Next-generation matrix

In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. In population dynamics it is used to compute the basic reproduction number for structured population models.¹ It is also used in multi-type branching models for analogous computations.²

The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann et al. (1990)³ and van den Driessche and Watmough (2002).⁴ To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into ${\displaystyle n}$ compartments in which there are ${\displaystyle m<n}$ infected compartments. Let ${\displaystyle x_{i},i=1,2,3,\ldots ,m}$ be the numbers of infected individuals in the ${\displaystyle i^{th}}$ infected compartment at time t. Now, the epidemic model is

${\displaystyle {\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)}$, 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 Jacobian matrix ${\displaystyle F(x)}$ and ${\displaystyle V(x)}$ are:

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

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 ${\displaystyle V={\frac {\partial V_{i}}{\partial x_{j}}}(x_{0})}$.

Now, the matrix ${\displaystyle FV^{-1}}$ is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of ${\displaystyle FV^{-1}}$ with the largest absolute value (the spectral radius of ${\displaystyle FV^{-1}}$). Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.⁵