Archive for May, 2020

R<1 is not enough: we need R<<1.

May 29, 2020

I am getting heartily fed up with the government’s daily ‘Number Theatre’ as David Spiegelhalter calls it. So many numbers, but so little meaningful insight into what is happening.

We are all repeatedly informed that we need the reproduction number to be less than unity (i.e. R< 1) in order to drive down the prevalence of the disease. But it is generally not stressed how extraordinarily hard this is, and how we really need R to be very much less than unity i.e. R<<1

Individuals are integers

R is expressed as a decimal fraction, but for any individual ill person, the number of people they infect is an integer not a fraction i.e. it could be:

  • 0 – ideally.
  • 1 – almost inevitably
  • 2 or more – quite easily

So to achieve R = 0.9 for ten ill people, the number of people infected could be:

  • 1,1,1,1,1,1,1,1,1,0 i.e. 9 people infect one other person, but one hero manages to infect no-one else.
  • 2,1,1,1,1,1,1,1,0,0 i.e. 1 person infects 2 other people – it’s easily done. Now to achieve R = 0.9 we need two zero-infection heroes.
  • 3,1,1,1,1,1,1,0,0,0 i.e. 1 person infects 3 other people – it’s easily done. Now to achieve R = 0.9 we need three zero-infection heroes.

In all of these cases – there are still many infectious people after a single ‘generation’ of transmission.

For a disease which can be infectious before people know they are ill, it is extraordinarily hard not to infect at least one other person. It has taken the disruption of our “lockdown” to achieve R in the range 0.7 to 0.9.

The situation above describes only transmission across one ‘generation’ of the virus. What happens when we trace infections down a chain? When we do this we find that chance fluctuations really matter.

Fluctuations really matter

To see the effect of fluctuations I created a simplified spread sheet model to track infections across 4 generations.

  • I started with a single infected person and tried to calculate the number of people who would be infected after 4 generations of transmission.
  • I simplistically assumed each infected person had a chance of infecting two other people, each with a probability R/2 to make a total transmission probability of R.
  • I then used Excel’s random number generator to produce a random number between 0 and 1.
  • If the random number generator produced a number less than R/2, then infection took place, otherwise no infection took place.

I then repeated this for three more generations and at the end I asked – how many people are infected now? I repeated the experiment 50 times for different values of R and the results are shown below.

On average, the numbers of people infected after 4 generations (red dots) were close to what would be expected i.e. R4, shown as a grey line. But the fluctuations were very significant. On the graph above I have shown in green the maximum number of people infected in at least 1 of the 50 simulations

  • During 50 experiments at R = 0.9, I twice observed that there were 4 infected patients after 4 generations of transmission when we would have expected (on average) only 0.65 of an infection.
  • During 50 experiments at R = 1.0, I observed on one occasion that there were 7 infected patients after 4 generations of transmission when we would have expected (on average) only a single infection .

This shows that even when the average R value is below unity, and infection clusters might be expected to die away on average. Fluctuations mean that it is possible – indeed likely – that some single infections can persist for many generations of transmission and in fact grow into clusters and seed a new outbreak.

Where we are now?

As Louis Armstrong might have said, I don’t get around much anymore. But on my expeditions out, I see many more people than at the height of the “lockdown”. It is also not hard to see some groups of people who do not appear to be social distancing.

If the “lockdown” achieved R = 0.7, then I would estimate that the R value now must be higher, probably close to 1. The point of this article is to say that that even for R values below 1 on average, fluctuations can allow a single infection to grow into a significant cluster over several generations.

The ONS estimates that there are roughly 133,000 people currently infected with the virus. Even with R=0.9 it will take until the end of the year to reduce the number of infections to 10,000 – still a massive number. Over that length of time, and as economic and social activities resume, there will be thousands of opportunities for small outbreaks to grow significantly.

In my opinion, with this infection rate and a likely transmission rate R close to unity, there is a strong likelihood of significant further outbreaks. 93% of our population is still susceptible and if the death rate were similar to what we have seen already we could potentially lose 10 times more people than we have already.

I understand the economic and social imperatives that are driving us towards re-opening the economy. But personally I am skeptical that the virus is sufficiently under control to allow re-opening without many further outbreaks. I do hope I am wrong.

Corona Virus UK: How will it end?

May 29, 2020

The state of the UK COVID-19 epidemic on day 148 of 2020


Thank you Lydia Denworth for providing me with answers to two questions I didn’t know I wanted answered:

  • Will the Corona Virus Disappear like SARS did in 2003?
  • Where did the 1918 Spanish flu go? 

The answers appeared in a Scientific American article which asked the one question to which we all want an answer:

  • How will the COVID-19 pandemic end?

Will the Corona Virus Disappear like SARS did?

  • SARS stands for Severe Acute Respiratory Syndrome.
  • CoV stands for Corona Virus

There are seven known corona viruses: four circulate widely and cause colds. Both our current pandemic and the SARS epidemic in 2003 were caused by altered strains of corona virus. The original strain was called SARS-CoV and the new strain is called SARS-CoV-2.

The original SARS-CoV was highly virulent but only gave rise to 8098 cases, and 774 deaths. And there have been no new cases since 2004.

This limited spread was achieved by aggressive tactics of quarantining and isolation. But with SARS-CoV people had an advantage.

  • SARS-CoV made people feel obviously unwell – they had difficulty breathing – almost immediately. But they didn’t spread the virus until they were quite severely unwell. This occurred typically one week after obvious symptoms. Thus people sought help before they became infectious.
  • SARS-CoV-2 affects us differently. Although the details are still unclear, it appears people may harbour and spread the virus before developing symptoms that cause them to seek help. Indeed, some people seem to spread the virus without becoming noticeably unwell.

This subtle difference – presumably arising from subtle differences in the interaction between the virus and our cells – has allowed SARS-CoV-2 to spread ahead of our awareness of it.

But viruses cannot exist indefinitely without a host, and so once all SARS infections were stopped and viral transmission ceased, the SARS-CoV virus ceased to exist.

Will SARS-CoV-2 disappear like SARS-CoV? No. It is too late for that. 

So humanity eliminated SARS-CoV by aggressive epidemiology. But what happened to the ‘Spanish’ Flu that killed perhaps 50 million people between 1918 and 1920?

The flu virus is completely different to the corona virus. The strain of flu that caused the 1918 pandemic is called H1N1, and because there was no vaccine, it’s progress around the world was only abated when almost everyone who could be infected, had been – the famous ‘herd’ (or ‘community’) immunity.

After the 1918 pandemic, H1N1 became endemic– circulating widely as a regular winter flu. The virus was not as lethal as in 1918-1919 because of community immunity, and a general tendency of new viruses to become less lethal over time.

As a virus evolves, the most virulent variants of the virus – i.e. those which spread the best –  tend to become more prevalent than more lethal strains. 

So the H1N1 variant evolved for virulence rather than lethality and circulated until the northern hemisphere winter of 1957, when it was eliminated from circulation by a new type of pandemic flu called H2N2.

Eliminated? Yes. The H2N2 variant replaced H1N1! If only we knew how to do that kind of trick.

So what will happen in the end?

In order to get rid of the SARS-CoV-2, we need to stop all active infections of host animals – ourselves and – apparently – bats. 

If we do not invent and deploy a vaccine for SARS-CoV-2, then this virus will become endemic and continue to spread around the world.

In communities like the UK where 93% of the population have not yet been exposed, it will likely represent an ongoing hazard. If exposure to 7% of the population has led to around 35,000 deaths so far, then exposure of the whole population would likely lead to around half a million deaths.

If the population prevalence of the virus is low – at the one sick individual in a million level – then the ongoing outbreak will probably be manageable though ongoing social-distancing, testing, and tracing. This is not where we are and it will be a challenge to reach this level.

We are currently at a prevalence of around 2 to 3 sick individuals in 1000 At this level, with many thousands of new infections every day then there exists the possibility for repeated outbreaks, particularly at large public events.

But over several years, the virus will likely evolve to become less lethal, and it may yet simply fade away like the 1918 pandemic flu – but I would not bet on that outcome.

The wildfire analogy

Viruses are frequently described as ‘spreading like wildfire’. The analogy is apt for understanding both how epidemics grow, but also how they die. 

  • Firstly, in the same way that a flame needs to be actively burning fuel (or smouldering somewhere) in order maintain the possibility of spreading, a virus needs to be actively infecting a host (possibly without symptoms) in order to maintain the possibility of spreading.
  • Secondly, as long as dry foliage exists, there is always the possibility that a wildfire might re-start. Similarly, as long as there exists a susceptible population, then there is always the possibility that an epidemic might re-start. 

Thinking about how a wildfire is contained, firebreaks are used to protect critical areas, but large fires can leap these firebreaks, sending burning embers to isolated locations far from the main fire.

Similarly, we use physical separation and vaccines to suppress the main virus outbreak, but smaller outbreaks can jump our barriers and lurk, ‘smouldering’ in a community.

COVID-19: Day 142: Population Prevalence Projections

May 22, 2020
Actual (black) and Projected (red) UK daily deaths

How will the population prevalence of COVID-19 develop?

This is a question about the future and so – of course – the answer is “We don’t know“. But we can make some estimates based on our understanding of viral transmission.

I approached the question of the population prevalence of COVID-19 using a projection from Worldometer. Downloading the data, I mapped out how we might expect the rate of daily deaths to decline.

I feel bad that I don’t know the basis of the Worldometer model, but then I am only looking at the results semi-quantitatively. They will help to guide my expectations as the summer progresses.

The graph at the head of the article shows the 7-day rolling average of daily deaths as a black line, and the projection as a dotted red line. There are two features to notice:

  • The current death rate is still high: more than 300 deaths every day.
  • As we proceed into the summer the death rate reduces, falling below 100 a day in mid-June. The uncertainty in the projection is shown shaded between two finely-dotted lines.

However it is difficult to see both the large numbers and the small numbers on the same graph. So, time to use a logarithmic vertical axis! The graph below shows the same data as the previous graph, but plotted on a logarithmic axis.

Actual (black) and Projected (red) UK daily deaths plotted on a logarithmic vertical scale.

Now we can see the behaviour in the tail of the graph.

  • We expect the death rate to fall to 30 deaths per day, a factor 10 lower than at present, in 6 to 7 weeks – around 45 days. If events proceed closer to the lower projection, this could happen in as little as 35 days.
  • Projecting further it will fall to around 3 deaths per day, a factor 100 lower than at present, in around 90 days – this is around the start of September and the new school term. If events proceed closer to the lower projection, this could happen in as little as 70 days.

However, the rate at which people die does not tell us about the hazard that we personally face.

A better indicator of personal hazard is the prevalence of ill people in the population.

Population Prevalence Projection

As shown on both figures in blue, a survey between 4th and 17th of May found a population prevalence of ill people of 1 in 400 – or 2500 people in every million people were ill with COVID-19.

Assuming that the population prevalence changes at the same rate as deaths, the graph below shows how the ill population might be expected to decline with time.

Estimated population prevalence of people actively ill with COVID-19

The coarsely dotted red line is based on the central projection from the first two graphs. The lower dotted red line is based on the more optimistic projection in the graphs above. Based on these slightly optimistic projections we expect:

  • Around the start of June, the population prevalence should be just less than 1000 per million.
  • Sometime in August we can expect the population prevalence to have fallen by a further factor 100 to around 10 per million.
  • At the start of the school term in September, the population prevalence might possibly be as low as 1 in a million.

These very low levels of population prevalence still hold the possibility for viral growth and so social distancing measures would still be required.

Additionally, as international travel resumes, new sources of viral transmission will fly into the country

But at these very low levels, the severity of restrictions on schools and large gatherings could be much more relaxed, especially if a strong contact tracing service was available at that time.

In the next article I will look at where the virus will go “In the end”!

COVID-19: Day 139: Are we ready to re-open schools?

May 19, 2020


Where are we now? 

We are now in the end-part of the first phase of the Corona virus 2020 tour of the UK.

The graph of ‘deaths in all settings’ is shown above. Today (day 139) the trend rate of deaths is roughly 350 deaths-per-day, and it is falling at about 125 deaths-per-day every week.

If the linear trend continued the death rate would fall close to zero deaths-per-day in mid-June. It is more likely that the rate of decline of the death rate will flatten off into a long tail as shown in the UK projection from Worldometer below.


Additionally, random testing amongst the UK population during the period 1 May to 10 May (day 121 to day 130). During this period researchers concluded that roughly 1 in 400 individuals were actively ill with COVID-19). This specifically excluded people with direct links to care homes or hospitals.

Full Re-opening of Schools

By 1st June the prevalence of sick individuals amongst the population is likely to have fallen further – it will probably be in the region of 1-in-1000 across the country.

At the 1-in-1000 level, with appropriate precautions, a large number of activities become very low risk. Why? Because the chance of meeting an infected individual is low, and social distancing means that even if an individual is infected, the chance that they will infect you is low.

But not all activities are low risk. And schools, where groups of roughly 1000 individuals gather joyously together, are one such place.

Schools contain people who are likely to practice social distancing only imperfectly. They also contain large numbers of shared touchable surfaces (hand rails, door knobs, gym equipment, laboratory kit, taps etc).

If schools re-opened fully on 1st June (Day 152 of 2020), then it would be more likely than not that every large school would contain an infected individual.

Personally, I would not consider this acceptable. Fully re-opening with a prevalence of infected individuals around the 1-in-1000 level would virtually guarantee that every school would seed new outbreaks that could then affect vulnerable people. When these inevitably occurred, the school would need to be shut in any case.

By September (another 92 days on from June 1st), with good fortune and continued efforts, the projection above indicates that the population incidence of corona virus might conceivably be more than 100 times lower (10 in a million). At this rate only 1 in 100 schools would be likely to contain an infected individual.

At this level, I think it would be possible to safely re-open schools with minimal risk and minimal precautions. One would probably seek to segment the population into smaller groups to enable contact tracing and isolation when the inevitable cases did occur.

Government Plans for 1st June

The government plan a partial school re-opening on 1st June. This will involve only between one quarter and one third of school places being filled. This reduces the chance that a school will contain an infected individual such that we could reasonably expect one infected individual in only every three or four schools.

Is that rate low enough? Personally I think not. And a Minister speaking in a pompous and condescending tone and implying that teachers do not have children’s interests at heart would not convince me. I doubt it convinces many teachers.

The judgment involves a balance of risks and benefits. As I see it:

  • The move would bring no benefit to the 67% to 75% of students who were not attending school.
  • For the 25% to 33% of the pupils who would attend, I would think there would  need to be some overwhelming and obvious benefit of the proposal in order to justify the extraordinary amount of trouble required to reconfigure schools. I don’t know what that benefit is nor how it could be delivered in 7 weeks.
  • At a population incidence of 1-in-1000, many schools would definitely harbour infected individuals, but the infrastructure for tracking and tracing people is not yet in place.

Personally, I think re-starting schools on 1st June has no overwhelming benefit. But at a population incidence of COVID-19 of 1-in-1000, it has many risks.

In September – if we all wash our hands and keep our distance – the population prevalence of corona virus should be low enough that near-normal operation of schools should be possible. And teachers and pupils can then focus mainly on teaching and learning. Wouldn’t that be nice :-).


Why did I leave NPL?

May 17, 2020


Me in July 2019

17 days in…

…and not working at NPL is everything I had hoped it would be. Even with COVID-anxiety and family illnesses, my time away from NPL has given me a lightness of spirit that I have not experienced for years.

Why do I feel so happy to have left NPL?

Earlier this year it became financially possible for me to leave NPL. And once it became possible to leave, the thought of staying suddenly felt unbearable and I handed in my notice within days.

Unbearable? Really?

One of the things I could not write about while I was still employed by NPL, was what it was like to work there.

Why couldn’t I write about NPL? Because saying even a single negative word about NPL would have resulted in my instant dismissal.

In truth, I still feel traumatised by my experiences at NPL. I expect it will take months to years to recover.

I feel like I have escaped from an abusive relationship. Even thinking about the place now still causes me to feel physically sick. And there is much that I can hardly bear to recall, let alone recount.

I stayed at NPL for 20 years, and there were good times. I worked with great colleagues and I believe I was well-regarded by them.

Without exception, every year my appraisal deemed I had ‘exceeded expectations’ or whatever daft phrase they had invented for ‘doing OK’. With colleagues I won two Rayleigh medals for the best scientific paper of the year, and in 2009 I was awarded an MBE!

And I worked with fantastic colleagues to deliver a UK contribution to the re-definition of the base units of the SI. I feel genuinely privileged to have taken part in such an activity.

But over this long period NPL as an institution has descended into a sad and chaotic state. If NPL were an individual I would describe them as suffering from a narcissistic delusional psychosis.

In my estimation it has become an organisation that has lost touch with the reality of what it is supposed to do and become obsessed with itself, its own processes, and its own self-regard.

In the worst of times, the bullying and intimidation that permeates the place left me feeling suicidal. I have considered whether to delete or moderate the previous sentence, but I have left it in because it is true. Indeed, I feel lucky to have escaped with as few ‘bruises’ as I have. Indeed, I feel lucky to still be alive.

I can’t bear to think back to the darkest times, but in order give you a flavour of the atmosphere at NPL, I will recount one small but telling incident that occurred earlier this year.

The ‘Leaders Blog’

Senior NPL management write a ‘blog’ on the intranet describing their reflections on various matters.

In March, after I had already handed in my notice, the Chief Scientist wrote a blog announcing a complete change of focus for the lab. The details don’t matter, but it was a very significant announcement.

As usual – despite the significance of the announcement – there was no response to the blog article. This was completely normal because for ‘Staff’ to comment on anything that ‘A Manager’ had written was to ‘put one’s head above a parapet’. I normally ignored these articles, but someone drew my attention to it, and – since I was leaving anyway – I thought I would comment.

The details of what I said don’t matter – they were respectfully worded – but raised the question of how this massive change might be achieved.

What does matter is that several colleagues contacted me afterwards to thank me for pointing out that there was a problem with the Chief Scientist’s ‘vision’.

And what matters more is that my colleagues – including several of the most senior scientists at NPL – were scared to comment themselves. Scared. Literally frightened. Some would not communicate via NPL e-mail for fear of eavesdropping. They had families to care for and they feared for their careers.

I thought this was serious enough to mention on the internal blog. I wrote a second comment saying that:

Several colleagues have contacted me privately to express similar concerns to those I expressed. But they felt inhibited from commenting because of the implicit threat of management retaliation if they were seen to disagree with a senior manager.”

Now in a healthy organisation, several things might have happened at this point.

  • HR might have contacted me to say they were concerned about these allegations of intimidation.
  • The Chief Scientist themselves might have expressed similar concerns.
  • A senior manager might have commented that in fact they welcomed dissenting views.

In fact what happened was… nothing.

The truth is that the air of intimidation and bullying at NPL was not news to anybody.

Indeed when I had previously mentioned the issue of bullying to my HR representative they indifferently responded that “ was HR’s job to enforce management policy…”.

Working in that kind of toxic atmosphere I felt like my soul was rotting.


This is what clarity looks like

May 11, 2020

At this difficult time, I thought I might offer my assistance to the UK government by showing them what clarity looks like. It looks like this (pdf here)


This is New Zealand’s summary of how they intend to respond to each level of threat.

The measures seem reasonable, but I am not advocating for or against them. My point is that in New Zealand everyone knows what they are!

They can look ahead and see what will and won’t be allowed in the future

One of the important advantages of clarity is that if there is a mistake in the guidance – too weak or too strong – it can be changed.

In contrast the UK’s instructions are clearly the product of confused and conflicted discussions – and so individuals are left unsure precisely what they are expected to do.

Here is the kiwi guidance in more detail.




  • These responses are cumulative i.e. All level 3 restriction apply at level 4.
  • The responses can be either local or national


  • People instructed to stay at home in their bubble other than for essential personal movement.
  • Safe recreational activity is allowed in local area.
  • Travel is severely limited.
  •  All gatherings cancelled and all public venues closed.
    Businesses closed except for essential services (e.g. supermarkets, pharmacies, clinics, petrol stations) and lifeline utilities.
  • Educational facilities closed.
  • Rationing of supplies and requisitioning of facilities possible.
  • Reprioritisation of healthcare services.


  • People instructed to stay home in their bubble other than for essential personal movement – including to go to work, school if they have to, or for local recreation.
  • Physical distancing of two metres outside home (including on public transport), or one metre in controlled environments like schools and workplaces.
  • People must stay within their immediate household bubble, but can expand this to reconnect with close family / whānau, or bring in caregivers, or support isolated people. This extended bubble should remain exclusive.
  •  Schools (years 1 to 10) and Early Childhood Education centres can safely open, but will have limited capacity. Children should learn at home if possible.
  • People must work from home unless that is not possible.
  •  Businesses can open premises, but cannot physically interact with customers.
    Low risk local recreation activities are allowed.
  •  Public venues are closed (e.g. libraries, museums, cinemas, food courts, gyms, pools, playgrounds, markets).
  • Gatherings of up to 10 people are allowed but only for wedding services, funerals and tangihanga. Physical distancing and public health measures must be maintained.
  • Healthcare services use virtual, non-contact consultations where possible.
  • Inter-regional travel is highly limited (e.g. for essential workers, with limited exemptions for others).
  • People at high risk of severe illness (older people and those with existing medical conditions) are encouraged to stay at home where possible, and  take additional precautions when leaving home. They may choose to work.


  • People can reconnect with friends and family, go shopping, or travel domestically, but should follow public health guidance.
  • Physical distancing of two metres from people you don’t know when out  in public is recommended, with one metre physical distancing in controlled environments like workplaces, unless other measures are in place.
  • A phased approach to gatherings – initially no more than 10 people at any gathering. This applies to funerals, tangihanga, weddings, religious ceremonies and gatherings in private homes. Restrictions reviewed regularly.
  • Sport and recreation activities are allowed, subject to conditions on gatherings and contact tracing requirements, and – where practical – physical distancing.
  • Public venues (museums, libraries, etc.) can open but must comply with public health measures. Gatherings rules do not apply to public venues as long as people are not intermingling.
  • Health and disability care services operate as normally as possible.
  • Most businesses can open to the public, but must follow public health guidance including in relation to physical distancing and contact tracing. Alternative ways of working encouraged where possible (e.g. remote  working, shift-based working, physical distancing, staggering meal breaks, flexible leave).
  • It is safe to send your children to schools, early learning services and  tertiary education. There will be appropriate measures in place.
  • People at higher-risk of severe illness from COVID-19 (e.g. those with underlying medical conditions, especially if not well-controlled, and seniors) are encouraged to take additional precautions when leaving home.
  • They may work, if they agree with their employer that they can do so safely.


  • Border entry measures to minimise risk of importing COVID-19 cases.
  • Intensive testing for COVID-19.
  • Rapid contact tracing of any positive case.
  • Self-isolation and quarantine required.
  • Schools and workplaces open, and must operate safely.
  • Physical distancing encouraged. No restrictions on gatherings.
  • Stay home if you’re sick, report flu-like symptoms.
  • Wash and dry hands, cough into elbow, don’t touch your face.
  • No restrictions on domestic transport – avoid public transport  or travel if sick.

To log-lin, to log-log, or not to log at all?

May 10, 2020

WarningDiscussing mathematics is difficult, and if you feel you will be offended by this discussion, please don’t read any further.

One oddity of our current COVID-19 epidemic, is the use of so-called logarithmic axes on graphical illustrations for the general public.

Normally, such axes are confined to arcane technical publications. So it is a measure of how essential they are for describing epidemic growth that they have become relatively common.

But while useful for making certain features of epidemic growth clear, the use of logarithmic axes can also hide other important trends.

Here is my guide to when to use each type of graph using images generated by the Epidemic Calculator website for an epidemic with a single source of origin in a population of 70 million. The data simulate:

  • An initial growth phase up to day 100 with R0 = 2.2
  • An intervention at day 100
  • An active epidemic phase  after day 100 with R0 = 0.73

Initial Epidemic Growth.

If we say a quantity is growing exponentially we mean that at each point, its rate of growth is proportional its value at that point. So as a quantity increases, so does its rate of growth.

In an epidemic, each infected person becomes a new source of infection, so the rate of spread of a virus is proportional to the number of people already infected.

In a population with no resistance, this leads to exponential growth in the early stages of an epidemic. 

In order to visualise this behaviour, a logarithmic vertical scale is especially helpful.

Let me first show you an example of epidemic behaviour plotted with a linear vertical scale.


On this graph it looks like almost nothing is happening for the first 80 days of the epidemic. There is a single death after 41 days… and then infections seem to ‘explode’ – apparently without warning. This is qualitatively my recollection of what happened in the UK.

However the same data plotted on a logarithmic scale show that even during its early days, the magnitude of what would eventually happen was apparent – it is exactly the same behaviour throughout this phase.


These so-called “Log-linear” graphs  – where the vertical axis is logarithmic and the horizontal axis is linear are helpful for visualizing the first phases of an epidemic where the epidemic growth is exponential.

Their key feature is that regular steps along a logarithmic axis correspond to regular multiplying factors. The effect is to allow us to see both small and large numbers on the same graph, without the small numbers ‘disappearing’ in comparison with the larger ones.

Active Epidemic Phase

In this epidemic simulation, action is taken after day 100. And after that point, the changes no longer cover such vast ranges of numbers.

In this phase – which is where we are in the UK at the moment – we want to take note of small changes in the numbers. In this phase, a linear vertical axis is the most appropriate for allowing us to see these small changes which are ‘squashed’ by the logarithmic axis.


Notice that even though the processes underlying the epidemic progress are highly complex, the trends are roughly linear over relatively extended periods – just as we see in the UK data. This is the reason I have allowed myself to linearly extrapolate trends from the data in my previous blog articles.

The Epidemic End Phase

While perusing these outstanding data visualisations of the Epidemic progress in New Zealand, I came across a final combination of axes that shows a way of summarising how an epidemic ends – something I am sure we all long to see.

In this representation,

  • The vertical axis shows the daily death rate
  • The horizontal shows the cumulative total number of deaths

Both axes are logarithmic and so this is called a log-log plot. Now we can see the full trajectory of an epidemic. Its initial exponential growth; the gritty struggle to control viral spread; and the final extinguishing of the viral flame.


The odd data points on the graph above arise from anomalies in the recording of daily deaths. This data is taken from John Hopkins University.

And after the End Phase…?

The movie of this epidemic would end as the infection rate reaches zero. But in reality, there are never any closing credits.

As long as this infection – or the prospect of another one like it  – exists somewhere on Earth, then we have to accept that, however rare, we must live with the possibility that it could all happen again.

I find this thought appalling.

  • Do we really want to live in a socially-distanced world where we forever fear to hug our loved ones?
  • Will we really have to quarantine ourselves on arrival at every foreign destination, and on our return?

Charting how an epidemic evolves helps us to understand the nature of what we are facing. But it does not tell us how we should face it.


Discussing mathematics is difficult, and if you have been offended by this discussion, I apologise. The reason I have written this is that I feel it is important that we all try to understand what is happening.

Farewell to my NPL colleagues

May 9, 2020

It’s been just a week now since I stopped working at NPL, and my sense of relief has not abated.

I was sad not able to say goodbye in person to many kind friends and colleagues. But I would like to thank the 60 or so friends who took the time to attend a ‘Zoom’ party (below) with me.

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It was not as weird as I feared! I had the chance (I think) to at least say “Hello and Goodbye” to everyone. And on the plus side I did not get as drunk as I might have done had we gone to a pub.

Anyway, it was lovely to see you all. And I wish you all the best for your futures – at NPL or beyond.



COVID-19: Day 127: I feel less optimistic

May 7, 2020

Warning: Discussing death is difficult, and if you feel you will be offended by this discussion, please don’t read any further.

In my last post (on day 121 of 2020) I indulged in a moment of optimism. I am already regretting it.

What caused my optimism?

My optimism arose because I had been focusing on data from hospitals: the so-called ‘Pillar 1’ data on cases diagnosed as people entered hospital, and the subsequent deaths of those people in hospital.

These were the data sets available at the outset, and they tell a story of a problem in the process of being solved.

My last post pointed out that each new ‘Pillar 1 case’ arose from an infection roughly 18 days previously. Applying a trend analysis to that data indicated that the actual rate of ongoing infection that gave rise to the Pillar 1 cases must currently be close to zero.

I think this conclusion is still correct. But elsewhere – particularly in care homes and peripheral settings – things are not looking so good.

Pillar 1 versus Pillar 2 Testing

Although each Pillar 1 or Pillar 2 ‘confirmed case’ designates a single individual with the corona-virus in their body, the two counts are not directly comparable.

  • Cases diagnosed by Pillar 1 testing correspond to individuals who have suffered in the community but their symptoms have become so bad, they have been admitted to hospital.
  • Cases diagnosed by Pillar 2 testing correspond to a diverse range of people who have become concerned enough about their health to ask for a test. This refers mainly to people working in ‘care’ settings.

Diagnosing Pillar 2 cases is important because they help to prevent the spread of the disease.

But whereas a Pillar 1 case is generally very ill – with roughly a 19% chance of dying within a few days – Pillar 2 cases are generally not so ill and are much less likely to lead to an imminent death


  • Around 19% of Pillar 1 ‘Cases’ will die from COVID-19.
  • In Pillar 2 ‘Cases’ the link is not so strong, but these cases give an indication of the general prevalence of the virus.

We should also note that as the number of tests increases, the indication of prevalence given by Pillar 2 diagnoses will slowly become more realistic.

What does the data say: 3 Graphs

Graph#1 shows the number of cases diagnosed by Pillar 1 and Pillar 2 testing.


Pillar 1 diagnosed cases are falling relatively consistently: this is what led to my aberrant optimism. However Pillar 2 cases are rising.

This rise in part reflects the higher number of tests. But it more closely reveals the true breadth of the virus’s spread. This rise is – to me – alarming.

Graph#2 below shows Pillar 1 and Pillar 2 cases lumped together. This shows no significant decline.


However, because deaths are more closely associated with Pillar 1 diagnoses, the number of daily deaths (Graph#3) is declining in a way more closely linked to the fall in Pillar 1 cases.



The NHS is coping – but the situation outside of hospitals looks like it is still not under control.

This reality is probably a consequence of the long-standing denial of the true importance of the care of elderly people, and the attempt to ‘relegate’ it from the ‘premier league’ of NHS care.

Considering the forthcoming lightening of regulations, it seems likely that viral spread in the community as a whole is currently very low. Thus a wide range of activities seem to me to be likely to be very safe.

But the interface between high risk groups – care workers in particular – and the rest of us, is likely to be area where the virus may spread into the general population.


Discussing death is difficult, and if you have been offended by this discussion, I apologise. The reason I have written this is that I feel it is important that we all try to understand what is happening.

COVID-19: Day 121: Reasons to be cheerful. One, Two, Three.

May 1, 2020

Warning: Discussing death is difficult, and if you feel you will be offended by this discussion, please don’t read any further.

Today – May 1st – is Day 121 of 2020 and I greet this day with a lightness of spirit I have not experienced for many years.

Why do I feel so good? Because yesterday I left NPL! That’s the first reason to be cheerful!

I’ll write more about my disaffection with NPL in due course, but for now let’s take another look at the data on the pandemic. And there we find two more reasons to be cheerful!

Back to the Pandemic

On Day 111 of 2020, the rate at which people were being admitted to hospital with COVID-19 (Pillar 1 test results) was declining, but slowly. A linear fit to the trend indicated that zero admissions would not be reached until roughly day 165 of 2020: 14th June:


Re-plotting the same data today, Day 121 of 2020, the same linear fit suggests that zero admissions will be reached around day 145 of 2020: 25th May 2020 – three weeks earlier!

So the decline in the rate of cases is steeper than it initially appeared – that is a second reason to be cheerful!


So when should we end the ‘Lock Down’?

Looking at the graph above it might seem that extending the lock-down out to day 145 would be appropriate. But in fact, it could make good sense to begin opening up well in advance of that. Why?

Yesterday (Day 120), 3059 people were Pillar-1 tested with COVID-19 as they entered hospital. These people were infected typically 18 days previously i.e. around day 102.

If the rate of Pillar-1 tested admissions is declining at 700 cases per week now, then this must be because roughly 18 days previously, new infections were declining at the same rate. So we can plot the implied rate of infection.


The implication of this analysis is that the rate of new infections across the entire UK is currently close to zero.

If, out of a sense of precaution, we allowed (say) 10 days more, then it seems to me that there would be very little risk in opening things up after, perhaps, day 137 – May 11th.


I have not included any analysis of care homes and similar care settings in this or any of my earlier blogs. But it seems that a disaster is still unfolding there.

Aside from the disaster of events in care homes in themselves, the presence of ‘hot’ infection sites leaves open the possibility of seeding further cases among residents, carers, and all who come into contact with them.


Discussing death is difficult, and if you have been offended by this discussion, I apologise. The reason I have written this is that I feel it is important that we all try to understand what is happening.

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