The construction of next-generation matrices for compartmental epidemic models

Abstract

The basic reproduction number R 0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of R 0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together ref lect the range of possibilities encountered in the literature for the characterization of R 0 . We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number R 0 . Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that R 0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that R 0 . 1 if and only if r . 0, and R 0 ¼ 1 if and only if r ¼ 0.

An introduction to compartmental modeling for the budding infectious disease modeler

“…This is straightforward for simple models, but as models increase in complexity they often contain multiple compartments with infectious individuals, making this process difficult. In this section we introduce a commonly used method for finding R 0 : the next generation matrix method (Diekmann, Heesterbeek, & Johan, 1990;Diekmann, Heesterbeek, & Roberts, 2009). Importantly, this method is centered on considering the dynamics near the DFE.…”

Limiting the spread of disease through altered migration patterns

“…Although R 0 is more tractable than τ , it is only useful if the resulting optimal strategy reduces the extent of the original epidemic in some sense. In the context of ODE models, Diekmann et al [10] notes that epidemics with high R 0 do not necessarily have a fast increase of incidence. Therefore, one might question the relevance of reducing R 0 if the threshold cannot be achieved and, if the threshold can be achieved, the advantage of reducing R 0 further.…”

Fungal disease dynamics in insect societies: Optimal killing rates and the ambivalent effect of high social interaction rates

“…Diekmann et al (2010) provide a procedure for calculating the basic reproduction number for any compartmental model as the spectral radius of the "next-generation matrix" K. Following their procedure to construct K shows that its spectral radius is…”