The Next-Generation Matrix and R₀

The basic reproduction number $R_0$ decides whether a pathogen invades, but for anything more structured than a textbook SIR model — multiple host types, stages, or transmission routes — the naive “ $\beta$ over $\gamma$ ” recipe breaks down. The next-generation matrix gives a rigorous, general way to compute $R_0$ as the dominant eigenvalue of a matrix built from the infection dynamics.

The idea

$R_0$ is the expected number of secondary cases produced by one typical infected individual in a fully susceptible population. In a structured model there are several kinds of infected individuals — say infants and adults, or exposed and infectious classes — and one infected of type $j$ produces a mixture of new infections across all types. We therefore need a matrix that bookkeeps “new infections in group $i$ caused by an infected in group $j$ ,” and $R_0$ becomes a summary of that matrix rather than a single ratio.

Construction

Work with the infected compartments only (for SIR-type models these are the $E$ , $I$ , and similar classes; $S$ and $R$ are excluded). Linearize their dynamics about the disease-free equilibrium (DFE), and split the linearized system into two parts:

$\mathcal{F}$ — the rate of appearance of new infections, and
$\mathcal{V}$ — the rate of all other transitions out of and between infected compartments (recovery, death, progression, waning).

Let $F$ and $V$ be the Jacobian matrices of $\mathcal{F}$ and $\mathcal{V}$ with respect to the infected variables, both evaluated at the DFE. The next-generation matrix is $K = F V^{-1},$ and the basic reproduction number is its spectral radius (largest-magnitude eigenvalue), $R_0 = \rho(K) = \rho\!\left(F V^{-1}\right).$ Here $V^{-1}$ requires the matrix inverse and determinant, and $\rho$ is found from an eigenvalue decomposition.

Interpretation

The pieces have a clean meaning. $V^{-1}$ is the matrix of expected times spent in each infected state: entry $(V^{-1})_{jk}$ is the expected time an individual who starts in state $k$ spends in state $j$ before leaving the infected classes. Multiplying by $F$ , which converts time-in-state into new infections, gives $K_{ij} = \text{expected number of new infections in group } i \text{ produced by one infected introduced into group } j.$ Because the total reproduction over generations is governed by repeated multiplication by $K$ , the long-run per-generation growth factor is the dominant eigenvalue $\rho(K)$ , and the epidemic can invade the DFE if and only if $R_0 > 1$ . This is the stability threshold of the disease-free equilibrium, restated as a spectral radius.

Worked example 1: simple SIR

For the SIR model the only infected compartment is $I$ , with $\frac{dI}{dt} = \beta \frac{S}{N} I - \gamma I.$ New infections appear at rate $\mathcal{F} = \beta \frac{S}{N} I$ and transitions (recovery) remove them at rate $\mathcal{V} = \gamma I$ . At the disease-free equilibrium $S = N$ , so the $1\times 1$ Jacobians are $F = \beta, \qquad V = \gamma.$ Then $K = F V^{-1} = \beta/\gamma$ and, since a $1\times 1$ matrix is its own eigenvalue, $R_0 = \rho(K) = \frac{\beta}{\gamma},$ recovering the familiar result.

Worked example 2: a two-type model

Suppose infection spreads in two host groups (say children and adults) that mix, and an infected in group $j$ transmits to group $i$ at rate $\beta_{ij}$ , while every infected recovers at rate $\gamma$ . Then $F = \begin{bmatrix} \beta_{11} & \beta_{12} \\ \beta_{21} & \beta_{22} \end{bmatrix}, \qquad V = \begin{bmatrix} \gamma & 0 \\ 0 & \gamma \end{bmatrix} = \gamma I_2,$ so $V^{-1} = \frac{1}{\gamma} I_2$ and $K = F V^{-1} = \frac{1}{\gamma}\begin{bmatrix} \beta_{11} & \beta_{12} \\ \beta_{21} & \beta_{22} \end{bmatrix}.$ $R_0$ is the spectral radius of this matrix. Take $\beta_{11} = 2,\ \beta_{12} = 1,\ \beta_{21} = 1,\ \beta_{22} = 1$ (per unit time) and $\gamma = 1$ . The eigenvalues of $K$ solve $\lambda^2 - (\operatorname{tr}K)\lambda + \det K = 0$ , i.e. $\lambda^2 - 3\lambda + 1 = 0$ , giving $\lambda = \frac{3 \pm \sqrt{5}}{2} \approx 2.618,\ 0.382.$ So $R_0 = \rho(K) \approx 2.62$ , larger than any single group’s own $\beta_{ii}/\gamma$ because cross-group transmission amplifies spread.

In code

We build $F$ and $V$ , form $K = FV^{-1}$ , and take the spectral radius from an eigen-decomposition.

R

gamma <- 1
F <- matrix(c(2, 1,
              1, 1), nrow = 2, byrow = TRUE)
V <- gamma * diag(2)

K <- F %*% solve(V)
R0 <- max(abs(eigen(K)$values))
$R0                       # ~2.618 = (3 + sqrt(5)) / 2

Python

import numpy as np

gamma = 1.0
F = np.array([[2., 1.],
              [1., 1.]])
V = gamma * np.eye(2)

K = F @ np.linalg.inv(V)
R0 = max(abs(np.linalg.eigvals(K)))
print(R0)               # ~2.618, the spectral radius of F V^{-1}

2.618033988749895

Julia

using LinearAlgebra

γ = 1.0
F = [2.0 1.0;
     1.0 1.0]
V = γ * I(2)

K = F * inv(Matrix(V))
R0 = maximum(abs.(eigvals(K)))
println(R0)             # ~2.618 = (3 + √5)/2

Why it matters

The next-generation matrix is the standard tool for computing $R_0$ in any structured epidemic model — age groups, spatial patches, vector-borne cycles, or staged infections like SEIR. It turns the vague notion of “average secondary cases” into a precise spectral quantity whose value above or below $1$ decides invasion, and its dominant eigenvector tells you which mix of host types the epidemic will settle into. Because $R_0 = \rho(FV^{-1})$ crosses $1$ exactly when the disease-free equilibrium loses stability, it is also the parameter that drives the transcritical bifurcation between elimination and endemicity.