Next-generation matrix

In epidemiology, the next-generation matrix is a method used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. This method is given by Diekmann et al. (1990) and Driessche and Watmough (2002). To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into $n$ compartments in which $m<n$, infected compartments. Let $x_{i},i=1,2,3,\ldots ,m$ be the numbers of infected individuals in the $i^{th}$ infected compartment at time t. Now, the epidemic model is

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

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

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

where

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

and

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

Let $x_{0}$ be the disease-free equilibrium. The values of the Jacobian matrices $F(x)$ and $V(x)$ are:

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

and

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

respectively.

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

Now, the matrix $FV^{-1}$ is known as the next-generation matrix. The largest eigenvalue or spectral radius of $FV^{-1}$ is the basic reproduction number of the model.

See also

• Mathematical modelling of infectious disease

Sources

• Zhien Ma and Jia Li, Dynamical Modeling and analysis of Epidemics, World Scientific, 2009.
• O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Disease, John Wiley & Son, 2000.
• J. M. Hefferenan, R. J. Smith and L. M. Wahl, "Prospective on the basic reproductive ratio", J. R. Soc. Interface, 2005