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 van den 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 there are $m<n$ 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

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

and

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

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

• Ma, Zhien; Li, Jia (2009). Dynamical Modeling and analysis of Epidemics. World Scientific. ISBN 978-981-279-749-0. OCLC 225820441.
• Diekmann, O.; Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Disease. John Wiley & Son.
• Hefferenan, J. M.; Smith, R. J.; Wahl, L. M. (2005). "Prospective on the basic reproductive ratio". J. R. Soc. Interface.