Archive for March, 2020

The plague priest of Annaberg

March 26, 2020

I find accounts of epidemics, whether documentary or fictional, fascinating. I appreciated texts of this kind by Camus (La Peste), Defoe (Journal of the Plague Year) and Giono (Le Hussard sur le Toit). This interest is reflected in a number of posts in this blog, for instance this one on the influenza pandemic of 1918. At the moment we all have the opportunity to experience what a pandemic is like, some of us more than others. In such a situation there are two basic points of view, depending on whether you see the events as concerning other people or whether you feel that you are yourself one of the potential victims. The choice of one of these points of view probably does not depend mainly on the external circumstances, except in extreme cases, and is more dependent on individual psychology. I do feel that the present COVID-19 pandemic concerns me personally. This is because Germany, where I live, is one of the countries with the most total cases at the moment, after China, Italy, USA and Spain. Every evening I study the new data in the Situation Reports of the WHO. The numbers to be found in the Internet are sometimes quite inconsistent. This can be explained by the time delays in reporting, the differences in the definitions of classes of infected individuals used by different people or organizations and unfortunately in some cases by poltically motivated lies. My strategy for extracting real information from this data is to stick to one source I believe to be competent and trustworthy (the WHO) and to concentrate on the relative differences between one day and the next and one country and another in order to be able to see trends. I find interesting the extent to which diagrams coming from mathematical models have found their way into the media reporting of this subject. Prediction is a high priority for many people at the moment.

Motivated by this background I started to read a historical novel by Gertrud Busch called ‘Der Pestpfarrer von Annaberg’ [the plague priest of Annaberg] which I got from my wife. The main character in the book is a person who really existed but many of the events reported there are fictional. Annaberg is a town in Germany, in the area called ‘Erzgebirge’, the literal English translation of whose name is ‘Ore mountains’. This mountain range lies on the border between Germany and the Czech republic. People were attracted there by the discovery of valuable mineral deposits. In particular, starting in the late fifteenth century, there was a kind of gold rush there (Berggeschrey), with the difference that the metal which caused it was silver rather than gold. My wife was born and grew up in that area and for this reason I have spent some time in Annaberg and other places close to there. The narrator of the book is Wolfgang Uhle, a priest in the Erzgebirge in the sixteenth century active in Annaberg during the outbreak of plague there. In fact in the end only a small part of the book concerns the plague itself but I am glad I read it. The author has created a striking picture of the point of view of the narrator, at a great distance from the modern world.

During Uhle’s first period as a priest there was a fire in a neighbouring village which destroyed many houses. He saved the life of a young girl, in fact a small child, who was playing in a burning house. Much to the amusement of the adults the girl said she would marry him when she was old enough. In fact she meant it very seriously and when she was old enough it did happen that after some difficulties she got engaged to him. The tragedy of Wolfgang Uhle is that he had a temper which was sometimes uncontrollable. Before the marriage took place he once got into a rage due to the disgraceful behaviour of the judge in his village. Unfortunately at that moment he was holding a large hammer in his hand. A young girl had asked him if a stone she had brought him was valuable. He had some knowledge of geology and he intended to use the hammer to break open the stone and find out more about its composition. In his sudden rage he hit the judge on the head with the hammer and killed him. He went home in a state of shock without any plan but his housekeeper brought him to flee over the border into Bohemia. He was sentenced to death in absentia and hid in the woods for five years. The girl who he was engaged to repudiated him, stamped on his engagement ring and quickly married another man. He partly lived from what he could find in nature, living at first in a cave. Later he started working together with a charcoal burner. I learned something about what that industry was like when I visited those woods myself a few years ago. Eventually he revealed his identity and had to leave.

In the woods he met a man who had got lost and asked him the way. The man wanted to go to Bärenstein, which is the town where my wife spent her childhood. He agreed to show him the way. The man told him that the plague had broken out in Annaberg and that the town was desperately searching for a priest to tend to the spiritual needs of the sick. Uhle decided that he should volunteer, despite the danger. He saw this as God giving him a chance to make amends for his crime. He wrote letters to the local prince and the authorities of the town. The prince agreed to grant him a pardon in return for his service as priest for the people infected with the plague. He then went to Annaberg and tended to the sick, without regard to the danger he was putting himself in. There is not much description of the plague itself in the book. There is a key scene where he meets his former love on her deathbed and it turns out that she had continued to love him and felt guilty for having abandoned him. Uhle survives the plague, gets a new position as a priest, marries and has children. This book was different from what I expected when I started reading it. Actually the fact that it was so different from things I otherwise encounter made it worthwhile for me to read it.


The Higgins-Selkov oscillator, part 3

March 17, 2020

Here I report on some further progress in understanding the dynamics of mathematical models for glycolysis. In a previous post I described some results on the Selkov model obtained in a paper by Pia Brechmann and myself. The main question left open there is whether this model admits unbounded oscillations. We have now been able to resolve this issue in a new paper, showing that there is precisely one value of the parameter \alpha for which solutions of this type exist. This progress was stimulated by a paper of Merkin, Needham and Scott (SIAM J. Appl. Math. 47, 1040). It was a piece of luck that we found this paper since its title gives no indication that it has any relation to the problem we were interested in. It concerns a model for certain chemical reactors which turns out to be mathematically identical to the Selkov model with the parameter \gamma equal to two. The authors claim that there are unbounded solutions for exactly one value of \alpha and present a derivation of this which is an intricate argument involving matched asymptotic expansions. I find this argument impressive but I have no idea how it could be transformed into a rigorous proof. Despite this it helped us indirectly. We had previously tried, without success, to find some way to transform the equations so as to obtain a well-behaved limit \alpha\to\infty. Combining some transformations in the paper of Merkin et al. with some we had used in our previous paper allowed this goal to be achieved. Once this has been done the dynamics is under control for \alpha sufficiently large. The next step is to do a shooting argument to show that there is a parameter value \alpha_1 for which a heteroclinic cycle at infinity exists. It had already been shown in our first paper that this is enough to conclude the occurrence of unbounded oscillations. The paper of Merkin et al. also helped us to understand what was the right set-up for the shooting argument. The uniqueness of \alpha_1 was obtained using a monotonicity property of the parameter dependence which appears to be new. At this point I think it is justified to say that the analysis of the main qualitative features of solutions of the Selkov oscillator is essentially complete. There is just one more thing I would like to know although I see it just as a curiosity. We showed that for \alpha a bit less than \alpha_1 there exists a stable periodic solution and that its diameter tends to infinity as \alpha\to\alpha_1. Merkin et al. give an expression for the leading order contribution to the diameter in this limit for \gamma=2. How could this be proved rigorously? How could an analogous expression be derived for other values of \gamma?

In the original paper of Selkov the system we have been discussing up to now is derived from a system with kinetics of Michaelis-Menten type by letting a parameter \nu tend to zero. He suggests that in the system with \nu>0 the unbounded oscillations are eliminated so that they could be thought of as an artefact of setting \nu=0. In our new paper we investigated this question although we were unfortunately only able to get partial results. One feature of the Selkov model not mentioned by Selkov in the original paper is that there are solutions which are unbounded and tend to infinity in a monotone manner. We showed that solutions of this type also exist for the Michaelis-Menten model and that they have exactly the same leading order asymptotics as in the case \nu=0. The periodic solutions of the Selkov oscillator are stable and arise in a supercritical Hopf bifurcation. That system admits no unstable periodic solutions whatsoever. By contrast, we found that the Michaelis-Menten system also exhibits subcritical Hopf bifurcations and correspondingly unstable periodic solutions.

I see a number of interesting directions in which this work could be extended but we will not attempt to do so in the foreseeable future since we have too many other projects to work on. I would only return to this if there turned out to be intriguing connections to other research projects or to the themes of master theses I was supervising.

Proving global stability of steady states of dynamical systems

March 13, 2020

In this post I want to discuss some techniques for proving that solutions of dynamical systems converge to steady states. One frequently used and powerful method of proving results of this type is to find a Lyapunov function. The problem with that is the lack of systematic ways of finding such functions even when they exist. Here I discuss a quite different approach. One way to try to prove that a solution converges to a steady state is to prove that it does not do anything else. Here I will restrict to solutions which are bounded (also away from the boundary of the domain of definition of the system, if there is one) so that they should converge to something. One obvious possibility is that the solution might converge to a periodic solution. Thus it is important to have criteria for ruling out periodic solutions. For systems of two equations there is a well-known criterion of this type due to Bendixson. A vector field on the plane whose divergence is everwhere positive (or everywhere negative) has no periodic solutions. The image of the solution is a Jordan curve and the integral of the divergence over its interior is positive (negative). On the other hand the divergence theorem shows that this is equal to a boundary integral and since the vector field is tangent to the boundary this integral is zero. Thus the existence of a periodic solution leads to a contradiction. This can be generalized in two ways. Instead of the whole plane the domain of the vector field could be a simply connected open subset. The vector field can be multiplied by a positive function, which does not affect the presence of periodic solutions. The resulting criterion is named after Dulac and the function which occurs is called a Dulac function. Other possibilities are homoclinic solutions and heteroclinic cycles. These define simple closed curves in the plane and can be ruled out by the argument just given in the presence of a Dulac function. Suppose now that all steady states are isolated. Then Poincaré-Bendixson theory shows that the possibilities we have considered are the only ones. To conclude, under the conditions we have considered on a dynamical system in two dimensions the existence of a Dulac function implies that every solution converges to a steady state.

The aim now is to find some generalization of this story to higher dimensions. The strategy consists of two parts. The first is a criterion for the absence of periodic solutions generalizing that of Bendixson. The second is to show that if an integral curve of a vector field returns arbitrarily close to its starting point there exists a small local perturbation of class C^1 of the vector field which admits a closed integral curve. This statement is known as Pugh’s closing lemma. If now the applicability of the Bendixson criterion is preserved under small perturbations this gives us a way of approaching the desired goal. From now on I concentrate on the Bendixson criterion. In the two-dimensional case the integral which was calculated can be thought of geometrically in the following way. Let S be the interior of the periodic solution. Now apply the flow of the vector field. The integral is the rate of change of the area of S under the flow. Since S does not move under the flow this rate of change is zero. Consider now a vector field on an open subset D of n-dimensional Euclidean space. A closed integral curve of this vector field can be thought of a mapping from the unit circle in the plane to D. If it is assumed that D is simply connected then (ignoring questions of regularity for the moment) this mapping extends to a mapping of the unit disc to D with image S. We now want to repeat what was done in the plane and consider how the area of S changes with the flow. On the one hand we would like to do a computation which shows that the rate of change of the area is non-zero. On the other hand we would like to choose S to be a surface of minimal area with the given boundary. These two elements together lead to a contradiction and rule out the existence of a periodic solution.

The rate of change of the area can be estimated using a suitable generalization of the divergence of the vector field. It is necessary to obtain information about the way in which two-dimensional area elements are affected by the flow. The necessary considerations can be found in a paper of Muldowney (Rocky Mountain Journal of Mathematics 20, 857). I cannot explain all details here and I will confine myself to mentioning some important ideas involved. We start with an autonomous system \dot x=f(x) of ODE. Linearizing about a solution gives a non-autonomous linear system \dot y=A(t)y, the variational equation. It describes how line elements are affected by the flow. To see how area elements are affected we must take the wedge product of two linearized solutions. It satisfies an equation \dot z=A^{[2]}(t)z where A^{[2]} is called the second additive compound of A. There is a direct algebraic formula which determines A^{[2]} from A. What is still needed is a way of estimating the solutions of these linear equations. This is done by using the Lozinskii logarithmic norm. I have discussed this in a previous post under the name ‘matrix measure’. Unfortunately I still do not have a good intuition for what it means. The definition is quite flexible due to the possibility of starting from different norms on the space of matrices. Concretely, one condition ensuring the absence of periodic solutions is that the Lozinskii norm of the second additive compound of Df is negative. With a suitable choice of norm on the space of matrices this is equal to the sum of the two largest eigenvalues of the symmetric part of Df.