## Archive for May, 2008

### In memoriam Jürgen Ehlers

May 30, 2008

Today I attended the funeral of Jürgen Ehlers who died recently and quite unexpectedly for those around him. Here I want to collect some personal thoughts about Jürgen. I was associated with him for twenty years during which time he was my colleague and mentor. I was first a member of his research group in Munich and later at the institute in Potsdam (Max Planck Institute for Gravitational Physics) in whose founding he was the key figure. This is all the more remarkable since his passion was for science and not for scientific politics.

This is not the place to write about the personal qualities which I admired in Jürgen and I will confine myself to talking about some of his scientific qualities. Although he decided to go into physics rather than mathematics at a very early stage of his career, he had a keen sense for mathematical questions and a wide mathematical knowledge. He was not the kind to devote himself to the production of long and complex mathematical proofs. What he did was to identify important physical questions and the challenging mathematical problems which lay hidden beneath them. Once he had identified them he was a master of exposition whose explanations were a pleasure to listen to. He came back again and again to statements which are frequently made more or less uncritically in textbooks on general relativity such as ‘Newtonian gravitational theory is the limit of general relativity as the speed of light tends to infinity’ or ‘It follows from the Einstein equations that small bodies move on geodesics’. He saw that these phrases were in need of not only a proof but of a precise statement. He contributed to the first of these questions by formulating his ‘frame theory’ and also worked hard on the second. His efforts are now beginning to bear mathematical fruits in the form of rigorous theorems on these questions.

In doing science Jürgen emphasised making distinctions, e.g. ‘distinguish between a mathematical model in physics and its physical interpretation’, ‘distinguish between an intuitive argument and a rigorous proof’, ‘insist on precision in terminology where it can bring clarification’. These things may seem obvious but they are often enough neglected. In communicating these things to me he was preaching to the converted but he encouraged me by his example to propagate these same ideas. I remember him once talking about the difference between the concept of proof in physics and mathematics. To establish a mathematical fact one proof suffices. Additional, more or less independent, proofs may bring more insight but they are not necessary. The physicist, on the other hand, often likes to have several proofs. He likes to reach the same endpoint by various routes. Each of the two procedures has its own advantages. Very roughly speaking, the physicist’s approach takes one fast and far. The mathematician’s approach has the advantage that the gains it achieves are lasting. What I have said about distinctions might sound like abstract philosophy but my experience is that these ideas are of great worth in improving communication between physics and mathematics. No doubt similar things are true for the relations between other sciences.

### How does interferon affect the immune system?

May 26, 2008

The positive effect of interferon $\beta$ as a therapy for multiple sclerosis was mentioned in a previous post. Its performance is judged by its reduction of the number of brain lesions seen on MRT scans. This is statistical data and the individual patient may ponder the question of what these statistics mean for her or his own case. The way in which interferon $\beta$ combats MS is not well understood. New insights on this question are presented in a recent paper of Prinz et. al. (Immunity 28, 675). My attention was drawn to this work by a talk given by one of the authors, Ulrich Kalinke, in Berlin on 15th May. He said little about EAE but the talk encouraged me to look further. An account of this work which is less technical than in the original paper can be found in a paper of Axtell and Steinman (Immunity 28, 600) This work concerns the model disease EAE in mice. It is found that in mice with the disease interferon $\beta$ is produced in the central nervous system but much less in other tissues. So it seems that the body itself is producing the interferon as a defence against inflammation of the central nervous system. This substance is known to have effects on many immune cells and it is a priori not clear which of these effects are relevant for disease progression in EAE. Prinz. et. al. find that the effects of the interferon on certain cell types (T-cells, B-cells and astrocytes) seem to be irrelevant while the important thing is its effect on microglia and macrophages. This could be an important step towards pinpointing the mechanism leading to the therapeutic effect of interferon $\beta$.

### Chemotaxis

May 25, 2008

Chemotaxis is the process by which cells move in response to gradients in the concentration of a chemical substance. There are many important examples in biology, for example embryonic development, wound healing, homing of white blood cells to a site of infection or the metastasis of cancer cells. Other examples include free-living organisms exemplified by the bacterium Escherichia coli or the cellular slime mould Dictyostelium discoideum. Both of these are favourite model organisms. Chemotaxis in the first is the subject of a book, ‘E. coli in motion‘ by Howard Berg while the second has its own website.

The modelling problems which arise in the study of chemotaxis can be divided into two types. One is to model how an individual cell reacts to a chemical gradient. The other is to model how whole populations of cells react. In the case of the first problem there is a big difference between the case of eukaryotic cells and that of bacteria. This has to do with the difference in size, which makes physics look very different in the two cases and restricts the mechanisms which are conceivable. The fact that eukaryotic cells move by deformation (extension of pseudopodia etc.) make the process much harder to describe theoretically than for a bacterium. Of course they are also more complicated on a biochemical level. When considering models for populations of cells it is particularly interesting to look at the case where the cells produce the chemical themselves. The model for chemotaxis which is probably most popular among mathematicians is the Keller-Segel model. An extensive review of the mathematical literature on this subject has been given by Dirk Horstmann (Jahresbericht der DMV, 105, 104-165; 106, 51-69).

In the case of E. coli, as described in the book of Berg, the mechanism by which the bacterium manages to move in a controlled way involves stochastic elements. There are flagella which move clockwise or anticlockwise and the motion is steered by the probabilities that they rotate in the two possible directions, depending on how many molecules of the substance being detected bind to receptors on the cell surface. Because the cell is so small it is not practical to use spatial differences in concentration to detect the motion. Instead temporal differences are used. In other words the bacterium, instead of asking the question ‘What is the direction of the gradient of the concentration where I am now?’ asks the question ‘how does the concentration change in time if I start moving in this direction’ for a sufficient sample of directions. On a mathematical level it is possible to start from a stochastic model encoding the behaviour of a single cell and derive a continuum model of the motion of a population of cells. For more details see the paper of Horstmann quoted above.

An indication how much (or how little) is understood about the mechanisms of chemotaxis of eukaryotic cells is provided by a talk of Michael Sixt from the Max Planck Institute of Biochemistry I heard in Berlin on 24th January. The following is based on some notes I made at that time. The central theme of the lecture of Sixt was chemotaxis of dendritic cells. These are immune cells which are responsible for collecting samples from tissues and transporting them to the lymph nodes where they are presented to other immune cells such as T-cells. He started by saying that chemotaxis of immune cells needs be fast in order to allow prompt immune responses. The usual explanation says that these cells use adhesion molecules called integrins in order to pull themselves through tissues. It turns out, however, that cells from knockout mice which have no functioning integrins can move as fast as normal cells in vivo. On a surface they cannot. The mechanism of their motion was studied in collagen gels. The motion at the front end took place by actin polymerization. When myosin was deactivated the front of the cells moved as fast as before but the back stayed where it was. The reason for this was that the nucleus, the most inflexible part of the cell gets stuck in the pores of the gel. Changing to a gel with larger pores increased the motility of the myosin-inhibited cells. To see more details other studies were done under agarose. This means that the cells are confined to move between a hard surface and a layer of gel which they cannot penetrate. Neutrophils were able to move faster than some other leukocytes. The reason has to do with their full name – neutrophil polymorphonuclear granulocytes. The irregular form of the nucleus allows it to be pushed more effectively through the pores. This kind of motion could also be very interesting from the point of view of metastasis. A therapy which is based on hitting adhesion molecules would not affect this kind of motion at all. What the adhesion molecules are needed for is extravasation (leaving the blood vessels). Sixt accompanied his talk by striking films taken under the microscope which illustrate the points just described. Recently a paper related to this talk where he was one of the authors appeared (Lammermann et. al., Nature 453, 51-55).

### Mathematics and multiple sclerosis

May 15, 2008

Multiple sclerosis (MS) is a serious chronic disease of the central nervous system whose cause is not understood. Some knowledge is available on mechanisms which play a role in the disease and it is believed to be autoimmune in nature. The myelin sheaths which insulate the nerve cells are damaged through being attacked by elements of the immune system. The majority view among experts seems to be that the disease is not caused by a pathogenic organism or virus. It may, however, bethat a virus (or more than one) plays an indirect role. There is, for instance, the idea of molecular mimicry. Here immune cells which recognize certain foreign proteins attack similar structures belonging to useful molecules in the body. For instance it could be that the legitimate target of the immune cells are parts of a bacterium while the analogous structures which are actually attacked belong to the myelin. Since the immune system is programmed not to attack molecules belonging to self there is a motivation for microorganisms to develop molecular mimicry – it can be more than an accident. Another indirect viral influence is the so-called innocent bystander mechanism. Here the virus causes an upregulation of the activity of the immune system. If the immune system is already defective then this may be enough to cause it to attack myelin.

What aspects of MS might usefully be modelled using mathematics? One possibility would be to model populations of immune cells, concentrations of signalling molecules or both by an ODE system. This has some similarity to what was discussed for the case of AIDS in earlier posts but might not involve any virus. It could be pure immune system dynamics. As far as I can see little has been done in this area up to now. I will discuss one example below. In fact, in looking for examples I have widened the field to include work on EAE. The background to this is that MS is confined to human beings and no other species is affected by a closely analogous naturally occurring disease. This restricts the opportunities for research. EAE is an artificial disease (typically of mice) which mimics some of the features of MS and can be used for research purposes. It played a big role in the development of one of the best presently available treatments for MS, Copaxone or Copolymer 1. This substance is also known by the name glatiramer acetate, which I find particularly ugly. It was tested on EAE with the expectation that it would lead to a worsening of the disease and actually had the opposite effect. It is now successfully used for the treatment of MS and it is clear that it is unlikely to have been developed without the help of the mice.

Returning to mathematical models, I want to discuss some work of Lev Bar-Or (Math. Biosci. 163, 35-58). To explain what it is about it is necessary to know that CD4+ T-cells or T-helper cells (the ones which were mentioned in the posts on AIDS) come in two varieties called Th1 and Th2. They can be distinguished by the different kind of cytokines (certain signalling molecules) which they produce. In fact there are two different phases of the immune system where these sets of cytokines dominate and they are also called Th1 and Th2. It seems that in general a Th2 immune response is associated with less intense disease activity in MS (or EAE) and that a Th1 response is associated with more intense activity. The immune system can shift from one phase to the other over time and it could be very valuable to understand more about the dynamics of this process. The cytokines associated with a Th1 immune response might reasonably be thought of as bad for MS patients. An example is interferon $\gamma$. This was once tried as a therapy for MS but the clinical trials were broken off when it was observed that it had a negative effect on the patients’ health. On the other hand similar trials with the compound interferon $\beta$ gave positive results and led to what is probably the most effective long-term therapy for MS at present. At this point a word of caution is in order concerning trusting simplified pictures and trusting the applicability of animal models. As mentioned in the paper of Lev Bar-Or, it has been found that reducing interferon $\gamma$ concentration can lead to a worsening of EAE in mice. A possible explanation for this is given there.

The mathematical model in the above paper is a system of four coupled ODE containing a large number of parameters. The unknowns are the cytokine production by helper T-cells and macrophages of types Th1 and Th2 respectively. In general cytokines of one of the types stimulate the cells which produce that type and inhibit the cells which produce the other type. If this were all that was contained in the model then it would probably not lead to very interesting dynamics. There is, however, an extra effect which can lead to a reversal of some of the signs in the coupling. This has to do with the antigen presentation activity of type II MHC molecules, a subject I do not want to enter into here. Depending on the choice of parameters the dynamical system has either one or two equilibria. (This conclusion is based on numerics.) Either one of the two types (Th1 or Th2) dominates for the given parameter set or there may be coexistence of Th1 and Th2 dominated equilbria. In reality there are a huge number of cytokines and different types of immune cells. A more realistic dynamical system would contain many more variables but would probably not be very useful. The system with four unknowns arises by some averaging and some process like this seems inevitable in bringing mathematics to bear on the problem of understanding the dynamics of complicated biological systems.

There are some interesting features of MS which seem to invite a dynamical systems approach. The first is that there is often a gap of many years between the first relevant symptom and the development of the full disease. This is reminiscent of the situation with HIV mentionedin a previous post but in the present case mutations of a virus cannot be the explanation. In any case it is possible to pose the question: is there a period of dormancy here or is there an active and rapid dynamics which finally leads to a qualitative change? The second feature is that there are two qualitative phases of MS, the relapsing-remitting and progressive. In the relapsing-remitting phase the symptoms repeatedly get worse and then better again while in the progressive case the symptoms steadily get worse. In many patients the disease makes the transition from relapsing-remitting to progressive at some time. This invites the comparison with a dynamical system with an attractor which changes from a limit cycle to an equilibrium point as a parameter is varied. But what should the dynamical system be?

If the case of the understanding of MS is compared with the successful example of HIV there are couple of evident differences. One is that AIDS research has attracted more money and more publicity than MS research. Here I want to concentrate on another, which is the lack of good quantitative diagnostic criteria in the case of MS. The powerful technique is the MRT scan which gives information about the number of lesions in the brain and which are active. This can at best be interpreted statistically. Moreover MRT scans are very expensive which limits the frequency with which they can be done in most cases. It seems to me that the ideal thing would be to have some chemical whose concentration in the blood can be measured relatively easily and which gives a reliable indication as to whether the state of the patient is getting better or worse. It is information of this kind which led to the breakthrough with AIDS. In principle it would be better to take samples from the central nervous system which is the scene of the action rather than the blood but doing this frequently is not practical since it requires a lumbar puncture. Perhaps it would be better to concentrate on doing a parallel analysis on some other autoimmune disease such as rheumatoid arthritis where sampling is easier so as to establish some basic theoretical understanding of the dynamical processes involved.

Balo’s concentric sclerosis, mentioned in the last post, may be a disease related to MS or may be a form of MS. For a long time the only observations of the rings of demyelination which occur there were from autopsies. Now MRT offers new possibilities and there are some indications that rings of demyelination may occur in more standard cases of MS, particularly in the very early stages. Perhaps this rare disease might offer a clue to understanding the dynamics of a much more common one – studying an extreme case may be the key to solving a scientific problem.

### Liesegang rings

May 9, 2008

Liesegang rings are a phenomenon which is widely known due to the work of Raphael Eduard Liesegang in 1896 although he was probably not the first one to make such a observation. An experimental set-up which can be used is the following. A Petri dish is covered with a layer of a gel containing potassium dichromate. Then a drop of silver nitrate solution is deposited at the centre of the dish. It is observed that over a period of hours coloured rings appear which are centred on the point where the drop of solution was. It turns out that similar observations can be made in many other chemical systems. Another typical set-up is to take a test tube filled with gel containing one chemical and put a solution of another chemical on top of the gel. In this case horizontal bands are produced. Part of the fascination of the Liesegang phenomenon is the striking visual patterns which accompany it. Of course it should not be forgotten that not everything which looks the same must have the same underlying mechanism. This kind of phenomenon appears to be widespread in chemistry and has also been invoked in connection with certain biological phenomena. See for instance the discussion and pictures of fungal growth in http://seedsaside.wordpress.com/2008/02/21/liesegang-rings/

It is tempting to try and find a mathematical explanation of the Liesegang phenomenon. It has been observed that the position of the rings follows a geometric progression, that the time of their appearance goes as the square root of the distance and empirical laws for their thickness have also been stated. Thus there are some definite things which a mathematical model could try to reproduce.
There have been many attempts to give a theoretical (and mathematical) account of the effect. Here, for reasons of personal preference, I will concentrate on models which can be formulated in terms of systems of partial differential equations.

On 6th May Arnd Scheel from the University of Minnesota gave a talk on Liesegang patterns at the Free University in Berlin. I had already been interested in the subject for several months and so I naturally wanted to attend. Due to an overlap with the time of my course on general relativity I was only able to hear about half of the lecture but fortunately the speaker took time to explain some of his insights into the subject to me afterwards. The models he uses are reaction diffusion equations, i.e. systems of the form $\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2}+f(u)$where the unknown $u$ is vector-valued, $D$ is a diagonal matrix and $f$ is some smooth function. Here the spatial dimension is one so that the model is describing bands rather than rings. There are models in the literature where the function $f$ is very irregular (in particular discontinuous). Here I want to restrict to the case that $f$ is smooth. Rather than trying to describe the dynamics of the formation of the rings it is easier to concentrate on the final steady state. Then the time derivative of $u$ vanishes and the equation reduces to a system of ordinary differential equations which can be studied using methods from the theory of dynamical systems. One of the main things I learned from Scheel’s talk was that the existence of Liesgang patterns can be associated to the presence of homoclinic orbits in this dynamical system. In other words, there is a time independent solution of the dynamical system and another solution which converges to it both in the past and in the future. The Liesegang bands correspond not to the homoclinic orbit itself (which would just give a single band) but to other solutions of the dynamical system which approach it asymptotically. Whether appropriate solutions exist depends on the form of $f$ and the chemical literature contains a huge variety of choices. Since the phenomenon is so widely observed it must expected that it has a certain stability. If $f$ is perturbed a little then the bands should survive. This is somewhat surprising due to the fact that homoclinic orbits are usually not stable under general perturbations of a dynamical system. The explanation proposed by Scheel is that the systems of relevance for chemistry have a special structure which has a definite chemical significance. Only perturbations should be considered which preserve this additional structure and it can happen that perturbations of this special type do not destroy the homoclinic solution.

The special structure of the dynamical systems just mentioned reminds me of a comment made by Karl Sigmund in a plenary talk he gave at the International Congress of Mathematicians in Berlin in 1998 This talk was one of the factors which got me interested in mathematical biology. Sigmund’s point was that the general theory of dynamical systems is not well adapted to many of the problems arising in population dynamics. For good reasons the general theory concentrates on generic systems, i.e. on those whose qualitative properties are preserved under small perturbations. Since experimental measurements are never exact this makes good sense from the point of view of the applicability of the results. It may, however, happen that there is some aspect of the system which should be exempted from perturbations since it has a definite meaning for the applications of interest. In other words, it makes sense to consider systems whose properties are stable under perturbations which are generic modulo an invariant manifold which is kept fixed during the perturbation. In population models whis has to do with the fact that the population of some species being zero has an absolute significance which is independent of the details of the population dynamics. This reminded me of a research interest of my own, cosmology, where the aim is to produce simple models of the dynamics of the universe which also give rise to dynamical systems. In that case if the extension of the universe in one spatial direction tends to zero as the big bang singularity is approached than this has an absolute significance like the extinction of a species in a population model. Restricting to perturbations which preserve some submanifold tends to make homoclinic and heteroclinic solutions more common.

My first contact with Liesegang rings came through a talk by Benoit Perthame I heard at a conference on chemotaxis which took place at the Radon Institute in Linz last December. He mentioned some work on modelling a rare neurological disease called Balo’s concentric sclerosis. In the literature on this subject I found that this disease is characterized by alternating concentric rings of damaged and relatively intact tissue and that a connection to Liesegang rings had been suggested. I intend to return to this topic in a future post.

### Using mathematics to understand AIDS better, part 3

May 4, 2008

In what way is mathematics applied in practise in the study of HIV? A first important fact is that there are measurable quantities which can be used to gauge the state of the patient. These are the densities of virus particles and CD4+ T-cells in the blood. The latter is actually used to give a practical definition of AIDS. After infection with HIV it usually takes a long time before the full symptoms of AIDS are observed, typically many years. The T-cell count falls gradually and the start of AIDS can be associated with this count falling under a certain value. It might be supposed that the long quiescent phase of the HIV infection resulted from a kind of dormant state of the virus. The new results involving the use of mathematics suggest a different picture, showing that this long asymptomatic phase is a period of evolution of virus and immune system. There is apparently still a lot to be learned about this evolution but at least some facts are known and the evidence for a dynamic as opposed to a quiescent picture is itself very important.

When some observable quantities are available for a dynamical process one way of trying to learn more about it is to perturb it. In other words, change some aspect of the situation and see what happens. An opportunity to do this in the case of AIDS was presented by the development of drugs which interfere with the development of the virus. Of particular importance were the protease inhibitors which cause the newly produced virions to be incapable of infecting new cells. Treating a patient with this type of substance leads to an exponential decay in the virus load over a period of two weeks and a rapid increase in the T-cell count during the same period. Of course a prerequisite for finding this out is measuring the relevant concentrations frequently.

The mathematical analysis is easiest in the case of a different type of drug, the reverse transcriptase inhibitor, which prevents virions, even those which existed before treatment, from productively infecting new cells. The effect on the basic model of virus dynamics as presented in the book of Nowak and May is the set one coefficient to zero and to cause two of the three equations to decouple from the remaining one. For this the simplifying assumption is made that the inhibitor is one hundred per cent effective. The resulting system is linear and can be solved explicitly. The conclusion is that the rate of decay of the virus population is controlled by the natural decay rate of free virus and the death rate of infected T-cells. The key parameter is the slower of the two rates and it is typically assumed that this is the death rate of the T-cells. Under this assumption the parameter representing this death rate can be computed from the data. This also provides information about the rate of infection of cells before treatment and shows convincingly that it is high.

The high turnover rate of the virus implies that it is able to mutate effectively to meet the challenge of a drug. It this which means that treatment with a single inhibitor only gives a short term effect. Using more than one drug simultaneously makes it more difficult for the virus to adapt. Combining three different substances and different inhibition mechanism is what has turned out to be the successful in keeping HIV under control for extended periods. There are still serious problems such as the cost of the treatment and the serious side effects. Nevertheless this was a turning point in the treatment of AIDS and theoretical analysis, and in particular mathematics, played a major role.

### Using mathematics to understand AIDS better, part 2

May 1, 2008

The paper of Ho et. al. quoted in my last post refers to a paper of Perelson et. al. (Mathematical Biosciences 114, 81-125). There a detailed mathematical analysis is carried out. The main type of cells infected by HIV are the CD4+ T-cells (also known as T-helper cells). In the model in the paper of Perelson et. al. the dynamics of the populations of virions and T-cells are included. The infection of the cells by the virus is taken into account but the immune response, by which the T-cells act back on the virus, is not. The basic mathematical object of study there is a system of four ordinary differential equations. The unknowns are the densities of uninfected T-cells, latently infected T-cells (i.e. those which are not yet producing new virions), actively infected T-cells and virions. The paper starts with the common procedure in looking at dynamical systems of finding the equilibrium solutions and determining their stability by linearization. Depending on the value of a particular parameter there are either one or two stationary solutions which may be biologically relevant. They change their stability properties at the critical value of the parameter where the second stationary solution appears. At the first equilbrium the virus density is zero and when the long-time behaviour is described by convergence to this equilibrium it means clearance of the virus. At the second equilibrium point both virus and T-cells are present and when the long-time behaviour is decribed by convergence to this equilibrium it corresponds to an endemic state where the infection persists. An interesting question is: are other types of long-time behaviour possible and if so can it be proved? This kind of ODE system can be treated rather successfully by numerical methods and very often this is considered enough. A conclusion of the paper of Perelson et. al. is that for certain values of the parameters in the system no other behaviour is possible and that this includes all parameter sets which are biologically relevant. This last conclusion depends on the experimental data. They also map out a parameter regime where there is a periodic solution which acts as an attractor. This means that the system shows persistent oscillations. Most of these conclusions are based on numerical work. For a certain set of parameters the conclusions are proved rigorously using a Lyapunov function. There is also some heuristic analysis of certain phases of the dynamics using a quasi-steady state assumption. From the point of view of the role of mathematical proofs the remaining questions are: can the statements about the late-time behaviour be proved rigorously and what advantages could result from doing so?

There is a paper by Shaw et. al. which is in some ways similar to that of Ho et. al. and which appeared back to back with it in the journal (Nature, 373, 117-122). It uses slightly different mathematical models and here some simple equations are included in the text. Nowak is one of the authors and here there is a close link to his book with May quoted previously. Neither the paper nor the book pay much attention to questions of rigorous proofs of the qualitative behaviour of solutions. More recently some papers have appeared which provide an anlysis of this kind. In a paper of De Leenheer and Smith (Siam J. Appl. Math. 63, 1313-1327) the theory of monotone dynamical systems is applied. This is a method which can be used to effectively reduce the dimension of a dynamical system by one if it has a special algebraic structure. If the original system is three-dimensional, and thus potentially involves strange attractors, the reduced system is two-dimensional. Much more powerful tools such as
Poincare-Bendixson theory are available in the two-dimensional case and rule out many complications. De Leenheer and Smith use this observation to give a rigorous global qualitative analysis of some systems arising in virus dynamics, including ones involving periodic solutions. Part of the results of this paper have also been proved in a more elementary way by Korobeinikov (Bull. Math. Biol. 66, 879-883). He uses a Lyapunov function. When a Lyapunov function is known apparently intractable problems can become simple. This can work for dynamical systems of any dimension. On the other hand there is usually no mechanical procedure forfinding Lyapunov functions. That is the point where inspiration is often necessary. It is also interesting to note that the analysis here was helped by using an analogy to a certain system arising in epidemiology (the SEIR model) which had previously been analysed rigorously. The possibility of establishing analogies between mathematical models coming from very different applications is a typical strength of mathematics. In this context I would like to mention something which Rupert Klein said in a talk “Multiple scales in weather and climate” he gave at the Berlin Mathematical School on 11th January 2008. He pointed out that working in climate modelling involves using information coming from many disciplines. Often the specialists in different areas use some of the same concepts but do not know it because these concepts are known by different names in the literature ofthe different research areas. He suggested that in this context mathematicians can play an important role as translators. Mathematicianshave the tendency to extract the essential concepts from a problem they study and this can make the relations between the activities in different fields manifest.