Next-generation matrix

${\displaystyle {\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)}$, where ${\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}$

In the above equations, ${\displaystyle F_{i}(x)}$ represents the rate of appearance of new infections in compartment ${\displaystyle i}$. ${\displaystyle V_{i}^{+}}$ represents the rate of transfer of individuals into compartment ${\displaystyle i}$ by all other means, and ${\displaystyle V_{i}^{-}(x)}$ represents the rate of transfer of individuals out of compartment ${\displaystyle i}$. The above model can also be written as

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=F(x)-V(x)}$

where

${\displaystyle F(x)={\begin{pmatrix}F_{1}(x),&F_{2}(x),&\ldots ,&F_{m}(x)\end{pmatrix}}^{T}}$

and

${\displaystyle V(x)={\begin{pmatrix}V_{1}(x),&V_{2}(x),&\ldots ,&V_{m}(x)\end{pmatrix}}^{T}.}$

Let ${\displaystyle x_{0}}$ be the disease-free equilibrium. The values of the parts of the

${\displaystyle F(x)}$

${\displaystyle V(x)}$

${\displaystyle DF(x_{0})={\begin{pmatrix}F&0\\0&0\end{pmatrix}}}$

and

${\displaystyle DV(x_{0})={\begin{pmatrix}V&0\\J_{3}&J_{4}\end{pmatrix}}}$

respectively.

Here, ${\displaystyle F}$ and ${\displaystyle V}$ are m × m matrices, defined as ${\displaystyle F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})}$ and ${\displaystyle V={\frac {\partial V_{i}}{\partial x_{j}}}(x_{0})}$.