The Susceptible–Antibody–Infectious–Removed (SAIR) model is a powerful analytical tool developed to capture the dynamics of self-immunization within an exposed population, addressing a critical challenge during epidemics like the COVID-19 pandemic: the substantial number of infected individuals who recover without formal diagnosis, thus acquiring immunity undetected. The SAIR model, and its extension (eSAIR), explicitly track this outcome, which is essential for determining true population immunity levels and estimating the actual burden of infection.
📐 Compartmental Structure and Flow Explanation
The basic SAIR model divides the total population (N) into four mutually exclusive compartments:
- Susceptible (S): Individuals vulnerable to infection. They also contribute to the Antibody compartment via self-immunization.
- Antibody (A): Individuals who gain immunity without entering the tracked infectious state—typically asymptomatic recoveries or naturally immune individuals.
- Infectious (I): Individuals actively transmitting disease.
- Removed (R): Individuals permanently removed from the transmission cycle (confirmed recoveries or fatalities).
A defining feature is the S → A flow, representing acquisition of immunity without documented infection.
🧮 Mathematical Formulation (ODEs)
Below are the SAIR equations rewritten in standard mathematical form.
Parameters: α = self-immunization rate, β = transmission rate, γ = recovery rate.
SAIR System
dS/dt = − α S − β S I / N
dA/dt = α S
dI/dt = β S I / N − γ I
dR/dt = γ I
The extended SAIR model (eSAIR) introduces time-varying rates α(t) and δ(t) and uses proportional terms (S̃ = S/N, etc.).
eSAIR System
dS̃/dt = − α(t) S̃ − β δ(t) S̃ Ĩ
dÃ/dt = α(t) S̃
dĨ/dt = β δ(t) S̃ Ĩ − γ Ĩ
dR̃/dt = γ Ĩ
⚗ Parameter Definitions and Ranges
| Parameter | Interpretation | Typical Range (day⁻¹) |
|---|---|---|
| α, α(t) | Self-immunization rate (S → A) | 0.001 – 0.1 |
| β | Transmission rate | 0.2 – 1.0 |
| γ | Recovery/removal rate (I → R) | 0.14 – 0.5 |
| δ(t) | Time-varying transmission modifier | 0 – 1.0 |
| R₀ | Basic reproduction number | 1.5 – 4.0 |
R₀ measures the initial intensity of contagion. γ corresponds to an infectious period of ~2–7 days. α depends on prevalence of undiagnosed cases.
🎯 Applicability and Limitations
| Use Case | Key Assumptions / Limitations |
|---|---|
| Quantifying True Immunity: Estimates the actual fraction immune (A + R) when many infections are undocumented. | Unidentifiable Parameters: α and α(t) require serological data to estimate reliably. |
| Spatio-Temporal Modeling: eSAIR can integrate with Cellular Automata (CA-eSAIR) to model geographic heterogeneity. | Homogeneous Mixing: Assumes uniform mixing without explicit contact networks. |
| Intervention Assessment: δ(t) enables incorporation of NPIs such as distancing or masks. | Simplified Progression: SAIR lacks detailed states such as latent or exposed compartments. |
📚 References
- Tang, L., Zhou, Y., Wang, L., et al. (2020). A review of multi-compartment infectious disease models.
- Zhou, Y., Wang, L., Zhang, L., et al. (2020). A spatiotemporal epidemiological prediction model to inform county-level COVID-19 risk in the USA.
- Liu, X., Takeuchi, Y., & Iwami, S. (2008). SVIR epidemic models with vaccination strategies.
- Kermack, W. O. & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics.
- Diekmann, O., Heesterbeek, J.A.P., & Metz, J.A.J. (1990). On the definition and computation of the basic reproduction ratio R₀.