Next-generation matrix – Wikipedia

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.^[1] It is also used in multi-type branching models for analogous computations.^[2]

The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann et al. (1990)^[3] and van den Driessche and Watmough (2002).^[4] To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into

${displaystyle n}$

compartments in which there are

${displaystyle m$

infected compartments. Let

${displaystyle x_{i},i=1,2,3,ldots ,m}$

be the numbers of infected individuals in the

${displaystyle i^{th}}$

infected compartment at time t. Now, the epidemic model is

${displaystyle {frac {mathrm {d} x_{i}}{mathrm {d} t}}=F_{i}(x)-V_{i}(x)}$

, where

${displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}$

In the above equations,

${displaystyle F_{i}(x)}$

represents the rate of appearance of new infections in compartment

${displaystyle i}$

.

${displaystyle V_{i}^{+}}$

represents the rate of transfer of individuals into compartment

${displaystyle i}$

by all other means, and

${displaystyle V_{i}^{-}(x)}$

represents the rate of transfer of individuals out of compartment

${displaystyle i}$

.
The above model can also be written as

${displaystyle {frac {mathrm {d} x}{mathrm {d} t}}=F(x)-V(x)}$

where

${displaystyle F(x)={begin{pmatrix}F_{1}(x),&F_{2}(x),&ldots ,&F_{m}(x)end{pmatrix}}^{T}}$

and

${displaystyle V(x)={begin{pmatrix}V_{1}(x),&V_{2}(x),&ldots ,&V_{m}(x)end{pmatrix}}^{T}.}$

Let

${displaystyle x_{0}}$

be the disease-free equilibrium. The values of the parts of the Jacobian matrix

${displaystyle F(x)}$

and

${displaystyle V(x)}$

are:

${displaystyle DF(x_{0})={begin{pmatrix}F&0\0&0end{pmatrix}}}$

and

${displaystyle DV(x_{0})={begin{pmatrix}V&0\J_{3}&J_{4}end{pmatrix}}}$

respectively.

Here,

${displaystyle F}$

and

${displaystyle V}$

are m × m matrices, defined as

${displaystyle F={frac {partial F_{i}}{partial x_{j}}}(x_{0})}$

and

${displaystyle V={frac {partial V_{i}}{partial x_{j}}}(x_{0})}$

.

Now, the matrix

${displaystyle FV^{-1}}$

is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of

${displaystyle FV^{-1}}$

with the largest absolute value (the spectral radius of

${displaystyle FV^{-1}}$

. Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.^[5]

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