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.

The Prime of Miss Jean Brodie

May 30, 2022

The novel ‘The Prime of Miss Jean Brodie’ by Muriel Spark has been on my bookshelf at home for many years. I bought it and read it when I was a student. I suppose I did so because I heard friends of mine who were studying English (including Ali Smith – see here for my relations with her) praising it. I do not know how much I liked it at the time. Now, for reasons not worth relating here, I took it off the shelf and opened it. I started reading a little and then could not stop until I had finished it. I had the strange impression that I liked the book much better and laughed a lot more than than the first time. Is this true or is it just my memory failing with increasing age? Now I am considering the possibility that among all novels by Scottish writers it is the one I like the best. What are other possible contenders for this role? The one which occurs to me is ‘Lanark’ by Alasdair Gray. I read it with admiration as a student. In a way Gray is to Glasgow as Joyce is to Dublin although I would not think of regarding Gray as being on the same level as Joyce. I once met Gray personally. The Aberdeen University Creative Writing Group invited him and Jim Kelman (who years later won the Booker Prize) to Aberdeen to give a reading. They came on the train from Glasgow and presumably consumed alcohol continuously during the whole journey. At any rate they were both very drunk when they arrived, whereby Gray appeared a bit less intoxicated than Kelman. I had no particular interest in Kelman; Gray was the one who interested me. I cannot remember much about our conversation, except that he pronounced ‘Proust’ incorrectly. I am not sure if that was ignorance or provocation. It was probably the latter. I have not read Lanark again since then and I think my copy did not survive my moves since then. I do not know what my impression would be if I started reading Lanark again but my intuition tells me that it would not captivate me like Miss Jean Brodie. This is because of the way I have changed over the past almost forty years.

As the title suggests the novel ‘The prime of Miss Jean Brodie’ is dominated by one character. She is strong and fascinating but it is wise to be cautious of having too much admiration for her. This is made clear at the latest by her professed admiration for Mussolini. (The novel is set in the years leading up to the Second World War.) She is a teacher at a private school, her pupils being girls of under the age of twelve. A group of these girls from one year come together to form ‘the Brodie set’. They are brought together not by any similarities between them but by their bond to Jean Brodie. This also keeps them together for the rest of their time at school. She is regarded by most of the teachers at the conventional school as too progressive and the headmistress is keen to find a reason to get rid of her. Eventually she succeeds in doing so. I cannot see Jean Brodie as a model for a teacher. She hatches out schemes to allow her pupils to avoid the work they should be doing and to listen to the stories she tells them. She has very definite ideas, for instance as to the relative importance of subjects: art first, philosophy second, science third. My ordering would be: science first, art second, philosophy third. The book is often very funny but I think there is also a lot in it which is very serious. Apart from the content, the use of language is very impressive. I enjoyed trying to imagine what it means for a jersey to be ‘a dark forbidding green’. There is even a little mathematics, although that is a subject which Jean Brodie has little liking for. Concerning the conflict of Miss Brodie with the Kerr sisters we read, ‘Miss Brodie was easily the equal of both sisters together, she was the square on the hypotenuse of a right-angled triangle and they were only the squares on the other two sides’. For me this is one of those few books which is able to bring some movement into my usual routine. I watched a documentary about Muriel Spark herself, which is also very interesting. Maybe I will soon read another of her novels. ‘Aiding and Abetting’ has also found its way from my distant past onto my bookshelf but I am not sure it will be the next one I read.

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.

The age of innocence

February 20, 2022

In a previous post I talked about my search for authors to read who are new to me. Now, following one suggestion there, I have read ‘The age of innocence’ by Edith Wharton. I did appreciate the book a lot. What are the reasons? The ‘psychological subtlety’ I mentioned in my previous post is also very much present in the novel. The society portrayed in the book is that of a small group of rich and elitist people in the New York of the late nineteenth century. In that group people tend to avoid talking about the things which are really important to them. This means that they have to be good at communicating without words. The novel is very successful in presenting the resulting non-verbal communication. Another aspect of the book I appreciate is the use of language. There are often phrases for which I felt the necessity to pause and savour them. The book is also rich in humour and irony.

What are the main themes of the novel? One is a certain society. We see a small number of people who are very rich, know each other well and are very resistent to letting anyone else into their circle. In some ways it is similar to the aristocracy in Europe at that time except that it lacks the long historical tradition and the codification of its rules. It consists mainly of people without special talents or great intelligence. In this society people are known because they are known and because certain other people in the same circle accept them. It reminds me a bit of the ‘celebrities’ of the present day who have similar characteristics including in many cases a lack of any obvious talent other than self-presentation. The main difference is that there are more chances to enter the circle of the celebrities. The two main characters in the book, Newland Archer and Ellen Olenska, are exceptions to the general rule. The Countess Olenska, as she is usually referred to in the book, comes from the small society described there but married a Polish count. She left him due to his bad treatment of her and came back to New York. What this treatment was is never really specified. In fact, it is not only the characters in the book who do not say openly what they mean. The author also presents many things in a way which is more suggestive than specific. The case of what really happened between the Countess Olenska and her husband is a good example but there are many more. In a way I found it a little frustrating to feel that I was waiting for information which never came. On the other hand I find that this allusive style has its own attraction. For me the Countess Olenska, who often ignores the strict rules of the society she is living in and is clearly highly intelligent and thoughtful, is the most attractive character in the novel. Newland Archer is the central character in the book in the sense that the story is told from his point of view, although in the third person. In his own way he questions the standards of the society around him. He often has impulses to act in a positive way but seems incapable of following them. He marries May Welland, a cousin of the Countess Olenska, who is the ideal partner for him in terms of her social standing and physical appearance. May is extremely conventional and not very intelligent. Her husband soon sees her as stupid, although in the end she has enough cunning to trick him and, in a sense, triumph over her clever cousin. There is a great deal of erotic tension between Archer and the Countess but, in the abstract and not only the literal sense, the orgasm never comes. May is treated cruelly in the text. Although it happens without any further comment the fact that she is given a painting of sheep as a present is a good example.

The two main characters are very different but in a way they mirror each other. I wonder to what extent the Countess Olenska is an image of the author and to what extent Newland Archer might be a better one. For me the most interesting aspect of the novel is what it has to say about love. I found the ending, whose content I will not reveal here, striking.

After I finished this post I remembered a question which I had asked myself and forgotten. This concerns the question of the relation of Marcel Proust, an author I much admire, to Edith Wharton. After all they did live in the same place at the same time. A short search led me to the following interesting answer.

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Persistence and permanence of dynamical systems

February 10, 2022

In mathematical models for chemical or biological systems the unknowns, which might be concentrations or populations, are naturally positive quantities. It is of interest to know whether in the course of time these quantities may approach zero in some sense. For example in the case of ecological models this is related to the question whether some species might become extinct. The notions of persistence and permanence are attempts to formulate precise concepts which could be helpful in discussing these questions. Unfortunately the use of these terms in the literature is not consistent.

I first consider the case of a dynamical system with continuous time defined by a system of ODE \dot x=f(x). The system is defined on the closure \bar E of a set E which is either the positive orthant in Euclidean space (the subset where all coordinates are non-negative) or the intersection of that set with an affine subspace. The second case is included to allow the imposition of fixed values of some conserved quantities. Let \partial E be the boundary of E, where in the second case we mean the boundary when E is considered as a subset of the affine subspace. Suppose that for any x_0\in \bar E there is a unique solution x(t) with x(0)=x_0. If x_0\in E let c_1(x_0)=\liminf d(x(t),\partial E), where x(t) is the solution with initial datum x_0 and d is the Euclidean distance. Let c_2(x_0)=\limsup d(x(t),\partial E). A first attempt to define persistence (PD1) says that it means that c_1(x_0)>0 for all x_0\in E. Similarly, a first attempt to define uniform persistence (PD2) is to say that it means that there is some m>0 such that c_1(x_0)\ge m for all x_0\in E. The system may be called weakly persistent (PD3) if c_2(x_0)>0 for all x_0\in E but this concept will not be considered further in what follows. A source of difficulties in dealing with definitions of this type is that there can be a mixing between what happens at zero and what happens at infinity. In a system where all solutions are bounded PD1 is equivalent to the condition that no positive solution has an \omega-limit point in \partial E. Given the kind of examples I am interested in I prefer to only define persistence for systems where all solutions are bounded and then use the definition formulated in terms of \omega-limit points. In that context it is equivalent to PD1. The term permanence is often used instead of uniform persistence. I prefer the former term and I prefer to define it only for systems where the solutions are uniformly bounded at late times, i.e. there exists a constant M such that all components of all solutions are bounded by M for times greater than some t_0, where t_0 might depend on the solution considered. Then permanence is equivalent to the condition that there is a compact set K\subset E such that for any positive solution x(t)\in K for t\ge t_0. A criterion for permanence which is sometimes useful will now be stated without proof. If a system whose solutions are uniformly bounded at infinity has the property that \overline{\omega(E)}\cap\partial E=\emptyset then it is permanent. Here \omega(E) is the \omega-limit set of E, i.e. the union of the \omega-limit sets of solutions starting at points of E. If there is a point in \overline{\omega(E)}\cap\partial E there is a solution through that point on an interval (-\epsilon,\epsilon) which is non-negative. For some systems points like this can be ruled out directly and this gives a way of proving that they are permanent.

Let \phi:[0,\infty)\times \bar E\to \bar E be the flow of the dynamical system, i.e. \phi(t,x_0) is the value at time t of the solution with x(0)=x_0. The time-one map F of the system is defined as F(x)=\phi(1,x). Its powers define a discrete semi-dynamical system with flow \psi(n,x)=F^n(x)=\phi(n,x), where n is a non-negative integer. For a discrete system of this type there is an obvious way to define \omega-limit points, persistence and permanence in analogy to what was done in the case with continuous time. For the last two we make suitable boundedness assumptions, as before. It is then clear that if the system of ODE is persistent or permanent its time-one map has the corresponding property. What is more interesting is that the converse statements also hold. Suppose the the time-one map is persistent. For a given E let K be the closure of the set of points F^n(x). This is a compact set and because of persistence it lies in E. Let K'=\phi (K\times [0,1]). This is also a compact subset of E. If t is sufficiently large then \phi([t],x)\in K where [t] is the largest integer smaller that t. This implies that \phi(t,x)\in K' and proves that the system of ODE is persistent. A similar argument shows that if the time-one map is permanent the system of ODE is permanent. We only have to start with the compact set K provided by the permanence of the time-one map.

Looking for a good read

January 9, 2022

In the recent past my reading has mostly been limited to mathematics and other scientific subjects. I have harldly found time to read literature. I have also tended to read only authors I already knew I liked. A few weeks ago I was in a second hand book shop and tried to do something against these tendencies. One of the things I did was to buy a book called ’50 Great Short Stories’ edited by Milton Crane. My motivation was less a desire to read short stories than to get to know authors I might like to read more of. Of course it was clear to me that reading a short story by an author may not give a very useful impression of what a novel by that author might be like. I was disappointed to find that there were few of the short stories in the book I liked very much. Those I did not appreciate much included ones by a number of authors who have written books I like very much, for instance E. M. Forster, Henry James, Guy de Maupassant, James Joyce, and Virginia Woolf. I long time ago I was keen on Aldous Huxley and read most of his books. This collection contains a short story by him ‘The Gioconda Smile’, which I am sure I had read before. I was not very enthusiastic about it this time but at least I did find it good. In the end there were only two stories in the collection which were by authors I had not previously read and which I liked enough so as to want to read more. The first is ‘The Other Two’ by Edith Wharton and I found the psychological subtlety of the writing attractive. I read a little about the author, who I found out has often been compared with Henry James. My superficial reading on this subject indicates to me that Wharton might have some of those qualities of James which I like while lacking some of those I dislike. Thus I am now motivated to read more of Wharton. The other story which made a very positive impression on me was ‘A Good Man is Hard to Find’ by Flannery O’Connor. I find it difficult to pin down what it was that I liked so much about it. In fact I think both aspects of the form and content were involved. I think a part of it was the impression of reading ‘something very different’. Here again I want to read more.

The other book I bought on that day was a collection of writings of Jean Paul, ‘Die wunderbare Gesellschaft in der Neujahrsnacht’. Unfortunately I found it completely inaccessible for me and I only read a small part of it.

Hopf bifurcations and Lyapunov-Schmidt theory

January 1, 2022

In a previous post I wrote something about the existence proof for Hopf bifurcations. Here I want to explain another proof which uses Lyapunov-Schmidt reduction. This is based on the book ‘Singularities and Groups in Bifurcation Theory’ by Golubitsky and Schaeffer. I like it because of the way it puts the theorem of Hopf into a wider context. The starting point is a system of the form \dot x+f(x,\alpha)=0. We would like to reformulate the problem in terms of periodic functions on an interval. By a suitable normalization of the problem it can be supposed that the linearized problem at the bifurcation point has periodic solutions with period 2\pi. Then the periodic solutions whose existence we wish to prove will have period close to 2\pi. To be able to treat these in a space of functions of period 2\pi we do a rescaling using a parameter \tau. The rescaled equation is (1+\tau)\dot x+f(x,\alpha)=0 where \tau should be thought of as small. A periodic solution of the rescaled equation of period 2\pi corresponds to a periodic solution of the original system of period 2\pi/(1+\tau). Let X_1 be the Banach space of periodic C^1 functions on [0,2\pi] with the usual C^1 norm and X_0 the analogous space of continuous functions. Define a mapping \Phi:X_1\times{\bf R}\times{\bf R}\to X_0 by \Phi(x,\alpha,\tau)=(1+\tau)\dot x+f(x,\alpha). Note that it is equivariant under translations of the argument. The periodic solutions we are looking for correspond to zeroes of \Phi. Let A be the linearization of f at the origin. The linearization of \Phi at (0,0,0) is Ly=\frac{dy}{dt}+Ay. The operator L is a (bounded) Fredholm operator of index zero.

Golubitsky and Schaeffer prove the following facts about this operator. The dimension of its kernel is two. It has a basis such that the equivariant action already mentioned acts on it by rotation. X_0 has an invariant splitting as the direct sum of the kernel and range of L. Moreover X_1 has splitting as the sum of the kernel of L and the intersection M of the range of L with X_1. These observations provide us with the type of splitting used in Lyapunov-Schmidt reduction. We obtain a reduced mapping \phi:{\rm ker}L\times{\bf R}\times{\bf R}\to {\rm ker}L and it is equivariant with respect to the action by translations. The special basis already mentioned allows this mapping to be written in a concrete form as a mapping {\bf R}^2\times{\bf R}\times{\bf R}\to {\bf R}^2. It can be written as a linear combination of the vectors with components [x,y] and [-y,x] where the coefficients are of the form p(x^2+y^2,\alpha,\tau) and q(x^2+y^2,\alpha,\tau). These functions satisfy the conditions p(0,0,0)=0, q(0,0,0)=0, p_\tau(0,0,0)=0, q_\tau(0,0,0)=-1. It follows that \phi is only zero if either x=y=0 or p=q=0. The first case corresponds to the steady state solution at the bifurcation point. The second case corresponds to 2\pi-periodic solutions which are non-constant if z=x^2+y^2>0. By a rotation we can reduce to the case y=0, x\ge 0. Then the two cases correspond to x=0 and p(x^2,\alpha,\tau)=q(x^2,\alpha,\tau)=0. The equation q(x^2,\alpha,\tau)=0 can be solved in the form \tau=\tau (x^2,\alpha), which follows from the implicit function theorem. Let r(z,\alpha)=p(z,\tau(z,\alpha)) and g(x,\alpha)=r(x^2,\alpha)x. Then \phi(x,y,\tau,\alpha)=0 has solutions with x^2+y^2>0 only if \tau=\tau(x^2+y^2,\alpha). All zeroes of \phi can be obtained from zeroes of g. This means that we have reduced the search for periodic solutions to the search for zeroes of the function g or for those of the function r.

If the derivative r_\alpha(0,0) is non-zero then it follows from the implicit function theorem that we can write \alpha=\mu(x^2) and there is a one-parameter family of solutions. There are two eigenvalues of D_xf of the form \sigma(\alpha)-i\omega(\alpha) where \sigma and \omega are smooth, \sigma (0)=0 and \omega (0)=1. It turns out that r_\alpha (0,0)=\sigma_\alpha(0,0) which provides the link between the central hypothesis of the theorem of Hopf and the hypothesis needed to apply the implicit function theorem in this situation. The equation r=0 is formally identical to that for a pitchfork bifurcation, i.e. a cusp bifurcation with reflection symmetry. The second non-degeneracy condition is r_z(0,0)=0. It is related to the non-vanishing of the first Lyapunov number.

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.