Biography of John D. Rockefeller by Ron Chernow

I have just read the biography ‘Titan’ of John D. Rockefeller by Ron Chernow. Rockefeller was a contemporary of Andrew Carnegie who I wrote about in a previous post, being just four years younger. Rockefeller became the richest man in the world after Carnegie had occupied that position. This book is a biography and not an autobiography and that is not an accident. While Carnegie had a talent and an inclination for writing and freely revealed many things about himself Rockefeller was extremely secretive. Given this it is surprising that so much is known about his life. At one time he gave extensive interviews to a journalist and Chernow was able to access the transcripts of these. He also engaged in a form of reverse engineering. Rockefeller wrote many letters but in doing so he tended to conceal the most important things. By contrast the writers of the letters he received were often less discrete and so Chernow could use those as a source of information about their recipient. Like Carnegie Rockefeller grew up in modest circumstances. However in another way his family background was very different. While Carnegie grew up in an atmosphere of honesty and hard work Rockefeller’s father ‘Big Bill’ was a swindler, quack doctor and bigamist. He used to abandon his wife and children for months at a time, although he did pay their bills when he returned after an unspecified length of time. The family frequently moved house due to the schemes of the father. It was very important for Rockefeller to achieve financial independence from his father. It was also important for him to fulfill moral standards which his father had violated. He was a dutiful father.

Rockefeller was very religious and his wife even more so. He belonged to the Baptist church and starting from a young age supported the church he went to with work, money and fundraising. He was strictly against drinking, smoking and even less obviously sinful things such as theatre and opera. The children were mostly confined at home, being taught by private tutors. The regime was very strict so that, for instance, a child who ate two pieces of cheese on one day received extensive reproaches. The mother stated that no woman needs more than two dresses. The children were encouraged to earn their own money. At a time when the family was already rich the parents concealed this fact from the children.

In his professional life Rockefeller was very civilized on the surface. On the other hand he was often very ruthless in secret. He usually observed the letter of the law although not always. On the other hand he did not hesitate to destroy the business of his competitors by all legal means when it suited him. Reading about these things reminded me of the methods of Bill Gates, which I read about in a biography some years ago. Rockefeller worked with all kinds of underhand tricks and perhaps he inherited this part of his character from his father. This means for me that I regard Rockefeller as a rather unpleasant character and in my judgment he is almost at the opposite pole from Carnegie. Rockefeller did not see his own acts as immoral, or at least he did not clearly admit it to himself. He believed that God was on his side and that he was working for good. In creating the monopoly of Standard Oil he believed he was working against the excessive instability of the oil market arising from unlimited competition. In his opinion he was not acting the way he did in order to become as rich as possible – his riches were just a byproduct of his doing the right thing. I find Rockefeller very strange character, with a complex mix of characteristics which I find positive or negative. There are aspects of his behaviour which I find admirable. There is the way in which he worked so consistently in order to achieve the things he believed in. There is his strict adherence to the religious principles which he believed to be valid. There is the fact that in a certain way he treated most people around him with respect. What I do not like is that he had little respect for science. He thought that a businessman should not waste his time with science since he could always hire a scientist if he needed one. His route to success and riches was through playing social games and I do not see that his professional activity led to technical advances, another contrast to Carnegie.

A well-known characteristic of Rockefeller is his philanthropy. He admired the corresponding activities of Carnegie but privately said that Carnegie was vain. He, Rockefeller, went to great lengths to stay in the background in the context of his philanthropic gifts. He generally did not want things named after him just because he had paid for them. He invested tremendous effort in trying to decide what were the most valuable causes he should contribute to. In this sense it seems that for him giving away money was more strenuous than earning it. Eventually, after the strain was damaging his health he delegated a lot of his philanthropic activity. An important principle of his was that he would only give money if it was matched by a certain sum raised for the same project from other sources. In the end he was often not too successful in implementing this policy – he was not able to stand up to the pressure from the beneficiaries. A good example of this were his huge contributions to the beginnings of University of Chicago. Rockefeller had a key influence on establishing medical research in the United States. He founded what is now called the Rockefeller University. He made a very important contribution to fighting hookworm in the southern US and later in other parts of the world. He himself believed in homeopathy but due to the fact that the people who managed his philanthropy were more enlightened than he was the money donated contributed essentially to establishing evidence-based medicine in the US and weakening the influence of homeopathy. Thus in a way he came to doing exactly the opposite of what his father had done.

This is a very high quality biography and contains a host of interesting things which I did not even mention.

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Talks about malaria

I recently heard two talks about malaria at the Mainzer Medizinische Gesellschaft. The first, by Michael Schulte, was historical in nature and the main theme was the role of quinine as a treatment. The second, by Martin Dennebaum, was about malaria and its therapy today. Both talks were not only useful sources of information about malaria but also contained more general insights about medicine and its relations to society. In German the tree which is the natural source of quinine is called Chinarinde (i.e. China bark, in English it is called cinchona) and this had left me with the impression that the tree was from China. The first thing I learned from the first talk is that this is false. The tree comes from the Americas and was first used for medicinal purposes in Peru. A few weeks ago Eva and I visited a botanical garden in Frankfurt (Palmengarten) and saw a lot of those tropical plants which are the sources of things we are familiar with in everyday life (e.g. chocolate, cocoa, tobacco) and we in particular saw a cinchona tree. However I did not pay enough attention to realise its geographical origin at that time. There were men in Peru who had to cross a river to get to work and shivered a lot after they came out of the water. The idea came up that the bark of this tree could be used to reduce the shivering. At that time there were Jesuit missionaries in Peru. The Jesuits had been instructed by their leader Ignatius Loyola to bring back interesting things such as animals and plants from the exotic places they visited. One of the Jesuits in Peru, knowing that malaria is often accompanied by intense shivering, thought that cinchona bark might also help against malaria. This quite random analogy turned out to lead to a great success. Cinchona bark was sent to Rome and used there to treat malaria. It is remarkable how successfully this was done although the doctors knew nothing about the mechanisms at work in malaria. Without knowing about the existence of quinine they developed a way of extracting it very effectively using alcohol. They determined the right time to give the drug in the cycle of symptoms. In order for the treatment to be successful the drug must be given just after the infected red blood cells burst and the organisms are in the open in the blood and not protected. Quinine, the element of the cinchona bark most active against malaria was isolated around 1820. The first industrial production took place in Oppenheim, a town on the Rhein not far from Mainz. At one time Oppenheim was reponsible for 60 per cent of the world production of the substance. Malaria was a big public health problem in that region at the time and that was what stimulated the development of the industry. There is a story that the British colonists in India used to drink gin and tonic because tonic water contains quinine and thus provides protection against malaria. The speaker left it until the end of his talk to say that while the colonists did use quinine in other forms the concentration in tonic water is too low to be useful against malaria.

The second talk started with an interesting case history. A flight attendant flew from Frankfurt to Equatorial Guinea. Ten days after she got back she developed fever. Her husband had fever at that time due to influenza and she assumed she had the same thing. For this reason she did not seek medical advice until three days later. That was a public holiday and she was told she should come back the next day so that all the necessary tests could be done. The next day she landed as an emergency in the University Hospital in Mainz. A blood test showed that 25 per cent of her blood cells were infected with the organism causing malaria. The amount which is considered life-threatening is 5 per cent. The speaker said that if she had waited longer she would probably not have lived another day. Fortunately she did get there on time and could be cured. There are very effective drugs to treat malaria, namely those based on artemisinin. These drugs have their origin in traditional Chinese medicine. During the Vietnam war malaria was a big problem for those on both sides of the conflict. On the US side four or five times as many soldiers died of malaria than in combat. Both sides were looking for a drug to help with this problem and both looked to herbal sources, at first with little success. In the case of the Vietnamese they had enlisted the help of the Chinese to do this work for them. In China a secret ‘Project 523’ was set up for this purpose. As a part of this project Tu Youyou led the search for a malaria drug based on traditional Chinese medicine. She was successful and eventually got a Nobel Prize in 2015 for the discovery of artemisinin. From traditional literature she obtained a list of candidate plants and then subjected them to modern scientific analysis, in particular using experiments on mice. Her first attempts with the plant which produces artemisinin were not successful and it was another hint from ancient literature which helped her to overcome that difficulty. In fact the active substance was being destroyed by an extraction process at high temperature and once she had developed an alternative process at lower temperature positive results were obtained. Once the right candidate drug had been obtained the further analysis proceeded using all the tools of modern (non-alternative) medicine. I am no friend of ‘alternative medicine’ and I cannot help comparing the phrase to ‘alternative facts’. One of the things I have against ‘alternative medicine’ is that I think that if some part of it was really effective then it would quickly be adopted by real medicine and thus leave the alternative region. Nevertheless the story of artemisinin shows how in exceptional cases there can be a valuable flow information from traditional to real medicine and that this may require a great amount of effort. The type of malaria which is sometimes deadly is that caused by Plasmodium falciparum. Other types, caused by other Plasmodium species are less deadly but can become chronic. I think of novels where a typical figure was an army officer who suffered from malaria because he had served in India. From the talk I learned that the other types of malaria can be prevented from becoming chronic – it is just necessary to give the right treatment. To emphasize that malaria should be taken seriously in a country like Germany today he mentioned that at that moment there was again a malaria patient in intensive care in the University Hospital in Mainz although that case had not been so critical as that of the flight attendant.

Another paper on hepatitis C: absence of backward bifurcations

In a previous post I wrote about a paper by Alexis Nangue, myself and others on an in-host model for hepatitis C. In that context we were able to prove various things about the solutions of that model but there were many issues we were not able to investigate at that time. Recently Alexis visited Mainz for a month, funded by an IMU-Simons Foundation Africa Fellowship. In fact he had obtained the fellowship a long time ago but his visit was delayed repeatedly due to the pandemic. Now at last he was able to come. This visit gave us the opportunity to investigate the model from the first paper further and we have now written a second paper on the subject. In the first paper we showed that when the parameters satisfy a certain inequality every solution converges to a steady state as . It was left open, whether this is true for all choices of parameters. In the second paper we show that it is not: there are parameters for which periodic solutions exist. This is proved by demonstrating the presence of Hopf bifurcations. These are obtained by a perturbation argument starting from a simpler model. Unfortunately we could not decide analytically whether the periodic solutions are stable or unstable. Simulations indicate that they are stable at least in some cases.

Another question concerns the number of positive steady states. In the first paper we showed under a restriction on the parameters that there are at most three steady states. This has now been extended to all positive parameters. We also show that the number of steady states is even or odd according to the sign of , where is a basic reproductive ratio. It was left open, whether the number of steady states is ever greater than the minimum compatible with this parity condition. If there existed backward bifurcations (see here for the definition) it might be expected that there are cases with and two positive solutions. We proved that in fact this model does not admit backward bifurcations. It is known that a related model for HIV with therapy (Nonlin. Anal. RWA 17, 147) does admit backward bifurcations and it would be interesting to have an intuitive explanation for this difference.

In the first paper we made certain assumptions about the parameters in order to be able to make progress with proving things. In the second paper we drop these extra restrictions. It turns out that many of the statements proved in the first paper remain true. However there are also new phenomena. There is a new type of steady state on the boundary of the positive orthant and it is asymptotically stable. What might it mean biologically? In that case there are no uninfected cells and the state is maintained by infected cells dividing to produce new infected cells. This might represent an approximate description of a biological situation where almost all hepatocytes are infected.

Advances in the treatment of lung cancer

I enjoy going to meetings of the Mainzer Medizinische Gesellschaft [Mainz Medical Society] but they have have been in digital form for a long time now due to the pandemic. Recently I attended one of these (digital) events on the subject of the development of the treatment of lung cancer. There was a talk by Roland Buhl about general aspects of the treatment of lung cancer and one by Eric Roessner on surgery in lung cancer. Before going further I want to say something about my own relation to cancer. When I was a schoolchild my mother got cancer. In Orkney, where we lived, there was no specialist care available and for that reason my mother spent a lot of time in the nearest larger hospital, Foresterhill in Aberdeen. During my first year as a student in Aberdeen there was an extended period where I visited my mother in hospital once a week. I was not intellectually engaged in this issue and I do not even know what type of cancer my mother had. I seem to remember that at one point her spleen was removed, which suggests to me that it was a cancer of the immune system, lymphoma or leukemia. After some months my mother had reached the point where no useful further therapy was possible. She returned to Orkney and died a few months later. I must admit that at that time I was also not very emotionally involved and that I was not a big help to my mother in those troubled times for her. While I was a student I was friends with two other students, Lynn Drever and Sheila Noble. At one time I frequently heard them talking about a book called the ‘The Women’s Room’ by Marilyn French. I was curious to find out more but they did not seem keen to talk about the book. After the end of my studies I read the book myself. It is a feminist book and I think a good and interesting one. The reason I mention Marilyn French here is another good and interesting book she wrote. It is called ‘A Season in Hell’, which is a translation of the Rimbaud title ‘Une saison en enfer’. In the book she gives a vivid inside view of her own fight with a cancer of the oesophagus. After very aggressive treatments she was eventually cured of her cancer but the side effects had caused extensive damage to her body (collapse of the spine, kidney failure etc.). Parallel to the story of her own illness she portrays that of a friend who had lung cancer and died from it quite quickly. This book gave me essential insights into what cancer means, objectively and subjectively, and what lung cancer means. My own most intensive contact with cancer was in 2013 when my wife was diagnosed with colon cancer. I do not want to give any details here except the essential fact that she was cured by an operation and that the disease has shown no signs of returning. Motivated by this history I recently did something which I would probably otherwise not have done, namely to have a coloscopy. I believe that this is really a valuable examination for identifying and preventing colon cancer and that it was my responsibility to do it, although I was anxious about how it would be. In fact I found the examination and the preparations for it less unpleasant than I expected and it was nice to have a positive result. It is also nice to know that according to present recommendations I only need to repeat the examination ten years from now. A few years ago in the month of November my then secretary got a persistent cough. After some time she went to the doctor and was very soon diagnosed with lung cancer. She only survived until February. I attended a small meeting organised by her family in her memory and there I learned some more details of the way her disease progressed.

Now let me come back to the lectures. The first important message is nothing new: most cases of lung cancer are caused by smoking. Incidentally, the secretary I mentioned above smoked a lot when she was young but gave up smoking very many years ago. The message is: if you smoke then from the point of view of lung cancer it is good to stop. However it may not be enough. In the first lecture it was emphasized that the first step these days when treating lung cancer is to do a genetic analysis to look for particular mutations since this can help to decide what treatments have a chance of success. In the case of the secretary the doctors did look for mutations but unfortunately she belonged to the majority where there were no mutations which would have been favourable for her prognosis under a suitable treatment. In the most favourable cases there are possibilities available such as targeted therapies (e.g. kinase inhibitors) and immunotherapies. These lectures are intended to be kept understandable for a general audience and accordingly the speaker did not provide many details. This means that since I have spent time on these things in the past I did not learn very much from that lecture. The contents of the second lecture, on surgical techniques, were quite unfamiliar to me. The main theme was minimally invasive surgery which is used in about 30% of operations for lung cancer in Germany. It is rather restricted to specialized centres due to the special expertise and sophisticated technical equipment required. It was explained how a small potential tumour in the lung can be examined and removed. In general the tumour will be found by imaging techniques and the big problem in a operation is to find it physically. We saw a film where an anaesthetised patient is lying on an operating table while the huge arms of a mobile imaging device do a kind of dance around them. The whole thing looks very futuristic. After this dance the device knows where the tumour is. It then computes the path to be taken by a needle to reach the tumour from outside. A laser projects a red point on the skin where the needle is to be inserted. The surgeon puts the point of the needle there and then rotates it until another red point coincides with the other end. This fixes the correct direction and he can then insert the needle. At the end of the needle there is a microsurgical device which can be steered from a computer. Of course there is also a camera which provides a picture of the situation on the computer screen. The movements of the surgeon’s hands are translated into movements of the device at the end of the needle. These are scaled but also subject to noise filtering. In other words, if the surgeon’s hands shake the computer will filter it out. There is also a further refinement of this where a robot arm connected to the imagining device automatically inserts the needle in the right way. The result of all this technology is that, for instance, a single small metastasis in the lung can be removed very effectively. One of the most interesting things the surgeon said concerned the effects of the pandemic. One effect has been that people have been more reluctant to go to the doctor and that it has taken longer than it otherwise would have for lung cancer patients to go into hospital. The concrete effect of this on the work of the surgeon is that he sees that the tumours he has to treat are on average in a more advanced state than they were than before the pandemic. Putting this together with other facts leads to the following stark conclusion which it is worth to state clearly, even if it is sufficiently well known to anyone who is wiling to listen. The reluctance of people to get vaccinated against COVID-19 has led to a considerable increase in the number of people dying of cancer.

My COVID-19 vaccination, part 3

Since my last post on this subject there have been further changes. In Germany 71% of people are now fully vaccinated against COVID-19 (two doses) which is good news. The precentage is higher than it was when I wrote about this the last time (53% at that time) but given that it has been about four months since then the rate of increase looks more like a trickle. A new focus of attention in now the booster and in that area there has been more dynamics. At the moment 38.5% of Germans have had the booster. Yesterday my wife and I had our third vaccination. This time it was with the vaccine of Moderna, which means that we have now tried all the usual flavours of the vaccination available here. The reason for our choice is similar to that for choosing AstraZeneca the first time. At the moment Moderna is unpopular compared to Biontech among many people and this has has led to differences in availability. We had an appointment in February but given the possible threat of the omicron variant and a change in the recommendations we decided to try to get vaccinated earlier. In terms of efficacy and safety we did not see a big difference between the Biontech and Moderna vaccines. The way to fulfilling the desire of an earlier appointment came through the local newspaper we subscribe to, the Allgemeine Zeitung. A lot of the news items I see there are old for me, since I have already read about the themes online. What the paper is useful for is local news. There I read about the initiative of Mathias Umlauf, a young doctor who has set up a private vaccination centre in Hechtsheim, a part of Mainz which is not far from where we live. It was easy to get an appointment there and we could choose the time freely. There only the vaccine of Moderna is used, due to its easy availability. The vaccination does not cost the patient anything. The whole process is very well organized and involves very little bureaucracy or waiting time. According to the information on the web page over 36000 vaccinations have been carried out there, which amounts to 16,9% of the population of Mainz. I am very impressed by what has been achieved there, especially in comparison with the official vaccination centre in Mainz, which does not seem to have been so dynamic. So let me say at this point: thank you Dr. Umlauf. Concerning side effects the only thing I noticed was a slight sensitivity of the arm to pressure during the night but it was really slight, significantly less than I experienced with the last vaccination and even that was a minor effect. Neither my wife nor I have experienced any other negative effects. We are happy to have reached this stage. Of course nobody knows what the further development of the pandemic will be but we are happy in the knowledge that we have done everything in our power to protect ourselves and to contribute to protecting others around us.

Uğur Şahin, Özlem Türeci and The Vaccine

 

I have just read the book ‘The Vaccine’ by Joe Miller, Uğur Şahin and Özlem Türeci. More precisely, I read the German version which is called ‘Projekt Lightspeed’ but I am assuming that the contents are not too different. The quality of the language in the version I read is high and I conclude from this that it is likely that both the quality of the language in the original and the quality of the translation are high. Miller is a journalist while Şahin and Türeci are the main protagonists of the story told in the book. It is the story of how the husband and wife team of researchers developed the BioNTech vaccine against COVID-19, a story which I found more gripping than fictional thrillers. The geographical centre of the story is Mainz. Şahin and Türeci live there and the headquarters of BioNTech, the company they founded, is also there. In fact when I moved to Mainz in 2013 I lived just a couple of hundred metres from what is now the area occupied by the BioNTech. Since I was interested in biotechnology the building was interesting for me. My first encounter with Şahin was a public lecture he gave about cancer immunotherapy in February 2015 and which I wrote about here. I heard him again in a keynote talk he gave at a conference at EMBL about cancer immunotherapy in February 2017. I was interested to hear his talk but it seems that it did not catch my attention since I did not mention it in the account I wrote of that meeting. One of the last lectures I attended live before the pandemic made such things impossible was at the university medical centre here in Mainz on 13th February 2020. Şahin was the chairman. The speaker was Melanie Brinkmann and the subject the persistence of herpes viruses in the host. I did not detect any trace of the theme COVID-19 in the meeting that day except for the fact that the speaker complained that she was getting asked so many questions on that subject on a daily basis. Later on she attained some public prominence in Germany in the discussion of measures against the pandemic. The book is less about the science of the subject than about the human story involved. I have no doubt that the scientific content is correct but it is not very deep. That is not the main subject of the book.

I now come to the story itself. Şahin and Türeci are both Germans whose parents came to Germany from Turkey. They studied medicine and they met during the practical part of their studies. They were both affected by seeing patients dying of cancer while medicine was helpless to prevent it. They decided they wanted to change the situation and have pursued that goal with remarkable consistency since then. They later came to the University of Mainz. They founded a biotechnology company called Ganymed producing monoclonal antibodies which was eventually sold for several hundred million Euros. They then went on to found BioNTech with the aim of using mRNA technology for cancer immunotherapy. An important role was played by money provided by the Strüngmann brothers. They had become billionaires through their company Hexal which sold generic drugs. They were relatively independent of the usual mechanisms of the financial markets and this was a big advantage for BioNTech. (A side remark: I learned from the book that the capital NT in the middle of the company name stands for ‘new technology’.) In early January 2020 Şahin foresaw the importance of COVID-19 and immediately began a project to apply the mRNA technology of BioNTech to develop a vaccine. The book is the story of many of the obstacles which he and Türeci had to overcome to attain this goal. In the US the vaccine is associated with the name Pfizer and it is important to mention at this point what the role of Pfizer was, namely to provide money and logistics. The main ideas came from Şahin and Türeci. Of course no important scientific development is due to one or two people alone and there are many contributions. In this case a central contribution came from Katalin Karikó.

How does the BioNTech vaccine work? The central idea of an mRNA vaccine is as follows. The aim is to introduce certain proteins into the body which are similar to ones found in the virus. The immune response to these proteins will then also act against the virus. What is actually injected is mRNA and that is then translated into the desired proteins by the cellular machinery. To start with the sequences of relevant proteins must be identified and corresponding mRNA molecules produced in vitro based on a DNA template. The RNA does not only contain the code for the protein but also extra elements which influence the way in which it behaves or is treated within a cell. In addition it is coated with some lipids which protect it from degradation by certain enzymes and help it to enter cells. Karikó played a central role in the development of this lipid technology. After the RNA has been injected it has to get into cells. A good target cell type are the dendritic cells which take up material from their surroundings by macropinocytosis. They then produce proteins based on the RNA template, cut them up into small peptides and display these on their surface. They also move to the lymph nodes. There they can present the antigens to T cells, which get activated. For T cells to get activated a second signal is also necessary and it is fortunate that mRNA can provide such a signal – in the language of vaccines it shows a natural adjuvant activity. In many more popular accounts of the role of the immune system in the vaccination against COVID-19 antibodies are the central subject. In fact according to the book many vaccine developers are somewhat fixated on antibodies and underestimate the role of T cells. There Şahin had to do a lot of convincing. It is nevertheless the case that antibodies are very important in this story and there is one point which I do not understand. Antibodies are produced by B cells and in order to do so they must be activated by the antigen. For this to happen the antigen must be visible outside the cells. So how do proteins produced in dendritic cells get exported so that B cells can see them?

I admire Şahin and Türeci very much. This has two aspects. The first is their amazing achievement in producing the vaccine against COVID-19 in record time. However there is also another aspect which I find very important. It is related to what I have learned about these two people from the book and from other sources. It has to do with a human quality which I find very important and which I believe is not appreciated as it should be in our society. This is humility. In their work Şahin and Türeci have been extremely ambitious but it seems to me that in their private life they have remained humble and this makes them an example to be followed.

 

My COVID-19 vaccination, part 2

Since my last post on this subject a few things have changed. In Germany 53% of people are fully vaccinated against COVID-19, which is good news. We are now in a situation where in this country any adult who wants the vaccination can get it. Of course this percentage is still a lot lower than what is desirable and the number of people being vaccinated per day has dropped to less than half what it was in mid June. I find it sad, and at the same time difficult to understand, that there are so many people who are not motivated enough to go out and get the vaccination.

Yesterday my wife and I got our second vaccination. In the meantime the relevant authority (STIKO) has recommended that those vaccinated once with the product of AstraZeneca should get an mRNA vaccine the second time. The fact that we waited the rather long time suggested to get our second injection meant that the new recommendation had already come out and we were able to get the vaccine of Biontech the second time around. There have not been many studies of the combination vaccination but as far as I have seen those that there are gave very positive results. So we are happy that it turned out this way. This time the arm where I got the injection was sensitive to pressure during the night but this effect was almost gone by this morning. The only other side effect I noticed was an increased production of endorphins. In other words, I was very happy to have reached this point although I know that it takes a couple of weeks before the maximal protection is there.

Every second year there is an event in Mainz devoted to the popularization of science called the Wissenschaftsmarkt. It has been taking place for the last twenty years. Normally it is in the centre of town but due to the pandemic it will be largely digital this year. This year it is on 11th and 12th September and has the title ‘Mensch und Gesundheit’ [rough translation: human beings and their health]. I will contribute a video with the title ‘Gegen COVID-19 mit Mathematik’ [against COVID-19 with mathematics]. The aim of this video is to explain to non-scientists the importance of mathematics in fighting infectious diseases. I talk about what mathematical models can contribute in this domain but also, which is just as important, about what they cannot do. If the public is to trust statements by scientists it is important to take measures against creating false expectations. I do not go into too much detail about COVID-19 itself since at the moment there is too little information available and too much public controversy. Instead I concentrate on an example from long ago where it is easier to see clearly. It also happens to be the example where the basic reproductive number was discovered. This is the work of Ronald Ross on the control of malaria. Ross was the one who demonstrated that malaria is transmitted by mosquito bites and he was rewarded for that discovery with a Nobel Prize in 1902. After that he studied ways of controlling the disease. This was for instance important in the context of the construction of the Panama Canal. There the first attempt failed because so many workers died of infectious diseases, mainly malaria and yellow fever, both transmitted by mosquitos. The question came up, whether killing a certain percentage of mosquitos could lead to a long-term elimination of malaria or whether the disease would simply come back. Ross, a man of many talents, set up a simple mathematical model and used it to show that elimination is possible and was even able to estimate the percentage necessary. This provided him with a powerful argument which he could use against the many people who were sceptical about the idea.

The basic reproduction number for infectious diseases, part 2

Here I continue the discussion of the previous post. There I mentioned that when defining a basic reproduction number it is necessary, in addition to defining a set of ODE, to make a choice of infected and uninfected compartments. This means partitioning the unknowns into two subsets. In addition it is necessary to distinguish processes regarded as new infections from others. This means splitting the right hand side of the equations into a sum of two terms. There are also some other biologically motivated assumptions on the right hand side. As indicated in the previous post, a reproduction number is a feature of a disease-free steady state of the model (i.e. a steady state where the unknowns belonging to the infected compartment are zero). In fact it is a feature of the linearization of the model about that point. Intuitively, it has to do with a situation where a population contains a very small number of infected individuals. The matrix defining the right hand side of the linearization is a sum of two terms, each of which is partitioned in a certain way. We now consider a situation where the disease-free steady state is perturbed by introducing a small number of individuals into each of the infected compartments while supposing that they cannot cause a significant number of secondary infections. The numbers of individuals in the infected compartments then satisfy a linear system. Under the given assumptions all eigenvalues of the matrix defining this system have negative real parts. Thus the solution tends to zero exponentially. We are interested in the average number of individuals in these compartments over all positive times. To understand this consider first the simpler example of a single compartment with an exponential behaviour . The average value of is . In fact what we are interested in is the number of infections expected from the individuals in the infectious compartment. We have the equation , where all eigenvalues of have positive real parts and we want to calculate , where describes the process of new infections. The matrix is the next generation matrix mentioned in the previous post. It is now a matter of some rather involved linear algebra to show that the modulus of the largest eigenvalue of this matrix is greater than one if and only if the greatest real part of an eigenvalue of the linearization of the system of ODE is greater than zero.

In the current reporting of epidemiological data the quantity is time-dependent. I am far away from being able to link this with the above discussion. I looked at the web page of the Robert Koch Institute to see what they say about how they produce their curves. This led me to a paper of Cintron-Arias et al. (Math. Biosci. Eng. 6, 261), which might be able to provide a gateway into this but it is clear that it involves many things which I do not understand at present. Perhaps I should first look at the classic book ‘Infectious Diseases of Humans’ by Anderson and May. Robert May died in April 2020 so that he did not have the opportunity to observe the present pandemic.

The basic reproduction number for infectious diseases

These days reproduction numbers for epidemiology are prominent in the popular media. Many people are familiar with the idea that stopping a resurgence of COVID-19 infections in a region has to do with making and keeping something called less than one. They may also be familiar with the informal definition of that it is the number of new infections caused by an infected individual. But how is (or as it is more commonly called by scientists) defined? A mathematician might not expect to find an answer in the media but it might be reasonable to expect one in the scientific literature on epidemiology. In the past I have been frustrated by the extent to which this fails to be the case. What is typically given is a description in words which I never found possible to convert into a precise mathematical account, despite considerable effort. Now, in the context of a project on hepatitis C which I have been working on with colleagues from Cameroon, my attention was drawn to a paper of van den Driessche and Watmough (Math. Biosci. 180, 29) which contains some answers of the type I was looking for. I was vaguely aware of this paper before but I had never seriously tried to read it because I did not realise its nature.

The context in which I would have liked to find answers is that of models given by systems of ordinary differential equations where the unknowns are the numbers of individuals in different categories (susceptible, infected, recovered etc.) as functions of time. How the numbers reported in the media are calculated (on the basis of discrete data) is something I have not yet tried to find out. At the moment I would like an answer in the context which bothered me in the past and this is the context treated in the paper mentioned above. A typical situation is that found in the basic model of virus dynamics, a system of three ODE describing the dynamics of a virus within a host, with the unknowns being uninfected cells, infected cells and virions. There is a quantity which can be expressed in terms of the coefficients of the system. If then the only non-negative steady state is virus-free. This is the uninfected state and it is globally asymptotically stable. If there is an uninfected state which is unstable and an infected state which is positive and globally asymptotically stable. This kind of situation is not unique to this example and similar things are seen in many models of infection. There is a reproductive number (or perhaps more than one) which defines a threshold between different types of late-time behaviour.

It is not obvious that the analysis of van den Driessche and Watmough applies to models of in-host dynamics of a pathogen since it is necessary to make a choice of infected and uninfected compartments which is related to the biological interpretation of the variables and not just to the mathematical structure of the model. Their analysis does apply to the basic model of virus dynamics if the infected compartments are chosen to be the infected cells and virions and the reaction fluxes are partitioned in a suitable way. The simple picture of the significance of the reproductive number given above does not always hold. There is also another scenario which can occur and does so in many practical examples and involves the notion of a backward bifurcation. It goes as follows. For sufficiently small the disease-free steady state is globally asymptotically stable but as is increased this property breaks down before is reached. A fold bifurcation occurs which creates a stable and an unstable positive steady state. The unstable steady state moves so as to meet the disease-free state when . For there are exactly two steady states and the positive one is globally asymptotically stable. There is bifurcation for but it has a different structure from that in the classical scenario (which is a transcritical bifurcation). It bears some resemblance to a sub-critical Hopf bifurcation.

The most useful insights I got from reading and thinking about the paper of van den Driessche and Watmough are as follows. The primary significance of concerns the disease-free steady state and its stability. The fact that it can sometimes characterise the stability of an infected steady state is a kind of bonus which does not apply to all models. What it can do is to provide information about the stability of a positive steady state in a regime where it is close to the bifurcation point where it separates from the disease-free steady state. This circumstance is analysed in the paper using centre manifold theory. The significance for the stability of a steady state which is far away is a weak one. Continuity arguments can be used to propagate information about stability through parameter space but only as long as no bifurcations happen. When this is the case depends on the details of the particular example being considered. What is the definition of given in the paper? It is the largest modulus of an eigenvalue of a certain matrix (the next generation matrix) constructed from the linearization of the system about the disease-free steady state, whereby the construction of this matrix incorporates information about the biological meaning of the variables. Consider the example of the basic model of virus dynamics with the choices of infected and uninfected compartments as above. There is more than way of partitioning the reaction fluxes. I first tried to put both the production of infected cells and the production of virions into the category of fluxes called in the paper. Applying the definition of the reproduction number given there leads to , where is the reproduction number usually quoted for this model. If instead only the production of infected cells is put into the category then the general definition gives the conventional answer . The two quantities defining the threshold are different but the definition of being above or below the threshold are the same. (). That this kind of phenomenon can occur is shown by example in the paper.

Herd immunity

I have a long term interest in examples where mathematics has contributed to medicine. Last week I heard a talk at a meeting of the Mainzer Medizinische Gesellschaft about vaccination. The speaker was Markus Knuf, director of the pediatric section of the Helios Clinic in Wiesbaden. In the course of his talk he mentioned the concept of ‘herd immunity’ several times. I was familiar with this concept and I have even mentioned it briefly in some of my lectures and seminars. It never occurred to me that in fact this is an excellent example of a case where medical understanding has benefited from mathematical considerations. Suppose we have a population of individuals who are susceptible to a particular disease. Suppose further that there is an endemic state, i.e. that the disease persists in the population at a constant non-zero level. It is immediately plausible that if we vaccinate a certain proportion of the population against the disease then the proportion of the population suffering from the disease will be lower than it would have been without vaccination. What is less obvious is that if exceeds a certain threshold then the constant level will be zero. This is the phenomenon of herd immunity. The value of depends on how infectious the disease is. A well-known example with a relatively high value is measles, where is about . In other words, if you want to get rid of measles from a population then it is necessary to vaccinate at least 95% of the population. It occurs to me that this idea is very schematic since measles does not occur as a constant rate. Instead it occurs in large waves. This idea is nevertheless one which is useful when making public health decisions. Perhaps a better way of looking at it is to think of the endemic state as a steady state of a dynamical model. The important thing is that this state is asymptotically stable in the dynamic set-up so that it recovers from any perturbation (infected individuals coming in from somewhere else). It just so happens that in the usual mathematical models for this type of phenomenon whenever a positive steady state (i.e. one where all populations are positive) exists it is asymptotically stable. Thus the distinction between the steady state and dynamical pictures is not so important. After I started writing this post I came across another post on the same subject by Mark Chu-Carroll. I am not sad that he beat me to it. The two posts give different approaches to the same subject and I think it is good if this topic gets as much publicity as possible.

Coming back to the talk I mentioned, a valuable aspect of it was that the speaker could report on things from his everyday experience in the clinic. This makes things much more immediate than if someone is talking about the subject on an abstract level. Let me give an example. He showed a video of a small boy with an extremely persistent cough. (He had permission from the child’s parents to use this video for the talk.) The birth was a bit premature but the boy left the hospital two weeks later in good health. A few weeks after that he returned with the cough. It turned out that he had whooping cough which he had caught from an adult (non-vaccinated) relative. The man had had a bad cough but the cause was not realised and it was attributed to side effects of a drug he was taking for a heart problem. The doctors did everything to save the boy’s life but the infection soon proved fatal. It is important to realize that this is not an absolutely exceptional case but a scenario which happens regularly. It brings home what getting vaccinated (or failing to do so) really means. Of course an example like this has no statistical significance but it can nevertheless help to make people think.

Let me say some more about the mathematics of this situation. A common type of model is the SIR model. The dependent variables are , the number of individuals who are susceptible to infection by the disease, , the number of individuals who are infected (or infectious, this model ignores the incubation time) and , the number of individuals who have recovered from the disease and are therefore immune. These three quantities depend on time and satisfy a system of ODE containing a number of parameters. There is a certain combination of these parameters, usually called the basic reproductive rate (or ratio) and denoted by whose value determines the outcome of the dynamics. If the infection dies out – the solution converges to a steady state on the boundary of the state space where . If, on the other hand, there exists a positive steady state, an endemic equilibrium. The stability this of this steady state can be examined by linearizing about it. In fact it is always stable. Interestingly, more is true. When the endemic steady state exists it is globally asymptotically stable. In other words all solutions with positive initial data converge to that steady state at late time. For a proof of this see a paper by Korobeinikov and Wake (Appl. Math. Lett. 15, 955). They use a Lyapunov function to do so. At this point it is appropriate to mention that my understanding of these things has been improved by the hard work of Irena Vogel, who recently wrote her MSc thesis on the subject of Lyapunov functions in population models under my supervision.