SEIRS

SEIRS is a compartmental model used in epidemiology to describe the spread of infectious diseases when immunity after recovery is temporary. It extends the SEIR framework by allowing recovered individuals to lose immunity and become susceptible again after a waning period.

The model comprises four compartments: Susceptible (S), Exposed (E), Infectious (I), and Recovered (R). Individuals move from S to E when they are exposed to infection, from E to I as they incubate, from I to R upon recovery, and from R back to S as immunity wanes. Some implementations also include births and natural deaths, creating a dynamic population with turnover.

In mathematical form, the system can be written as a set of ordinary differential equations. With demography (birth and death rate μ) and total population N = S + E + I + R, one common formulation is:

dS/dt = μN − β S I / N + ω R − μ S

dE/dt = β S I / N − (σ + μ) E

dI/dt = σ E − (γ + μ) I

dR/dt = γ I − (ω + μ) R

Here, β is the transmission rate, σ is the rate at which exposed individuals become infectious (1/incubation period), γ is the recovery rate (1/infectious period), ω is the rate of waning immunity (1/immunity duration), and μ is the natural death (and birth, when population size is balanced). In closed-population models without demography, the μ terms are omitted.

SEIRS is used to study diseases where immunity is not permanent and recurrent outbreaks are possible, highlighting how waning immunity interacts with transmission dynamics and seasonality.