SEIRS Epidemiological Model

Disease States

SEIRS is a standard compartmental model that divides the population into Susceptible (S), Exposed (E), Infectious (I), and Recovered (R) groups. Susceptible individuals become Exposed after a transmissive contact with someone Infectious, then progress to the Infectious state following an incubation period, and eventually move to the Recovered state. Over time, Recovered individuals may lose immunity and become Susceptible again.

Differential Equations

The SEIRS model's dynamics are governed by a system of differential equations in a deterministic, mean-field framework, following standard epidemiological compartment models. This approach assumes a homogeneous population (everyone shares the same rates and parameters) and uniform mixing (every individual is equally likely to interact with every other), which together simplify how disease spread is modeled.

$$\begin{align*} \frac{d\class{varS}{S}}{dt} &= -\beta \class{varS}{S} \class{varI}{I} + \omega \class{varR}{R} - \mu \class{varS}{S} + \mu(1-p) \\ \frac{d\class{varE}{E}}{dt} &= \beta \class{varS}{S} \class{varI}{I} - \sigma \class{varE}{E} - \mu \class{varE}{E} \\ \frac{d\class{varI}{I}}{dt} &= \sigma \class{varE}{E} - \gamma \class{varI}{I} - (\mu + \alpha) \class{varI}{I} \\ \frac{d\class{varR}{R}}{dt} &= \gamma \class{varI}{I} - \omega \class{varR}{R} - \mu \class{varR}{R} + \mu p \\ \end{align*}$$

Mean Parameters

Disease can spread when a Susceptible person contacts someone Infectious. The chance that this contact leads to infection is proportional to the infectious person's transmissibility, β. This β parameter is linked to the basic reproduction number, R0 — the expected number of new infections caused by one Infectious individual in a fully Susceptible population.

$$\begin{align*} \text{transmissibility} &= \beta \\ \text{basic reproduction number} &= R_0 = \frac{\sigma}{\sigma + \mu} \frac{\beta}{\gamma + \mu + \alpha} \\\\ \text{infection-induced mortality rate} &= \alpha \\ \text{infection-to-death period} &= 1 / \alpha \\\\ \text{progression rate} &= \sigma \\ \text{incubation period} &= 1 / \sigma \\\\ \text{recovery rate} &= \gamma \\ \text{infectious period} &= 1 / \gamma \\\\ \text{re-susceptibility rate} &= \omega \\ \text{immunity duration} &= 1 / \omega \\\\ \text{natural mortality rate} &= \mu \\ \text{life expectancy} &= 1 / \mu \\\\ \text{vaccination rate} &= p \\\\ \text{susceptible fraction} &= S \\ \text{exposed fraction} &= E \\ \text{infectious fraction} &= I \\ \text{recovered fraction} &= R \\ \end{align*}$$

Initial Parameters

$$\begin{align*} S_0 &= S_i - p \\ E_0 &= 1 - S_i \\ I_0 &= 0 \\ R_0 &= p \\ S + E + I + R &= 1 \\ \end{align*}$$

References

Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: The SEIRS model for infectious disease dynamics. (2020) Nature Methods 17:557–558.