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 formula_1 compartments in which there are formula_2 infected compartments. Let formula_3 be the numbers of infected individuals in the formula_4 infected compartment at time "t". Now, the epidemic model is
In the above equations, formula_7 represents the rate of appearance of new infections in compartment formula_8. formula_9 represents the rate of transfer of individuals into compartment formula_8 by all other means, and formula_11 represents the rate of transfer of individuals out of compartment formula_8.
The above model can also be written as
where
and
Let formula_16 be the disease-free equilibrium. The values of the Jacobian matrices formula_17 and formula_18 are:
and
respectively.
Here, formula_21 and formula_22 are "m" × "m" matrices, defined as
formula_23 and formula_24.
Now, the matrix formula_25 is known as the next-generation matrix. The largest eigenvalue or spectral radius of formula_26 is the basic reproduction number of the model.