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. It is also used in multi-type branching models for analogous computations. The method to compute the basic reproduction ratio using the next-generation matrix 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 {\displaystyle n} compartments in which there are m < n {\displaystyle m<n} infected compartments. Let x i , i = 1 , 2 , 3 , … , m {\displaystyle x_{i},i=1,2,3,\ldots ,m} be the numbers of infected individuals in the i t h {\displaystyle i^{th}} infected compartment at time t. Now, the epidemic model is
d x i d t = F i ( x ) − V i ( x ) {\displaystyle {\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)} , where V i ( x ) = [ V i − ( x ) − V i + ( x ) ] {\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}
In the above equations, F i ( x ) {\displaystyle F_{i}(x)} represents the rate of appearance of new infections in compartment i {\displaystyle i} . V i + {\displaystyle V_{i}^{+}} represents the rate of transfer of individuals into compartment i {\displaystyle i} by all other means, and V i − ( x ) {\displaystyle V_{i}^{-}(x)} represents the rate of transfer of individuals out of compartment i {\displaystyle i} . The above model can also be written as
d x d t = F ( x ) − V ( x ) {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=F(x)-V(x)}
where
F ( x ) = ( F 1 ( x ) , F 2 ( x ) , … , F m ( x ) ) T {\displaystyle F(x)={\begin{pmatrix}F_{1}(x),&F_{2}(x),&\ldots ,&F_{m}(x)\end{pmatrix}}^{T}}
and
V ( x ) = ( V 1 ( x ) , V 2 ( x ) , … , V m ( x ) ) T . {\displaystyle V(x)={\begin{pmatrix}V_{1}(x),&V_{2}(x),&\ldots ,&V_{m}(x)\end{pmatrix}}^{T}.}
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