How is this different from SEIR/SEIRS?

SEIR adds a latent exposed class before infectiousness. SIRS here is three compartments plus waning, boosters, and seasonal β(t), with the endemic equilibrium written on the page.

What does a booster pulse do?

At scheduled times a fraction of susceptibles is moved into R, as if a campaign restored immunity. It does not change ω; it only resets who is currently protected.

When is there an endemic equilibrium?

For constant β, an endemic state with I*>0 exists when R0>1 and ω>0. If immunity is permanent (ω=0) the long-run infected fraction returns to 0 after a single outbreak, as in classic SIR.

SIRS with Waning Immunity

SIRS compartments with waning ωR→S, booster pulses, seasonal β(t), endemic equilibrium, and R0 vs fade-out.

Presets

SIRS rates (N = 1, time in days)

β₀ (baseline transmission)0.4 /d

γ (recovery)0.2 /d

ω (waning R→S)0.01 /d

ε (seasonal amplitude)0

T (seasonal period)365 d

I(0) initial infected0.02

R(0) initial recovered0

Three compartments only: S+I+R=1, no latent E. Waning ωR→S can hold an endemic level when ℛ₀>1. Seasonal β(t) modulates transmission; booster pulses instantly move a coverage fraction of S into R. Reset to apply new I(0) and R(0).

Booster pulses

Enable periodic boosters

Booster coverage0.25

Booster period365 d

Shortcuts

Space / Enter — play / pause
R — reset time series

Measured values

ℛ₀ = β₀/γ2.000

S* = 1/ℛ₀0.500

I* endemic0.0238

S (now)0.980

I (now)0.0200

R (now)0.000

peak I0.0200

time0.0 d

About this model

The SIRS model tracks susceptible, infectious, and recovered fractions with S+I+R=1. Infection is β(t)SI, recovery is γI, and immunity wanes at rate ω (R→S). Optional seasonal forcing uses β(t)=β0(1+ε sin(2πt/T)). Periodic booster pulses move a coverage fraction from S into R. When R0=β0/γ>1 the endemic equilibrium is S*=1/R0 and I*=ω(1−1/R0)/(ω+γ); otherwise the disease fades. This page has no latent E compartment — that is the separate SEIR/SEIRS simulator.

Who it's for: Epidemiology, public-health modeling, and mathematical biology courses on waning immunity and seasonality.

Key terms

SIRS
Waning immunity
Booster
Seasonal forcing
Endemic equilibrium
R0
Herd immunity

How it works

SIRS with waning immunity: fractions S+I+R=1 and no latent E. Infection β(t)SI, recovery γI, return ωR→S. Constant-β endemic state (ε=0, ω>0, ℛ₀=β₀/γ>1) is **S*=1/ℛ₀ and I*=ω(1−1/ℛ₀)/(ω+γ); otherwise I*=0 (fade-out or a single SIR outbreak). Optional seasonality β(t)=β₀(1+ε sin(2πt/T)) and periodic booster pulses that move coverage·S** from S into R.

Key equations

S' = −β(t) S I + ω R

I' = β(t) S I − γ I

R' = γ I − ω R

β(t) = β₀ (1 + ε sin(2π t / T))

ℛ₀ = β₀/γ · S* = 1/ℛ₀ · I* = ω(1 − 1/ℛ₀)/(ω+γ) (ε=0, ω>0, ℛ₀>1; else I*=0)

Frequently asked questions

How is this different from SEIR/SEIRS?: SEIR adds a latent exposed class before infectiousness. SIRS here is three compartments plus waning, boosters, and seasonal β(t), with the endemic equilibrium written on the page.
What does a booster pulse do?: At scheduled times a fraction of susceptibles is moved into R, as if a campaign restored immunity. It does not change ω; it only resets who is currently protected.
When is there an endemic equilibrium?: For constant β, an endemic state with I*>0 exists when R0>1 and ω>0. If immunity is permanent (ω=0) the long-run infected fraction returns to 0 after a single outbreak, as in classic SIR.

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