The model is described on the Compartmental Models Wikipedia Page.
I wrote a quick-and-dirty Dash app to eyeball good initial starting values for the parameters.
Quick & Dirty Dash App Code to visualize a continuous time SEIR model
import requests
import dash, json
import numpy as np
import pandas as pd
import datetime as dt
from functools import lru_cache
import dash_core_components as dcc
import dash_html_components as html
import dash_bootstrap_components as dbc
from pyswarm import pso
from plotly.graph_objs import *
from scipy.integrate import odeint
from plotly import graph_objs as go
from dash.dependencies import Input, Output
# @lru_cache(maxsize = 3)
def get_data(country = 'Spain'):
assert country in ['Spain', 'United Kingdom']
data = requests.get('https://pomber.github.io/covid19/timeseries.json')
data = pd.DataFrame(data.json()[country])
data = data.loc[data.deaths >= 10, :]
data.date = pd.to_datetime(data.date)
data.reset_index(inplace = True)
if country == 'Spain':
lockdown_date = dt.datetime(2020, 3, 15)
if country == 'United Kingdom':
lockdown_date = dt.datetime(2020, 3, 24)
return data, lockdown_date
def dydt(y, t, *args):
S, E, I, R = y; N = np.sum(y);
b_0, b_1, a, y, t_l = args
dxdt = np.zeros(4)
b = b_0 * np.exp(-b_1 * np.max([0, t - t_l])) # b_0 if t <= t_l else b_1
dxdt[0] = -b*S*I/N
dxdt[1] = b*S*I/N - a*E
dxdt[2] = a*E - y*I
dxdt[3] = y*I
return dxdt
def simulate(b_0 = 0.99, b_1 = 0.1, t_l = 50, n = 1e6, e_0 = 10, i_0 = 0, t = 100, infec_prd = 5, recov_prd = 7):
theta = (b_0, b_1, 1/infec_prd, 1/recov_prd, t_l)
initial_state = [n - e_0 - i_0, e_0, i_0, 0]
simuation = odeint(dydt, initial_state, np.arange(t), args = theta)
return simuation
app = dash.Dash(__name__, external_stylesheets = [dbc.themes.YETI])
app.layout = html.Div(children=[
html.Div(children=[
html.H1(children='SEIR Visualization'),
html.Label('Select country:'),
dbc.ListGroup([
dbc.ListGroupItem("United Kingdom", id="country_uk", n_clicks=0),
dbc.ListGroupItem("Spain", id="country_sp", n_clicks=1)]
),
dcc.Markdown('---'),
html.Label('β before intervention:'),
dcc.Slider(
id='b_0',
min=0,
max=5,
marks={i: 'Label {}'.format(i) if i == 1 else str(np.round(i, 1)) for i in np.linspace(0, 5, 10)},
value=1.15,
step=0.01
),
html.Label('q: (β after intervention = β exp[-q * time since intervention])'),
dcc.Slider(
id='b_1',
min=0,
max=1,
marks={i: 'Label {}'.format(i) if i == 1 else str(np.round(i, 1)) for i in np.linspace(0, 1, 10)},
value=0.125,
step=0.01
),
html.Label('Log of initial population exposed:'),
dcc.Slider(
id='e_0',
min=-7,
max=np.log(47e6),
marks={i: 'Label {}'.format(i) if i == 1 else str(int(np.round(i))) for i in np.linspace(-7, np.log(47e6), 10)},
step=0.01
),
html.Label('Infectious Period:'),
dcc.Slider(
id='infec_prd',
min=1,
max=14,
marks={i: 'Label {}'.format(i) if i == 1 else str(int(np.round(i))) for i in np.linspace(1, 14, 7)},
value=5.1,
step=0.1
),
html.Label('Removal Period:'),
dcc.Slider(
id='recov_prd',
min=1,
max=14,
marks={i: 'Label {}'.format(i) if i == 1 else str(int(np.round(i))) for i in np.linspace(1, 14, 7)},
value=7.1,
step=0.1
)
], style = {
"position": "fixed",
"top": 0,
"left": 0,
"bottom": 0,
"width": "19rem",
"padding": "2rem 1rem",
"background-color": "#f8f9fa",
}),
html.Div(children = [
dbc.Alert(id = "prop-rem", color="primary"),
dcc.Graph(id='viz-graph')], style = {
"margin-left": "18rem",
"margin-right": "2rem",
"padding": "2rem 1rem",
})
])
@app.callback(
[Output("country_uk", "active"),
Output("country_sp", "active")],
[Input("country_uk", "n_clicks"),
Input("country_sp", "n_clicks")]
)
def update_country(ncl_uk, ncl_sp):
print((ncl_uk, ncl_sp))
if ncl_uk is None:
return False, True
if (ncl_sp is None) or (ncl_sp >= ncl_uk):
return False, True
return True, False
@app.callback(
Output("e_0", "value"),
[Input("country_uk", "active"),
Input("country_sp", "active")]
)
def update_initial_exposed(country_uk, country_sp):
if country_uk:
e_0 = 9
if country_sp:
e_0 = 10
return e_0
@app.callback(
[Output("viz-graph", "figure"),
Output("prop-rem", "children")],
[Input("country_uk", "active"),
Input("country_sp", "active"),
Input("b_0", "value"),
Input("b_1", "value"),
Input("e_0", "value"),
Input("infec_prd", "value"),
Input("recov_prd", "value")]
)
def update_histogram(country_uk, country_sp, b_0, b_1, e_0, infec_prd, recov_prd):
if country_uk:
country = 'United Kingdom'
if country_sp:
country = 'Spain'
data, lockdown_date = get_data(country)
ifr = 0.01
n = len(data.deaths)
t_l = np.argwhere(np.array(data.date == lockdown_date))[0, 0]
simulated_seir = simulate(b_0 = b_0, b_1 = b_1, t_l = t_l, n = 47e6, t = n, e_0 = np.exp(e_0), i_0 = 0, infec_prd = infec_prd, recov_prd = recov_prd)
timestamps = data.loc[1:, 'date']
simulated_removals_diff = np.diff(simulated_seir[:, 3])
actual_removals_diff = np.diff(data.deaths)
return {
'data': [
{'x': timestamps, 'y': simulated_removals_diff, 'type': 'line', 'name': 'Predicted'},
{'x': timestamps, 'y': actual_removals_diff/ifr, 'type': 'line', 'name': country},
],
'layout': {
'title': 'Removed Population',
'yaxis': {'range': [1, 7], 'type': 'log'}
}
}, str('Proportion removed: ' + str(int(1000 - simulated_seir[-1, 0]/47e3)/10) + '%.')
if __name__ == "__main__":
app.run_server(debug=True, port=8080, host='0.0.0.0')A Discrete Time SEIR Model in JAGS
This is an implementation of the discrete time epidemiological (SEIR) model based on:
P. E. Lekone and B. F. Finkenstädt, “Statistical Inference in a Stochastic Epidemic SEIR Model with Control Intervention”, 2006.
I’ve made some changes to it, e.g. below, an intervention does not lead to an exponential decay of exposure probabilities - rather, the intervention considered here (a lockdown) just leads to lower exposure probabilities. If the population is large, the paths are very close to the model’s continuous time counterpart (the binomial variance is pretty small), so perhaps the stochastic treatment of the paths (and so many hidden states) isn’t necessary.
JAGS Code
model {
b_i ~ dexp(1)
b_m ~ dunif(0, b_i)
S[1] = N
E[1] = E_0
I[1] = 0
R[1] = 0
for(t in 1:(T - 1)) {
S[t + 1] = S[t] - B[t]
E[t + 1] = E[t] + B[t] - C[t]
I[t + 1] = I[t] + C[t] - D[t]
R[t + 1] = R[t] + D[t]
B[t] ~ dbin(Pr[t], S[t])
C[t] ~ dbin(1 - exp(-p), E[t])
D[t] ~ dbin(1 - exp(-y), I[t])
b[t] = ifelse(t <= T_l, b_m, b_i)
Pr[t] = 1 - exp(-b[t] * I[t] / N)
}
}2021
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