Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Authors: van den Driessche and Watmough, 2005.


This paper defines the basic reproductive number as the spectral radius of a certain matrix,

$R_0 = \rho (FV^{-1})$.

The matrices $F$ and $V$ emerge in the computation of the Jacobian. To compute the Jacobian, the authors decompose the state equations into two parts, one corresponding to new infections and the other corresponding to 'everything else':

$\displaystyle{x^{\prime}_{i} = \hat{G}_i(x) - \hat{V}_i(x)}$

If the state has the form $x = (x_1, \cdots, x_m, x_{m+1}, \cdots, x_n)$, where only $x_1$ through $x_m$ admit infection, then the Jacobian breaks into blocks:

$\displaystyle{ J \hat{G} = \left( \begin{array}{cc} F + V & 0 \\ J_3 & J_4 \end{array} \right)}$

Although this decomposition is not unique, in many cases it is apparently intuitive, and yields satisfying results in the sense that the corresponding $R_0$ is actually a functional threshhold parameter (i.e. you get one kind of behavior for $R_0 >1$, another for $R_0 < 1$.)

To Do:

1. Sections 1-4 are transparent, but section 5 contains a bunch of stuff having to do with "center manifolds". This is apparently standard fare in dynamical systems--have a look at the relevant text, see if you can make head or tail of it.

2. Starting with the examples, see if you can calculate the corresponding number $R_0$ without looking at the derivations in the paper.

3. Can you get your hands on some data and try estimating this number? If so, it would be interesting to compare the estimates to those obtained from other methodologies.