The paper by Diekmann, Heesterbeek, and Roberts focuses on the construction of the next-generation matrix (NGM) for compartmental epidemic models, which is crucial for defining and calculating the basic reproduction number ($\mathcal{R}_0$). The authors clarify the confusion surrounding the construction of the NGM, particularly in the context of compartmental models. They present a detailed recipe for constructing the NGM from basic model ingredients and introduce two related matrices: the NGM with large domain and the NGM with small domain. These matrices reflect the range of possibilities encountered in the literature for characterizing $\mathcal{R}_0$. The authors show that these matrices are connected and have the same non-zero eigenvalues, with the largest eigenvalue being $\mathcal{R}_0$. They encourage the construction of the NGM through direct epidemiological reasoning, emphasizing the clear interpretation of the elements of the NGM and the model ingredients. The paper includes examples to illustrate the methods and provides formal recipes for constructing the NGMs, including linear algebra-based approaches and epidemiological reasoning. The authors also prove that $\mathcal{R}_0$ defined as the dominant eigenvalue of the NGM is connected to the Malthusian parameter $r$, with $\mathcal{R}_0 > 1$ if and only if $r > 0$, and $\mathcal{R}_0 = 1$ if and only if $r = 0$.The paper by Diekmann, Heesterbeek, and Roberts focuses on the construction of the next-generation matrix (NGM) for compartmental epidemic models, which is crucial for defining and calculating the basic reproduction number ($\mathcal{R}_0$). The authors clarify the confusion surrounding the construction of the NGM, particularly in the context of compartmental models. They present a detailed recipe for constructing the NGM from basic model ingredients and introduce two related matrices: the NGM with large domain and the NGM with small domain. These matrices reflect the range of possibilities encountered in the literature for characterizing $\mathcal{R}_0$. The authors show that these matrices are connected and have the same non-zero eigenvalues, with the largest eigenvalue being $\mathcal{R}_0$. They encourage the construction of the NGM through direct epidemiological reasoning, emphasizing the clear interpretation of the elements of the NGM and the model ingredients. The paper includes examples to illustrate the methods and provides formal recipes for constructing the NGMs, including linear algebra-based approaches and epidemiological reasoning. The authors also prove that $\mathcal{R}_0$ defined as the dominant eigenvalue of the NGM is connected to the Malthusian parameter $r$, with $\mathcal{R}_0 > 1$ if and only if $r > 0$, and $\mathcal{R}_0 = 1$ if and only if $r = 0$.