Infection-Age Structured Epidemic Model

• Infection-age structured models are mathematical frameworks that track time since infection to capture non-Markovian dynamics and variable infectious periods.
• They are formulated using first-order partial differential equations, such as the McKendrick–von Foerster equation, often coupled with ODEs for non-infected compartments.
• These models enhance epidemic prediction and intervention design by incorporating heterogeneous transmission, oscillatory behavior, and realistic sojourn-time distributions.

Infection-age structured epidemic models are mathematical frameworks in which the temporal dynamics of infectious disease spread are explicitly modeled by keeping track of the time since infection (“infection age”) for each infected individual or cohort. This approach extends classical compartmental models by resolving the non-Markovian, time-varying effects of infectiousness and removal, enabling precise representation of non-exponential sojourn-time distributions, multi-stage progression, and heterogeneity in transmission. The infection-age structure is mathematically formulated via first-order partial differential equations (most commonly in the class of McKendrick–von Foerster equations), often coupled to ODEs for non-infected compartments, and underpins both deterministic and stochastic analysis across host–pathogen systems, public health intervention design, and parametric inference from incidence data.

1. Mathematical Formulation and McKendrick–von Foerster PDE

The core object is the infection-age density $n(t,a)$, where $n(t,a)\,da$ is the expected number of individuals at calendar time $t$ whose “age of infection” lies in $[a,a + da]$. In the absence of demographic turnover and assuming large $N$, the evolution of this density follows a linear transport-removal equation: $$(\partial_t + \partial_a)\,n(t,a) = -\mu(a)\,n(t,a),\quad a > 0,\ t > 0$$ accompanied by an age-zero boundary condition: $n(t, 0) = \int_0^\infty \beta(a)\,n(t,a)\,da$ where $\mu(a)$ is the infection-age dependent removal (recovery or death) rate, and $\beta(a)$ is the per-capita transmission (infectivity) at age $a$. This McKendrick–von Foerster structure allows for straightforward representation of arbitrary sojourn-time distributions and time-varying infectivity (Foutel-Rodier et al., 2020).

This framework generalizes easily to more complex compartmental structures (e.g., SIR, SEIR, multi-stage infection), control variables (such as vaccination and social contact modulation (d'Onofrio et al., 10 May 2024)), and spatial or demographic heterogeneity.

2. Connection to Compartmental Models and Renewal Equations

Classical ODE SIR-type models are embedded as special cases of the infection-age PDE with exponential removal and infectivity:

• For SIR, choosing $\mu(a) = \gamma$, $\beta(a) = \beta e^{-\gamma a}$, and defining $S(t) = 1 - \int_0^\infty n(t,a)\,da$, the dynamics reduce exactly to

$$\dot{S} = -\beta S I,\quad \dot{I} = \beta S I - \gamma I,\quad \dot{R} = \gamma I$$

• For SEIR or Erlang/Gamma compartments, non-constant $\mu(a)$ and structured $\beta(a)$ recover the ODE chain in higher dimensions (Foutel-Rodier et al., 2020).

The infection-age PDE is also equivalent, via characteristics, to nonlinear renewal equations for the incidence $C(t)$: $C(t) = S(t) \int_0^\infty \beta(a) F(a) C(t-a)\,da$ where $F(a) = \exp\left(-\int_0^a \mu(s)\,ds\right)$ is the survival function in the infectious class. This duality underpins analytical and numerical techniques and permits systematic ODE approximations for integer-valued Erlang or Gamma infectious periods (Scarabel et al., 12 Nov 2025).

3. Thresholds, Reproduction Numbers, and Stability

The infection-age setting provides a clear and general definition of the basic reproduction number: $R_0 = \int_0^\infty \beta(a)\,F(a)\,da$ where $\beta(a)$ is per-infective infectivity and $F(a)$ is the probability of surviving infective removal up to age $a$. This generalizes the classical $R_0 = \beta/\gamma$ and is directly computable for arbitrary sojourn distributions (Foutel-Rodier et al., 2020, Scarabel et al., 12 Nov 2025, d'Onofrio et al., 10 May 2024).

Standard threshold results hold: the disease-free equilibrium is globally asymptotically stable if $R_0 < 1$, and endemic equilibria exist for $R_0 > 1$. In models with multiple strains or structured heterogeneity, competitive exclusion principles and the structure of endemic attractors persist, with extensions such as coexistence continua for $R_0^x = R_0^y > 1$ (Richard, 2019).

Infection-age modeling also naturally encodes oscillatory (Hopf) bifurcations and bistability: non-trivial periodic solutions can arise (e.g., under waning immunity, or via high-variance infectious durations) and are tracked through bifurcation analysis—genuine bistability owing to infection-age distributions cannot be replicated by single-parameter Markovian ODE chains (Scarabel et al., 12 Nov 2025, Zhang et al., 2017).

4. Generalizations: Networks, Spatial Diffusion, Multitype and Control

Spatial structure and network heterogeneity are incorporated by coupling infection-age PDEs with spatial diffusion and/or interaction kernels:

• Reaction-diffusion infection-age PDE systems model epidemic fronts and spatial pattern formation (Fitzgibbon et al., 2017, Walker, 2022, Cherniha et al., 12 Nov 2024).
• On networks or random graphs, each node or individual tracks infection age, and contact-dependent infection rates are modulated via network adjacency; limiting systems (graphon limits) again reduce to McKendrick–von Foerster type equations with integral “force of infection” terms (Chen et al., 2016, Pang et al., 6 Feb 2025).
• Multi-population or vector-host systems admit cross-infection with infection-age structures, requiring joint PDEs for each host, vector, or strain (Fitzgibbon et al., 2017, Richard et al., 2020, Richard, 2019).
• Control mechanisms such as vaccination, social distancing, and contact tracing can be optimized directly in infection-age models, exploiting their ability to represent time-varying intervention effects and infer parameters from incidence data alone (d'Onofrio et al., 10 May 2024, Huo, 2013).

5. Analytical and Numerical Approaches

Infection-age structured models, due to their nonlocality in both time and age, require tailored analytical and computational techniques:

• Semigroup theory provides global existence, uniqueness, and long-time asymptotics even for nonlinear or spatially extended systems (Foutel-Rodier et al., 2020, Richard, 2019, Walker, 2022).
• Volterra-type Lyapunov functionals and next-generation operator methods yield threshold and global stability results, leveraging integral operator representations (Richard, 2019, Walker, 2022).
• Numerical resolution is often based on grid discretization in $(t,a)$ via method-of-characteristics schemes, which preserve the structure and allow efficient calculation of incidence and prevalence profiles (Brinks et al., 2020). Spectral and pseudospectral discretizations are employed for efficient bifurcation and stability analysis in infinite-dimensional settings (Scarabel et al., 12 Nov 2025).

6. Epidemiological Implications and Applications

Infection-age structured models capture key epidemiological phenomena that are inaccessible to simple ODE frameworks:

• Non-exponential infectious period distributions, incubation phases, or variable infectivity profiles are directly incorporated via age-dependent parameter functions, altering epidemic peaks, final sizes, and control requirements (Foutel-Rodier et al., 2020, Webb et al., 2015).
• Realistic extinction time distributions in subcritical or waning phases are substantially longer than those predicted by Markovian models, impacting public health planning (Mougabe-Peurkor et al., 2023).
• The analytic structure permits rigorous inference of transmission and removal rates from incidence data under Poisson likelihoods, as exemplified in the fitting of COVID-19 and Ebola datasets (Foutel-Rodier et al., 2020, Webb et al., 2015, Huo, 2013).
• In the context of multiple strains or intervention strategies, infection-age frameworks facilitate comparison of policy efficacy and elucidate competitive dynamics, especially when time-varying or age-specific interventions are relevant (Richard, 2019, Huo, 2013).

In summary, the infection-age structured epidemic model constitutes a mathematically rigorous, flexible, and data-compatible generalization of compartmental epidemic modeling, justifying its centrality in both theoretical and applied infectious disease dynamics (Foutel-Rodier et al., 2020, Scarabel et al., 12 Nov 2025, d'Onofrio et al., 10 May 2024).