Next Generation Matrix in Epidemiology

• Next Generation Matrix (NGM) is a structured approach that separates infection generation (F) and removal dynamics (V) to quantify epidemic thresholds.
• The methodology uses Jacobian evaluations at the disease-free equilibrium to extract key spectral properties like the basic reproduction number (R₀) and gross reproduction number (Rg).
• NGM extends to Petri net models and multi-strain frameworks, providing insights into stability analysis and extinction probabilities in complex epidemics.

The next generation matrix (NGM) is a central analytical construct in mathematical epidemiology, providing a unified framework for quantifying key epidemic thresholds—including the basic reproduction number (R₀), gross reproduction number (Rg), and the asymptotic fraction of susceptibles (s*)—as well as for characterizing local and global stability properties in compartmental models and, more recently, Petri net representations. The NGM encapsulates the interplay between transmission and transition processes by decomposing model equations into infection-generating and removal dynamics, whose matrix representations are then evaluated at the disease-free equilibrium (DFE) to extract spectral properties that govern epidemic phases and outcomes.

1. Foundational Structure and Definition of the NGM

The next generation matrix arises from a compartmental epidemic model described by a system of ordinary differential equations (ODEs), wherein the population is partitioned into variables representing different disease states. The evolution of the infected compartments can generally be written as:

$i' = F(X) - V(X)$

where $i$ is the vector of infectious variables, $F(X)$ encodes the rate of appearance of new infections (transmission terms), and $V(X)$ accounts for internal transitions or exits from infection (recovery, death, progression, etc.). By evaluating these functions at the DFE and (when necessary) computing the corresponding Jacobians, one obtains two matrices, $F$ and $V$, leading to the definition:

$\text{NGM} = F V^{-1}$

This construction generalizes naturally to multi-strain models and to discrete-event modeling frameworks such as Petri nets (Reckell et al., 3 Jul 2025).

2. Algorithmic Construction and the (F,V) Gradient Decomposition

A recent refinement introduces an algorithmic, universal approach for NGM construction based on minimal user input: the specification of a set of infectious (disease) compartments (Avram et al., 2023). The algorithm proceeds by:

• Calculating the Jacobian of the model's dynamics ($M = \nabla_x[dyn]$) restricted to the infectious equations.
• Identifying positive interaction terms for new infections (forming $F$), setting syntactic negatives to zero in a computer algebra system.
• Defining $V$ as $F - M$.
• Establishing $\text{NGM} = F V^{-1}$.

Mathematica scripts automate this decomposition, including disease-free equilibrium (DFE) computation, rational univariate reduction, and Jacobian factorization. The construction of (F,V) is fundamental not only to NGM computation but also to associated stochastic branching process calculations and extinction probability equations.

3. Epidemic Thresholds: Basic and Gross Reproduction Numbers

The NGM methodology provides explicit formulae for key epidemic thresholds. Depending on the construction of $F$ and $V$, two important thresholds emerge in complex models (Yang, 2020):

• Gross reproduction number (Rg): Defines the initial epidemic potential and is derived from the spectral properties of a diagonally constructed $V$. In two-strain TB models, Rg is given by:

$$Rg = \max \left\{ R_1, R_2, \frac{R_3}{(1-R_1)(1-R_2)} \right\}$$

with $R_1, R_2, R_3$ capturing direct and inter-strain transmission channels and transitions.

• Basic reproduction number ($R_0$): In most single-strain or simplified models, $R_0$ coincides with the spectral radius of $F V^{-1}$ at the DFE. For SIR-type Petri net models, both continuous and discrete-time representations agree:

$R_0 = \rho(F V^{-1}) = \frac{\beta}{\gamma}$

where $\beta$ is the transmission rate and $\gamma$ is the removal rate (Reckell et al., 3 Jul 2025).

4. Fraction of Susceptibles and Final Size Relationships

The NGM also yields the steady-state fraction of the population that remains susceptible after an epidemic, denoted $s^*$. In SIR-like models, the classical relation holds:

$s^* \approx \frac{1}{R_0}$

But in multi-strain or networked epidemic models, $s^*$ results from solving characteristic polynomials derived from the NGM, e.g.:

$$s^* = \frac{[R_{10}(1-R_{21}) + R_{20}(1-R_{11}) + R_{31}] - \sqrt{\Delta}}{2 R_{10} R_{20}}$$

where $\Delta$ is the discriminant (Yang, 2020). In models with a “full” $V$ construction, the inverse of the NGM’s spectral radius directly yields $s^*$:

$s^* = \frac{1}{p}$

with $p$ as the spectral radius.

5. Extinction Probabilities and Fixed Point Equations

A further generalization ties NGM structure to extinction probabilities in corresponding branching process approximations. In multitype branching scenarios, extinction probabilities $q_j$ satisfy equations (e.g., Equation 8 in (Avram et al., 2023)):

$$\prod_{k=1}^J (1 - q_k)^{V_{kj}} f_j = 0, \quad j = 1, ..., J$$

where $f_j$ is related to the column sums of $F$ and $V_{kj}$ is a matrix element of $V$. This results in interlinked conditions for both deterministic thresholds (e.g., $R_0$) and stochastic fade-out probabilities.

6. Extensions to Petri Net Models

The NGM approach has been formalized for Petri nets, where places represent compartments and transitions enact inter-compartmental movement (Reckell et al., 3 Jul 2025). Token flows and arc weights correspond to transmission and transition terms:

• Transmission matrix ($\mathcal{F}$): Aggregates transitions introducing new infections, computed using arc weights from non-infected to infected places and their partial derivatives at the DFE.
• Transition matrix ($\mathcal{V}$): Aggregates removal processes and transitions between infected places, also via arc weights.

The corresponding $F$ and $V$ matrices yield:

$R_0 = \rho(F V^{-1})$

Comparisons across models (SIRS, SEIR, vector-borne models) show perfect algebraic equivalence between Petri net-based and classical ODE-based $R_0$ calculations.

7. Stability Analysis and Spectral Properties

The spectral radius of the NGM ($p$) governs stability at the DFE:

• If $p < 1$, disease-free equilibrium is locally asymptotically stable; epidemic dies out.
• If $p > 1$, endemic equilibrium may arise; epidemic persists.

Global stability analysis further employs the left eigenvector associated with $p$ to weigh compartmental sensitivities and construct suitable Lyapunov functions or employ matrix-theoretical guarantees (Yang, 2020). These eigenvectors dictate directional perturbation effects in the state space.

---

The next generation matrix thus constitutes a core analytic device in epidemic modeling. It bridges deterministic and stochastic model analyses, integrates a broad spectrum of model architectures (ODEs, Petri nets, multi-strain networks), and grounds epidemic phase quantification in rigorous spectral theory. Automated algebraic procedures and computational implementations (principally in Mathematica) ensure generalizability and reproducibility across modeling applications, affirming the NGM's universal significance in quantitative epidemiology.