Another perspective on Coronavirus – Prof. Michael Levitt

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

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

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

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

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

Report of Michael Levitt’s forecast for the China outbreak

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

Michael’s interview with UnHerd

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

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

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

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

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

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

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

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

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

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

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

Model update for the latest UK Coronavirus numbers

Introduction and summary

This is a brief update to my UK model predictions in the light of a week’s published data regarding Covid-19 cases and deaths in all settings – hospitals, care homes and the community – rather than just hospitals and the community, as previously.

In order to get the best fit between the model and the published data, I have had to reduce the effectiveness of interventions (lockdown, social distancing, home working etc) from 85% last week ( in my post immediately following the Government change of reporting basis) to 84.1% at present.

This reflects the fact that care homes, new to the numbers, seem to influence the critical R0 number upwards on average, and it might be that R0 is between .7 and .9, which is uncomfortably near to 1. It is already higher in hospitals than in the community, but the care home figures in the last week have increased R0 on average. See my post on the SIR model and importance of R0 to review the meaning of R0.

Predicted cases are now at 2.8 million (not reflecting the published data, but an estimate of the underlying real cases) with fatalities at 42,000.

Possible model upgrades

The Government have said that they are to sample people randomly in different settings (hospital, care homes and the community), and regionally, better to understand how the transmission rate, and the influence on the R0 reproductive number, differs in those settings, and also in different parts of the UK.

Ideally a model would forecast the pandemic growth on the basis of these individually, and then aggregate them, and I’m sure the Government advisers will be doing that. As for my model, I am adjusting overall parameters for the whole population on an average basis at this point.

Another model upgrade which has already been made by academics at Imperial College and at Harvard is to explore the cyclical behaviour of partial relaxations of the different lockdown components, to model the response of the pandemic to these (a probable increase in growth to some extent) and then a re-tightening of lockdown measures to cope with that, followed by another fall in transmission rates; and then repeating this loop into 2021 and 2022, showing a cyclical behaviour of the pandemic (excluding any pharmaceutical (e.g. vaccine and medicinal) measures). I covered this in my previous article on exit strategy.

This explains Government reluctance to promise any significant easing of lockdown in any specific timescales.

Current predictions

My UK model (based on the work of Prof. Alex Visscher at Concordia University in Montreal for other countries) is calibrated on the most accurate published data up to the lockdown date, March 23rd, which is the data on daily deaths in the UK.

Once that fit of the model to the known data has been achieved, by adjusting the assumed transmission rates, the data for deaths after lockdown – the intervention – is matched by adjusting parameters reflecting the assumed effectiveness of the intervention measures.

Data on cases is not so accurate by a long way, and examples from “captive” communities indicate that deaths vs. cases run at about 1.5% (e.g. the Diamond Princess cruise ship data).

The Italy experience also plays into this relationship between deaths and actual (as opposed to published) case numbers – it is thought that a) only a single figure percentage of people ever get tested (8% was Alex’s figure), and b) again in Italy, the death rate was probably higher than 1.5% because their health service couldn’t cope for a while, with insufficient ICU provision.

In the model, allowing for that 8%, a factor of 12.5 is applied to public total and active cases data, to reflect the likely under-reporting of case data, since there are relatively few tests.

In the model, once the fit to known data (particularly deaths to date) is made as close as possible, then the model is run over whatever timescale is desired, to look at its predictions for cases and deaths – at present a short-term forecast to June 2020, and a longer term outlook well into 2021, by when outcomes in the model have stabilised.

Model charts for deaths

The fit of the model here can be managed well, post lockdown, by adjusting the percentage effectiveness of the intervention measure, and this is currently set at 84.1%. This model predicts fatalities in the UK at 42,000. They are reported currently (8th May 2020) at 31241.

Model charts for cases

As we can see here, the fit for cases isn’t as good, but the uncertainty in case number reporting accuracy, owing to the low level of testing, and the variable experience from other countries such as Italy, means that this is an innately less reliable basis for forecasting. The model prediction for the outcome of UK case numbers is 2.8 million.

If testing, tracking and tracing is launched effectively in the UK, then this would enable a better basis for predictions for case numbers than we currently have.

Conclusions?!

I’m certainly not at a concluding stage yet. A more complex model is probably necessary to predict the situation, once variations to the current lockdown measures begin to happen, likely over the coming month or two in the first instance.

Models are being developed and released by research groups, such as that being developed by the RAMP initiative at https://epcced.github.io/ramp/

Academics from many institutions are involved, and I will take a look at the models being released to see if they address the two points I mentioned here: the variability of R0 across settings and geography, and the cyclical behaviour of the pandemic in response to lockdown variations.

At the least, perhaps, my current model might be enhanced to allow a time-dependent interv_success variable, instead of a constant lockdown effectiveness representation.

Re-modelling after changes to UK Coronavirus data collection and reporting

Change of reporting basis

The UK Government yesterday changed the reporting basis for Coronavirus numbers, retrospectively (since 6th March 2020) adding in deaths in the Care Home and and other settings, and also modifying the “Active Cases” to match, and so I have adjusted my model to match.

This historic information is more easily found on the Worldometer site; apart from current day numbers, it is harder to find the tabular data on the UK.gov site, and I guess Worldometers have a reliable web services feed from most national reporting web pages.

The increase in daily and cumulative deaths over the period contrasts with a slight reduction in daily active case numbers over the period.

Understanding the variations in epidemic parameters

With more resources, it would make sense to model different settings separately, and then combine them. If (as it is) the reproduction number R0<1 for the community, the population at large (although varying by location, environment etc), but higher in hospitals, and even higher in Care Homes, then these scenarios would have different transmission rates in the model, different effectiveness of counter-measures, and differences in several other parameters of the model(s). Today the CSA (Sir Patrick Vallance) stated that indeed, there is to be a randomised survey of people in different places (geographically) and situations (travel, work etc) to work out where the R-value is in different parts of the population.

But I have continued with the means at my disposal (the excellent basis for modelling in Alex Visscher’s paper that I have been using for some time).

Ultimately, as I said I my article at https://www.briansutton.uk/?p=1595, a multi-phase model will be needed (as per Imperial College and Harvard models illustrated here:-

Repeated peaks with no pharmaceutical intervention

and I am sure that it is the Imperial College version of this (by Neil Ferguson and his team) that will be to the forefront in that advice. The models looks at variations in policy regarding different aspects of the lockdown interventions, and the response of the epidemic to them. This leads to the cyclicity illustrated above.

Model adjustments

In my model, the rate of deaths is the most accurately available data, (even though the basis for reporting it has just changed) and the model fit is based on that. I have incorporated that reporting update into the model.

Up to lockdown (March 23rd in the UK, day 51), an infection transmission rate k11 (rate of infection of previously uninfected people by those in the infected compartment) and a correction factor are used to get this fit for the model as close as possible prior to the intervention date. For example, k11 can be adjusted, as part of a combination of infection rates; k12 from sick (S) people, k13 from seriously sick (SS) people and k14 from recovering (B, better) people to the uninfected community (U). All of those sub-rates could be adjusted in the model, and taken together define the overall rate of transition of people from from Unifected to Infected.

After lockdown, the various interventions – social distancing, school and large event closures, restaurant and pub closures and all the rest – are represented by an intervention effectiveness percentage, and this is modified (as an average across all those settings I mentioned before) to get the fit of the model after the lockdown measures as close as possible to the reported data, up to the current date.

I had been using an intervention effectiveness of 90% latterly, as the UK community response to the Government’s advice has been pretty good.

But with the UK Government move to include data from other settings (particularly the Care Home setting) I have had to reduce that overall percentage to 85% (having modelled several options from 80% upwards) to match the increased reported historic death rate.

It is, of course, more realistic to include all settings in the reported numbers, and in fact my model was predicting on that basis at the start. Now we have a few more weeks of data, and all the reported data, not just some of it, I am more confident that my original forecast for 39,000 deaths in the UK (for this single phase outlook) is currently a better estimate than the update I made a week or so ago (with 90% intervention effectiveness) to 29,000 deaths in the Model Refinement article referred to above, when I was trying to fit just hospital deaths (having no other reference point at that time).

Here are the charts for 85% intervention effectiveness; two for the long term outlook, into 2021, and two up until today’s date (with yesterday’s data):

Another output would be for UK cases, and I’ll just summarise with these charts for all cases up until June 2020 (where the modelled case numbers just begin to level off in the model):-

As we can see, the fit here isn’t as good, but this also reflects the fact that the data is less certain than for deaths, and is collected in many different ways across the UK, in the four home countries, and in the conurbations, counties and councils that input to the figures. I will probably have to adjust the model again within a few days, but the outlook, long term, of the model is for 2.6 million cases of all types. We’ll see…

Outlook beyond the Lockdown – again

I’m modest about my forecasts, but the methodology shows me the kind of advice the Government will be getting. The behavioural “science” part of the advice (not in the model) taking the public “tiredness” into account, was the reason for starting partial lockdown later, wasn’t it?

It would be more of the same if we pause the wrong aspects of lockdown too early for these reasons. Somehow the public have to “get” the rate of infection point into their heads, and that you can be infecting people before you have symptoms yourself. The presentation of the R number in today’s Government update might help that awareness. My article on R0 refers

Neil Ferguson of Imperial College was publishing papers at least as far back as 2006 on the mitigation of flu epidemics by such lockdown means, modelling very similar non-pharmaceutical methods of controlling infection rates – social distancing, school closures, no public meetings and all the rest.  Here is the 2006 paper, just one of 188 publications over the years by Ferguson and his team.  https://www.nature.com/articles/nature04795 

The following material is very recent, and, of course, focused on the current pandemic. https://www.imperial.ac.uk/…/Imperial-College-COVID19…

All countries would have been aware of this from the thinking around MERS, SARS and other outbreaks. We have a LOT of prepared models to fall back on.

As other commentators have said, Neil Ferguson has HUGE influence with the UK Government. I’m not sure how quickly UK scientists themselves were off the mark (as well as Government). We have moved from “herd immunity” and “flattening the curve” as mantras, to controlling the rate of infection by the measures we currently have in place, the type of lockdown that other countries were already using (in Europe, Italy did that two weeks before we did, although Government is saying that we did it earlier in the life of the epidemic here in the UK).

One or two advisory scientists have broken ranks (John Edmunds reported at https://www.reuters.com/…/special-report-johnson… ) on this to say that the various Committees should have been faster with their firm advice to Govenment. Who knows?

But what is clear from the public pronouncements is that Governments are now VERY aware of the issue of further peaks in the epidemic, and I would be very surprised to see rapid or significant change in the lockdown (it already allows some freedoms here in the UK, not there in some other countries, for example exercise outings once a day). I wouldn’t welcome being even more socially distanced than others, as a fit 70+ year-old person, through the policy of shielding, but if it has to be done, so be it.

The SIR model and importance of the R0 Reproductive Number

In the recent daily UK Government presentations, the R0 Reproductive Number has been mentioned a few times, and with good reason. Its value is as a commonly accepted measure of the propensity of an infectious disease outbreak to become an epidemic.

It turns out to be a relatively simple number to define, although working back from current data to calculate it is awkward if you don’t have the data. That’s my next task, from public data.

If R0 is below 1, then the epidemic will reduce and disappear. If it is greater than 1, then the epidemic grows.

The UK Government and its health advisers have made a few statements about it, and I covered these in an earlier post.

This is a more technical post, just to present a derivation of R0, and its consequences. I have used a few research papers to help with this, but the focus, brevity(!) and mistakes are mine (although I have corrected some in the sources).