next-generation matrix next-generation matrix 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 compartments in which there are m < n infected compartments. let x i , i = 1 , 2 , 3 , … , m ,i=1,2,3,\ldots ,m} be the numbers of infected individuals in the i t h infected compartment at time t. now, the epidemic model is d x i d t = f i ( x ) − v i ( x ) x_ t}}=f_(x)-v_(x)} , where v i ( x ) = [ v i − ( x ) − v i + ( x ) ] (x)=} in the above equations, f i ( x ) (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 ) ^(x)} represents the rate of transfer of individuals out of compartment i . the above model can also be written as d x d t = f ( x ) − v ( x ) x} t}}=f(x)-v(x)} where f ( x ) = ( f 1 ( x ) , f 2 ( x ) , … , f m ( x ) ) t f_(x),&f_(x),&\ldots ,&f_(x)\end^ and v ( x ) = ( v 1 ( x ) , v 2 ( x ) , … , v m ( x ) ) t . v_(x),&v_(x),&\ldots ,&v_(x)\end^.} let x 0 be the disease-free equilibrium. the values of the parts of the jacobian matrix f ( x ) and v ( x ) are: d f ( x 0 ) = ( f 0 0 0 ) )=f&0\\0&0\end and d v ( x 0 ) = ( v 0 j 3 j 4 ) )=v&0\\j_&j_\end respectively. here, f and v are m × m matrices, defined as f = ∂ f i ∂ x j ( x 0 ) (x_)} and v = ∂ v i ∂ x j ( x 0 ) (x_)} . now, the matrix f v − 1 is known as the next-generation matrix. the basic reproduction number of the model is then given by the eigenvalue of f v − 1 with the largest absolute value (the spectral radius of f v − 1 . next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.