Nobel lecture of Harvey Alter

In 2020 the Nobel prize for medicine was awarded to Harvey Alter, Michael Houghton and Charles Rice for their role in the discovery of the hepatitis C virus. I now watched the videos of the corresponding Nobel lectures. For my taste the lecture of Alter was by far the most interesting of the three. I think that he was also the one who played the most fundamental role in this discovery. At the beginning of his lecture he emphasizes the point that the most important discoveries in science often come as a complete surprise and not as a result of planned research programmes. Alter was 85 when he got the prize and so he had to wait a long time for it. The papers documenting his fundamental contributions were published in 1989. A central part of this work was the collection and preservation of blood samples from patients undergoing open heart surgery. Why was this group chosen? One of the most important modes of infection with hepatitis B and C used to be blood transfusions. This continued to be the case until tests were available to screen donors for these diseases. This kind of surgery involves extensive blood transfusions and so the chances of infection were relatively high in these patients. Also these patients suffered from relatively few other diseases which could have been confounding factors. These blood samples were an invaluable resource in the search for the virus. They were the basis of painstaking analysis over many years.

One important feature of hepatitis C is that it becomes chronic in 70 per cent of cases. This looks like a failure of the immune system to handle this disease. What are the reasons for this failure? One concerns quasispecies. The hepatitis C virus has an RNA genome and the copying of RNA is very error-prone. This leads to a huge variety in the genomes of virions in a single patient. This in turn results in rapid mutations of the virus. If an antibody has developed to combat the virus then selective pressure will quickly cause a new form to become dominant which is not vulnerable to that antibody. It seems to me that if this type of effect is to be captured using mathematical model it will require a stochastic model. Deterministic models of the type I have studied in the past are probably not helpful for that. In the lecture it is also mentioned that the number of T cells (CD4+ and CD8+) declines very much in chronically infected hepatitis C patients. No explanation is offerred as to why that is the case. Deterministic mathematical models might be able to contribute some understanding in that case.

The lecture contains the following interesting story. There was a time at which liver cancer was much more common in Japan than in the West. The reason for this was that that cancer was in many cases a late stage effect of hepatitis C. During wars in the early part of the 20th century many Japanese soldiers injected drugs with shared needles and this was what spread the disease. It was observed that there were many cases of jaundice (the most striking symptom of hepatitis) on the battlefield. Decades later many of these men developed serious liver disease, including cancer. Japanese doctors predicted that a similar phenomenon would be seen in the West when the effects of recreational drug use became manifest. They were right.

Fighting homeopathy

Most health systems in the world have financial problems and Germany is no exception to this. As a scientifically educated person it seems to me outrageous that the public health system here in Germany should spend money on homeopathic treatments. There is hardly anything in the area of ‘alternative medicine’ which is based on ideas which so blatantly contradict scientific reasoning. I was thus very glad to see that the health minister Karl Lauterbach has announced that he wants to ban the public health system from paying for homeopathic treatments. It is not the first time that a politician has tried this and suggestions of this type are usually greeted by a storm of opposition. I am not very optimistic that the idea of stopping public funds being used to finance homeopathy will actually find a sufficiently strong political consensus so as to result in legislation. If the idea actually succeeded I would be very happy. Of course the loudest opposition comes from those who earn money with homeopathy. One argument used is that the money involved is a small part of the total costs of the public health system. Of course it is the duty of those responsible to make sure that government money spent on health is well spent. The standards are usually very high. Often people cannot get treatments paid which are probably valuable but where the level of evidence required has not been reached. Thus I see the issue of homeopathy is one of principle. It is an insult to someone who is seriously ill and has to accept that the health insurance cannot pay for a treatment which would probably be effective (but is not yet provably so) while it pays for quackery. Germany has made a lot of important contributions to medicine but it should not be forgotten that although the disease homeopathy (as I see it) is widespread in the world it had its origin in Germany.

Karl Lauterbach is quite open about the fact that it is not the amount of money which is the central point. He is a scientist and qualified as a medical doctor and acts according to the ethical principles of these disciplines. He became publicly known through his role in the COVID-19 pandemic. (His academic speciality is epidemiology and public health.) During the pandemic he tended towards recommending strict measures and in this way he made himself unpopular with many people. At the same time his role was appreciated by many others (including myself) and this led to his appointment as health minister. He is a member of the SPD, having switched a long time ago from the CDU. Some politicians of other parties have accused him of using the issue of homeopathy to direct people’s attention away from other parts of his health policy which are seen as unsuccessful. I feel sure that this is not true. It is typical of Lauterbach, during the pandemic and otherwise, that he publicly says what he believes to be the truth, even when that results in unpopularity and attacks on him in the media. Although there are many aspects of Lauterbach’s politics I do not agree with I tend to identify with him as a person. I see him as a representative of honesty and rationalism in the public domain. The fact that many of the attacks on him involve personal antipathy and making fun of his appearance only tend to strengthen my feeling of solidarity with him.

It might be said that politicians have more important things which they should concentrate on than homeopathy but I am not prepared to accept that. I think that in the long term the struggle between science and superstition is a matter of key importance for the future of our civilization.

Nobel lectures of Katalin Karikó and Drew Weissman

Yesterday I listened to the Nobel lectures of Katalin Karikó and Drew Weissman, describing their work on mRNA vaccines. The lecture of Karikó mainly described the history of their discoveries while that of Weissman went into more technical details and discussed the prospects for the future applications of this technology. One interesting aspect of Karikó’s lecture was what she said about the difficulties she experienced during her scientific career. She was repeatedly unable to obtain funding for her research and lost several jobs due to her lack of success in this endeavour. For many years her work was supported by sympathetic colleagues. She could not afford assistants and had to do the menial jobs in the lab herself, down to thawing out the fridge where she kept her samples. In her talk she did not complain loudly about the injustice done to her in this way but restricted herself to making brief comments along the way. Eventually her career was saved when she was given a good job at the then obscure company Biontech.

Now let me come to the science, following Karikó’s account. After mRNA was discovered it took about twenty years before it could be synthesized artificially. mRNA is the template for the production of proteins and this gives rise to the idea that it might be made to cause cells to produce desirable proteins, for instance drugs. It turned out that there are several problems with this. The first is that within a living organism mRNA is attacked by the immune system and destroyed. The second is that artifical mRNA seemed to give poor protein yields. The third is that mRNA is a rather unstable molecule and thus only survives for a short time after it has been introduced into the body. Artificial mRNA is like the molecule is described in the textbooks. It consists of a string of nucleotides each of which contains one of the bases adenine, cytosine, guanine and uracil. Natural mRNA as it occurs in the human body is very different since many of the nucleotides containing the bases have been chemically modified. At the beginning of the work described in the lecture the enzymes responsible for these modifications were not known so that this process could not be understood, let alone controlled. A key type of experiment done by Karikó and Weissman was to feed dendritic cells with nucleotides, natural or modified, and look at whether they showed an inflammatory reaction, producing cytokines. It turned out that the dendritic cells reacted much less strongly to mRNA including certain modified nucleotides occuring naturally than to the textbook mRNA. It is easy to guess why this should be the case. (This is my speculation, not a statement from the lecture.) mRNA occurring in the body could be from a pathogen such as a virus and then the immune system should eliminate it. The modifications could be a way the body could signal to the immune system that an mRNA molecule is made by the host and should be left alone. In any case it was found by trying many examples that one powerful way of suppressing the immunogenicity of the RNA was to replace uridine by the modified molecule pseudouridine. This provided an avenue to removing the first of the difficulties in applying RNA therapeutically. It turns out that the modified RNA produces higher protein yields and is more stable than the textbook RNA. In other words, it can contribute to the solution of the other two problems as well.

If RNA is to be used as a vaccine then while the immune system should ignore the RNA it should react strongly to the corresponding protein. This seemed to work well in the case of RNA vaccines but this was paradoxical. Normally a protein is not enough to make a vaccine. It must be accompanied by another substance, an adjuvant, which activates the innate immune system. The RNA vaccine contained no known adjuvant. The solution to this problem is as follows. In order to get the RNA into a cell it has to be coated in lipids. It turns out that these lipids act as an adjuvant. In the end they activate the so-called follicular helper T cells. This kind of vaccine is remarkable in that it can stimulate the immune system more strongly than the pathogen it is intended to be a vaccine against. For instance the RNA vaccines against COVID-19 cause a production of antibodies which is several times higher than an infection with the virus itself.

Now a lot is known about the use of different kinds of RNA to achieve different effects. Various aspects of this were explained in the lecture of Weissman. He discussed a variety of different applications which appear within reach: improved vaccines against infectious diseases, vaccines against cancer, production of drugs. Appart from their flexibility and effectiveness the RNA techniques have the potential to replace the extremely expensive processes required for the therapy of certain diseases by rather cheap ones. Weismann’s talk gave the impression that the RNA techniques could soon lead to revolutionary advances in medicine. He is not at all the type of person who comes across as an advertiser. Instead he makes an impression of someone who is modest and trustworthy. He discussed a wide variety of examples. Let me concentrate on one. This is the idea of a universal influenza vaccine. The influenza virus mutates frequently with the result that it is necessary to develop a new vaccine each year to be effective for the new dominant variant. A dream is to develop a vaccine which would be effective for all types. It has been found that RNA vaccines can be effective against many antigens simultaneously, for instance for all types of influenza. This is being tested in practice now. I was excited by what I heard in this lecture. Of course there will no doubt be many unexpected difficulties in implementing these ideas but I think that there is a good chance that they could bring a major improvement in medicine as a whole.

Talk on personalized tumour therapy in pediatrics

Yesterday I heard a talk at the Mainzer Medizinische Gesellschaft by Alexandra Russo from the university hospital in Mainz. She is responsible for treating children with cancer in that hospital. She talked about progress in cancer therapy for children, focussing on the situation in Germany. She did also mention international collaborations of her group in Mainz. A positive fact is that 80% of childhood cancers can be cured definitively. In other words the tumour can be eliminated in such a way that it does not return. This sounds much better than the situation for adults. A negative fact is that this percentage has not changed significantly in the last twenty years. Another negative fact is that among those people cured many have long-time side effects resulting from the chemotherapy they had. There was a group of former patients described who had had chemotherapy with a particular substance and many of these young people had hearing aids. Hearing loss is a known side effect of the drug they were treated with. The speaker is enthusiastic and optimistic about being able to change these things by applying personalized therapies. At the moment most treatments are still based on the classical methods: surgery, radiotherapy and chemotherapy. One special feature is that radiotherapy is not used in children under the age of three since the side effects would be too dramatic.

Some special features of treating children were mentioned. One is that it is often necessary to use drugs which are not approved for the treatment of children. There is a law in Germany that new drugs must be tested for their use in children as well as adults. However there are many exceptions to this based on the argument that the disease concerned does not occur in children. For instance children do not get lung cancer. The necessity of off-label use leads to problems with getting the drugs payed for and getting permission from an ethics commission. Even when these problems have been overcome in a given case there are still the problems that for children, due to lack of studies, the correct dosage and the safety profile are not known. The commercially available drugs are not suitable for use in children. For instance a single pill may contain a dose which is much too high for a small child. This means that the drugs must be specially processed by the clinic before they can be used. The speaker proudly showed a picture of a new machine they just got which is a 3-d printer for pills. It is still at the prototype stage but she expects it to bring them great benefits in the near future.

The speaker said that the amount of money spent on therapy in Germany is one of the highest in the world although not as much as in the US. In Britain only about half as much is spent. What is not so good is that much less sequencing is carried out in Germany than in many countries. So when she needs a lot of sequencing for her patients it can be difficult. A method for trying to improve the problems with the studies and approval would be to do studies linked not to a particular type of cancer cell but to a particular type of genetic defect. To make this possible sufficient genetic data about the patients must be obtained. She mentioned the example of Pembrolizumab, which was the first cancer drug to be approved for certain genetic situations rather a specific disease.

In the talk two examples of patients were discussed in some detail. The first was of a boy with a brain tumour who was six years old at diagnosis. It was first diagnosed as an astrocytome but this was later revised to glioblastome. A genetic irregularity was found which is best known from lung cancer. Then the idea was to try to use a drug which had worked in a lung cancer of this type to treat the brain tumour. An extra difficulty was to find a drug which would cross the blood-brain barrier. A drug was found and did have a positive effect. Unfortunately an infection of the patient meant that the treatment had to be discontinued for some time. This led to the tumour getting out of control and the death of the patient. Perhaps if the detailed genetic information had been available more quickly so that a good treatment could have been started more quickly the story would have had a better end. The second example was that of a nine-month baby with apparently swollen lymph nodes which turned out to have a tumour in the neck region. Despite different types of chemotherapy the tumour grew very much until it was acutely life-threatening. In looking for a targetted drug it was again important to have enough genetic information so as to see what signalling pathways were involved in the pathology. It turned out that the MAPK cascade was involved. (My attention is always drawn by the MAPK cascade since I did some work on mathematical models for it.) One way of obtaining information was to culture pieces of tumour in the presence of different drugs to see which ones might be helpful. In this case a suitable drug was found, a MEK inhibitor called cobimetinib. After three weeks treatment the size of the tumour had decreased by 94% and the child could live a relatively normal life. Unfortunately it looks as if the drug must be given indefinitely in order to control the tumour. At the moment the strategy is to try and reduce the dose so as to reduce the side effects.

I have a lot of admiration for someone doing a job like this. It involves intellectual challenges, emotional difficulties and the need for strong practical qualities. In any case, I found the talk fascinating.

Conference on mathematics and immunology in Blagoevgrad

I recently attended a conference on mathematics and immunology in Blagoevgrad in Bulgaria. I must admit that I knew very little about Bulgaria. I knew the name, I knew where it was and I knew that it had a Slavic language. That was about all. Thus it was educational for me to be in the country and also have the chance to talk to Bulgarians. It was in particular interesting to learn that there was a large Bulgarian empire centuries ago. I flew to Sofia and took a train south to the site of the conference. The train was quite old and slow but it passed through interesting countryside, some of which was extremely green and lush. I saw several Bee-eaters from the train. On the evening I arrived there were a lot of swallows flying around. They were flying quite high and fast and I was not able to identify them definitely. They looked and sounded different to the species I was familiar with. While they were eating mosquitos the mosquitos were eating me and that made it difficult to concentrate. A couple of days later I was able to convince myself that one suspicion I had had at the beginning was correct – they were Red-rumped Swallows.

The conference itself was small and I much prefer small and thematically focussed meetings such as this to large conferences. It was very pleasant to be among people who are interested in and knowledgeable about both mathematics and immunology. The first speaker was Becca Asquith who talked about KIRs, the inhibitory receptors found on natural killer cells. What I did not know before was that KIRs are also presented on T cells. It is well known that the susceptibility of individuals to infectious or auto-immune diseases is dependent on which MHC molecules they have. I was aware that, for instance, certain HLA-DR types are associated to an increased risk of multiple sclerosis. In fact HLA molecules may have positive or negative effects on disease risk. It turns out that the types of KIRs a given individual has can also have analogous effects. A central theme of the talk was how these statistically observed effects can be explained mechanistically and how mathematical models can help to distinguish between different mechanisms. A number of talks were related to epidemiology, considering in-host models, population models and the coupling of the two. One concerned a virus I did not know previously, the Usutu virus. Stanca Ciupe described modelling for experiments where house sparrows were exposed to mosquitos carrying this virus. The virus (and its name) originated in Africa but it has been spread to parts of Europe by migratory birds. Doing some more reading on the subject I discovered that it is reponsible for some cases where a lot of blackbirds have been seen to die. Humans can become infected with the virus but it does not seem to be directly harmful to us. The virus is nevertheless interesting as a candidate for one which might mutate and cause a serious disease of humans in the future. Jonathan Forde gave an interesting talk on the question of the best way to invest limited (financial or human) resources in fighting COVID-19 or other infectious diseases. Sometimes there is an optimal balance between vaccination and testing.

There were also talks on subjects other than infectious diseases. Vladimira Suvandjieva talked about some work she has done on modelling the role of NETs in lupus. Here mathematics is applied to a biological subject which I wrote about in this blog a long time ago. In this phenomenon the immune cells involved are neutrophils. Doron Levy talked about using mathematical models to better understand cancer immunotherapy. There were talks on the ways in which the time of administration can influence the effectiveness of chemotherapy against cancer and the strength of side effects. What is notable is a strong dependence on the sex of the patient. Apart from the science it was interesting to talk to the other participants from various parts of the world about political themes, thus being exposed to different facts and opinions. I have not had so many opportunities for this kind of conversation since the pandemic.

Fomites and backward bifurcations

I first met the curious word ‘fomite’ a few months ago. It means a carrier of infection which is not a human being or an animal. It evolved in an interesting way, as recounted in the Wikipedia article on this concept. It started out as the Latin word ‘fomes’. The plural of that is ‘fomites’, which was taken over into English. There it was interpreted as an English plural and gave rise to the corresponding singular ‘fomite’. Let me now get away from the word and consider the concept it designates. During the COVID-19 pandemic there was a lot of discussion about the modes of transmission of infection. In particular people wanted to understand the importance of the disinfection of surfaces (and of hands). These inanimate surfaces, where the virus might be present and act as a source of infection, are then (the surfaces of) fomites. As far as I can judge the consensus developed that this mode of transmission was not the key factor in the case of COVID-19 with attention being concentrated on aerosols. However important they may be in that particular case there is no doubt that there are diseases for which transmission by fomites is very important. The theme of hospital infections is a very important one these days and there fomites are likely to play a central role.

I came across the word fomite in the course of some research I did recently with Aytül Gökce and Burcu Gürbüz on some population models for infectious diseases. We just produced a preprint on that. One aspect of infection which was included in our model and which is not very common in the literature is that of infections coming from virus in the environment and here we can think of fomites. In modelling this phenomenon a choice has to be made of a function which describes the force of infection. We can consider an expression for the rate of infection of the from , where is the number of susceptibles and is the concentration of virus in the environment which may cause infection. What should we choose for the function ? This type of issue comes up in other modelling settings, in particular in those related to biology, where a choice of response function has to be made. In a biochemical reaction could describe the reaction rate as a function of the concentration of the substrate. Typical choices are the Michaelis-Menten function and the Hill function where is replaced in this expression by a power . The Michaelis-Menten function has a mechanistic basis, introduced by the people who it is now named after. Hill also presented a mechanistic basis for his function. It incorporates the phenomenon of cooperative binding, with the original example being the binding of oxygen to haemoglobin. Another example of the choice of response functions occurs in predator-prey models in ecology. Here describes the dependence of the rate of uptake of prey as a function of the density of prey. The functions which are mathematically identical to the Michaelis-Menten and Hill functions are called Holling type II and Holling type III. Holling gave mechanistic interpretations of these. Type II is associated with the fact that the predator needs some time to process prey and that that time is lost for the search for prey. Type III is associated with learning by the predator. In this list type I is missing. Type I refers to a linear function, except that to account for the fact that the rate of uptake of prey is in reality limited the linear function is usually cut off at some level, after which it is constant.

It seems that in epidemiology these issues have not been studied so much. There is an extended discussion of the subject in Chapter 10 of the book ‘Mathematical Epidemiology of Infectious Diseases’ by Diekmann and Heesterbeek. I mentioned this book in a previous post but unfortunately forgot to mention the names of the authors. At the beginning of the chapter it is stated clearly that it will not give a definitive answer to the questions it raises. I appreciate this kind of modesty. In that chapter there is a mechanistic derivation of a response function in a particular epidemiological setting. It is more complicated that the conventional ones and is not a rational function. The function corresponding to Holling type II has been been discussed in relation to epidemiology, apparently for the first time by Dietz. His discussion is purely phenomenological as opposed to mechanistic. In other words he chooses the simplest function satisfying some basic desired properties. In our paper we consider phenomenological models containing a power as in Holling type II () and Holling type III (). We found that the case gives rise to backward bifurcations while the case does not. Thus here the details of the response function can be of considerable mathematical or even medical importance. (The presence of backward bifurcations has implications for the strategy of therapies.) Another topic of the paper is the different ways in which imperfect vaccination can be described mathematically but I will not go any deeper into this theme here.

The formalism of van den Driessche and Watmough

Here I continue the discussion of the paper of van den Driessche and Watmough from the point I left it in the last post. We have a system of ODE with unknowns , . The first of these have the status of infected variables. The right hand side of the evolution equation for is the sum of three terms , and where , and are all non-negative. This is condition (A1) of the paper. On the right hand side of this type of system there are positive and negative terms. If we take to be the total contribution of the negative terms then it is uniquely defined by the system of ODE. On the other hand splitting the positive terms involves a choice. The notation is also used. As is usual in reaction networks when . This is condition (A2) of the paper. The represent new infections and hence must be zero for . This is condition (A3) of the paper. It restricts the ambiguity in the splitting. In condition (A4) the set of disease-free steady states is supposed to be invariant and this leads to the vanishing of some non-negative terms at those points. In general this condition restricts the choice of the infected compartments. In the fundamental model of virus dynamics the infected variables are a subset of . However choosing it to be a proper subset would violate condition (A4). Thus for this model there is no choice at this point. There does remain a choice in splitting the positive terms, as already discussed in a previous post. It remains to consider the last condition (A5). Its interpretation in words is that if the infection is stopped the disease-free steady state is asymptotically stable. Mathematically, stopping the infection means setting some parameters in the model to zero.

Consider now the linearization of the system at a disease-free steady state. The choice of infectious compartments induces a block structure. Splitting the RHS of the system into and also splits the linearization into a sum. It is an elementary consequence of the assumptions made that some of the entries of the matrices occurring in these decompositions are zero. It can also be shown that some entries are non-negative. To describe a new infection we can take an initial datum close to the disease-free steady state. Then we may reasonably suppose (leaving aside questions of proof) that the actual dynamics is well-approximated by the linearized dynamics during an initial period. Under the given assumptions the linearization is block lower triangular. The linearized evolution is partially decoupled and the uninfected variables decay exponentially. So the dynamics essentially reduces to that of the infected variables and is generated by the upper left block, called in the paper. We get an exponential phase of decay with populations proportional to . During this phase the number of susceptibles is approximately constant, equal to its value at the disease-free steady state. Hence we get an explicit expression for the rate of infection in this phase. Integrating this with respect to time gives an expression for the ratio of total number of individuals infected in that phase as a function of the original number of individuals in the different compartments. It is given by the matrix , which is the NGM previously mentioned.

I have not succeeded in making a connection between the formalism of van den Driessche and Watmough and the discussion on p. 19 of the book of May and Nowak. Towards the bottom of that page it is easy to see the connection with the unique positive eigenvalue of the linearization at the boundary steady state if is simply interpreted as the known combination of parameters. After I had written this I noticed a source cited by van den Driessche and Watmough, the book ‘Mathematical Epidemiology of Infectious Diseases’ by Diekmann and Heesterbeek and, in particular, Theorem 6.13 of that book. (To avoid any confusion, I am talking about the original edition of this book, not the later extended version.) I had never seen the book before and I borrowed it from the library. I found a text which pays attention to the relations between mathematical rigour and applications in an exemplary way. I feel that if I read the whole book carefully and did all the exercises (which is the recommendation of the authors) I might be able to get rid of all the problems which are the subject of this and the previous related posts. Since this ideal engagement with the book is something I will not achieve soon, if ever, I restrict myself to some limited comments. On p. 105 the authors write ‘Many modellers … even distrust whether the threshold value one of … corresponds exactly to their stability condition. … The remainder of this section is intended for modellers who recognise themselves in the above description.’ I recognise myself in this sense. The section referred to contains Theorem 6.13.

The basic reproduction number, yet again

In this post I come back to the basic reproduction number, a concept which seems to me like a thorn in my flesh. Some of what I write here might seem rather like a repetition of things I have written in previous posts but I feel a strong need to consolidate my knowledge on this issue so as to possibly finally extract the thorn. The immediate reason for doing so is that I have been reading the book ‘Infectious diseases of humans’ by Anderson and May. I find this book very good and pleasant to read but my progress was halted when I came to Section 2.2 on p. 17 whose title is ‘The basic reproductive rate of a parasite’. (The question of whether it is better to use the word ‘rate’ or the word ‘number’ in this context is not the one that interests me.) What is written there is ‘The basic reproductive rate, is essentially the average number of successful offspring a parasite is capable of producing’. It is clear that this is not a precise mathematical definition, as shown in particular by the use of the word ‘essentially’. In that section there is further discussion of the concept without a definition. As a mathematician I am made uneasy by a discussion using what is apparently a mathematical concept without giving a definition. A possible reaction to this unease would be to say, ‘Be patient, maybe a definition will be given later in the book’. Unfortunately I have not been able to find a definition elsewhere in the book. I also checked whether it might not be included in some supplementary material or appendices. Later in the book this quantity is calculated in some models. This might provide an opportunity for reverse engineering the definition. This book is not aimed at mathematicians, which is an excuse for the lack of a definition. Next I turned to the book ‘Virus dynamics’ by Nowak and May which is related but a bit more mathematical. Here the place I get stuck is on p. 19. Again there is a definition of in words, ‘the number of newly infected cells that arise from any one infected cell when almost all cells are uninfected’. Just before that a specific model, the ‘basic model of virus dynamics’ has been introduced.

The basic model is clearly defined in the book. It is a system of ordinary differential equations depending on some parameters. Depending on the values of these parameters there is one positive steady state or none. These two cases can be characterized by a certain ratio of the parameters. If this ratio is greater than one there is a positive steady state. If it is less than or equal to one there is none. We can introduce the notation for this quantity and call it the basic reproduction number. This is a clear and permissible mathematical definition and it is an interesting statement about the model that such a quantity exists and can be written down explicitly in terms of the parameters of the system. This definition does, however, have two disadvantages. There are many models similar to this one in the literature. Thus the question comes up: if I have a specific model does there exist a quantity analogous to and if so how can I calculate it? The other is the idea that is not just a mathematical quantity. It potentially contains useful biological insights and that is something I would like to understand.

Is there more to be learned about this issue from the book of Nowak and May? They do have a kind of derivation in words of the expression for . It is obtained as the product of three quantities, each of which comes with a certain interpretation. Thus a strategy would be to try and understand these three quantities individually and then understand why it is relevant to consider their product. At this point I will introduce some insights which I mentioned in a previous post. For many models the existence of a positive steady state is coupled to the stability of a steady state on the boundary of the non-negative orthant (disease-free steady state). I believe that is really a quantity related to the disease-free steady state and that in a sense it is just a coincidence that it is related to the question of the existence of a positive steady state. This is illustrated by the fact that there are models exhibiting a so-called backward bifurcation where there do exist positive steady states in some cases with . Beyond this, I believe that depends only on the linearization of the system about the disease-free steady state.

As I have discussed elsewhere, the best source I know for understanding mathematically is a paper of van den Driessche and Watmough. There it is explained how, subject to making some choices, the linearization of the system at the disease-free steady state gives rise to something called the ‘next generation matrix’, let me abbreviate it by NGM. In that paper is defined to be the spectral radius of the NGM. Thus we obtain a rigorous definition of provided we have a rigorous definition of the NGM. In addition it is proved in that paper that the linearization at the disease-free steady state has an eigenvalue with positive real part precisely when , a fact with obvious consequences for stability of that steady state. It is clearly stated in the paper that the NGM does not only depend on the system of ODE defining the model but also on the choices already mentioned. The first choice is that of the variables are supposed to correspond to infected individuals. A disease-free steady state is then by definition one at which these variables vanish. This means that if we have a specific model and a specific choice of boundary steady state where some unknowns vanish and we want to apply this approach the infected variables must be a subset of the unknowns which vanish there. It is not ruled out that they might be a proper subset. This discussion has made it clear to me what strategy I should pursue to make progress in this area. I should look in great detail at the part of the paper of van den Driessche and Watmough where the method is set up. Since I think this post is already long enough I will do that on another occasion.

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.

Is mathematics being driven out by computers?

In the past two weeks I attended two conferences. The first was the annual meeting of the Deutsche Mathematikervereinigung (DMV, the German mathematical society) in Berlin. The second was the joint annual meeting of the ESMTB (European Society for Mathematical and Theoretical Biology) and the SMB (Society for Mathematical Biology) in Heidelberg. I had the impression that the participation of the SMB was relatively small compared to previous years. (Was this mainly due to the pandemic or due to other problems in international travel?) There were about 500 participants in total who were present in person and about another 100 online. I was disappointed with the plenary talks at both conferences. The only one which I found reasonably good was that of Benoit Perthame. One reason I did not like them was the dominance of topics like machine learning and artificial intelligence. This brings me to the title of this post. I have the impression that mathematics (at least in applied areas) is becoming ever weaker and being replaced by the procedure of developing computer programmes which could be applied (and sometimes are) to the masses of data which our society produces these days. This was very noticeable in these two conferences. I would prefer if we human beings would continue to learn something and not just leave it to the machines. The idea that some day the work of mathematicians might be replaced by computers is an old one. Perhaps it is now happening, but in a different way from that which I would have expected. Computers are replacing humans but not because they are doing everything better. There is no doubt there are some things they can do better but I think there are many things which they cannot. The plenary talks at the DMV conference on topics of this kind were partly critical. There occurred examples of a type I had not encountered before. A computer is presented with a picture of a pig and recognizes it as a pig. Then the picture is changed in a very specific way. The change is quantitatively small and is hardly noticeable to the human eye. The computer identifies the modified picture as an aeroplane. In another similar example the starting picture is easily recognizable as a somewhat irregular seven and is recognized by the computer as such. After modification the computer recognizes it as an eight. This seems to provide a huge potential for mistakes and wonderful opportunities for criminals. I feel that the trend to machine learning and related topics in mathematics is driven by fashion. It reminds me a little of the ‘successes’ of string theory in physics some years ago. Another aspect of the plenary talks at these conferences I did not like was that the speakers seemed to be showing off with how much they had done instead of presenting something simple and fascinating. At the conference in Heidelberg there were three talks by young prizewinners which were shorter than the plenaries. I found that they were on average of better quality and I know that I was not the only one who was of that opinion.

In the end there were not many talks at these conferences I liked much but let me now mention some that I did. Amber Smith gave a talk on the behaviour of the immune system in situations where bacterial infections of the lung arise during influenza. In that talk I really enjoyed how connections were made all the way from simple mathematical models to insights for clinical practise. This is mathematical biology of the kind I love. In a similar vein Stanca Ciupe gave a talk about aspects of COVID-19 beyond those which are common knowledge. In particular she discussed experiments on hamsters which can be used to study the infectiousness of droplets in the air. A talk of Harsh Chhajer gave me a new perspective on the intracellular machinery for virus production used by hepatitis C, which is of relevance to my research. I saw this as something which is special for HCV and what I learned is that it is a feature of many positive strand RNA viruses. I obtained another useful insight on in-host models for virus dynamics from a talk of James Watmough.

Returning to the issue of mathematics and computers another aspect I want to mention is arXiv. For many years I have put copies of all my papers in preprint form on that online archive and I have monitored the parts of it which are relevant for my research interests for papers by other people. When I was working on gravitational physics it was gr-qc and since I have been working on mathematical biology it has been q-bio (quantitative biology) which I saw as the natural place for papers in that area. q-bio stands for ‘quantitative biology’ and I interpreted the word ‘quantitative’ as relating to mathematics. Now the nature of the papers on that archive has changed and it is also dominated by topics strongly related to computers such as machine learning. I no longer feel at home there. (To be fair I should say there are still quite a lot of papers there which are on stochastic topics which are mathematics in the classical sense, just in a part of mathematics which is not my speciality.) In the past I often cross-listed my papers to dynamical systems and maybe I should exchange the roles of these two in future – post to dynamical systems and cross-list to q-bio. If I succeed in moving further towards biology in my research, which I would like to I might consider sending things to bioRxiv instead of arXiv.

In this post I have written a lot which is negative. I feel the danger of falling into the role of a ‘grumpy old man’. Nevertheless I think it is good that I have done so. Talking openly about what you are unsatisfied with is a good starting point for going out and starting in new positive directions.