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{{more footnotes|date=June 2013}}

In [[epidemiology]], the '''next-generation matrix''' is a method used to derive the [[basic reproduction number]], for a [[Compartmental models in epidemiology|compartmental model]] of the spread of [[infectious disease]]s. 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 <math> n </math> compartments in which <math> m<n </math>, infected compartments. Let <math> x_i , i=1,2,3,\ldots,m </math> be the numbers of infected individuals in the <math> i ^ {th}</math> infected compartment at time&nbsp;''t''. Now, the [[epidemic model]] is
:<math> \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x)</math>, where  <math> V_i(x)= [V^-_i(x)-V^+_i(x)] </math>
In the above equations, <math> F_i(x)</math> represents the rate of appearance of new infections in compartment <math> i </math>.  <math>V^+_i</math> represents the rate of transfer of individuals into compartment <math> i </math> by all other means, and <math>V^-_i (x)</math> represents the rate of transfer of individuals out of compartment <math> i </math>.
The above model can also be written as

: <math>\frac{\mathrm{d} x_i}{\mathrm{d}t}= F(x)-V(x)</math> 

where

: <math> F(x) = \begin{pmatrix}
     F_1(x), & F_2(x), & \ldots, & F_n(x)
    \end{pmatrix}^T </math>

and

: <math> V(x) = \begin{pmatrix}
     V_1(x), & V_2 (x), & \ldots, & V_n(x)
    \end{pmatrix}^T. </math>

Let <math> x_0 </math> be the disease-free equilibrium. The values of the [[Jacobian matrix|Jacobian matrices]] <math> F(x) </math> and <math> V(x) </math> are:

: <math>DF(x_0) = \begin{pmatrix}    
    F & 0 \\
    0 & 0 
  \end{pmatrix} </math>

and

: <math>
 DV(x_0) = \begin{pmatrix}
     V & 0 \\
     J_3 & J_4  
  \end{pmatrix} </math>
respectively.

Here, <math>F</math> and <math> V </math> are ''m''&nbsp;×&nbsp;''m''  matrices, defined as 
<math> F= \frac{\partial F_i}{\partial x_j}(x_0) </math> and <math> V=\frac{\partial V_i}{\partial x_j}(x_0) </math>.

Now, the matrix <math> FV^{-1}</math> is known as the next-generation matrix. The largest [[eigenvalue]] or [[spectral radius]] of <math> FV^{-1} </math> is the basic reproduction number of the model.

==See also==
*[[Mathematical modelling of infectious disease]]

==References==
{{Reflist}}

==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

[[Category:Matrices]]
[[Category:Epidemiology]]