Simulating Discrete models

from epimodels.discrete import models as DM
from IPython.display import Markdown as md

import matplotlib.pyplot as P

SIS model

The SIS (Susceptible-Infectious-Susceptible) model for diseases that do not confer immunity after recovery.

modelsis = DM.SIS()
md(str(modelsis))

modelsis([0, 1, 1000], [0,50], 1001, {'beta': 2, 'gamma': 1})
modelsis.plot_traces()

SIR Model

The SIR (Susceptible-Infectious-Removed) model is a classic compartmental model.

sirmodel = DM.SIR()
md(str(sirmodel))

sirmodel([1000, 1, 0], [0,500], 1001, {'beta': .2, 'gamma': .1})
sirmodel.plot_traces()

SEIS Model

The SEIS (Susceptible-Exposed-Infectious-Susceptible) model includes an exposed compartment but no permanent immunity.

model = DM.SEIS()
md(str(model))

model([1000, 0, 1], [0, 100], 1001, {'beta': 0.5, 'e': 0.3, 'r': 0.1, 'b': 0})
model.plot_traces()

SEIR Model

The SEIR (Susceptible-Exposed-Infectious-Removed) model includes an exposed compartment for diseases with incubation period.

model = DM.SEIR()
md(str(model))

model([1000, 0, 1, 0], [0, 100], 1001, {'beta': 0.5, 'e': 0.3, 'r': 0.1, 'b': 0, 'alpha': 0.01})
model.plot_traces()

SIRS Model

The SIRS (Susceptible-Infectious-Removed-Susceptible) model accounts for waning immunity.

model = DM.SIRS()
md(str(model))

model([1000, 1, 0], [0, 200], 1001, {'beta': 0.3, 'r': 0.1, 'b': 0, 'w': 0.02})
model.plot_traces()

SIpRpS model

This model assumes that a fraction \(\delta\) of infectious individuals acquire full immunity while the remaining \((1-\delta)\) returns to the susceptible stage.

model = DM.SIpRpS()
md(str(model))

model([1000, 1, 0], [0, 100], 1001, {'beta': 0.5, 'r': 0.2, 'b': 0, 'delta': 0.7})
model.plot_traces()

SEIpRpS model

Similar to SIpRpS but with an exposed compartment. A fraction \(\delta\) of infectious individuals acquire full immunity while the remaining \((1-\delta)\) returns to the susceptible stage.

model = DM.SEIpRpS()
md(str(model))

model([1000, 0, 1, 0], [0, 50], 1001, {'beta': 1, 'r': .3, 'e': 1, 'b': 0, 'delta': 0.6})
model.plot_traces()

model.parameter_table()

SIpR model

This model allows recovered individuals to be reinfected at a reduced rate \(p\beta\).

model = DM.SIpR()
md(str(model))

model([1000, 1, 0], [0, 150], 1001, {'beta': 0.3, 'r': 0.1, 'b': 0, 'p': 0.3})
model.plot_traces()

SEIpR model

Similar to SIpR but with an exposed compartment. Recovered individuals can be reinfected at a reduced rate \(p\beta\).

model = DM.SEIpR()
md(str(model))

model([1000, 0, 1, 0], [0, 150], 1001, {'beta': 0.3, 'e': 0.2, 'r': 0.1, 'b': 0, 'alpha': 0.01, 'p': 0.3})
model.plot_traces()

SEQIAHR model

A complex model with asymptomatic and hospitalized compartments, suitable for modeling diseases like COVID-19.

model = DM.SEQIAHR()
md(str(model))

params = {
    'chi': 0.3,    # Quarantine reduction factor
    'phi': 0.1,    # Hospitalization rate from I
    'beta': 0.5,   # Transmission rate
    'rho': 0.1,    # Recovery rate from H
    'delta': 0.2,  # Recovery rate from I
    'gamma': 0.1,  # Recovery rate from A
    'alpha': 0.3,  # Incubation rate
    'mu': 0.01,    # Mortality rate in H
    'p': 0.5,      # Proportion asymptomatic
    'q': 20,       # Quarantine start day
    'r': 30        # Quarantine duration
}
inits = [10000, 0, 1, 0, 0, 0, 0, 0]
model(inits, [0, 100], 10001, params)
model.plot_traces()

Influenza model

A complex age-structured model for influenza with four age classes (0-2, 3-14, 15-59, >60 years). Each age class has compartments for susceptible, incubating, subclinical, clinical, and complicated individuals.

model = DM.Influenza()
md(str(model))

inits = [
    100, 0, 0, 0, 0,  # Age group 1 (0-2 years): S1, E1, Is1, Ic1, Ig1
    200, 0, 0, 0, 0,  # Age group 2 (3-14 years): S2, E2, Is2, Ic2, Ig2
    500, 0, 1, 0, 0,  # Age group 3 (15-59 years): S3, E3, Is3, Ic3, Ig3
    200, 0, 0, 0, 0   # Age group 4 (>60 years): S4, E4, Is4, Ic4, Ig4
]
params = {
    'beta': 0.0003,  # Transmission rate
    'r': 0.3,        # Recovery rate
    'e': 0.5,        # Incubation rate
    'c': 0.5,        # Clinical progression rate
    'g': 0.2,        # Complication rate
    'd': 0.1,        # Death/recovery rate from complications
    'pc1': 0.3,      # Clinical proportion age group 1
    'pc2': 0.3,      # Clinical proportion age group 2
    'pc3': 0.4,      # Clinical proportion age group 3
    'pc4': 0.6,      # Clinical proportion age group 4
    'pp1': 0.05,     # Complication proportion age group 1
    'pp2': 0.05,     # Complication proportion age group 2
    'pp3': 0.1,      # Complication proportion age group 3
    'pp4': 0.3,      # Complication proportion age group 4
    'b': 0           # Birth rate
}
model(inits, [0, 100], 1001, params)
model.plot_traces()