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 x_i, 1\le i\le n. The first m of these have the status of infected variables. The right hand side of the evolution equation for x_i is the sum of three terms {\cal F}_i, {\cal V}_i^+ and -{\cal V}_i^- where {\cal F}_i, {\cal V}_i^+ and {\cal V}_i^- 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 {\cal V}_i^- 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 {\cal V}_i={\cal V}_i^--{\cal V}_i^+ is also used. As is usual in reaction networks {\cal V}_i^-=0 when x_i=0. This is condition (A2) of the paper. The {\cal F}_i represent new infections and hence must be zero for i>m. 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 \{y,v\}. 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 {\cal F}_i and {\cal V}_i 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 -V in the paper. We get an exponential phase of decay with populations proportional to e^{-tV}. 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 FV^{-1}, 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 R_0 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 R_0 … 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.


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