The assumption of exponentially distributed waiting times in classic compartmental models leads to the mathematically convenient, but often biologically restrictive, memoryless property. Age-of-Infection Models (also known as Time-Since-Infection, TSI models) address this by explicitly incorporating the time spent in an infected state (τ) as a determinant of contagiousness, infectivity profile, and probability of recovery. This approach is essential for modeling diseases where infectivity changes significantly after exposure, such as HIV/AIDS.
📐 Compartmental Structure and Flow Explanation
The rigorous representation of time-since-infection requires functional differential equations, often expressed as integral equations, which track the infectivity kernel A(τ) across the continuous age of infection.
To retain tractability within an Ordinary Differential Equation (ODE) framework, this continuous dependency is approximated by introducing a linear chain of k sequential infectious sub-compartments:
I₁, I₂, …, Iₖ.
This structure allows the model to approximate general, non-exponential residence times (such as Erlang or Gamma distributions) while modeling varying levels of infectiousness (ωⱼ) within each stage.
Infection Flow:
Susceptible individuals (S) become infected and enter the first stage I₁.
Progression:
Individuals progress sequentially through infectious stages Iⱼ → Iⱼ₊₁ at rate kγ.
Removal:
Individuals in the final stage Iₖ transition into the Removed class R.
The instantaneous infection rate depends on the weighted contribution of all infectious stages, reflecting how infectivity varies with time-since-infection.
🧮 Mathematical Formulation (Linear Chain Approximation)
Let S denote Susceptible, Iⱼ the number in infectious stage j (out of k total stages), and R the Removed class.
The system of ODEs is:
dS/dt = − β S ( Σⱼ ωⱼ Iⱼ )
dI₁/dt = β S ( Σⱼ ωⱼ Iⱼ ) − kγ I₁
dIⱼ/dt = kγ Iⱼ₋₁ − kγ Iⱼ for j = 2, …, k
dR/dt = kγ Iₖ
The term Σⱼ ωⱼ Iⱼ is the generalized force of infection, aggregating the contributions of all infectious stages.
⚗ Parameter Definitions and Typical Ranges
| Parameter | Definition | Typical Range |
|---|---|---|
| β | Baseline transmission coefficient | 0.2 – 1.0 per day |
| γ | Inverse of mean infectious period | 0.1 – 0.5 per day |
| kγ | Progression rate between stages | Depends on duration and number of stages |
| k | Number of sequential infectious stages | Integer ≥ 1 |
| ωⱼ | Relative infectivity in stage j | 0.0 – 1.0 |
Higher values of k imply lower variance in the infectious duration distribution.
🎯 Applicability and Limitations
Applications
- Modeling non-Markovian processes
Appropriate when latency or infectivity duration clearly follows non-exponential distributions. Avoids unrealistic memoryless behavior. - Tracking variable infectivity
Essential for diseases like HIV/AIDS, where infectivity A(τ) is highly dependent on time since infection. The weights ωⱼ approximate this variation. - Capturing richer dynamics
Age-of-infection structure can generate oscillations, multi-wave behavior, or even chaos—dynamics impossible in classical SIR/SEIR models.
Assumptions and Weaknesses
- High dimensionality
Adding k stages increases model size, making analysis and simulation more demanding. - Difficult parameterization
Infectivity weights ωⱼ are often hard to measure and must be estimated or assumed. - Approximation of true delay equation
The linear-chain ODE approach approximates the true distributed delay integral equation, trading accuracy for tractability.
📚 Selected References
- Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics.
- Hethcote, H. W. (2000). The mathematics of infectious diseases.
- Lloyd, A. L. (2001). Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods.
- Hyman, J. M., & Stanley, E. A. (1988). Using mathematical models to understand the AIDS epidemic.
- Li, M. Y., & Liu, X. (2014). An SIR epidemic model with time delay and general non-linear incidence rate.