## Archive for the ‘public health’ Category

### The basic reproduction number for infectious diseases, part 2

July 9, 2020

Here I continue the discussion of the previous post. There I mentioned that when defining a basic reproduction number it is necessary, in addition to defining a set of ODE, to make a choice of infected and uninfected compartments. This means partitioning the unknowns into two subsets. In addition it is necessary to distinguish processes regarded as new infections from others. This means splitting the right hand side of the equations into a sum of two terms. There are also some other biologically motivated assumptions on the right hand side. As indicated in the previous post, a reproduction number is a feature of a disease-free steady state of the model (i.e. a steady state where the unknowns belonging to the infected compartment are zero). In fact it is a feature of the linearization of the model about that point. Intuitively, it has to do with a situation where a population contains a very small number of infected individuals. The matrix defining the right hand side of the linearization is a sum of two terms, each of which is partitioned in a certain way. We now consider a situation where the disease-free steady state is perturbed by introducing a small number of individuals into each of the infected compartments while supposing that they cannot cause a significant number of secondary infections. The numbers of individuals in the infected compartments then satisfy a linear system. Under the given assumptions all eigenvalues of the matrix defining this system have negative real parts. Thus the solution tends to zero exponentially. We are interested in the average number of individuals in these compartments over all positive times. To understand this consider first the simpler example of a single compartment with an exponential behaviour $x(t)=x_0e^{-at}$. The average value of $x$ is $\int_0^\infty x_0e^{-at}dt=\left[-a^{-1}x_0e^{-at}\right]_0^\infty=x_0a^{-1}$. In fact what we are interested in is the number of infections expected from the individuals in the infectious compartment. We have the equation $\psi'(t)=-V\psi (t)$, where all eigenvalues of $V$ have positive real parts and we want to calculate $F\int_0^\infty e^{-Vt}dt=FV^{-1}\psi (0)$, where $F$ describes the process of new infections. The matrix $FV^{-1}$ is the next generation matrix mentioned in the previous post. It is now a matter of some rather involved linear algebra to show that the modulus of the largest eigenvalue of this matrix is greater than one if and only if the greatest real part of an eigenvalue of the linearization of the system of ODE is greater than zero.

In the current reporting of epidemiological data the quantity $R$ is time-dependent. I am far away from being able to link this with the above discussion. I looked at the web page of the Robert Koch Institute to see what they say about how they produce their curves. This led me to a paper of Cintron-Arias et al. (Math. Biosci. Eng. 6, 261), which might be able to provide a gateway into this but it is clear that it involves many things which I do not understand at present. Perhaps I should first look at the classic book ‘Infectious Diseases of Humans’ by Anderson and May. Robert May died in April 2020 so that he did not have the opportunity to observe the present pandemic.

### The basic reproduction number for infectious diseases

June 12, 2020

These days reproduction numbers for epidemiology are prominent in the popular media. Many people are familiar with the idea that stopping a resurgence of COVID-19 infections in a region has to do with making and keeping something called $R$ less than one. They may also be familiar with the informal definition of $R$ that it is the number of new infections caused by an infected individual. But how is $R$ (or $R_0$ as it is more commonly called by scientists) defined? A mathematician might not expect to find an answer in the media but it might be reasonable to expect one in the scientific literature on epidemiology. In the past I have been frustrated by the extent to which this fails to be the case. What is typically given is a description in words which I never found possible to convert into a precise mathematical account, despite considerable effort. Now, in the context of a project on hepatis C which I have been working on with colleagues from Cameroon, my attention was drawn to a paper of van den Driessche and Watmough (Math. Biosci. 180, 29) which contains some answers of the type I was looking for. I was vaguely aware of this paper before but I had never seriously tried to read it because I did not realise its nature.

The context in which I would have liked to find answers is that of models given by systems of ordinary differential equations where the unknowns are the numbers of individuals in different categories (susceptible, infected, recovered etc.) as functions of time. How the numbers reported in the media are calculated (on the basis of discrete data) is something I have not yet tried to find out. At the moment I would like an answer in the context which bothered me in the past and this is the context treated in the paper mentioned above. A typical situation is that found in the basic model of virus dynamics, a system of three ODE describing the dynamics of a virus within a host, with the unknowns being uninfected cells, infected cells and virions. There is a quantity $R_0$ which can be expressed in terms of the coefficients of the system. If $R_0\le 1$ then the only non-negative steady state is virus-free. This is the uninfected state and it is globally asymptotically stable. If $R_0>1$ there is an uninfected state which is unstable and an infected state which is positive and globally asymptotically stable. This kind of situation is not unique to this example and similar things are seen in many models of infection. There is a reproductive number (or perhaps more than one) which defines a threshold between different types of late-time behaviour.

It is not obvious that the analysis of van den Driessche and Watmough applies to models of in-host dynamics of a pathogen since it is necessary to make a choice of infected and uninfected compartments which is related to the biological interpretation of the variables and not just to the mathematical structure of the model. Their analysis does apply to the basic model of virus dynamics if the infected compartments are chosen to be the infected cells and virions and the reaction fluxes are partitioned in a suitable way. The simple picture of the significance of the reproductive number given above does not always hold. There is also another scenario which can occur and does so in many practical examples and involves the notion of a backward bifurcation. It goes as follows. For $R_0$ sufficiently small the disease-free steady state is globally asymptotically stable but as $R_0$ is increased this property breaks down before $R_0=1$ is reached. A fold bifurcation occurs which creates a stable and an unstable positive steady state. The unstable steady state moves so as to meet the disease-free state when $R_0=1$. For $R_0>1$ there are exactly two steady states and the positive one is globally asymptotically stable. There is bifurcation for $R_0=1$ but it has a different structure from that in the classical scenario (which is a transcritical bifurcation). It bears some resemblance to a sub-critical Hopf bifurcation.

The most useful insights I got from reading and thinking about the paper of van den Driessche and Watmough are as follows. The primary significance of $R_0$ concerns the disease-free steady state and its stability. The fact that it can sometimes characterise the stability of an infected steady state is a kind of bonus which does not apply to all models. What it can do is to provide information about the stability of a positive steady state in a regime where it is close to the bifurcation point where it separates from the disease-free steady state. This circumstance is analysed in the paper using centre manifold theory. The significance for the stability of a steady state which is far away is a weak one. Continuity arguments can be used to propagate information about stability through parameter space but only as long as no bifurcations happen. When this is the case depends on the details of the particular example being considered. What is the definition of $R_0$ given in the paper? It is the largest modulus of an eigenvalue of a certain matrix (the next generation matrix) constructed from the linearization of the system about the disease-free steady state, whereby the construction of this matrix incorporates information about the biological meaning of the variables. Consider the example of the basic model of virus dynamics with the choices of infected and uninfected compartments as above. There is more than way of partitioning the reaction fluxes. I first tried to put both the production of infected cells and the production of virions into the category of fluxes called $\cal F$ in the paper. Applying the definition of the reproduction number given there leads to $\sqrt{R_0}$, where $R_0$ is the reproduction number usually quoted for this model. If instead only the production of infected cells is put into the category $\cal F$ then the general definition gives the conventional answer $R_0$. The two quantities defining the threshold are different but the definition of being above or below the threshold are the same. ($\sqrt{x}<1\Leftrightarrow x>1$). That this kind of phenomenon can occur is shown by example in the paper.

### Herd immunity

February 14, 2019

I have a long term interest in examples where mathematics has contributed to medicine. Last week I heard a talk at a meeting of the Mainzer Medizinische Gesellschaft about vaccination. The speaker was Markus Knuf, director of the pediatric section of the Helios Clinic in Wiesbaden. In the course of his talk he mentioned the concept of ‘herd immunity’ several times. I was familiar with this concept and I have even mentioned it briefly in some of my lectures and seminars. It never occurred to me that in fact this is an excellent example of a case where medical understanding has benefited from mathematical considerations. Suppose we have a population of individuals who are susceptible to a particular disease. Suppose further that there is an endemic state, i.e. that the disease persists in the population at a constant non-zero level. It is immediately plausible that if we vaccinate a certain proportion $\alpha$ of the population against the disease then the proportion of the population suffering from the disease will be lower than it would have been without vaccination. What is less obvious is that if $\alpha$ exceeds a certain threshold $\alpha_*$ then the constant level will be zero. This is the phenomenon of herd immunity. The value of $\alpha_*$ depends on how infectious the disease is. A well-known example with a relatively high value is measles, where $\alpha$ is about $0.95$. In other words, if you want to get rid of measles from a population then it is necessary to vaccinate at least 95% of the population. It occurs to me that this idea is very schematic since measles does not occur as a constant rate. Instead it occurs in large waves. This idea is nevertheless one which is useful when making public health decisions. Perhaps a better way of looking at it is to think of the endemic state as a steady state of a dynamical model. The important thing is that this state is asymptotically stable in the dynamic set-up so that it recovers from any perturbation (infected individuals coming in from somewhere else). It just so happens that in the usual mathematical models for this type of phenomenon whenever a positive steady state (i.e. one where all populations are positive) exists it is asymptotically stable. Thus the distinction between the steady state and dynamical pictures is not so important. After I started writing this post I came across another post on the same subject by Mark Chu-Carroll. I am not sad that he beat me to it. The two posts give different approaches to the same subject and I think it is good if this topic gets as much publicity as possible.

Coming back to the talk I mentioned, a valuable aspect of it was that the speaker could report on things from his everyday experience in the clinic. This makes things much more immediate than if someone is talking about the subject on an abstract level. Let me give an example. He showed a video of a small boy with an extremely persistent cough. (He had permission from the child’s parents to use this video for the talk.) The birth was a bit premature but the boy left the hospital two weeks later in good health. A few weeks after that he returned with the cough. It turned out that he had whooping cough which he had caught from an adult (non-vaccinated) relative. The man had had a bad cough but the cause was not realised and it was attributed to side effects of a drug he was taking for a heart problem. The doctors did everything to save the boy’s life but the infection soon proved fatal. It is important to realize that this is not an absolutely exceptional case but a scenario which happens regularly. It brings home what getting vaccinated (or failing to do so) really means. Of course an example like this has no statistical significance but it can nevertheless help to make people think.

Let me say some more about the mathematics of this situation. A common type of model is the SIR model. The dependent variables are $S$, the number of individuals who are susceptible to infection by the disease, $I$, the number of individuals who are infected (or infectious, this model ignores the incubation time) and $R$, the number of individuals who have recovered from the disease and are therefore immune. These three quantities depend on time and satisfy a system of ODE containing a number of parameters. There is a certain combination of these parameters, usually called the basic reproductive rate (or ratio) and denoted by $R_0$ whose value determines the outcome of the dynamics. If $R_0\le 1$ the infection dies out – the solution converges to a steady state on the boundary of the state space where $I=0$. If, on the other hand, $R_0>1$ there exists a positive steady state, an endemic equilibrium. The stability this of this steady state can be examined by linearizing about it. In fact it is always stable. Interestingly, more is true. When the endemic steady state exists it is globally asymptotically stable. In other words all solutions with positive initial data converge to that steady state at late time. For a proof of this see a paper by Korobeinikov and Wake (Appl. Math. Lett. 15, 955). They use a Lyapunov function to do so. At this point it is appropriate to mention that my understanding of these things has been improved by the hard work of Irena Vogel, who recently wrote her MSc thesis on the subject of Lyapunov functions in population models under my supervision.

### ECMTB 2018 in Lisbon

July 24, 2018

I am attending a meeting of the ESMTB in Lisbon. I am happy that the temperatures are very moderate, much more moderate than those at home in Mainz. The highest temperatures I encountered so far were due to body heat, for the following reason. The first two sessions on reaction networks were in rooms much too small for the number of participants with people sitting on extra chairs brought in from outside, windowsills and the floor, as available. This caused a rise in temperature despite the open windows. In any case, the size of those audiences is a good sign for the field.

Another plenary talk, by Eörs Szathmáry, was about relationships between learning processes in the brain and evolution. I only understood a limited amount of the content but I nevertheless enjoyed the presentation, which combined a wide variety of different ideas. One specific point I did pick up concerns the Darwin finches. The speaker explained that the molecular (genetic) basis of the different beak shapes of these birds is now understood. This evolution takes place in a three-parameter space and it was indicated how this kind of space can be established. A similar process has been modelled in the case of RNA by Uri Alon. In the talk there was a nice partially implicit joke. One of the speaker’s main collaborators in this work is called Watson and he hinted at the idea of himself as Sherlock Holmes. Apart from the content I found the delivery of the talk very accomplished. I found it good in a way I cannot make precise.

### Herbal medicine and its dangers

December 18, 2016

I recently heard a talk by Thomas Efferth of the Institute for Pharmacology of the University of Mainz on herbal medicine. There is a common point of view that substances derived from plants are harmless and good while the chemical drugs of standard medicine are evil. The speaker emphasized that plants have good reasons for not being good to those who eat them. They do not have immune systems of the type we do and they cannot run away and so it is natural that they use poisons to defend themselves. Herbal medicines are effective in some cases but they need to be subject to controls as much as do substances obtained by artificial chemical means. In the talk a number of examples of the dangers of ‘natural’ medicines were presented and I will write about some of them here.

The first example is that of Aristolochia. This a large genus of plants, some of which are poisonous. One of these, Aristolochia clematitis, has been extensively used in herbal medicine. It was used extensively in the west in ancient times and is used in traditional Chinese medicine until today. In the talk the story was told of an incident which happened in Belgium. There was a product sold as a means of losing weight which contained a Chinese plant. It sold so well that the manufacturer’s supplies of the plant were running out. When more was ordered a fateful mistake took place. There are two plants which have the same name in China. The one is that which was originally contained in the weight-loss product. The other is the poisonous Aristolochia fangchi and it was the one which was delivered. This led to more than 100 cases of kidney failure in the people using the product. Another way in which plants can be dangerous is as weeds in crop fields. In the Balkans contamination of grain with Aristolochia clematitis led to a kidney disease called Balkan nephropathy, with 35000 recorded cases. The substance, aristolochic acid, which is responsible for the kidney toxicity is also known to be a strong carcinogen. Interestingly, this substance is not poisonous for everyone and its bad effects depend a lot on the variability in liver enzymes among individuals.

A class of substances used by many plants to protect themselves against insects are the pyrrolizidine alkaloids. These substances are hepatotoxic and carcinogenic. They may move through the food chain being found, for instance, in honey. It has been noted that there may be risks associated to the amount of these substances contained in medicinal herbs used both in the West and in China. It was mentioned in the talk that drinking too much of certain types of herbal tea may be damaging to health. The problem is usually not the plants that are the main components of the teas but other plants which may be harvested with them in small quantities. There is at least one exception to this, namely coltsfoot (Tussilago farfara). In one case the death of an infant due to liver disease is believed to be due to the mother drinking this type of tea during pregnancy. After that the sale of coltsfoot was banned in Germany.

There were some remarks in the talk on heavy metals which I found quite suprising. One concerned ayurvedic medicine which has an aura of being gentle and harmless. In fact in many of these substances certain heavy metals are added delibrately (lead, mercury and arsenic). According to Wikipedia more than 80 cases of lead poisoning due to ayurvedic ‘medicines’ have been recorded. Another remark was that there can be significant concentrations of heavy metals in tobacco smoke. The negative health effects of smoking are sufficiently well known but this aspect was new to me.

Another theme in the talk was interactions between herbal medicines and normal drugs. Apparently it is often the case that patients who use herbal remedies are afraid to mention this to their doctors since they think this may spoil the relationship to their practitioner. Then it can happen that a doctor is suprised by the fact that a drug he prescribes is not working as expected. Little does he know that the patient is secretly taking a ‘natural’ drug in parallel. An example is St. John’s wort which is sometimes taken as a remedy for depression. It may work and it has no direct negative effects but it can be problematic because it reduces the effects of other drugs taken at the same time, e.g. the contraceptive pill. It changes the activity of liver enzymes and causes them to eliminate other drugs from the body faster than would normally happen, thus causing an effective reduction of the dose.

We are surrounded by poisonous plants. I was always sceptical of the positive effects of ‘natural’, plant-derived medicines. Now I have realised how seriously the dangers of these substances should be taken.

### Hepatitis C

May 29, 2016

I once previously wrote something about hepatitis C in this blog which was directed to the mathematical modelling aspects. Here I want to write about the disease itself. This has been stimulated by talks I heard at a meeting of the Mainzer Medizinische Gesellschaft. The speakers were Ralf Bartenschlager from Heidelberg and and Stefan Zeuzem from Frankfurt. The first speaker is a molecular biologist who has made important contributions to the understanding of the structure and life cycle of the virus. For this work he got the 2015 Robert Koch prize together with Charles Rice from the Rockefeller University. The second speaker is a clinician.

Hepatitis C is transmitted by blood to blood contact. According to Zeuzem the main cause of the spread of this disease in developed countries is intravenous drug use. Before there was a test for the disease it was also spread via blood transfusions. (At one time the risk of infection with hepatitis due to a blood transfusion was 30%. This was mainly hepatitis B and by the time of discovery of hepatitis C, when the risk from hepatitis B had essentially been eliminated, it had dropped to 5%.) He also mentioned that there is a very high rate of infection in certain parts of Egypt due to the use of unsterilized needles in the treatment of other diseases. Someone asked how the disease could have survived before there were injections. He did not give a definitive answer but he did mention that while heterosexual contacts generally carry little risk of infection with this virus homosexual contacts between men do carry a significant risk. The disease typically becomes chronic and has few if any symptoms for many years. It does have dramatic long-term effects, namely cirrhosis and cancer of the liver. He showed statistics illustrating how public health policies have influenced the spread of the disease in different countries. The development in France has been much more favourable (with less cases) than in Germany, apparently due to a publicity campaign as a result of political motives with no direct relevance to the disease. The development in the UK has been much less favourable than it has even in Germany due an almost complete lack of publicity on the theme for a long time. The estimated number of people infected in Germany is 500000. The global number is estimated as 170 million.

There has been a dramatic improvement in the treatment of hepatitis C in the past couple of years and this was the central theme of the talks. A few years ago the situation was as follows. Drugs (a combination of ribavirin and interferon $\alpha$) could be used to eliminate the virus in a significant percentage of patients, particularly for some of the sub-types of the virus. The treatment lasted about a year and was accompanied by side effects that were so severe that there was a serious risk of patients breaking it off. Now the treatment only lasts a few weeks, it cures at least 95% of the patients and in many situations 99% of them. The side effects of the new treatments are moderate. There is just one problem remaining: the drugs for the best available treatment are sold for extremely high prices. The order of magnitude is 100000 euros for a treatment. Zeuzem explained various aspects of the dynamics which has led to these prices and the circumstances under which they might be reduced in the future. In general this gave a rather depressing picture of the politics of health care relating to the approval and prescription of new drugs.

Let me get back to the scientific aspects of the theme, as explained by Bartenschlager. A obvious question to ask is: if hepatitis C can essentially be cured why does HIV remain essentially incurable despite the huge amount of effort and money spent on trying to find a treatment? The simple answer seems to be that HIV can hide while HCV cannot. Both these viruses have an RNA genome. Since the copying of RNA is relatively imprecise they both have a high mutation rate. This leads to a high potential for the development of drug resistance. This problem has nevertheless been overcome for HCV. Virus particles are continually being destroyed by the immune system and for the population to survive new virus particles must be produced in huge numbers. This is done by the liver cells. This heavy burden kills the liver cells after a while but the liver is capable of regenerating, i.e, replacing these cells. The liver has an impressive capability to survive this attack but every system has its limits and eventually, after twenty or thirty years, the long-term effects already mentioned develop. An essential difference between HIV and HCV is that the RNA of HCV can be directly read by ribosomes to produce viral proteins. By contrast, the RNA of HIV is used as a template to produce DNA by the enzyme reverse transcriptase and this DNA is integrated into the DNA of the cell. This integrated DNA (known as the provirus) may remain inactive, not leading to production of protein. As long as this is the case the virus is invisible to the immune system. This is one way the virus can hide. Moreover the cell can divide producing new cells also containing the provirus. There is also another problem. The main target of HIV are the T-helper cells. However the virus can also infect other cells such as macrophages or dendritic cells and the behaviour of the virus in these other cells is different from that in T-helper cells. It is natural that a treatment should be optimized for what happens in the typical host cell and this may be much less effective in the other cell types. This means that the other cells may serve as a reservoir for the virus in situations where the population is under heavy pressure from the immune system or drug treatment. This is a second sense in which the virus can hide.

Some of the recent drugs used to treat HCV are based on ideas developed for the treatment of HIV. For instance a drug of this kind may inhibit certain of the enzymes required for the reproduction of the virus. There is one highly effective drug in the case of HCV which works in a different way. The hepatitis C virus produces one protein which has no enzymatic activity and it is at first sight hard to see what use this could be for the virus. What it in fact does is to act as a kind of docking station which organizes proteins belonging to the cell into a factory for virus production.

The hepatitis C virus is a valuable example which illustrates the relations between various aspects of medical progress: improvement in scientific understanding, exploitation of that information for drug design, political problems encountered in getting an effective drug to the patients who need it. Despite the negative features which have been mentioned it is the subject of a remarkable success story.

### Symposium in Mainz on controversies in biomedicine

October 26, 2014

Last Friday I attended a symposium on controversies in biomedicine at the Academy of Science and Literature in Mainz. There were a number of talks and a round table discussion at the end. The event itself was not the scene of much controversy. It seems that most of the people attending had a positive attitude to biomedical research. At least there was not much sign of the contrary in the questions after the talks. I thought that an event like this might have attracted more participants with a critical view of the subject but that does not seem to have been the case. The one vein of controversial discussion was between some journalists who had been invited for the round table (from Bavarian TV, Süddeutsche Zeitung and Frankfurter Allgemeine Zeitung) and the majority of the participants who were presumably scientific researchers in one form or another. The journalists expressed the opinion that scientists did not take part actively enough in public debates and the scientists suggested that journalists often sensationalized scientific subjects of public interest.

The first talk, by Christof von Kalle was about gene therapy. This taught me a number of things concerning this subject which I did not previously know much about. One example he discussed was that of X-linked SCID (severe combined immunodeficiency). The first choice of therapy for this fatal condition is a bone marrow transplant but this is dependent on the availability of a suitable donor. In other cases gene therapy was tried. It often cured the SCID but a high proportion of patients later got leukemia. A promoter in the inserted DNA had not only activated the gene it was supposed to but an oncogene as well. My one criticism of the talk is that the speaker packed in much too much information, sometimes flashing slides for just a couple of seconds. The next talk was by Bernhard Fleckenstein on pathogenic viruses. His main theme was biosecurity and biosafety. The first has to do with preventing voluntary misuse (such as bioterrorism) and the second with preventing accidents. I learned that this is still the subject of lively discussion. One amazing story is that there was an attempt to stop a paper written in Holland being published in the US by claiming that it was an export and that therefore an authority in Holland responsible for exports of goods had the right to forbid it. There is no doubt that experimental work on influenza viruses which are both highly virulent and highly infectious could be dangerous. However in my opinion it makes no sense to ban such research or to try to keep the results secret. This is because I think that somebody will do the research anyway, despite bans, and any important results will leak out. I think that the danger is minimized if the research is legal and open rather than illegal and secret. The next talk, by Martin Lohse, was on the necessity of animal experiments. One aspect which came out clearly was the tension between the legislation limiting research on animals and that requiring a certain amount of such research in the form of testing before a new drug can be approved.

Over lunch I had some stimulating conversations with other participants. The first talk after lunch was by Jörg Michaelis on the benefit or otherwise of screening, in particular for cancer. He recounted his own experience in organizing a large study on the use of screening of small children for neuroblastoma, with negative results. He then surveyed what is known about the value of various other types of screening. In particular he stated that screening for skin cancer, as paid for by the German public health service is not justified by any scientific evidence. Nobody in the audience contradicted this. The last talk of the day, by Uwe Sonnewald was about green genetic engineering. Among other things he presented statistics on the huge difference in the level of the use of these techniques in the Americas and in Europe, particularly Germany. If the bar representing Germany had not been a different colour it would have been invisible. The meeting ended with the round table. I just want to mention one point which arose there. In a recent post I mentioned the negative attitude to science and technology, particular in the area of biomedicine, which I notice in Germany. (This was one motivation for me to attend the event I am writing about here, with the idea of collecting arguments in support of science.) Of course this was a recurrent theme in the symposium in one form or another, particularly during the round table. An idea which appeared repeatedly, implicitly or explicitly, during the day was that the troubled relationship of the Germans to this subject could have to do with thinking of the abuses carried out by certain German doctors during the time that the Nazis were in power. This is maybe an obvious point but in the discussion someone (I think it was Christof Niehrs) introduced another idea, one which was new to me. He asked if it was possible that this troubled relationship perhaps goes back much further, namely to the period of romanticism when there was a reaction in Germany against the rationalism of the Enlightenment. I found this symposium very informative and it provided me with a lot of material which I can use in the future in discussions on this type of subject.