Matched asymptotic expansions and the Kaplun-Lagerstrom model

In a post I wrote a long time ago I described matched asymptotic expansions as being like black magic. Now I have understood some more about how to get from there to rigorous mathematics. My main guide in doing so has been Chapter 7 of the book ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. There they give an extensive treatment of a model problem by Kaplun and Lagerstrom. The ultimate source of this is work of Stokes on hydrodynamics around 1850. In his calculations he found some paradoxical phenomena. Roughly speaking, attempting to obtain an asymptotic expansion for a solution led to inconsistencies. These things remained a mystery for many years. A big step forward came in the work of Kaplun and Lagerstrom in 1957. There they introduced an ODE model which, while having no direct physical interpretation, provides a relatively simple mathematical context in which to understand these phenomena. It is this model problem which is treated in detail by Hastings and McLeod. The model is a boundary value problem for the equation . We look for a solution with and . The first two terms look like the expression for the Laplacian of a spherically symmetric function in dimensions and for this reason the motivation is strong to look at the cases and which are vaguely related to fluid flow around a cylinder and flow around a sphere, respectively. It turns out that the case is a lot harder to analyse than the case . When the problem has a unique solution for . We would like to understand what happens to this solution as . It is possible to find an asymptotic expansion in but it is not enough to use powers of when building the expansion. There occurs a so-called switchback term containing . This is a singular limit although the parameter in the equation only occurs in a lower order term. This happens because the equation is defined on a non-compact region.

Consider the case . In applying matched asymptotic expansions to this problem the first step is to do a straightforward (formal) expansion of the equation in powers of . This gives differential equations for the expansion coefficients. At order zero there is no problem solving the equation with the desired boundary conditions. At order one this changes and it is not possible to implement the desired boundary condition at infinity. This has to do with the fact that in the correct asymptotic expansion the second term is not of order but of order . This extra term is the switchback term. Up to this point all this is formal. One method of obtaining rigorous proofs for the asymptotics is to use GSPT, as done in two papers of Popovic and Szmolyan (J. Diff. Eq. 199, 290 and Nonlin. Anal. 59, 531). There is an introduction to this work in the book but I felt the need to go deeper and I looked at the original papers as well. To fit the notation of those papers I replace by . Reducing the equation to first order by introducing as a new variable leads to a non-autonomous system of two equations. Introducing as a new dependent variable and using it to eliminate from the right hand side of the equations in favour of leads to an autonomous system of three equations. This allows the original problem to be reformulated in the following geometric way. The -axis consists of steady states. The point is denoted by . The aim is to find a solution which starts at a point of the form and tends to as . A solution of this form for small and positive is to be found by perturbation of a corresponding solution in the case . For the centre manifold of is two-dimensional and given explicitly by . In the case it is more degenerate and has an additional zero eigenvalue. To prove the existence of the desired connecting orbit we may note that for this is equivalent to showing that the manifold consisting of solutions starting at points of the form and the manifold consisting of solutions converging to intersect. The first of these manifolds is obviously a deformation of a manifold for . We would like the corresponding statement for the second manifold. This is difficult to get because of the singularity of the limit. To overcome this is introduced as a new dynamical variable and a suitable blow-up is carried out near the -axis. In this way it is possible to get to a situation where there are two manifolds which exist for all and depend smoothly on . They intersect for and in fact do so transversely. It follows that they also intersect for small and positive. What I have said here only scratches the surface of this subject but it indicates the direction in which progress could be made and this is a fundamental insight.

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Talks about malaria

I recently heard two talks about malaria at the Mainzer Medizinische Gesellschaft. The first, by Michael Schulte, was historical in nature and the main theme was the role of quinine as a treatment. The second, by Martin Dennebaum, was about malaria and its therapy today. Both talks were not only useful sources of information about malaria but also contained more general insights about medicine and its relations to society. In German the tree which is the natural source of quinine is called Chinarinde (i.e. China bark, in English it is called cinchona) and this had left me with the impression that the tree was from China. The first thing I learned from the first talk is that this is false. The tree comes from the Americas and was first used for medicinal purposes in Peru. A few weeks ago Eva and I visited a botanical garden in Frankfurt (Palmengarten) and saw a lot of those tropical plants which are the sources of things we are familiar with in everyday life (e.g. chocolate, cocoa, tobacco) and we in particular saw a cinchona tree. However I did not pay enough attention to realise its geographical origin at that time. There were men in Peru who had to cross a river to get to work and shivered a lot after they came out of the water. The idea came up that the bark of this tree could be used to reduce the shivering. At that time there were Jesuit missionaries in Peru. The Jesuits had been instructed by their leader Ignatius Loyola to bring back interesting things such as animals and plants from the exotic places they visited. One of the Jesuits in Peru, knowing that malaria is often accompanied by intense shivering, thought that cinchona bark might also help against malaria. This quite random analogy turned out to lead to a great success. Cinchona bark was sent to Rome and used there to treat malaria. It is remarkable how successfully this was done although the doctors knew nothing about the mechanisms at work in malaria. Without knowing about the existence of quinine they developed a way of extracting it very effectively using alcohol. They determined the right time to give the drug in the cycle of symptoms. In order for the treatment to be successful the drug must be given just after the infected red blood cells burst and the organisms are in the open in the blood and not protected. Quinine, the element of the cinchona bark most active against malaria was isolated around 1820. The first industrial production took place in Oppenheim, a town on the Rhein not far from Mainz. At one time Oppenheim was reponsible for 60 per cent of the world production of the substance. Malaria was a big public health problem in that region at the time and that was what stimulated the development of the industry. There is a story that the British colonists in India used to drink gin and tonic because tonic water contains quinine and thus provides protection against malaria. The speaker left it until the end of his talk to say that while the colonists did use quinine in other forms the concentration in tonic water is too low to be useful against malaria.

The second talk started with an interesting case history. A flight attendant flew from Frankfurt to Equatorial Guinea. Ten days after she got back she developed fever. Her husband had fever at that time due to influenza and she assumed she had the same thing. For this reason she did not seek medical advice until three days later. That was a public holiday and she was told she should come back the next day so that all the necessary tests could be done. The next day she landed as an emergency in the University Hospital in Mainz. A blood test showed that 25 per cent of her blood cells were infected with the organism causing malaria. The amount which is considered life-threatening is 5 per cent. The speaker said that if she had waited longer she would probably not have lived another day. Fortunately she did get there on time and could be cured. There are very effective drugs to treat malaria, namely those based on artemisinin. These drugs have their origin in traditional Chinese medicine. During the Vietnam war malaria was a big problem for those on both sides of the conflict. On the US side four or five times as many soldiers died of malaria than in combat. Both sides were looking for a drug to help with this problem and both looked to herbal sources, at first with little success. In the case of the Vietnamese they had enlisted the help of the Chinese to do this work for them. In China a secret ‘Project 523’ was set up for this purpose. As a part of this project Tu Youyou led the search for a malaria drug based on traditional Chinese medicine. She was successful and eventually got a Nobel Prize in 2015 for the discovery of artemisinin. From traditional literature she obtained a list of candidate plants and then subjected them to modern scientific analysis, in particular using experiments on mice. Her first attempts with the plant which produces artemisinin were not successful and it was another hint from ancient literature which helped her to overcome that difficulty. In fact the active substance was being destroyed by an extraction process at high temperature and once she had developed an alternative process at lower temperature positive results were obtained. Once the right candidate drug had been obtained the further analysis proceeded using all the tools of modern (non-alternative) medicine. I am no friend of ‘alternative medicine’ and I cannot help comparing the phrase to ‘alternative facts’. One of the things I have against ‘alternative medicine’ is that I think that if some part of it was really effective then it would quickly be adopted by real medicine and thus leave the alternative region. Nevertheless the story of artemisinin shows how in exceptional cases there can be a valuable flow information from traditional to real medicine and that this may require a great amount of effort. The type of malaria which is sometimes deadly is that caused by Plasmodium falciparum. Other types, caused by other Plasmodium species are less deadly but can become chronic. I think of novels where a typical figure was an army officer who suffered from malaria because he had served in India. From the talk I learned that the other types of malaria can be prevented from becoming chronic – it is just necessary to give the right treatment. To emphasize that malaria should be taken seriously in a country like Germany today he mentioned that at that moment there was again a malaria patient in intensive care in the University Hospital in Mainz although that case had not been so critical as that of the flight attendant.