seir Module

SEIR compartmental epidemic model with incubation period.

The SEIR model adds an **E**xposed (latent) compartment for diseases with an incubation period where individuals are infected but not yet infectious.

Equations

\[\begin{split}\\frac{dS}{dt} &= -\\frac{\\beta S I}{N} \\\\ \\frac{dE}{dt} &= \\frac{\\beta S I}{N} - \\sigma E \\\\ \\frac{dI}{dt} &= \\sigma E - \\gamma I \\\\ \\frac{dR}{dt} &= \\gamma I\end{split}\]

where: - \(\\beta\) = transmission rate (day⁻¹) - \(\\sigma\) = progression rate E→I (day⁻¹) - \(\\gamma\) = recovery rate (day⁻¹) - \(1/\\sigma\) = mean incubation period - \(1/\\gamma\) = mean infectious period - \(R_0 = \\beta / \\gamma\)

Class

class episia.models.seir.SEIRModel(parameters)

Bases: CompartmentalModel

SEIR epidemic model with explicit incubation compartment.

Parameters:

parameters (SEIRParameters) – SEIRParameters instance.

Example:

from episia.models import SEIRModel
from episia.models.parameters import SEIRParameters

params = SEIRParameters(
    N=1_000_000,
    I0=1,
    E0=10,
    beta=0.35,
    sigma=1/5.2,   # COVID-19-like incubation
    gamma=1/14,
    t_span=(0, 365),
)
result = SEIRModel(params).run()
result.plot().show()

__init__(parameters)

Parameters:

parameters (SEIRParameters)

property compartment_names: List[str]

Ordered list of compartment names, e.g. [‘S’, ‘I’, ‘R’].

Examples

COVID-19 modeling:

from episia.models import SEIRModel
from episia.models.parameters import SEIRParameters

params = SEIRParameters(
    N=1_000_000,
    I0=10,
    E0=50,               # Initial exposed
    beta=0.35,
    sigma=1/5.2,         # 5.2 days incubation
    gamma=1/14,          # 14 days infectious
    t_span=(0, 365)
)

model = SEIRModel(params)
result = model.run()

print(f"R₀: {params.r0:.2f}")
print(f"Peak infected: {result.peak_infected:,.0f}")
print(f"Peak exposed: {result.metadata['metrics']['peak_exposed']:,.0f}")

result.plot().show()

Comparing with SIR:

from episia.models import SIRModel, SEIRModel
from episia.models.parameters import SIRParameters, SEIRParameters

# SIR (no incubation)
sir_params = SIRParameters(N=1e6, I0=10, beta=0.3, gamma=0.1)
sir_result = SIRModel(sir_params).run()

# SEIR (5 days incubation)
seir_params = SEIRParameters(N=1e6, I0=10, E0=50, beta=0.3,
                              sigma=0.2, gamma=0.1)
seir_result = SEIRModel(seir_params).run()

print(f"SIR peak: {sir_result.peak_time:.0f} days")
print(f"SEIR peak: {seir_result.peak_time:.0f} days")