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 {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 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 {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 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