After weeks of tight COVID-19 restrictions we could all do with a breather. Many businesses are being hit hard by a lack of pre-Christmas trade and the rest of us are longing for some time with friends and family. Surely we could afford a stretch of time with more relaxed rules, and then tighten up restrictions again later? What difference can a few days make?

As we recently reported on *Plus * magazine, the sad, but mathematically unshakeable, truth is “a lot”. The graph below illustrates why. It shows, for realistic assumptions, how milder rules can allow a steep exponential growth of cases over a relatively short period of time (the rising part of the red curve), which can’t be undone in the same amount of time using stricter rules (the declining part of the red curve).

The red curve shows that rise in relative incidence under milder restrictions, and then the slow decline after tougher measures are brought in after seven days (on day *D*). The coloured intervals denote the various time periods involved in bringing the relative incidence down to a target value. The exact meaning of this graph is explained in this article on *Plus* magazine.

A few days of relative freedom come at the price of a longer time spent under tough interventions. This will lead to all the health costs for those who catch COVID-19 as a result. And even just focussing on the economic case, a delay to intervention is only beneficial in the very short term: a longer time will be probably be needed in a higher tier, or worse, another lockdown in future.

You can find out more about this graph and where it comes from on *Plus* magazine. It illustrates just how essential mathematics is when it comes to understanding the pandemic.

Put briefly, the graph uses an *exponential model* to predict the course of the disease — such models are suitable for the kind of time period we are looking at, and more than enough to illustrate the principles involved. Of course, the prediction also depends on assumptions on the level of transmission under mild rules and under stricter rules — which brings us another idea we have explained on *Plus* and which has now become a household concept: the reproduction number *R.*

One very important parameter in disease modelling is the *basic reproduction number *of a disease, denoted by R_{0}: that’s the average number of people an infected person goes on to infect, assuming that everyone in the population is susceptible to catching the disease. As people recover from the disease and (hopefully) become immune, and as interventions such as lockdowns take effect, the number of susceptible people decreases.

The *effective reproduction number* *R* of the disease, that is the average number of people an infected person goes on to effect in practice, becomes smaller than R_{0} over time as the effect of interventions takes hold. A relatively easy calculation (see this article on *Plus* magazine) shows that, in order to force the epidemic into decline, one needs to get the effective reproduction number down to less than 1. The model that produces the graph above assumes that *R*=1.2 for the mild set of rules and *R*=0.9 for the strict rules. With these values one week of the milder restrictions needs to be paid for with almost two weeks under stricter rules if we want to get the relative incidence of the disease back to where it was when the mild set of rules was introduced.

As we now all too well by now, it’s possible to get *R* to less than 1 with interventions such as lockdowns, but it’s hard keeping it there. This is where vaccination will (hopefully) come in. The over-acrching long term aim of vaccination programmes will be to permanently get *R* down to below 1. When that’s the case then we have so-called *herd immunity:* not enough people can catch the disease for an epidemic to take hold. Another reasonably straight-forward calculation (see this article on *Plus* magazine) shows that, in theory, we can get herd immunity by vaccinating a proportion of 1-1/R_{0 }of the population. For an R_{0} of 2.5, the higher end of the estimates for COVID-19, this means that we need to get at least a proportion of 0.6, so that’s at least 60%, of the population immune.

While *R* is a useful measure of how the epidemic is behaving in many ways, one thing it doesn’t tell us is how *quickly* things are changing. This is because *R* is not a rate, there is no timescale involved. For diseases like HIV or TB, where there can be months or years between one person infecting the next person, even *R*=2 means slow growth over time. However for influenza or measles, where the infection is much faster, *R*=2 means very rapid growth.

A measure that does capture the time dimension is the *growth rate* of a disease. If the growth rate is positive, the number of new cases each day is increasing, if the growth rate is 0, the number of new cases stays constant. What is needed to keep the epidemic under control is for the growth rate to be negative and hence the number of new cases to be decreasing. As an example, if the number of new cases has decreased by 3% since yesterday, then the growth rate *r *is, approximately, *r* = -0.03 per day. Our graph above corresponds to a growth rate of roughly 0.03 for the milder set of rules, and -0.02 or the stricter rules.

To find out more about the growth rate and how it is related to *R*, see this article on *Plus* magazine.

Finally, we have a quick look at epidemiological models. Our example above uses an exponential model (find out more here), which is perfectly suitable for the time period involved. However, when we are considering longer time periods we need to take into account that the pool of susceptible people will shrink.

This is why the leading paradigm in longer term disease modelling is the so-called SIR model, which has been around for over 100 years. The general idea here is to divide a population up into classes (for example susceptible (S), infected (I) and recovered (R)) and to describe the way people pass from one class to another by mathematical equations that depend on particular parameters.

By linking up many SIR models representing geographical locations, such as towns, or other sub-populations, such as schools, one can then model the spread of the disease in a whole country. Crucial in determining the parameters are the contact patterns between people — who meets whom and how often — and these are inferred from social mixing studies. By adjusting those contact rates, one can then also simulate the effect of interventions such as school closures or social distancing measures. You can find out more about the maths of epidemiological modelling in a crisis in this *Plus *article.

These are just some of the topics related to COVID-19 we have explored on *Plus* magazine. To read more about these and others — from using artificial intelligence for diagnosis to deciding football leagues when matches have been cancelled— head over to the *Plus* magazine COVID page.

**Rome, January 21****, 2021****.** Pop Math is online! It is the calendar of the European Mathematical Society for all the outreach events in mathematics in Europe and beyond.

Today is the starting day for the new website Pop Math __http://popmath.eu/__ created by the Raising Public Awareness (RPA) Committee of the European Mathematical Society.

The website displays on a world map the announcements of all maths outreach events for a general audience and also academic or professional events about math outreach. The events can be commercial or non-profit, online or in-person, and have a limited duration.

Everybody can submit an event from this page __https://www.popmath.eu/submit-event__*.*

“*This is the simplest way to find and share events of mathematical outreach”* says Roberto Natalini, director of the IAC-Cnr in Rome and Chair of the RPA Committee.

“*Pop Math not only lets you discover new math events, but it is also an archive of everything happening in the field*”, states Sylvie Benzoni, director of Institut Henri Poincaré in Paris and member of the committee.

Give it a try! Just this week you can choose between a conference on Optimal Transport in France, a talk on the number 1 in Germany, an event on AI and maths in Italy, and don’t miss the presentation on Indra’s Pearls in Singapore! The website also features workshops (for example, a week-long one in the UK) and festivals, like the upcoming maths marathon in Spain and maths week in Ireland!

Please share the website, announce your own events and enjoy popular mathematics!

Contact: Roberto Natalini, __roberto.natalini@cnr.it__

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European Women in Mathematics (EWM) is an international association of women working in the field of mathematics in Europe. Founded in 1986, EWM has several hundred members and coordinators in 33 European countries. Twice a year, the association publishes a newsletter for its members. The 34th newsletter just appeared. We repost here Anna Cherubini’s foreword. Enjoy!

]]>This issue is entirely devoted to the ongoing pandemic and its consequences, from the point of view of mathematicians.

Starting with research on COVID-19, we interviewed four researchers who apply mathematics to the analysis and mitigation of the pandemic and its consequences:

Ilaria Dorigatti(Imperial College London),Gabriela Gomes(Strathclyde),Michelle Kendall(Warwick) andGanna Rozhnova(UMC Utrecht).Moreover,

Iulia Bulai(Basilicata) reflected on mathematical models and their power when applied to crises such as the current pandemic, whileLaura Tedeschini Lalli(Roma Tre) focuses on a completely different and equally fascinating topic: the ‘silence’ caused by the lockdown, which made it possible to listen to and analyse sounds normally hidden, in particular those of the Fontana di Trevi in Rome.The impact of COVID-19 was tragic for many and from many points of view. The pandemic claimed more than a million victims and triggered an economic crisis that impoverished and generally put many people in difficulty. It had less tragic but still serious effects on academic work.

EWM set up a working group, coordinated by

Maria G Westdickenberg(RWTH Aachen) on the impact of the crisis triggered by the pandemic on the work (and life) of women mathematicians, especially junior researchers: the open letter has been so-far signed by more than 870 researchers and endorsed by 11 organisations and committees.Mia Jukic(Leiden),Rebecca Waldecker(Martin-Luther Halle-Wittenberg), and Maria present the results of the group’s work.In these times of crisis, when it is more difficult for many people to do research work and publish, a transition in publishing modes is underway that will have a major impact on our work: with

open access, in some of its declinations, publishing in prestigious journals may end up being very difficult to those who do not have substantial research funds.Susanna Terracini(Torino) has analysed for us the characteristics ofopen accessin its various forms.Another pervasive aspect of the times we are living in are online seminars and conferences: we have, temporarily, lost the possibility to meet to talk about our work while we have ‘gained’ the opportunity to access a huge offer of seminars from all over the world. The organisers of the

and theOne World seminar seriestell us about their initiatives.Stochastic programming society virtual seminar seriesAnd finally, we asked for personal testimonies of life in the past months.

We received two impressions from

Elisabetta Strickland(Rome Tor Vergata) andElena Resmerita(Alpen-Adria Universitaet Klagenfurt), two ‘snapshots’ of their life in lockdown.Marie Françoise Ouedraogo(Ouagadougou),Sibusiso Moyo(Durban University of Technology),Fadipe-Joseph Olubunmi(Ilorin),Entisar Alrasheed(Bahri), andSelma Negzaoui(Monastir) from theAfrican Women in Mathematics Association, write about their experiences. AndSylvie Paycha(Potsdam) shares with us her thoughts about how the line between private and public life is getting even more blurred.We half-believed, when we planned this issue in summer, that life would have returned to normal in Autumn, when this Newsletter was due. Unfortunately, this is not the case.

This issue is also a way to connect with each other in this complex time.

Anna Maria Cherubini

The Cambridge University Mathematical Society, *The Archimedeans*, has published a magazine called “Eureka” ( https://archim.org.uk/eureka/ ) since 1939. Here some news by **Adam Atkinson**.

Here is the cover of issue number 46, from 1986. It contains an article by John Conway on “audioactive decay”, a surprisingly detailed analysis of the “look and say sequence” 1, 11, 21, 1211, …

According to the Eureka website, articles by Dirac and Hardy have appeared, and elsewhere I read that there is an article by Freeman Dyson in issue 8, published in 1944.

The appearance of the magazine has changed somewhat since 1986:

My Eureka collection is far from complete:

I didn’t imagine that it ever would be complete. For decades many of the back issues have, fairly obviously, not been for sale. I didn’t even imagine I would ever see some of the earlier issues.

However, I recently learned that a digital archive of Eureka has been created, containing all the issues from 3 to 65, available at https://archim.org.uk/eureka/archive/ – I have of course asked the obvious question and am told they have a source for issues 1 and 2 and hope to put those online soon as well.

I hope that this news is of some interest or use to visitors to Mathematics in Europe.

**Adam Atkinson**

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Let us suppose we travel from Earth to the furthest observable point in the universe. We have seven satellites on our spacecraft, used to keep communications between us and the Earth. Let’s suppose that the speed of the satellites coincides with that of light, or in any case equal to a speed whose difference with is negligible, while the speed of the spacecraft is . The satellite, once it reaches Earth orbit, transmits the information we have loaded into its memory, then heads back to us to collect the new information. Meanwhile, within 24 hours of each other, we launch all the satellites.

The time each probe takes will be given by the formula

where is the distance traveled on the outward journey (or if you prefer the relative position of the spacecraft respect to the Earth at the time the first probe was launched), the distance of the return (or the position of the spacecraft when the first probe returns) and is the speed of the probe.

In the meantime, the spacecraft has also moved by the segment and the time taken by the spacecraft is given by

These two times, however, are the same, so it is easy to derive the relationship between the initial and final position of the spacecraft during the travel period of the first probe:

So the path that the first probe will travel on its second journey will be

and on the umpteenth journey will be

Suppose, then, that the first probe is sent after two days, that is

The second probe will be launched after 3 days, the third after 4 and so on. So in general the initial starting point for probe will be

At this point, setting and the time in days, we can obtain a table like the one in the image below:

Now let’s try to ask ourselves this question: **how many times do I have to launch a probe before reaching, for example, Proxima Centauri b, the closest exoplanet, at a distance of about 4 light years, or 1460 days?**

If we look at the table, we conclude that, before reaching

If instead we don’t want to use the table, but to apply a formula, we will have to invert the formula of using logarithms:

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**Creative and innovative challenges in education worldwide amidst the COVID-19 Crisis** were in focus at the **Global Online Conference **organized by the **UNESCO Chairs of the University of Jyväskylä** and **Council for Creative Education, Finland**. The goal of the event was to celebrate the **United Nations’ World Creativity and Innovation Day** on 21 April 2020. The online educational conference attracted 12,220 unique visitors from more than 60 countries, across five continents. **Please find more information on the event here**.

**Reidun Twarock: Virus Structure through a Mathematical Microscope.
**Link: https://youtu.be/kDht2M0_q24

**Andreas Daniel Matt: Celebrating first International UNESCO Day of Mathematics amidst Covid-19 Crisis.**

Link: https://youtu.be/K77ASNsplKw

**Paul Hildebrandt: Viruses and Hyperspace.**

Link: https://youtu.be/QAxaGW1PUpI

**Mike Acerra: Hands-on Modeling and STEAM Learning about viruses, including COVID-19 with LUX BLOX.**

Link: https://youtu.be/hxr1CPqUlxM

**Thierry (Noah) Dana-Picard: E-Teaching of Differential Geometry in a time of COVID-19 crisis.**

Link: https://youtu.be/kHGX-CmeeTQ

**Christopher S. Brownell: STEAM Education Efforts Include Playful Mathematics.**

Link: https://youtu.be/iAZtks57c88

**Zsolt Lavicza: Developing technological and pedagogical innovations in STEAM education.**

Link: https://youtu.be/Q_LCyTyIVls

**Kristof Fenyvesi: Creativity and Innovation: Developing Critical Skills for Critical Times through STEAM Learning.**

Link: https://youtu.be/47KBNN9Cdmw

**Zhao Yuqian: Distance Education of Playful Coding for Children in Hubei Province During the COVID-19 Crisis.**

Link: https://youtu.be/Ms4v0hMhw9I

**Pamela Burnard: Revisioning New Ideas of STEAM, Enacting ‘Art-Science Creativity’, Attending to the Present (and Post) COVID-19 Outbreak.
** Link: https://youtu.be/MvaSiUJeX0Y

**Janika Leoste: Robotics Education During and After the COVID-19 Crisis.** Link: https://youtu.be/uywpF9XZGH0

**Huei Syuan Lin & Ing Ru Chen: STEAM Movement and Developing Critical Skills for Critical Times in Taiwan for the Youth by the Youth.**

Link: https://youtu.be/uTUl7DqFGyI

**Anne M. Harris: Creative Ecologies in the time of COVID-19.**

Link: https://youtu.be/27M6pnJkFpo

**Pekka Neittaanmäki: UN’s Day of Creativity and Innovation: Critical Skills in Critical Times.
**Link: https://youtu.be/wwDHZyUq3kY

**Heikki Lyytinen: UN’s Day of Creativity and Innovation: Critical Skills in Critical Times.
**Link: https://youtu.be/2kbi3y10r98

*Click here to view the embedded video.*

**Ben Sparks: The Coronavirus Curve – Numberphile.**

*Click here to view the embedded video.*

**Michel Rigo: Modèles mathématiques et confinement (in French).**

Sometimes I wish I hadn’t invented that game.

–John Horton Conwayabout theGame of life^{[3]}

The most famous mathematical invention by **John Conway** was undoubtedly the *Game of life*, that was popolirized by **Martin Gardner** on his column published on *Scientific American* ^{[1]}, but today I want to tell you something about the free will theorem.

The theorem was proposed by Conway with **Simon Kochen**, inspired by the question about the interpretation of quantum mechanics. The statement is:

If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responsesof the particles are equally not functions of the information accessible to them.

^{[2]}

That is, the theorem states that the outcome of the experiment is independent of the choices made by the experimenters. Or if you prefer that the eyes of the experimenters does not influence the outcome of the experiment (and therefore does not change the universe).

In the following the two mathematicians are concerned with showing the consistency of the theorem with quantum mechanics, first of all showing that this is only one of the last arguments against the hidden variables of **David Bohm**, proposed to restore the determinism lost by quantum mechanics.

The other interesting effort of the two mathematicians is to reconcile everything, at least from a logical-philosophical point of view, with relativity. In their judgment, after introducing a series of concepts, such as actual randomness (the universe would appear random from any reference system), Conway and Kochen show a series of errors of interpretation related to the EPR paradox that would push physicists to believe the existence of violations of the principle of randomness (and therefore of relativity). The point is that if in the system of the two connected states A, B the spin A is measured and it is down, it is much more correct to say that if the measurement is performed on B its spin will be up and not the spin of B is instantly up.

The situation is undoubtedly much more complex and detailed than this, but overall the feeling is that the two mathematicians wrote an explicit invitation to physicists to take the question more relaxed. Which seems confirmed by the phrase that I think perfectly summarizes the work of Conway and Kochen:

God lets the world run free.

**Card Colm Mulcahy**, an irish mathematician and Conway’s friend, on 11 april 2020 published on twitter the news of the death of Conway. His source was *a close associate of his* and confirmed by the family.

I written this little post in his honour: good bye, Mr. Conway.

- Gardner, M. (1970). Mathematical games: The fantastic combinations of John Conway’s new solitaire game “life”.
*Scientific American*, 223(4), 120-123. (pdf|html) - Conway, J., & Kochen, S. (2006). The free will theorem.
*Foundations of Physics*, 36(10), 1441-1473. doi:10.1007/s10701-006-9068-6 (arXiv) - Dierk Schleicher (2013). Interview with John Horton Conway.
*Notices of American Mathematica Society*, vol. 60, n. 5, pp.567-575 (pdf)

Infectious diseases are much on everybody’s mind at the moment, as frantic efforts are going into stopping the spread of the coronavirus and developing a vaccine. Medical research is obviously important in this, but so is mathematics. It is used extensively in modelling infectious diseases: finding out how rapidly they can be expected to spread, how many people will be affected, and also what proportion of a population should be vaccinated, if a vaccine exists. Here we republish, by courtesy of the author and the original hosting site Plus Magazine, an article by Marianne Freiberger, where these topics are discussed.

This article is published by courtesy of

A basic mathematical model, developed back in the 1920s but still used today, is called the *SIR model*. To understand how it works, imagine you are playing a computer game. In it there’s a population of people (of a city, country, continent or the world) divided into three classes: those that are not yet sick, but susceptible to the disease (class S), those that are sick and infectious (class I), and those that have been removed from the disease (class R), either because they have recovered and become immune, or because they have died. There also is a set of equations which describes how many people pass from one class to another in a given time step, say in a day, or in a month. You now click “go” and watch the computer simulate the disease developing over time. This, in essence, is how scientists use the SIR model.

Of course, everything depends on the equations that govern the transition from one class into the other. In the basic SIR model, these equations depend crucially on the likelihood that an infected person infects someone else and on the average length of time someone is sick for before they recover or die. When scientists use the SIR model to predict the evolution of a disease, they estimate these parameters by observing how the disease behaves in real life. By figuring out the impact interventions, such as travel restrictions or improvements in hygiene, have on the important parameters, they can also predict how useful those interventions are likely to be.

It turns out that much hinges on one special number, called the *basic reproduction ratio*, usually denoted by *R _{0}*. It measures the number of people an infectious person goes on to infect, on average, in a totally susceptible population. For measles, which is airborne and spreads easily,

The SIR model as we have described it here is of course too simple to apply to all real-life diseases and you may have to modify it to get more accurate predictions. For example, some diseases, such as childhood diseases and AIDS, affect some people more than others, so we may have to subdivide the population into further classes. The infection rate may also vary over time, for example it’s higher among school children during term time than during the holidays, and a more sophisticated model needs to take account of that. And some diseases, like malaria, are transmitted by animals and so the model must include animal as well as human populations. But still, the SIR model is a good starting point for building those more complex models.

To find out more about the SIR model, the basic reproduction number, and the maths of infectious diseases in general, read these *Plus* articles:

- The mathematics of diseases describes the SIR model in detail.
- Protecting the nation explores how scientists decide whether a vaccine and a vaccination strategy are effective and safe.
- Swine flu uncertainty describes how scientists went about getting those vital first estimates on numbers of infected and dead in the case of swine flu.
- Build your own disease is a classroom activity exploring epidemiological models using basic probability theory.

To see all our content on epidemiology, see our infectious disease package.

**Marianne Freiberger**

The International Day of Mathematics (IDM) is a worldwide celebration. Each year on March 14 all countries will be invited to participate through activities for both students and the general public in schools, museums, libraries and other spaces.

On November 26, 2019, the 40th session of the General Conference, UNESCO proclaimed March 14 as the International Day of Mathematics. The first official celebration will be on March 14, 2020.

March 14 is already celebrated in many countries as Pi Day because that date is written as 3/14 in some countries and the mathematical constant Pi is approximately 3.14.

The International Day of Mathematics is a project led by the International Mathematical Union with the support of numerous international and regional organizations.

Please, go to the page of the event to discover all the events which will take place in your region.

Will you organize an event in your city to celebrate the International Day of Mathematics 2020? Tell us more!

We are preparing an interactive map of celebrations of the International Day of Mathematics around March 14 2020 for our official website (https://www.idm314.org/).

Will you organize a celebration, large or small, in a park, museum, school, library or any other public venue? Tell us more about your plans and we will include your city or country in the first version of the map.

Please fill this FORM.

]]>If you are a reader of the *Hitchhiker’s guide to the galaxy*, you probably know that 42 is the answer to the *Ultimate Question of Life, the Universe, and Everything*. The choice of the number by **Douglas Adams** was quite random, excluding the simple fact that the number liked the writer. Yet the 42 was the protagonist of a recent news related to one of the open problems of mathematics:

Is there a number that is not 4 or 5 modulo 9 and that cannot be expressed as a sum of three cubes?

To find an answer to this question, mathematicians used numerical methods. In particular, **Andreas-Stephan Elsenhans** and **Jorg Jahnel** ^{[2]} using a particular vector space ^{[1]}, searched solutions of the following diophantine equation:

for with . This method was later developed by **Sander Husiman** ^{[3]} to . In the end all numbers, except for 33 and 42, below 100 that are not 4 or 5 module 9 have solutions.

The cubic decomposition of these two numbers come in 2019. In both cases, the protagonist was **Andrew Booker** ^{[4]}. In the case of 33 the solution arrives in march:

It’s interesting to observe that Booker, in abstract, write that he was inspired by a *Numperphile*‘s youtube video:

The solution for 42 arrived at the beginning of september:

In this case Booker obtains his result in collaboration with **Andrew Sutherland**: in this way the list of all numbers less than 100 that are not 4 or 5 module 9 is completed.

Now, in the list of numbers between 100 and 1000, the numbers without a cubic decomposition are: 114, 165, 390, 579, 627, 633, 732, 921, e 975.

- Forgive me for the excessive simplification.
- Elsenhans, Andreas-Stephan; Jahnel, Jörg (2009), New sums of three cubes,
*Mathematics of Computation*, 78 (266): 1227–1230, doi:10.1090/S0025-5718-08-02168-6 - Huisman, Sander G. (2016), Newer sums of three cubes, arXiv:1604.07746
- Booker, A.R. Cracking the problem with 33.
*Res. number theory*(2019) 5: 26. doi:10.1007/s40993-019-0162-1 (arXiv)