Infection-Age Structured Epidemic Models

• Infection-age structured epidemic models stratify the infected population by elapsed time since infection, enabling realistic non-exponential infectious period modeling.
• They employ coupled PDEs and renewal equations to accurately capture age-dependent transmission, removal rates, and the impact of targeted interventions.
• Advanced numerical methods, including Galerkin spectral schemes and Erlang approximations, provide robust tools for simulating and optimizing epidemic control strategies.

Infection-age structured epidemic models are a class of mathematical frameworks for infectious disease dynamics in which the infectious population is stratified by the time elapsed since infection ("infection age"). This explicit treatment of infection age enables the incorporation of complex features such as non-exponential infectious periods, age-dependent transmission and removal rates, behavioral or immunological changes over the course of infection, and refined representations of intervention strategies. These models generalize the classical Kermack–McKendrick approach and are formulated as coupled systems of partial differential equations (PDEs), renewal equations, or measure-valued stochastic processes, with boundary conditions reflecting the renewal of infection and nonlocal transmission/intervention mechanisms. Infection-age structure is essential for rigorously capturing phenomena such as multi-wave epidemic cycles, effects of contact tracing, intervention optimization, and detailed within-host/pathogen dynamics.

1. Mathematical Formulation of Infection-Age Structured Models

Infection-age structured models describe the population by epidemiological status (e.g., susceptible S, infected I, removed R) and, for the infectious class, stratify by the duration since infection. Let $i(a, t)$ denote the density of individuals of infection age $a$ at time $t$. The fundamental system, in the homogeneous (well-mixed) case, is a transport–decay PDE coupled to an ODE for susceptibles: $$\frac{\partial}{\partial t}i(a,t) + \frac{\partial}{\partial a}i(a,t) = -\gamma(a)\, i(a,t),$$

$$i(0, t) = S(t) \int_0^\infty \beta(a)\, i(a,t)\, da,$$

$$\frac{d}{dt}S(t) = -S(t) \int_0^\infty \beta(a)\, i(a, t)\, da.$$

Here, $\beta(a)$ is the infection-age dependent transmission kernel, and $\gamma(a)$ the removal (recovery or death) rate. This general framework includes extensions for birth/death (demography), spatial diffusion, network and trait heterogeneities, loss of immunity, and reinfection (Foutel-Rodier et al., 2020, Brinks et al., 2020, Peterson et al., 2020).

The infection-age PDE class is unified by the McKendrick–von Foerster operator for aging and the nonlocal boundary condition for new infections, corresponding to the total incidence generated by the current infectious distribution (Foutel-Rodier et al., 2020). The formulation allows arbitrary infection-age-dependent transmission and removal, in contrast to classical compartmental models which typically assume memoryless exponential transitions.

2. Basic Reproduction Number and Threshold Criteria

The basic reproduction number $R_0$ in infection-age structured models is computed via the next-generation operator, incorporating the infection-age profile of transmission and survival: $$R_0 = \int_0^\infty \beta(a)\, \exp\left( -\int_0^a \gamma(s)\, ds \right)\, da.$$ This formula expresses, for a fully susceptible population, the expected number of secondary infections generated by a typical infected individual, integrating over their entire infectious life history (Brinks et al., 2020, Chen et al., 2016, Peterson et al., 2020, Webb et al., 2015). For SIR/SIRS models with demography or loss of immunity, and in network-structured or graphon limits, $R_0$ generalizes to a spectral radius of an integral operator or a matrix-valued kernel (see, e.g., (Pang et al., 6 Feb 2025, Chen et al., 2016)), or is expressed as a product of transmission, network, and infection-age terms.

The epidemic threshold is sharp: if $R_0 \leq 1$, the disease-free equilibrium is globally asymptotically stable; if $R_0 > 1$, an endemic equilibrium exists and is unique and globally stable under broad conditions (Chen et al., 2016, Brinks et al., 2020, Richard, 2019). These results also extend to competitive (multi-strain) infection-age models (with competitive exclusion principles and explicit global stability Lyapunov functionals) (Richard, 2019). Infection-age structure fundamentally alters thresholds and endemic levels compared to Markovian models, with the entire distribution of $\beta(a)$ and survival weighting the threshold.

3. Numerical Methods and Model Reduction Strategies

Efficient numerical schemes for infection-age models involve discretization of the infection-age axis (method of lines), explicit or implicit Euler time-stepping, and upwind finite differences for the age-transport equation. Advanced discretizations include Galerkin/spectral schemes (expansion in orthogonal polynomials) and artificial multi-stage (Erlang or hypoexponential) ODE approximations to the infection-age structure (Peterson et al., 2020, Brinks et al., 2020, Scarabel et al., 12 Nov 2025).

• The method of lines with explicit time-step Δt must satisfy the Courant–Friedrichs–Lewy (CFL) condition Δt ≤ Δa to ensure numerical stability.
• k-stage Erlang approximations yield ODE chains with analytically tractable properties but can induce spurious bifurcation phenomena if not matched to observed distributions (Scarabel et al., 12 Nov 2025).
• Galerkin/spectral methods are spectrally accurate with modest ODE system size and applicable to smooth kernels.

These schemes support control optimization, parametric inference, and simulation of high-dimensional or multi-population models, as in the open-source pyross implementation (Peterson et al., 2020).

4. Applications and Examples: Network, Stochastic, Reinfection, and Optimal Control

Epidemics on Networks

Infection-age structure has been incorporated into SIS dynamics on static heterogeneous networks with arbitrary degree distributions. Age-dependent infection $\beta(a)$ and recovery $\gamma(a)$ are weighted by the contact degree, yielding threshold parameters: $$R_0 = \frac{1}{\langle k \rangle} \sum_{k=1}^n k\, \varphi(k)\, P(k)\, S_k^0\! \int_0^\infty \beta(a) \exp\bigl[ -\mu a - \int_0^a \gamma(\xi) d\xi\bigr] da,$$ where $\varphi(k)$ modulates per-link infectivity (Chen et al., 2016). The dynamics reveal combined effects of network heterogeneity and infection-age, with precise intervention implications for transmission and removal rates.

Stochastic Individual-Based Models and Graphons

Measure-valued stochastic SIR models track individual infection ages and traits, converging in the large-population limit to infection-age PDEs on random networks (graphon limits). The effective threshold is the spectral radius of a next-generation operator incorporating individual heterogeneity, network structure, and age-dependent infectivity (Pang et al., 6 Feb 2025).

Reinfection and Waning Immunity

Dual-age structured SIS/SIRS models explicitly couple infection age and immunity age, enabling analysis of waning immunity and reinfection. These form high-dimensional transport–reaction PDEs with renewal boundary conditions. The basic reproduction number is given by an integral over the joint infection-immunity age kernel, and the models predict rich oscillatory and multi-wave phenomena, including genuine bistability if infection duration distributions are sufficiently peaked (Scarabel et al., 12 Nov 2025, Kovacevic et al., 7 Nov 2025).

Control, Contact Tracing, and Intervention

Age-structured models allow for precise modeling of interventions targeted by infection age, such as contact tracing efficacy as a function of delay, quarantine rates, and age-dependent social distancing or vaccination. Optimal control problems are formulated as infinite-dimensional Pontryagin-type systems, with costates and Hamiltonians adapted for infection-age dynamics. These models have been applied to design optimal control strategies for COVID-19, combining bang–bang and continuous controls for social distancing and vaccination (d'Onofrio et al., 2024, Huo, 2013).

5. Modeling Advantages, Theoretical Properties, and Inference

Infection-age structured models possess several critical advantages over classical ODE compartmental frameworks:

• The one-dimensional age structure enables accommodation of arbitrary sojourn time distributions, variable infectivity profiles, and individual/trait/network heterogeneities without exponential blowup of state-space dimension (Foutel-Rodier et al., 2020, Peterson et al., 2020).
• Every finite-state compartmental model with rank-one transmission can be recast as an infection-age PDE by integrating over the Markov process on sojourns (Foutel-Rodier et al., 2020).
• Existence, uniqueness, positivity, and global well-posedness have been established for a wide range of settings, including competitive multi-strain systems, spatial diffusion, and nonlocal integral couplings (Richard, 2019, Walker, 2022, Fitzgibbon et al., 2017).
• Analyses of stability, threshold phenomena, and phase transitions (i.e., Hopf bifurcations, bistability of periodic orbits and endemic equilibria) are explicit and tractable within this formalism (Scarabel et al., 12 Nov 2025).
• Statistical inference is naturally facilitated via Poisson likelihoods over reconstructed expected incidence, and age-structured models enable robust fitting to observed case data, including complex delays such as hospitalizations, ICU, or deaths (Foutel-Rodier et al., 2020).

6. Biological and Epidemiological Implications

Infection-age structure modulates epidemic thresholds, prevalence, and effectiveness of interventions. Specific conclusions from application papers include:

• Control measures targeting early infection ages (such as prompt isolation or acceleration of recovery) disproportionately reduce $R_0$ and epidemic growth (Chen et al., 2016, Webb et al., 2015).
• Multi-wave dynamics and periodic outbreaks emerge naturally from waning immunity and infection-age structure, even with constant parameters (Kovacevic et al., 7 Nov 2025).
• Contact tracing and ring vaccination are most effective when rapid and targeted according to infection age, supported by explicit model-based assessment (Huo, 2013).
• Network structure interacts multiplicatively with infection-age effects, and high-degree nodes sustain persistent infection unless targeted (Chen et al., 2016).
• Model-based inference on real data (e.g., SARS or Ebola) can estimate age-dependent R(t,a) surfaces, inform the timing and strength of interventions, and quantify under-reporting or delays (Li et al., 5 Dec 2025, Webb et al., 2015).

These findings demonstrate the necessity and flexibility of infection-age structured epidemic models for capturing the mechanistic underpinnings and control of infectious disease outbreaks in realistic settings.