## 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

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

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 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.

.

## 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

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

October 4, 2021