🦠The SEIRS Model: Susceptible–Exposed–Infectious–Recovered–Susceptible

🦠 Overview: Capturing Temporary Immunity. The SEIRS compartmental model extends the classic SIR framework by incorporating an Exposed (E) state (latency) and allowing Recovered (R) individuals to lose immunity and return to the Susceptible (S) pool after some time. In the standard SIR model, recovered individuals are permanently immune, which means SIR can describe one self-limited epidemic but cannot explain endemic disease transmission or recurring outbreaks. In contrast, the SEIRS model assumes immunity is temporary – a realistic feature for diseases like seasonal influenza where infection confers only short-term protection.

Key idea: By cycling recovered individuals back to the susceptible class, SEIRS can capture recurrent epidemic waves and endemic behavior. This makes SEIRS especially relevant for infections that do not confer lifelong immunity. For example, in seasonal influenza, immunity wanes as the virus evolves antigenically, causing susceptible populations to replenish over time. The SEIRS structure is thus able to model annual or multi-year influenza outbreaks that repeat as immunity from prior infection diminishes.

🥮 SEIRS Dynamics: Differential Equations In an SEIRS model, the population is divided into compartments for each disease stage: S(t), E(t), I(t), R(t) at time t. Each compartment’s size changes according to a system of ordinary differential equations (ODEs) capturing the flow of individuals between states. A simplified SEIRS model (assuming a fixed population and no births or non-disease deaths during the epidemic) can be written as:

dS/dt = – β * S * I + ζ * R
dE/dt = β * S * I – σ * E
dI/dt = σ * E – γ * I
dR/dt = γ * I – ζ * R

Here, β is the transmission rate, σ is the incubation progression rate, γ is the recovery rate, and ζ is the immunity waning rate. The term β S I represents new infections. Exposed individuals become infectious at rate σ. Infectious individuals recover at rate γ. Recovered individuals lose immunity at rate ζ. The R → S transition is critical for capturing temporary immunity.

Notably, if the epidemic unfolds over a long period or if demographic processes are important, one can include natality and mortality terms. However, for short-term epidemics like a single influenza season, births/deaths are often negligible and the simpler equations above suffice.

📊 Parameters and Typical Values (Seasonal Influenza) To concretely understand SEIRS, it helps to consider typical parameter values for a real disease. Seasonal influenza is a prime example: infection grants only transient immunity, and the illness exhibits annual epidemic cycles. Below are the key SEIRS parameters with representative values (per person, per day):

• Transmission rate (β): ~0.3 (range 0.2–0.6)
• Incubation rate (σ): ~0.5–1 (latent period 1–2 days)
• Recovery rate (γ): ~0.2–0.33 (infectious period 3–5 days)
• Immunity waning rate (ζ): ~0.001–0.005 (immunity duration several months to ~1 year)
• Basic reproduction number (R₀): ~1.3 (range 1.1–1.6)

These rates are normalized to population size in models. Influenza transmissibility varies seasonally, and immunity waning may differ by age or viral mutation rate.

🔄 Endemic Equilibrium and Seasonal Oscillations A notable outcome of the SEIRS model is the possibility of an endemic equilibrium – a steady state where the disease persists in the population. In SEIRS, the continual return of recovered individuals to susceptibility can balance the infection process, leading to a non-zero endemic infection level for R₀ > 1.

Real-world diseases like influenza rarely settle to a steady prevalence. Instead, oscillatory behavior is observed due to:

• Intrinsic oscillations: SEIRS systems may exhibit damped or sustained oscillations depending on the immunity duration.
• Seasonal forcing: Transmission rate β varies seasonally (e.g., β = β0[1 + ε cos(2πt)]), producing annual waves synchronized with high-β seasons.

Seasonal forcing explains annual influenza epidemics. The SEIRS model with seasonal β captures these limit-cycle behaviors, and the disease counts oscillate around a non-zero equilibrium due to periodic changes in contact rates.

📚 Foundational and Contemporary References

• Kermack, W.O. & McKendrick, A.G. (1927): “A Contribution to the Mathematical Theory of Epidemics.” Proc. Roy. Soc. A 115.
• Liu, W.M., Levin, S.A. & Iwasa, Y. (1986): “Influence of Nonlinear Incidence Rates upon the Behavior of Simple SIRS Models.” J. Math. Biol. 23.
• Korobeinikov, A. (2006): “Lyapunov Functions and Global Stability for SIR and SIRS Models.” Bull. Math. Biol. 68(3).
• Tang, B. et al. (2020): “Deterministic SEIR/SEIRS Modeling of COVID-19.” Infectious Disease Modelling 5.