Next-generation matrix

The next generation matrix is a method used to derive the basic reproduction number, for a compartmental model. This method is given by Diekmann et al (1990) and Driessche and Watmough (2002). To calculate the basic reproduction number by using next generation matrix, the whole population is divided into $n$ compartments in which $m<n$ , infected compartments. Let $x_{i},i=1,2,3.........m$ be the numbers of infected individulas 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 appearence 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),&..........,F_{n}(x)\end{pmatrix}}^{T}

and

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

Let

x_{0}

be the disease free equilibrium. The value 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 next generation matrix. The largest eigenvalue or spectral radius of $FV^{-1}$ is the basic reproduction number of the model.

References

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