The paper discusses the construction of next-generation matrices (NGMs) for compartmental epidemic models, which are essential for calculating the basic reproduction number $ R_0 $. $ R_0 $ is a key quantity in infectious disease epidemiology, representing the average number of new infections caused by one infected individual in a fully susceptible population. The NGM is a matrix that captures the transmission and transition dynamics of the model, and its dominant eigenvalue gives $ R_0 $. The paper clarifies the construction of the NGM, showing that two related matrices exist: the NGM with large domain ($ K_L $) and the NGM with small domain ($ K_S $). These matrices are derived from the model's transmission ($ T $) and transition ($ \Sigma $) matrices. $ K_L $ is obtained by $ -T\Sigma^{-1} $, while $ K_S $ is derived by reducing the dimensionality when the determinant of $ K $ is zero. Both matrices share the same non-zero eigenvalues, with the largest being $ R_0 $. The paper emphasizes the importance of using epidemiological reasoning to construct the NGM, rather than purely linear algebraic methods. It provides examples, including an SEI model with two latent categories and a sexually transmitted infection model, to illustrate the process. The examples show how to identify states-at-infection and construct the NGM accordingly. The paper also demonstrates that $ R_0 $ is connected to the Malthusian parameter $ r $, with $ R_0 > 1 $ if and only if $ r > 0 $, and $ R_0 = 1 $ if and only if $ r = 0 $. The paper concludes that the NGM provides a rigorous and biological way to calculate $ R_0 $, which is crucial for understanding the potential for initial spread of an infectious disease. The methods described allow for the construction of the NGM from the model's specifications, making it easier to compute $ R_0 $ and understand the dynamics of infectious diseases.The paper discusses the construction of next-generation matrices (NGMs) for compartmental epidemic models, which are essential for calculating the basic reproduction number $ R_0 $. $ R_0 $ is a key quantity in infectious disease epidemiology, representing the average number of new infections caused by one infected individual in a fully susceptible population. The NGM is a matrix that captures the transmission and transition dynamics of the model, and its dominant eigenvalue gives $ R_0 $. The paper clarifies the construction of the NGM, showing that two related matrices exist: the NGM with large domain ($ K_L $) and the NGM with small domain ($ K_S $). These matrices are derived from the model's transmission ($ T $) and transition ($ \Sigma $) matrices. $ K_L $ is obtained by $ -T\Sigma^{-1} $, while $ K_S $ is derived by reducing the dimensionality when the determinant of $ K $ is zero. Both matrices share the same non-zero eigenvalues, with the largest being $ R_0 $. The paper emphasizes the importance of using epidemiological reasoning to construct the NGM, rather than purely linear algebraic methods. It provides examples, including an SEI model with two latent categories and a sexually transmitted infection model, to illustrate the process. The examples show how to identify states-at-infection and construct the NGM accordingly. The paper also demonstrates that $ R_0 $ is connected to the Malthusian parameter $ r $, with $ R_0 > 1 $ if and only if $ r > 0 $, and $ R_0 = 1 $ if and only if $ r = 0 $. The paper concludes that the NGM provides a rigorous and biological way to calculate $ R_0 $, which is crucial for understanding the potential for initial spread of an infectious disease. The methods described allow for the construction of the NGM from the model's specifications, making it easier to compute $ R_0 $ and understand the dynamics of infectious diseases.