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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. and van den Driessche and Watmough. 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 In the above equations, F i {\displaystyle F_{i}} 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 − {\displaystyle V_{i}^{-}} represents the rate of transfer of individuals out of compartment i {\displaystyle i}.The above model can also be written as where