The Susceptible–Infectious–Recovered–Susceptible (SIRS) model extends the classic SIR framework by incorporating temporary immunity. After recovering, individuals transition to the Recovered (R) class, where they are immune for a period before returning to the Susceptible (S) pool. This reinfection cycle is ideal for modeling diseases like seasonal influenza, respiratory syncytial virus (RSV), or the common cold, which recur over time due to waning immunity.
🧬 1. Core Compartmental Structure
The population of constant size N is divided into:
- S (Susceptible): Individuals at risk of infection
- I (Infectious): Individuals who are currently contagious
- R (Recovered): Individuals with temporary immunity
A hallmark of this model is the R → S transition, driven by the immunity waning rate (δ), which returns recovered individuals to the susceptible pool. The total population satisfies:
N = S(t) + I(t) + R(t)
🔢 2. Deterministic Equations
The standard SIRS model with births (Λ) and deaths (μ) is governed by:
dS/dt = Λ − β·S·I − μ·S + δ·R
dI/dt = β·S·I − (γ + μ)·I
dR/dt = γ·I − (δ + μ)·R
Parameter meanings:
- β: Transmission rate
- γ: Recovery rate (1 / infectious period)
- δ: Immunity waning rate (1 / duration of immunity)
- μ: Natural death rate
- Λ: Birth rate (usually set to μ·N for constant population)
📊 3. Typical Parameter Ranges
| Parameter | Description | Typical Range | Units |
|---|---|---|---|
| β | Transmission rate | 0.2 – 0.5 | per day |
| γ | Recovery rate | 0.1 – 0.5 | per day |
| δ | Immunity waning rate | 0.001 – 0.005 | per day |
| μ | Natural death rate | 0.00003 – 0.00005 | per day |
| 1/γ | Infectious period | 2 – 10 | days |
| 1/δ | Duration of immunity | ~6 months – 3 years | days |
| Λ | Birth rate | Equal to μ·N | per day |
These values are consistent with human diseases like influenza and common cold viruses.
📈 4. Epidemiological Dynamics
- The basic reproduction number, R₀ = β / (γ + μ), determines whether an outbreak can persist.
- If R₀ ≤ 1, the disease dies out and the Disease-Free Equilibrium (DFE) is globally stable.
- If Râ‚€ > 1, the system stabilizes at an Endemic Equilibrium (EE) where infections persist at a constant level.
- The fraction of susceptibles at endemic equilibrium is approximately S = 1 / Râ‚€*.
Unlike in SIR models, the recycling of individuals via waning immunity allows the infection to persist indefinitely, even in the absence of births.
🌀 5. Seasonal Forcing and Recurring Outbreaks
If the transmission rate β varies seasonally (e.g., increases in winter), the SIRS model can exhibit oscillatory dynamics:
- Regular epidemics with periodic peaks
- Irregular outbreaks based on transmission strength and immunity duration
- Phenomena such as multi-year epidemic cycles
This behavior makes the SIRS model highly applicable for diseases with seasonal waves, like influenza or RSV.
📚 Reference Papers
- Kermack, W. O. & McKendrick, A. G. (1932). Contributions to the Mathematical Theory of Epidemics. II. Proceedings of the Royal Society A, 138, 55–83.
- Korobeinikov, A. (2006). Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bulletin of Mathematical Biology, 68(3), 615–626.
- Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653.
- Lu, Q. (2009). Stability of SIRS system with random perturbations. Physica A, 388(18), 3677–3686.