If the population is classified into $\mathbf{S}$, $\mathbf{E}$, $\mathbf{I}$ and $\mathbf{R}$ compartments such that \begin{equation} \label{eq4} \begin{aligned} \mathbf{S} &=\dfrac{S_{1}N_{1}+S_{2}N_{2}+ \dotsb + S_{k}N_{k}}{N_{1}+N_{2}+\dots +N_{k}} = \dfrac{1}{N} \sum_{n=1}^{k} S_{n} N_{n} \nonumber \\ \mathbf{E} &=\dfrac{E_{1}N_{1}+E_{2}N_{2}+ \dotsb + E_{k}N_{k}}{N_{1}+N_{2}+\dots +N_{k}} = \dfrac{1}{N} \sum_{n=1}^{k} E_{n} N_{n} \nonumber \\ \mathbf{I} &=\dfrac{I_{1}N_{1}+I_{2}N_{2}+ \dotsb + I_{k}N_{k}}{N_{1}+N_{2}+\dotsb +N_{k}} = \dfrac{1}{N} \sum_{n=1}^{k} I_{n} N_{n} \nonumber \\ \mathbf{R} &=\dfrac{R_{1}N_{1}+R_{2}N_{2}+ \dotsb + R_{k}N_{k}}{N_{1}+N_{2}+\dotsb +N_{k}} = \dfrac{1}{N} \sum_{n=1}^{k} R_{n} N_{n} . \end{aligned} \end{equation} The total normalized population of a region (state) or country is given by \begin{equation} \label{eq5} \mathbf{S}+\mathbf{E}+\mathbf{I}+\mathbf{R}=1 \end{equation} The dynamics of SEIR epidemic spreading model that describe the transmission of disease in a population without demography in one region is given by the following system of differential equations : \begin{align} \label{eq6} \dfrac{d\mathbf{S}}{dt}&= - \dfrac{1}{N} \sum_{n=1}^{k} \left[ \mathbb{S} \mathbb{K}_{\kappa \lambda} \right]_{n} N_{n} \mathbf{I} \nonumber \\ \dfrac{d\mathbf{E}}{dt}&=\dfrac{1}{N} \sum_{n=1}^{k} \left[ \mathbb{S} \mathbb{K}_{\kappa \lambda} \right]_{n} N_{n} \mathbf{I} - \frac{1}{T_{inc.}} \mathbf{E} \nonumber \\ \dfrac{d \mathbf{I}}{dt}&= \frac{1}{T_{inc.}} \mathbf{E} -\frac{1}{T_{inf.}} \mathbf{I} \nonumber \\ \dfrac{d \mathbf{R}}{dt}&= \frac{1}{T_{inf.}} \mathbf{I} \end{align} Where $\mathbb{S}=diag(S_{1},\dots,S_{k})$ and $\mathbb{K}_{\kappa \lambda}$ is $(k\times k)$ mobility infectiousness matrix, with $\mathbf{N}=\mathbf{S}+\mathbf{E}+\mathbf{I}+\mathbf{R}=1$ such that $\mathbf{N}=(N_{1},\dots,N_{k})^T$ and initial condition of dynamics \begin{align}\label{eq7} \mathbf{S_{0}}&=\dfrac{S^0_{1}N_{1} +S^0_{2}N_{2} + \dots S^0_{k}N_{K} }{N} , ~~~` \mathbf{E_{0}}=\dfrac{E^0_{1}N_{1} +E^0_{2}N_{2} + \dots E^0_{k}N_{K} }{N} \nonumber \\ \mathbf{I_{0}}&=\dfrac{I^0_{1}N_{1} +I^0_{2}N_{2} + \dots I^0_{k}N_{K} }{N}, ~~~~ \mathbf{R_{0}}=\dfrac{R^0_{1}N_{1} +R^0_{2}N_{2} + \dots R^0_{k}N_{K} }{N} \end{align}
Since the $\mathbf{S}$, $\mathbf{E}$, $\mathbf{I}$ and $\mathbf{R}$ compartments are scalar by above formula, and the family mobility infectiousness $ \left[ \mathbb{S} \mathbb{K}_{\kappa \lambda} \right]$ is $k\times k$ matrix, I don't know how to determine the next-generation matrix of this dynamics. Please, I need help!