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.

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