go from the “final size” of the unmitigated epidemic to a size which just gives herd immunity.

If interventions are “too” effective during the mitigation period (like CI_HQ_SD), they reduce transmission to the extent that herd immunity isn’t reached when they are lifted, leading to a substantial second wave. Put another way, there is an optimal effectiveness of mitigation interventions which is <100%.

That is CI_HQ_SDOL70 for the range of mitigation measures looked at in the report (mainly a green shaded column in the table above).

While, for suppression, one wants the most effective set of interventions possible.

All of this is predicated on people gaining immunity, of course. If immunity isn’t relatively long-lived (>1 year), mitigation becomes an (even) worse policy option.

Herd Immunity

The impact of very effective lockdown on immunity in subsequent phases of lockdown relaxation was something I hadn’t included in my own (single phase) modelling. My model can only (at the moment) deal with one lockdown event, with a single-figure, averaged intervention effectiveness percentage starting at that point. Prior data is used to fit the model. It has served well so far, until the point (we have now reached) at which lockdown relaxations need to be modelled.

But in my outlook, potentially, to modelling lockdown relaxation, and the potential for a second (or multiple) wave(s), I had still been thinking only of higher % intervention effectiveness being better, without taking into account that negative feedback to the herd immunity characteristic, in any subsequent more relaxed phase, other than through the effect of the changing comparative compartment sizes in the SIR-style model differential equations.

I covered the 3-compartment SIR model in my blog post on April 8th, which links to my more technical derivation here, and more complex models (such as the Alex de Visscher 7-compartment model I use in modified form, and that I described on April 14th) that are based on this mathematical model methodology.

In that respect, the ability for the epidemic to reproduce, at a given time “t” depends on the relative sizes of the infected (I) vs. the susceptible (S, uninfected) compartments. If the R (recovered) compartment members don’t return to the S compartment (which would require a SIRS model, reflecting waning immunity, and transitions from R back to the S compartment) then the ability of the virus to find new victims is reducing as more people are infected. I discussed some of these variations in my post here on March 31st.

My method might have been to reduce the % intervention effectiveness from time to time (reflecting the partial relaxation of some lockdown measures, as Governments are now doing) and reimpose it to a higher % effectiveness if and when the R_t (the calculated R value at some time t into the epidemic) began to get out of control. For example, I might relax lockdown effectiveness from 90% to 70% when R_t reached R_t<0.7, and increase again to 90% when R_t reached R_t>1.2.

This was partly owing to the way the model is structured, and partly to the lack of disaggregated data I would have available to me for populating anything more sophisticated. Even then, the mathematics (differential equations) of the cyclical modelling was going to be a challenge.

In the Imperial College paper, which does model the potential for cyclical peaks (see below), the “trigger” that is used to switch on and off the various intervention measures doesn’t relate to R_t, but to the required ICU bed occupancy. As discussed above, the intervention effectiveness measures are a much more finely drawn range of options, with their overall effectiveness differing both individually and in different combinations. This is illustrated in the paper (a slide presented in the April 17th Cambridge Conversation I reported in my blog article on Model Refinement on April 22nd):

What is being said here is that if we assume a temporary intervention, to be followed by a relaxation in (some of) the measures, the state in which the population is left with regard to immunity at the point of change is an important by-product to be taken into account in selecting the (combination of) the measures taken, meaning that the optimal intervention for the medium/long term future isn’t necessarily the highest % effectiveness measure or combined set of measures today.

The phrase “herd immunity” has been an ugly one, and the public and press winced somewhat (as I did) when it was first used by Sir Patrick Vallance; but it is the standard term for what is often the objective in population infection situations, and the National Geographic articles are a useful reminder of that, to me at least.

The arithmetic of herd immunity, the R number and the doubling period

I covered the relevance and derivation of the R₀ reproduction number in my post on SIR (Susceptible-Infected-Recovered) models on April 8th.

In the National Geographic paper by Cathleen O’Grady, a useful rule of thumb was implied, regarding the relationship between the herd immunity percentage required to control the growth of the epidemic, and the much-quoted R₀ reproduction number, interpreted sometimes as the number of people (in the susceptible population) one infected person infects on average at a given phase of the epidemic. When R_t reaches one or less, at a given time t into the epidemic, so that one person is infecting one or fewer people, on average, the epidemic is regarded as having stalled and to be under control.

Herd immunity and R₀

One example given was measles, which was stated to have a possible starting R₀ value of 18, in which case almost everyone in the population needs to act as a buffer between an infected person and a new potential host. Thus, if the starting R₀ number is to be reduced from 18 to R_t<=1, measles needs A VERY high rate of herd immunity – around 17/18ths, or ~95%, of people needing to be immune (non-susceptible). For measles, this is usually achieved by vaccine, not by dynamic disease growth. (Dr Fauci had mentioned over 95% success rate in the US previously for measles in the reported quotation above).

Similarly, if Covid-19, as seems to be the case, has a lower starting infection rate (R₀ number) than measles, nearer to between 2 and 3 (2.5, say (although this is probably less than it was in the UK during March; 3-4 might be nearer, given the epidemic case doubling times we were seeing at the beginning*), then the National Geographic article says that herd immunity should be achieved when around 60 percent of the population becomes immune to Covid-19. The required herd immunity H% is given by H% = (1 – (1/2.5))*100% ~= 60%.

Whatever the real Covid-19 innate infectivity, or reproduction number R₀ (but assuming R₀>1 so that we are in an epidemic situation), the required herd immunity H% is given by:

H%=(1-(1/R₀))*100% (1)

(*I had noted that 80% was referenced by Prof. Chris Whitty (CMO) as loose talk, in an early UK daily briefing, when herd immunity was first mentioned, going on to mention 60% as more reasonable (my words). 80% herd immunity would correspond to R₀=5 in the formula above.)

R₀ and the Doubling time

As a reminder, I covered the topic of the cases doubling time T_Dhere; and showed how it is related to R₀ by the formula;

R₀=d(log_e2)/T_D (2)

where d is the disease duration in days.

Thus, as I said in that paper, for a doubling period T_Dof 3 days, say, and a disease duration d of 2 weeks, we would have R₀=14×0.7/3=3.266.

If the doubling period were 4 days, then we would have R₀=14×0.7/4=2.45.

As late as April 2nd, Matt Hancock (UK secretary of State for Health) was saying that the doubling period was between 3 and 4 days (although either 3 or 4 days each leads to quite different outcomes in an exponential growth situation) as I reported in my article on 3rd April. The Johns Hopkins comparative charts around that time were showing the UK doubling period for cases as a little under 3 days (see my March 24th article on this topic, where the following chart is shown.)

In my blog post of 31st March, I reported a BBC article on the epidemic, where the doubling period for cases was shown as 3 days, but for deaths it was between 2 and 3 days ) (a Johns Hopkins University chart).

Doubling time and Herd Immunity

Doubling time, T_D(t) and the reproduction number, R_t can be measured at any time t during the epidemic, and their measured values will depend on any interventions in place at the time, including various versions of social distancing. Once any social distancing reduces or stops, then these measured values are likely to change – T_D downwards and R_t upwards – as the virus finds it relatively easier to find victims.

Assuming no pharmacological interventions (e.g. vaccination) at such time t, the growth of the epidemic at that point will depend on its underlying R₀ and duration d (innate characteristics of the virus, if it hasn’t mutated**) and the prevailing immunity in the population – herd immunity.

(**Mutation of the virus would be a concern. See this recent paper (not peer reviewed)

The doubling period T_D(t) might, therefore, have become higher after a phase of interventions, and correspondingly R_t < R₀, leading to some lockdown relaxation; but with any such interventions reduced or removed, the subsequent disease growth rate will depend on the interactions between the disease’s innate infectivity, its duration in any infected person, and how many uninfected people it can find – i.e. those without the herd immunity at that time.

These factors will determine the doubling time as this next phase develops, and bearing these dynamics in mind, it is interesting to see how all three of these factors – T_D(t), R_t and H(t) – might be related (remembering the time dependence – we might be at time t, and not necessarily at the outset of the epidemic, time zero).

Eliminating R from the two equations (1) and (2) above, we can find:

H=1-T_D/d(log_e2) (3)

So for doubling period T_D=3 days, and disease duration d=14 days, H=0.7; i.e. the required herd immunity H% is 70% for control of the epidemic. (In this case, incidentally, remember from equation (2) that R₀=14×0.7/3=3.266.)

(Presumably this might be why Dr Fauci would settle for a 70-75% effective vaccine (the H% number), but that would assume 100% take-up, or, if less than 100%, additional immunity acquired by people who have recovered from the infection. But that acquired immunity, if it exists (I’m guessing it probably would) is of unknown duration. So many unknowns!)

For this example with 14 day infection period d, and exploring the reverse implications by requiring R_t to tend to 1 (so postulating in this way (somewhat mathematically pathologically) that the epidemic has stalled at time t) and expressing equation (2) as:

T_D (t)= d(log_e2)/R_t (4)

then we see that T_D(t)= 14*log_e(2) ~= 10 days, at this time t, for R_t~=1.

Thus a sufficiently long doubling period, with the necessary minimum doubling period depending on the disease duration d (14 days in this case), will be equivalent to the R_t value being low enough for the growth of the epidemic to be controlled – i.e. R_t <=1 – so that one person infects one or less people on average.

Confirming this, equation (3) tells us, for the parameters in this (somewhat mathematically pathological) example, that with T_D(t)=10 and d=14,

H(t) = 1 – (10/14*log_e(2)) ~= 1-1 ~= 0, at this time t.

In this situation, the herd immunity H(t) (at this time t) required is notionally zero, as we are not in epidemic conditions (R_t~=1). This is not to say that the epidemic cannot restart – it simply means that if these conditions are maintained, with R_t reducing to 1, and the doubling period being correspondingly long enough, possibly achieved through social distancing (temporarily), across whole or part of the population (which might be hard to sustain) then we are controlling the epidemic.

It is when the interventions are reduced, or removed altogether that the sufficiency of % herd immunity in the population will be tested, as we saw from the Imperial College answer to my question earlier. As they say in their paper:

“Once interventions are relaxed (in the example in Figure 3, from September onwards), infections begin to rise, resulting in a predicted peak epidemic later in the year. The more successful a strategy is at temporary suppression, the larger the later epidemic is predicted to be in the absence of vaccination, due to lesser build-up of herd immunity.“

Herd immunity summary

Usually herd immunity is achieved through vaccination (eg the MMR vaccination for Rubella, Mumps and Measles). It involves less risk than the symptoms and possible side-effects of the disease itself (for some diseases at least, if not for chicken-pox, for which I can recall parents hosting chick-pox parties to get it over and done with!)

The issue, of course, with Covid-19, is that no-one knows yet if such a vaccine can be developed, if it would be safe for humans, if it would work at scale, for how long it might confer immunity, and what the take-up would be.

Until a vaccine is developed, and until the duration of any CoVid-19 immunity (of recovered patients) is known, this route remains unavailable.

Hence, as the National Geographic article says, there is continued focus on social distancing, as an effective part of even a somewhat relaxed lockdown, to control transmission of the virus.

Is there an uptick in the UK?

All of the above context serves as a (lengthy) introduction to why I am monitoring the published figures at the moment, as the UK has been (informally as well as formally) relaxing some aspects of it lockdown, imposed on March 23rd, but with gradual changes since about the end of May, both in the public’s response and in some of the Government interventions.

My own forecasting model (based on the Alex de Visscher MatLab code, and my variations, implemented in the free Octave version of the MatLab code-base) is still tracking published data quite well, although over the last couple of weeks I think the published rate of deaths is slightly above other forecasts, as well as my own.

Worldometers forecast

The Worldometers forecast is showing higher forecast deaths in the UK than when I reported before – 47924 now vs. 43,962 when I last posted on this topic on June 11th:

**Worldometers UK deaths forecast based on Current projection scenario by Oct 1, 2020**

My forecasts

The equivalent forecast from my own model still stands at 44,367 for September 30th, as can be seen from the charts below; but because we are still near the weekend, when the UK reported numbers are always lower, owing to data collection and reporting issues, I shall wait a day or two before updating my model to fit.

But having been watching this carefully for a few weeks, I do think that some unconscious public relaxation of social distancing in the fairer UK weather (in parks, on demonstrations and at beaches, as reported in the press since at least early June) might have something to do with a) case numbers, and b) subsequent numbers of deaths not falling at the expected rate. Here are two of my own charts that illustrate the situation.

In the first chart, we see the reported and modelled deaths to Sunday 28th June; this chart shows clearly that since the end of May, the reported deaths begin to exceed the model prediction, which had been quite accurate (even slightly pessimistic) up to that time.

Model vs. reported deaths, to June 28th 2020 — Model vs. reported deaths, linear scale, to June 28th 2020

In the next chart, I show the outlook to September 30th (comparable date to the Worldometers chart above) showing the plateau in deaths at 44,367 (cumulative curve on the log scale). In the daily plots, we can see clearly the significant scatter (largely caused by weekly variations in reporting at weekends) but with the daily deaths forecast to drop to very low numbers by the end of September.

Model vs. reported deaths, cumulative and daily, to Sep 30th 2020 — Model vs. reported deaths, log scale, cumulative and daily, to Sep 30th 2020

I will update this forecast in a day or two, once this last weekend’s variations in UK reporting are corrected.

Tags Coronavirus, Covid-19, herd immunity, Imperial College, modeling, modelling

Coronavirus Covid-19 Michael Levitt

Coronavirus model tracking, lockdown and lessons

Post author By Brian
Post date June 11, 2020
No Comments on Coronavirus model tracking, lockdown and lessons

Model chart June-11th, showing UK deaths data daily and cumulatively, both modeeled and reported, to September 30th 2020

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.

Tags Coronavirus, Covid-19, lockdown, modeling, modelling

Coronavirus Covid-19 Michael Levitt Reproductive Number Uncategorized

Current Coronavirus model forecast, and next steps

Post author By Brian
Post date June 2, 2020
2 Comments on Current Coronavirus model forecast, and next steps

Reported Deaths vs. Modelled deaths, both cumulative and daily,

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.”

Tags Coronavirus, Covid-19, Michael Levitt, modeling, modelling, SARS-CoV-2

Coronavirus Covid-19 Michael Levitt

Michael Levitt’s analysis of European Covid-19 data

Post author By Brian
Post date May 29, 2020
No Comments on Michael Levitt’s analysis of European Covid-19 data

Deaths graphs from EurMoMo allow the calculation of Excess deaths

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 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.

Tags Coronavirus, Covid-19, Data, modeling, modelling, SARS-CoV-2

Coronavirus Covid-19

My model calculations for Covid-19 cases for an earlier UK lockdown

Post author By Brian
Post date May 25, 2020
1 Comment on My model calculations for Covid-19 cases for an earlier UK lockdown

Chart 6 UK cases reported vs. model log y-axis 9th March lockdown, 84% effectiveness

Introduction

A little while ago (14th May), I published a post entitled What if UK lockdown had been 2 weeks earlier? where I explored the possible impact of a lockdown intervention date of 9th March instead of 23rd March, the actual UK lockdown date.

That article focused more on the impact on the number of deaths in those two scenarios, rather than the number of Covid-19 cases, where the published data is not as clear, or as complete, since so few people have been tested.

That post also made the point that this wasn’t a proper forecast, because the calibration of the model for that early an intervention date would have been compromised, as there was so little historic data to which to fit the model at that point. That still applies here.

Therefore the comparisons are not valid in detail against reported data, but the comparative numbers between the two models show how a typical model, such as mine (derived from Alex de Visscher’s code as before), is so dependent on (early) input data, and, indeed, responds in a very non-linear way, given the exponential pattern of pandemic growth.

Cases

I present below the two case numbers charts for the 9th March and 23rd March lockdown dates (I had covered the death data in more detail in my previous post on this topic, but will return to that below).

In the charts for cases here, we see in each chart (in orange) the same reported data, to date (24th May) but a big difference in the model predictions for cases. For the 9th March lockdown, the model number for cases by the 23th March is 14,800.

The equivalent model number for cases for 23rd March lockdown (i.e. modelled cases with no prior lockdown) is 45,049 cases, about 3 times as many.

The comparative reported number (the orange curve above) for 23rd March is 81,325 (based on multiplying up UK Government reported numbers (by 12.5), using Italy’s and other data concerning the proportion of real cases that might ever be tested (about 8%), as described in my Model Update post on May 8th). Reported case numbers (in other countries too, not just in the UK) underestimate the real case numbers by such a factor, because of the lack of sufficient public Coronavirus testing.

As I said in my previous article, a reasonable multiple on the public numbers for comparison might, then, be 12.5 (the inverse of 8%), which the charts above are reflect for the orange graph curve.

Deaths

For completeness, here are the comparative charts showing the equivalent model data for deaths, for the two lockdown dates.

On the right, my live model matches reported deaths data, using 84% lockdown intervention effectiveness, for the actual March 23rd lockdown, quite accurately. The model curve and the reported data curve are almost coincident (The reported data is the orange curve, as always).

On the left, the modelled number of deaths is lower from the time of lockdown. By 23rd March, for 9th March lockdown, it is 108, lower than it is for lockdown at the 23rd March (402) (with no benefit from lockdown at all in the latter case, of course).

These compare with the model numbers for deaths at the later date of May 13th, reported in my May 13th post, of 540 and 33,216 for March 9th and March 23rd lockdowns respectively (at virtually the same 84.1% intervention effectiveness).

As for the current date, at 84% effectiveness, of 24th May, the numbers of deaths on the right, for the actual 23rd March lockdown data and model is 36,660 (against the reported 36,793), and for the 9th March lockdown, on the left, would have been, in the model, 570 deaths.

That seems a very large difference, but see it as an internal comparison of model outcomes on those two assumptions. whatever the deficiencies of the availability of data to fit the model to an earlier lockdown, it is clear that, by an order of magnitude, the model behaviour over that 2 month period or so is crucially dependent on when that intervention (lockdown) happens.

This shows the startling (but characteristic) impact of the exponential pandemic growth on the outcomes from the different lockdown dates, for an outcome reporting date, 13th May, just 51 days later than the March 23rd reporting date, and for an outcome reporting date, 24th May, 62 days after March 23rd.

The model shows deaths multiplying by 5 in that 51 day period for 9th March lockdown, but 82 times as many deaths in that period for the 23rd March lockdown. For the 62 day period (11 days later), the equivalent multiples are 5.2 and 339 for 9th March and 23rd march lockdown respectively.

My 9th March lockdown modelled numbers are lower than those from Professor Rowland Kao’s research group at Edinburgh, if their Scottish numbers are scaled up for the UK. Indeed, I think my absolute numbers are too low for the March 9th lockdown case. But remember, this is about model comparisons, it’s NOT an absolute forecast.

In terms of the long term outlook (under the somewhat unrealistic assumption that 84% lockdown effectiveness continues, and in the (possibly more realistic assumption of) absence of a vaccine) deaths plateau at 42,500 for the actual March 23rd lockdown, but would have plateaued at only 625 in my model if the lockdown had been March 9th (as covered in my previous post).

Conclusions

For cases, the modelled March 9th lockdown long-term plateau, under similar assumptions) would have been 41,662 cases; but for the actual 23rd March lockdown, the model shows 2.8 million cases, a vastly higher number showing the effect of exponential behaviour, with only a 2 week difference in the timing of the intervention measures taken (at 84% effectiveness in both cases). That’s how vital timing is, as is the effectiveness of measures taken in the pandemic situation.

These long-term model outcomes reflect the observation of a likely deaths/cases ratio (1.5%) from the “captive” community on the cruise ship Diamond Princess.

But as I said earlier, these are comparisons within my model, to assess the nature and impact of an earlier lockdown, with the main focus in this post being the cases data.

It is a like-for-like comparison of modelled outcomes for two assumptions, one for the actual lockdown date, 23rd March, where the model fits reported data quite well (especially for deaths), and one for the earlier, postulated 9th March lockdown date (where the model fit must be questionable) that has been discussed so much.

Tags Coronavirus, Covid-19, modeling, modelling

Introduction

Updated interventions effectiveness and dates

Model adjustments

Comparative model charts

Alternative % settings in red

Hotspots and local lockdowns

The excess deaths measure

Discussion

Concluding comments

Introduction

Forecast comparisons

Comparison of my forecast with reported data

Lockdown relaxations

Model consequences

Lockdown, intervention effectiveness and herd immunity

Conclusions

Introduction

Types of mechanistic SIR models

Phenomenological curve-fitting

The problem with data

Curve fitting my UK model forecast

My current forecast data

The curve-fitting Gompertz function

Discussion

Closing remarks

Introduction

Research literature

Twitter and publishing

Preamble

Categorization of modelling approaches

Phenomenological models

Mechanistic and statistical models

Metapopulation spatial models

a) Cross-coupled metapopulation models

b) Mobility metapopulation models

Summary of model types

Concluding comments

References

Introduction

Re-forecast

ONS & PHE work on trends, and concluding comments

Introduction

The USA

The UK, and implications of the Imperial College modelling

Herd Immunity

The arithmetic of herd immunity, the R number and the doubling period

Herd immunity and R0

R0 and the Doubling time

Doubling time and Herd Immunity

Herd immunity summary

Is there an uptick in the UK?

Worldometers forecast

My forecasts

Introduction

My model forecast

The ONS work

The June 10th Government briefing

Parliamentary Science & Technology Select Committee

The lockdown timing debate

Concluding comment

Introduction

Curve fitting and forecasting approaches

Curve-fitting approach

Forecasting approach

The June 2nd UK reported and modelled data

**June 3rd update for 83.5% intervention effectiveness

Concluding remarks

Introduction

EuroMOMO

The Excess Deaths measure

The UK background during March 2020

Prof. Michael Levitt

Michael’s EuroMOMO analysis

The EuroMOMO charts and Michael’s analysis

Michael exposes an error in the EuroMOMO charts

Reconciling the numbers, and age range data

Michael’s concluding statement

Additional note on the age dependency of Covid-19 risk

Introduction

Cases

Herd immunity and R₀

R₀ and the Doubling time