The basic reproduction number for infectious diseases, part 2

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.

One Response to “The basic reproduction number for infectious diseases, part 2”

1. Paper on hepatitis C | Hydrobates Says:

[…] of my understanding of the concept of the basic reproductive number as discussed here and here. During this project I have learned a lot of new things, mathematical and biological, and I feel […]

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