# A changing world

In an earlier post, I discussed the ‘stringency index’, which has recently been developed as a way of measuring how severe – stringent – a country’s response has been to the Coronavirus epidemic.

The Financial Times, as part of its live coronavirus coverage, has now produced the following animated world map of the stringency index from the start of the year up to 24 March:

It’s striking how most of the world outside of China stays blue for most of February – arguably time thrown away – and how rapidly most of the world turns red and purple from the middle of March.

As an aside, the tweet below contains a great video where John Burn-Murdoch of the FT explains several of the decisions made by his team in the way they have chosen to present graphs showing the scale of the epidemic across countries:

# Sex and the Coronavirus

Actually, not in that sense, but you can find relevant information here.

For good and for bad, the Coronavirus epidemic is generating a large amount of data. And as more data become available, Statistics plays its part in understanding the virus in terms of its mechanisms of transmission and spread.

One very obvious aspect of the original Chinese data – described in an academic paper in the Lancet – which has subsequently been confirmed as data from other countries became available, is a difference in death rates for infected males and females. The rate of contagion for males and females is broadly similar, as shown in the following diagram

The slight difference in rate of infection between the sexes has also been subsequently observed in other countries – males always having a slightly higher infection rate – so although the difference is slight, it’s likely to be a genuine phenomenon rather than a random effect due to small amounts of data.

But in any case, this difference in infection rates pales into comparison when comparing death rates for males and females. In the original Lancet paper the ratio of male to female Coronavirus deaths is reported as 73% : 27%. So if you’re a male, does this place you in a higher risk category?

Not necessarily. In pre-coronavirus days, various posts in this blog – for example here – discussed the way that an apparent effect, such as death rates varying according to an individual’s sex, could actually be due to an entirely different phenomenon. In particular, smoking rates among men in China are very much higher than those of women. And since almost all deaths due to Coronavirus occur via failures of the respiratory system, it was hypothesised that the increased death rates among men was actually a consequence of smokers being at higher risk.

Unfortunately for men – though not for smokers – this hypothesis has been found to be unsupported by data from other countries. Based on the latest available data from all countries, the death rates for males and females who contract COVID-19 are given by the following table:

The fatality rates are different depending on whether you look at confirmed or unconfirmed cases, but in each case the ratio of fatalities of males to females is around 62% : 38%. This is a less extreme ratio than was found from the Chinese data, but since this now includes data from countries where the difference in smoking rates between males and females is much smaller than for China, it implies that smoking is not the only issue. It might explain why the ratio is worse for China than elsewhere, but it can’t be the whole story.

This New York Times article based on the Italian data points out that previous coronavirus epidemics such as SARS and MERS also led to higher fatality rates among males, and argues this is likely to be due to women having generally stronger immune defence systems due to genetics.

1. Various newspaper articles have discussed this phenomenon: here, for a discussion of the Italian data; here for a discussion of the Spanish data.
2. The Lancet paper referred to above was published on 29 January. It concluded

We have to be aware of the challenge and concerns brought by 2019-nCoV to our community. Every effort should be given to understand and control the disease, and the time to act is now.

Of course, it’s easy to be wise after the event. But the Lancet paper was wise before any of the events outside of China had taken place.

# Stay strong, stay at home

This is a quick follow-up to yesterday’s post, ‘Reasons to be cheerful’. I suggested there that looking at the same data in different ways can give you an alternative perspective on things. Specifically, I showed how looking at the rate of change in the number of new Coronavirus cases leads to a more optimistic view of how the epidemic is being brought under control, compared to just looking at the cases.

The following graph is like that of the previous post, but now showing the number of deaths due to Coronavirus through time in the worst affected countries.

Again, for most countries, there is some slight flattening of the curves, but if you live in a country like Italy, it’s difficult to see much encouragement that things are actually improving, despite the country now having been in total lockdown for 3 weeks.

But, in a series of very helpful tweets, Julia Steinberger who is professor in social ecology and ecological economics at the University of Leeds presented the data in different way, shedding a different light on things. Her graph, shown below, plots the current doubling rate of new Coronavirus deaths  against the total number of deaths. The doubling rate is the number of days it will take the number of deaths to double if the current rate of deaths is maintained. So, the higher the value, the better the epidemic is being contained.

Looked at this way:

• Improvements in Spain, Italy and especially China, where social restrictions have been in place longest, are evident.
• In the US, where the potential scale of the epidemic was initially underestimated, the doubling rate decreased for some time, and has only recently started to climb.
• In the UK, after initially climbing, the doubling rate has actually been declining, though the number of deaths in the last couple of days since the graph was produced have been lower, so the doubling rate has actually increased in recent days. Based on today’s numbers, the current doubling rate is around 4.9 days for the UK.

Updating to the most recent numbers for other countries as well, we find the current doubling rate for the US is around 4.6 days, for Spain it’s 5.4 days and for Italy it’s 9.53 days. In other words, it’s improving everywhere.

Admittedly, the picture is a bit more noisy than that of the previous post,  partly because there are fewer deaths than cases, and also because these are daily values rather than weekly averages. But in any case the message is clear, especially once numbers are updated using the most recent data: social restrictions are working and numbers are improving, even if it’s difficult to see from the original plot. Re-interpreting the numbers in terms of doubling rates gives a much more optimistic picture of how the epidemic is being brought under control.

In summary: stay strong, stay at home. It does work.

It’s probably best to be a little cautious when interpreting the recent improvement in the UK numbers. Legally binding social restrictions have only been in place for a week, which is too short a time for effects to show up in the numbers of fatalities. So, whatever improvements there have been in numbers in the last couple of days is not due to government restrictions. It’s possible, however, that people’s behaviour patters had changed in advance of the formal government restrictions being announced, and this is what’s driving the improvement in numbers. It’s also possible, however, that the improvement is due to a combination of noise and changes in the way the data are being collated. We’ll get a clearer picture in the next few days once more data become available.

# Reasons to be cheerful…

Ok, not cheerful exactly, but optimistic.

Often, looking at the same data in a different way can give a completely different perspective on things. The following graph is the updated number of reported Coronavirus cases country-by-country through time.

• The graph for each country is shifted so that time is measured from the first date on which 1000 cases were reported in that country. In this way the graph for each country is starting at roughly the same level.
• The graph is on a logarithmic scale, meaning that exponential growth as discussed in earlier posts, would show up as a straight line on this graph.
• Almost all countries display exponential growth at the start of the epidemic followed by a flattening, Both the rate of exponential growth and tendency to flatten varies from country to country.
• Despite the lockdowns and other restrictions imposed in many countries in recent weeks, it’s hard to convince yourself that there’s been any noticeable improvement.

And yet… based on the same data – albeit half a day later or so – the following graph shows the percentage increase in new cases – averaged over the previous week to minimise the effect of random day-to-day changes.

For almost all of the countries, the daily percentage increase in cases has fallen and is continuing to fall. In Italy, for example, the daily increase has gone down from around 19% to 8% in the space of a couple of weeks. The trend in the UK is also downwards, but by a smaller amount. However, enforced social controls have only been in place in the UK for less than a week.

One slight caveat is that the information from these graphs is limited to confirmed cases. This means that:

1. The numbers themselves are bound to be an underestimate of the number of infected individuals in a country;
2. Comparisons between countries are complicated by the fact that some countries are testing many more individuals than others;
3. And the trajectory for each country is also complicated by possible changes in testing protocols as the epidemic has evolved.

Nonetheless, the overall trends in these graphs are likely to be broadly indicative of a slowing of the epidemic in almost all countries. The picture for the US is especially complicated however due to wide scale state-by-state differences in testing protocols, that are also changing rapidly in time.

Of course, what we’re seeing in terms of changes in growth rate is also present in the graph above on case numbers. The almost linear reduction in growth rates is due to the slight flattening of the curves of the case numbers on a logarithmic scale. It’s simply that looking at the data this way, the daily changes are highlighted and we get a more realistic – even optimistic – picture of how, despite daily numbers of cases that seem persistently high, the projection for a couple of weeks time is that the rate of new cases will be totally manageable.

So, be optimistic, cheerful even, that the social restrictions are having an effect on the epidemic growth and that there is light at the end of the tunnel.

And if you need them,  here are many more reasons to be cheerful, curated by David Byrne, no less.

# Response

The World Health Organisation officially declared the current Coronavirus outbreak a pandemic on 12 March.  A pandemic is technically defined as:

… new disease for which people do not have immunity spreads around the world beyond expectations…

though this is largely subjective, which is why the declaration for the current outbreak was not made till 12 March. But even before that date, most countries realised the Coronavirus epidemic was already on their doorsteps and needed some kind of response.

But how rapid and how stringent have different countries been in their responses?

This is the subject of a new tracker which monitors how different governments have responded to the crisis according to the number of cases they presently have in their country. Specifically, they define something called a stringency index which records, on a scale of 0 to 100, how stringent a country’s measures are. Full details of the definition of the stringency index and the methodology used are available here. Broadly speaking, the more restrictive and widespread a country’s measures, the greater the value of the index. However, the index does not measure how effective the measures are, nor how strictly they are applied or followed.

The tracker is live, which means it is regularly updated. However, as of 24 March, a summary of the way 6 different countries have responded to the crisis is contained in the following figure:

For each country, time is measured in days since the first case appeared in that country, and the black curve shows the trajectory of the epidemic in terms of number of cases. (Bear in mind though that the number of cases is also related to the number of tests carried out, so direct comparison of these curves across countries may not be entirely valid).

The red dots show the value of the stringency index on the same timescale. You need to look at the right-hand axis to read-off the actual values of the index. For all countries the stringency index has generally risen as the epidemic has grown: countries have responded to the crisis by bringing in measures to control the virus spread. But there are significant differences across the different countries:

• In France and especially Italy, the stringency index follows the trajectory of the epidemic very closely. In other words, governments there have responded quickly to the scale of the epidemic as it has grown.
• In South Korea, where the epidemic has been largely controlled, the stringency measure values increase ahead of the growth of the epidemic. That’s to say, the government has anticipated the growth of the epidemic and brought disease control measures in quickly to stop the epidemic growth before it occurred.
• The United Kingdom’s first use of restrictive measures was very slow, and they have since been playing catch-up relative to the size of the epidemic.
• In the US, there was almost no attempt at control until long after the start of the epidemic. Belatedly, more stringent measures have been applied, but these are still substantially less restrictive than those of France or Italy.
• China’s pattern is more complicated. Since they were the first country affected by the outbreak, it’s perhaps understandable that their initial response was slow. Their subsequent response was rapid, though, enabling a subsequent reduction in stringency, which has more recently been raised again – presumably in an attempt to prevent a second wave of the epidemic. The maximum stringency index is considerably lower than that of France or Italy, presumably because although their measures were more restrictive, they were localised in severity to the hardest-hit province of Hubei.

One might quibble about the actual definitions used for the stringency index, but these conclusions broadly chime with common perceptions about the efficacy of different government responses to the epidemic.

# #iorestoacasa

“Io resto a casa” translates as “I’m staying home” and is the latest message of solidarity against Coronavirus here in Italy. As you’ll know, Italy went into lockdown a couple of weeks before the UK. Based on model and expert predictions, we should start to see some improvement in numbers round-about now. Actually, numbers did improve for a couple of days, but the most recent numbers have suggested stability, rather than a downturn. Some random variation in numbers is inevitable, even if the trend now is for things to get better, but still it’s disheartening when numbers don’t improve as quickly as you’d hoped.

With this in mind, although the models show that the reduction in transmission rates that result from a lockdown will actually result in an improvement, what evidence is there that this approach works in practice?

In part, there’s the evidence from China, who managed to bring their epidemic under almost total control in a fairly short space of time. Given both the size and spread of the population in China, this has been an incredible achievement. But the social-distancing and quarantining methods used there were considerably more restrictive than those used in western countries, so how can we be sure that the measures applied in Italy and the UK will have a similar impact?

I also gave an answer to this in a previous post, which compared the trajectory of the epidemic in two provinces of Lombardia – Lodi, which introduced an early lockdown and Bergamo which did so much later – and showed that an early lockdown led to a much flatter subsequent trajectory of the epidemic.

But there’s similar statistical evidence also available from earlier epidemics. An academic paper published in 2007 by  Markel et al. compared the trajectory of the 1918-19 influenza epidemic in different cities of the United States, relating the trajectories to the methods of social control used to limit the epidemic, which were generally different from city to city. The following graph, for example, shows how the time to introduce social control measures affected the overall number of fatalities. Generally, the quicker the response time (d), the fewer the fatalities.

The following sets of graphs are also relevant in understanding how the timing and nature of social interventions impacted on the trajectory of the epidemic in 4 different cities (the ones marked with a solid black dot in the graph above). The curve shows the number of excess deaths per 100,000 of population as time progresses. The triangle in each case shows the date of the first identified case in that city. The horizontal bars beneath each graph show the periods in which each type of intervention was applied.

The main conclusions are:

• Each of the cities applied social restrictions of one sort or another, and each was successful in bringing the epidemic under control;
• St Louis and Denver relaxed some restrictions soon after an improvement in numbers, but then had a second peak and had to re-introduce them;
• New York chose not to close schools, but had a higher peak than St Louis and Denver. On the other hand, by not relaxing the restrictions that they did apply, they had only a very slight second peak.
• Denver and St Louis had quite similar strategies, but Denver’s were introduced later (see previous figure) and consequently had higher peaks and almost double the number of fatalities. This emphasises the importance of timing.
• Pittsburgh’s restrictions were more limited and introduced later. As a result, their peak and total number of excess fatalities was greater than the other cities, though they also avoided a second peak.

So, it’s not just about models. There is hard statistical evidence that social restrictions do work, and that fine tuning the timing and nature of interventions will have an effect on the trajectory of an epidemic. Of course, no two epidemics – or indeed countries – are identical, so what happened in the 1918-19 flu epidemic in the US won’t repeat identically for the current epidemic in the UK or Italy. But, the evidence is that social controls do work; that the type of controls applied can make a difference; and that timing is also critically important.

In summary, it might not always be easy to “stare a casa” – stay at home – but if everyone follows the rules of restriction, the effect on the course of the epidemic will be dramatic.

Though this post is based on the work by Markel et al., I also drew material from this summary article by Alex Tabarrok.

# Epidemic calculator

The picture above is a screenshot from a brilliant online epidemic calculator. It’s based on a standard infectious disease model, SEIR  (Susceptible-Exposed-Infected-Removed), for which references are included in the same link. The calculator lets you choose various settings for an epidemic, and shows you how the epidemic is expected to progress, both before an intervention – shown by the vertical black dashed line – and after. The default settings for the calculator seem to be set at current best estimates for the current Coronavirus outbreak. The screenshot above is based on these default settings, but assumes a population close to 60 million. The different shaded regions in the graph correspond to counts of different aspects – fatalities, hospitalisations and so on – which you can choose to display or hide.

One of the main settings in the model is something called the ‘basic reproduction number’, written $R_0$.  This is the average number of people an infected person will infect, and is given by

$R_0= E \times p$.

where E is the number of contacts made and p is the probability of transmission.

I discussed in an earlier post how the value of $R_0$ determines the trajectory of the epidemic: if it is greater than 1 the epidemic grows exponentially; if it’s less than 1 the epidemic fades out. For the COVID-19 outbreak a reasonable estimate for $R_0$ is thought to be 2.2, which is the default setting in the calculator.

So, one thing you can do is change the value of $R_0$and see how it changes the epidemic evolution. Other settings in the model are likely to be less well estimated, but by  experimenting with values in the calculator it’s interesting and useful to see which aspects cause the trajectory to change dramatically and which have very little impact.

The other interesting feature of the calculator is that it allows you to see the effect of an intervention such as social-distancing. You simply choose by what percentage $R_0$ is reduced. For example, in the screenshot above, it’s assumed that$R_0$ is reduced by 2/3 to 0.73. The light-blue shaded region shows how – under the given settings – the number of hospitalised individuals continues to grow for a further month or so, before tailing off. Obviously, whether social distancing results in a 2/3 reduction in $R_0$ is anybody’s guess, but again you can experiment with this value to see how different degrees of conformity to the guidelines are likely to impact on the epidemic trajectory.

Question:

Was the UK government too slow in introducing a full-scale lockdown? You can slide the vertical black line left or right to see what the effect of introducing social control measures just a few days earlier or later would have been.

# Life comes at you fast

A few posts back I tried to explain the concept of herd immunity, since that seemed to be a cornerstone of the UK policy to handle the Coronavirus epidemic. Now, just a short time later, that approach seems to be off the table, and the UK is catching up with other European countries in applying measures that restrict social contact and therefore limit the rate of transmission of the virus. The previous post also described – loosely – how if an infected person passes the virus to an average of less than one other person, then the epidemic will fade out; otherwise it will grow exponentially.

So, what forced the change in government policy? Actually, not very much – the basic scientific modelling had been around for some time. But evidence from Italy suggested that demand for ICU support in hospitals for infected individuals – both in terms of number of patients, and length of treatment – would be greater than originally assumed. And the effect of this recalibration meant that the NHS capacity for ICU would have been woefully inadequate without some kind of intervention.

The change is strategy was based on work carried out at Imperial College and summarised in this report. As academic papers go it’s fairly readable, but I thought it might still be useful to give a brief summary here. So, I’ll give an outline of the methodology used, and then a picture-trail of the main conclusions.

The techniques used can be summarised as follows:

1. A standard model for transmission of flu-type epidemics was adopted. This basically assumes that anyone having the disease has a probability of passing the disease on to anyone they have contact with. So the rate of transmission depends on the probability of transmission and the average number of contacts a person has. (See this post for discussion on these types of models.)
2. The parameters for this model – things like the transmission rate of the disease – were estimated using data from China and Italy, where the current epidemic already has a longer history;
3. The model also requires country-specific demographic information extracted from the population census, so that the numbers of infections within households, between work colleagues and so on, can be reasonably predicted.
4. Simulations from the model were generated under alternative strategies for population restriction, leading to probability estimates of the number of infections and fatalities under each strategy.

Two broad types of strategy were considered:

• Mitigation strategies, in which the average transmission rate is reduced, but stays greater than 1. In this case there is exponential growth of the epidemic until the herd immunity effect kicks in and the epidemic dies out.
• Suppression strategies, in which the average transmission rate is reduced to a level below 1, so that the exponential growth phase of the epidemic is shortened considerably.

And here’s the picture-trail giving the conclusions (for the UK):

Picture 1:

Based on the input demographics and the estimated transmission rates, this graph shows the expected number of daily fatalities – both for the UK and US – if no population restrictions were applied. For the UK the peak number of fatalities per day would occur towards the end of May, with around half a million fatalities in total. This is a large number of fatalities, but the epidemic would be effectively over by July, at which point the acquired immunity in the population as a whole would prevent further epidemic outbreak.

Picture 2:

This graph shows the effect on ICU beds of various forms of mitigation strategy, ranging from school closures only (green) to isolating cases, quarantining affected households and social-distancing of over-70’s (blue). Also shown again, for comparison, is the ‘do nothing’ curve (black). The red line is current capacity for ICU beds, while the shaded light blue area is the time period over which it is assumed the restriction measures are in place. So, just as with a ‘do nothing’ policy, each of these strategies leads to the epidemic being extinguished due to the herd immunity effect, albeit a few weeks later towards the end of July. And each of the strategies does reduce the peak demand on ICU facilities. But, even the most stringent of these strategies leads to a demand on ICU beds that is still around 12 times current capacity. This is considered unsustainable.

Picture 3:

This graph considers suppression strategies. Again, the demand on ICU beds is plotted through time, assuming a suppression strategy is adopted for the time window shaded in blue. The second panel is just a zoomed-in section of the first graph, focusing on the lower part of the graph. Both suppression strategies offer a massive improvement over doing nothing (again shown in black) up until July. The version which includes school closures as well as social distancing is actually predicted to keep ICU demand well below capacity right through to October, while a loser version without school closures leads to a 50% shortfall in resources, which I imagine to be manageable.

So in the short term these suppression approaches are far superior to mitigation in keeping ICU demand below reasonable levels. The problem, as you see from the graph, is that once the restrictions are removed, the epidemic starts all over again in the autumn. Indeed, the most stringent approach, including school closures, leads to demand in the winter of 20/21 that is higher than what the ‘do nothing’ strategy would have led to in the summer of 2020.

Picture 4:

To get round the problem of the epidemic re-starting, the report looks at various strategies of containment based on the idea of relaxing restrictions when pressure on ICU units is low, and then placing them back when numbers grow back to a specified level. In this picture, the blue rectangles correspond to periods where restrictions are applied. In each such period, after a short period of further growth, the epidemic is controlled and brought back down to very low-levels. Then the restrictions are relaxed again, and the pattern repeats itself. In this way, some semblance of normal life is maintained by having periods with no restrictions, while the level of the epidemic is always contained by having periods with restrictions. As you can see in this final picture though, it’s estimated that the periods with restrictions would need to be about twice as long as those without.

So, there are no easy solutions.

• Mitigation would allow the epidemic to run its course and fade in the space of just a few months. But it would lead to very many fatalities, and unsustainable pressures on the NHS;
• Suppression through social distancing, quarantining and school closures will reduce short-term fatalities and ease pressure on health services, but does little to alter the long-term trajectory of the epidemic;
• On-off versions of suppression can be used to contain the epidemic to sustainable levels, but will require long periods of restrictions, well into 2021 at least.

Of course, none of this is especially cheerful, but it’s obviously important to know the science when planning. It seems that the UK government’s original approach was a version of mitigation, until the recalibrated version of the model used in the Imperial College report set out what the short-term consequences of that would imply. So, like most other Europeans countries, the government moved to the current – and still evolving – suppression strategy based on social distancing, quarantining and school closures. Exactly as unfolded in Italy, it became imperative to control the first wave of the epidemic; concerns about potential future waves will have to be addressed, but by then more will be understood about the spread of the epidemic.

There are, moreover, a number of issues which may make the picture less gloomy than it seems.

1. Though the report has used the very best expert opinion available when building models and estimating unknowns, it’s possible that things are better than the model assumes;
2. A big unknown is the number of asymptomatic carriers in the population. If there are many people who have the virus without realising it – and there is some evidence to suggest that’s the case – then the natural build-up to a ‘herd immunity’ effect may be much more advanced than the model assumes, and the epidemic may die out quickly after a first wave, even with a suppression-based restrictions;
3. It may be that the virus is more strongly seasonal than the model assumes, and that summer in the northern hemisphere causes a slowdown of the virus;
4. Trials for vaccines are already underway. If a successful vaccine can be brought developed quickly and distributed, it may also eliminate the need for further rounds of restrictions;
5. Tests that can assess whether someone has previously had the virus are also under development. At the moment, social distancing is required of all individuals. But there may be many people who have had the virus without realising and who are now immune. Identifying such individuals through testing would enable them to return safely to work.
6. There are promising signs that certain existing anti-viral treatments, perhaps used in combination, will prove to be an effective cure to the Coronavirus disease, at least for some groups of critically ill patients.

In summary: the statistically-based Imperial College analysis shows how the government can implement social-interaction strategies to keep fatalities and pressure on health service facilities to tolerable levels. The time bought by these strategies – admittedly at a large economic and social cost – can then be used to enable other sciences to develop tests and vaccines to stem the epidemic entirely. It’s a battle, but understanding the statistics and adhering to the strategies adopted are key to winning it.

The Imperial College report contains considerably more detail than I’ve included here.

Other summaries of the report can be found here and here. Thanks to Michael.Freeman@Smartodds.co.uk for pointing me to the second of those.

# The Rules of Contagion

With uncanny timing, Adam Kucharski, who is a professor at the London School of Hygiene and Tropical Medicine, has just published a book titled ‘The Rules of Contagion. Why Things Spread – and Why They Stop’. It deals, among other things, with the spread of epidemics, but shows how the science of epidemics applies equally to many other phenomena. As the Sunday Times says:

This is a hell of a moment for a book like this to come out … the principles of contagion, which, Kucharski argues, can be applied to everything from folk stories and financial crises to itching and loneliness, are suddenly of pressing interest to all of us.

The book is written in a way that’s both interesting and accessible, and if you’re at all interested in the mathematical/statistical aspects of the current epidemic, this is a very good place to learn something.

You can currently get the Kindle version at Amazon for less than the price of a coffee.

Incidentally, Adam contacted me a few years ago asking for an interview as he was writing a book about the history of gambling, all the way through to modern-day companies, like Smartodds, that are connected to the gambling industry. After taking advice I declined, so as to avoid any potential disclosure of proprietary information. I did write to Adam though, setting out some of my own thoughts about the industry and the background to my involvement at Smartodds. His book on this subject, ‘The Perfect Bet: Taking the Luck out of Gambling‘ is also a good read.

# Numbers and pictures

Statistics is playing a fundamental role in supporting decision-makers by providing predictions of the Coronavirus epidemic spread and of the likely impact of possible courses of action they could take. Nothing is certain – from the transmission of the disease, to the way individuals will behave – which is why probability theory plays such an important role. We can’t be sure certain things will happen, but we can reasonably assign probabilities to them.

But at a more elementary level, clear presentation of data in both numerical and graphical form, is also important for understanding many characteristics of the epidemic. There are now various sources of well-presented information, and I thought it might be helpful to provide a list here of the best one’s I’ve found so far. If anyone has alternative sources, please send them to me or include them in the comments below and I’ll add them to the list

This page gives current counts of various types – including new cases – per country. It also includes simple graphics that track the epidemic evolution. There are links to each individual country, where a country-specific history of numbers is available, and also links to look at effects by age, sex and so on. Graphs and so on are updated daily, but the numbers themselves are updated every time a country releases new daily data.

This page is updated daily and gives very clear graphics of a number of aspects of the epidemic. It shows, for example, slight differences in the age distribution of mortalities for Italy and China and also compares the mortality and contagion rate for this epidemic against those of other epidemics and diseases.

This is a dashboard giving numbers and a geo-graphical display of current cases. A more detailed UK-specific version of the dashboard is also hosted here.

This is a similar country-specific dashboard, but for Italy.

5. ft.com

The Financial Times gives this comparison of the epidemic growth across countries. It’s updated daily, though sometimes I can’t get past a paywall. Similar figures are available anyway in the dashboards above.

Like I say, please let me know of any other useful sources and I’ll add them to the list.