## Archive for the ‘diseases’ Category

### Is mathematics being driven out by computers?

September 28, 2022

In the past two weeks I attended two conferences. The first was the annual meeting of the Deutsche Mathematikervereinigung (DMV, the German mathematical society) in Berlin. The second was the joint annual meeting of the ESMTB (European Society for Mathematical and Theoretical Biology) and the SMB (Society for Mathematical Biology) in Heidelberg. I had the impression that the participation of the SMB was relatively small compared to previous years. (Was this mainly due to the pandemic or due to other problems in international travel?) There were about 500 participants in total who were present in person and about another 100 online. I was disappointed with the plenary talks at both conferences. The only one which I found reasonably good was that of Benoit Perthame. One reason I did not like them was the dominance of topics like machine learning and artificial intelligence. This brings me to the title of this post. I have the impression that mathematics (at least in applied areas) is becoming ever weaker and being replaced by the procedure of developing computer programmes which could be applied (and sometimes are) to the masses of data which our society produces these days. This was very noticeable in these two conferences. I would prefer if we human beings would continue to learn something and not just leave it to the machines. The idea that some day the work of mathematicians might be replaced by computers is an old one. Perhaps it is now happening, but in a different way from that which I would have expected. Computers are replacing humans but not because they are doing everything better. There is no doubt there are some things they can do better but I think there are many things which they cannot. The plenary talks at the DMV conference on topics of this kind were partly critical. There occurred examples of a type I had not encountered before. A computer is presented with a picture of a pig and recognizes it as a pig. Then the picture is changed in a very specific way. The change is quantitatively small and is hardly noticeable to the human eye. The computer identifies the modified picture as an aeroplane. In another similar example the starting picture is easily recognizable as a somewhat irregular seven and is recognized by the computer as such. After modification the computer recognizes it as an eight. This seems to provide a huge potential for mistakes and wonderful opportunities for criminals. I feel that the trend to machine learning and related topics in mathematics is driven by fashion. It reminds me a little of the ‘successes’ of string theory in physics some years ago. Another aspect of the plenary talks at these conferences I did not like was that the speakers seemed to be showing off with how much they had done instead of presenting something simple and fascinating. At the conference in Heidelberg there were three talks by young prizewinners which were shorter than the plenaries. I found that they were on average of better quality and I know that I was not the only one who was of that opinion.

In the end there were not many talks at these conferences I liked much but let me now mention some that I did. Amber Smith gave a talk on the behaviour of the immune system in situations where bacterial infections of the lung arise during influenza. In that talk I really enjoyed how connections were made all the way from simple mathematical models to insights for clinical practise. This is mathematical biology of the kind I love. In a similar vein Stanca Ciupe gave a talk about aspects of COVID-19 beyond those which are common knowledge. In particular she discussed experiments on hamsters which can be used to study the infectiousness of droplets in the air. A talk of Harsh Chhajer gave me a new perspective on the intracellular machinery for virus production used by hepatitis C, which is of relevance to my research. I saw this as something which is special for HCV and what I learned is that it is a feature of many positive strand RNA viruses. I obtained another useful insight on in-host models for virus dynamics from a talk of James Watmough.

Returning to the issue of mathematics and computers another aspect I want to mention is arXiv. For many years I have put copies of all my papers in preprint form on that online archive and I have monitored the parts of it which are relevant for my research interests for papers by other people. When I was working on gravitational physics it was gr-qc and since I have been working on mathematical biology it has been q-bio (quantitative biology) which I saw as the natural place for papers in that area. q-bio stands for ‘quantitative biology’ and I interpreted the word ‘quantitative’ as relating to mathematics. Now the nature of the papers on that archive has changed and it is also dominated by topics strongly related to computers such as machine learning. I no longer feel at home there. (To be fair I should say there are still quite a lot of papers there which are on stochastic topics which are mathematics in the classical sense, just in a part of mathematics which is not my speciality.) In the past I often cross-listed my papers to dynamical systems and maybe I should exchange the roles of these two in future – post to dynamical systems and cross-list to q-bio. If I succeed in moving further towards biology in my research, which I would like to I might consider sending things to bioRxiv instead of arXiv.

In this post I have written a lot which is negative. I feel the danger of falling into the role of a ‘grumpy old man’. Nevertheless I think it is good that I have done so. Talking openly about what you are unsatisfied with is a good starting point for going out and starting in new positive directions.

### Another paper on hepatitis C: absence of backward bifurcations

June 13, 2022

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 $t\to\infty$. 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 $R_0-1$, where $R_0$ 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 $R_0<1$ 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

May 1, 2022

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.

### Another conference on biological oscillators at EMBL in Heidelberg

March 11, 2022

I recently attended a conference at EMBL in Heidelberg and I very much enjoyed experiencing a live conference for the first time in a couple of years. I heard similar sentiments expressed by many of the other participants at the meeting. This conference at EMBL was a sequel to one which I previously wrote about here. The present event was in hybrid form with many of the speakers remote. There were nevertheless more than a hundred people attending on site. The conference started with a presymposium. This was intended to teach some mathematics to biologists. I attended it since I saw it as an opportunity to learn more about what kind of mathematics is really of interest to biologists. Among the main themes discussed were the relationships between positive feedback and multistability and between negative feedback and oscillations. First there was a one-hour talk by Hanspeter Herzel. Then there was a practical part where we were supposed to play with a computer programme. I had downloaded the necessary programmes (R and RStudio) as recommended but this part of the event was a failure for me. I am simply lacking in basic computer competence. It was not explained to us how to begin using the programme and I was not able to supply this missing information on my own. The first part, the lecture, was more interesting for me. The speaker mentioned a paper which he wrote with others about circadian oscillations in the number of lymphocytes in different tissues (D. Druzd et al., Immunity 46, 120). I had previously wondered about the possible roles of oscillations in immunology but I never thought of that direction. I spoke to Herzel about this in a coffee break. This demonstrates a huge advantage of live versus online conferences. I am sure that the information he and I exchanged over coffee would never have been communicated if the conference had been only online. There is a standard picture in immunology in which antigen is being continuously transported to lymph nodes, where it can activate lymphocytes. A key point of the paper is that this does not happen at a constant rate. Instead the process is highly oscillatory. Lymphocytes reach their highest level in the lymph nodes mainly at the beginning of the active phase  (i.e. the beginning of the dark phase in the mice in which these observations were carried out). This means that the effectiveness of a vaccination or another chemical intervention may depend strongly on the time at which it is administered. Herzel told me about an example where this has been seen in practise in cancer immunotherapy. I decided that I wanted to investigate this more closely. Before I could do that I heard the talk of Francis Levi, which was exactly on this topic. Returning to the paper quoted above, according to Herzel the mathematical content was very elementary, using a linear model. I am happy that simple mathematical models and ideas can lead to useful biological insights. What I do not find so good is that the information on the mathematics presented in the paper is so minimal, even in the supplementary material. There is one aspect of this story which is unclear to me. It is important for the functioning of the immune system that a given T cell visits many lymph nodes in a day. Thus the delays to the entrance or exit from lymph nodes which are supposed to implement the rhythm must act in some kind of averaged sense.

I also had a chance to talk to Levi over coffee and get some additional insights about some aspects of his lecture. He has been working on chronotherapy in oncology for many years. This means the idea that the effectiveness of a cancer therapy can be very dependent on the time of day it is administered. He has applied these ideas in practise but the ideas have not gained wide acceptance in the community of oncologists. There is a chance that this may change soon due to the appearance of two papers on this subject in the prestigious journal ‘The Lancet Oncology’ in November 2021. One of the papers (22, 1648) is by Levi, the other (22, 1777) by Qian et al.

Now let me mention a couple of the other contributions I liked best. On Monday there was a (remote) talk by Albert Goldbeter on the coupling between the cell cycle and the circadian clock. Here, as elsewhere in this conference, entrainment was a central theme. There was a discussion of the role of multiple limit cycles in these models. There was also a (remote) talk by Jim Ferrell. His subject was cataloguing certain aspects of an organism called the mouse lemur. The idea was to have a list of cell types and hormones and to know which cell types produce and are affected by which hormones. There is a preprint on this subject on BioRxiv. One feature of these primates which I found striking is the following. They are much fatter in winter than in summer and this is related to a huge difference in thyroid hormones. If I remember correctly it is a factor of ten. For comparison, in humans thyroid hormones also vary with the time of year but only on the scale of a couple of per cent. In a talk by Susan Golden (live) on the Kai system in cyanobacteria I was able to experience one of the pioneers in that field.

### My COVID-19 vaccination, part 3

December 31, 2021

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

October 4, 2021

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

August 5, 2021

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.

### Paper on hepatitis C

November 13, 2020

Together with Alexis Nangue and other colleagues from Cameroon we have just completed a paper on mathematical modelling of hepatitis C. What is being modelled here is the dynamics of the amount of the virus and infected cells within a host. The model we study is a modification of one proposed by Guedj and Neumann, J. Theor. Biol. 267, 330. One part of it is a three-dimensional model, which is related to the basic model of virus dynamics. It is coupled with a two-dimensional model which gives a simple description of the way in which the virus replicates inside a host cell. This intracellular process is related to the mode of action of the modern drugs used to treat hepatitis C. I gave some information about the disease itself and its treatment in a previous post. In the end the object of study is a system of five ODE with many parameters.

For this system we first proved global existence and boundedness of the solutions, as well as positivity (positive initial data lead to a positive solution). One twist here is a certain lack of regularity of the coefficients of the system. When some of the unknowns become zero the right hand side of the equations is discontinuous. This means that it is necessary to prove that this singular set is not approached during the evolution of a positive solution. The source of the irregularity is the use of something called the standard incidence function instead of simple mass action kinetics. The former type of kinetics has a long history in epidemiology and I do not want to try to explain the background here. In any case, there are arguments which say that mass action kinetics leads to unrealistic results in within-host models of hepatitis and that the standard incidence function is better.

We show that the model has up to two virus-free steady states and determine their stability. The study of positive steady states is more difficult and, at the moment, incomplete. We have proved that there cannot be more than three steady states but we do not know if there is ever more than one. Under increasingly restrictive assumptions on the parameters (restrictions which are unfortunately not all biologically motivated) we show that there is at least one positive steady state, that there is exactly one and that that one is asymptotically stable. Under certain other assumptions we can show that every solution converges to a steady state. This last proof uses the method of Li and Muldowney discussed in a previous post. Learning about this method was one of the (positive) side effects of working on this project. Another was an improvement of my understanding of the concept of the basic reproductive number as discussed here and here. During this project I have learned a lot of new things, mathematical and biological, and I feel that I am now in a stronger position to tackle other projects modelling hepatitis C and other infectious diseases.

This

### Talk by David Ho on COVID-19

August 22, 2020

On Thursday David Ho gave a keynote lecture at the SMB conference. He talked about work to develop monoclonal antibodies against SARS-CoV-2. He started by apologising, in view of the given audience, that there would be no mathematics in his talk but he did make make clear his continuing belief in the importance of applying mathematics to biology. He has been leading an effort with a precise medical goal – to find effective neutralising antibodies against this new virus. Antibodies were obtained from five patients severely ill with COVID-19. Four of them survived while one later died of the disease. These antibodies were then analysed by biochemical and bioinformatic means to find those which bound best to the spike protein of the virus. In this context I learned some basic things about the virus. The spike, which is used by the virus to enter cells is considered the number one target for antibodies which could be effective in combating the disease. More precisely there are two different subdomains which are possible targets, one more at the tip of the spike (the receptor-binding domain) and another more on the sides (the N-terminal domain), which is a trimer. A number of antibodies were found which bind to the first subdomain or to one of the subunits of the second. Another was found whose binding site is somewhat less local. This whole process was carried out in just few weeks, a remarkable achievement.

The antibodies just mentioned are the therapeutic candidates. The idea is to either produce monoclonal antoibodies with these sequences or possibly versions which are improved so as to be longer-lived. Monoclonal antibodies are known to be extremely expensive when used to treat other diseases, such as cancer. They are also expensive in the present context, but the speaker said that the retail cost depends very much on the quantity produced. In other applications the number of patients is relatively small and the cost correspondingly high. If the antobodies were being used for a very large number of patients the cost would be lower. It would remain problematic for low and middle income countries. It has been discussed that the Gates foundation might make it possible to offer this treatment in poorer countries for fifty dollars a dose. The main advantage of this method compared with that of trying to use antibodies from the serum of patients directly is that it is much more practical to apply on a very large scale. The effectiveness of the antibodies against the disease has been tested in hamsters. Ho made the impression of someone tackling a major problem of humanity head on with some of the best tools available. He said that an article giving an account of the work had appeared in Nature (Potent neutralizing antibodies against multiple epitopes on SARS-CoV-2 spike, Nature 584, 450). In response to one question on one aspect of the treatment he said that the answer was not known but he would just be continuing to a Zoom meeting of researchers leading the attempts to develop therapies which was to discuss exactly that question.

### The basic reproduction number for infectious diseases, part 2

July 9, 2020

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 $x(t)=x_0e^{-at}$. The average value of $x$ is $\int_0^\infty x_0e^{-at}dt=\left[-a^{-1}x_0e^{-at}\right]_0^\infty=x_0a^{-1}$. In fact what we are interested in is the number of infections expected from the individuals in the infectious compartment. We have the equation $\psi'(t)=-V\psi (t)$, where all eigenvalues of $V$ have positive real parts and we want to calculate $F\int_0^\infty e^{-Vt}dt=FV^{-1}\psi (0)$, where $F$ describes the process of new infections. The matrix $FV^{-1}$ 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 $R$ 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.