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. and van den Driessche and Watmough. To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into compartments in which there are infected compartments. Let be the numbers of infected individuals in the infected compartment at time t. Now, the epidemic model is
In the above equations, represents the rate of appearance of new infections in compartment. represents the rate of transfer of individuals into compartment by all other means, and represents the rate of transfer of individuals out of compartment.
The above model can also be written as
where
and
Let be the disease-free equilibrium. The values of the Jacobian matrices and are:
and
respectively.
Here, and are m × m matrices, defined as
and.
Now, the matrix is known as the next-generation matrix. The largest eigenvalue or spectral radius of is the basic reproduction number of the model.