Apps

## Apps.MeaslesVirus History

----

!!!! Solution (Python GEKKO)

%width=550px%Attach:measles_dynamics.png

(:toggle hide gekko button show="Show GEKKO (Python) Code":)
(:div id=gekko:)
Thanks to Cameron Price for providing the GEKKO Python code.

'''Estimate 26 beta values and gamma'''

(:source lang=python:)
import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt

# Import Data
# write csv data file
t_s = np.linspace(0,78,79)
# case data
cases_s = np.array([180,180,271,423,465,523,649,624,556,420,\
423,488,441,268,260,163,83,60,41,48,65,82,\
145,122,194,237,318,450,671,1387,1617,2058,\
3099,3340,2965,1873,1641,1122,884,591,427,282,\
174,127,84,97,68,88,79,58,85,75,121,174,209,458,\
742,929,1027,1411,1885,2110,1764,2001,2154,1843,\
1427,970,726,416,218,160,160,188,224,298,436,482,468])

# Initialize gekko model
m = GEKKO()

# Number of collocation nodes
nodes = 4

# Number of phases (years in this case)
n = 3

#Biweek periods per year
bi = 26

# Time horizon (for all 3 phases)
m.time = np.linspace(0,1,bi+1)

# Parameters that will repeat each year
N        = m.Param(3.2e6)
mu      = m.Param(7.8e-4)
rep_frac = m.Param(0.45)
Vr      = m.Param(0)

beta    = m.MV(2,lb = 0.1)
beta.STATUS = 1

gamma = m.FV(value=0.07)
gamma.STATUS = 1
gamma.LOWER = 0.05
gamma.UPPER = 0.5

# Data used to control objective function
casesobj1 = m.Param(cases_s[0:(bi+1)])
casesobj2 = m.Param(cases_s[bi:(2*bi+1)])
casesobj3 = m.Param(cases_s[2*bi:(3*bi+1)])

# Variables that vary between years, one version for each year
cases = [m.CV(value = cases_s[(i*bi):(i+1)*(bi+1)-i],lb=0) for i in range(n)]
for i in cases:
i.FSTATUS = 1
i.WMODEL = 0
i.MEAS_GAP = 100

S = [m.Var(0.06*N,lb = 0,ub = N) for i in range(n)]
I = [m.Var(0.001*N, lb = 0,ub = N) for i in range(n)]
V = [m.Var(2e5) for i in range(n)]

# Equations (created for each year)
for i in range(n):
R = m.Intermediate(beta*S[i]*I[i]/N)
m.Equation(S[i].dt() == -R + mu*N - Vr)
m.Equation(I[i].dt() == R - gamma*I[i])
m.Equation(cases[i] == rep_frac*R)
m.Equation(V[i].dt() == -Vr)

# Connect years together at endpoints
for i in range(n-1):
m.Connection(cases[i+1],cases[i],1,bi,1,nodes)#,1,nodes)
m.Connection(cases[i+1],'CALCULATED',pos1=1,node1=1)
m.Connection(S[i+1],S[i],1,bi,1,nodes)
m.Connection(S[i+1],'CALCULATED',pos1=1,node1=1)
m.Connection(I[i+1],I[i],1,bi,1,nodes)
m.Connection(I[i+1],'CALCULATED',pos1=1, node1=1)

# Solver options
m.options.IMODE = 5
m.options.NODES = nodes
m.EV_TYPE = 1
m.options.SOLVER = 1

# Solve
m.Obj(2*(casesobj1-cases[0])**2+(casesobj3-cases[2])**2)

m.solve()

# Calculate the start time of each phase
ts = np.linspace(1,n,n)

# Plot
plt.figure()
plt.subplot(4,1,1)
tm = np.empty(len(m.time))
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,cases[i].value,label='Cases Year %s'%(i+1))
plt.plot(tm,cases_s[(i*bi):(i+1)*(bi+1)-i],'.')
plt.legend()
plt.ylabel('Cases')

plt.subplot(4,1,2)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,beta.value,label='Beta Year %s'%(i+1))
plt.legend()
plt.ylabel('Contact Rate')

plt.subplot(4,1,3)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,I[i].value,label='I Year %s'%(i+1))
plt.legend()
plt.ylabel('Infectives')

plt.subplot(4,1,4)
for i in range(n):
tm = m.time + ts[i]
plt.plot(tm,S[i].value,label='S Year %s'%(i+1))
plt.legend()
plt.ylabel('Susceptibles')
plt.xlabel('Time (yr)')

plt.show()
(:sourceend:)
(:divend:)
February 16, 2019, at 02:15 PM by 173.117.225.218 -
Changed line 7 from:
Understanding the spread of the measles virus from historical data of major metropolitan areas will help researchers understand the fundamentals of disease spread through a population.  This may help guide policy for school closure, travel restrictions, and other measures intended to slow disease spread until a suitable vaccine can be developed.
to:
Understanding the spread of the measles virus from historical data of major metropolitan areas will help researchers understand the fundamentals of disease spread through a population.  This may help guide policy for school closure, travel restrictions, and other measures intended to slow disease spread until a vaccine can be developed or deployed.
February 16, 2019, at 02:03 PM by 173.117.225.218 -
Changed lines 76-78 from:
!!! Reference

Daniel P. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.
to:
!!! References

*
Daniel P. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, pp. 5137-5142, Baltimore, MD, June 29 - July 2, 2010.

* Daniel P. Word, James K. Young, Derek Cummings, Carl D. Laird, "Estimation of seasonal transmission parameters in childhood infectious disease using a stochastic continuous time model", Computer Aided Chemical Engineering, pp. 229-234, 28, 2010
.
February 16, 2019, at 01:59 PM by 173.117.225.218 -
Deleted lines 3-4:

Changed line 27 from:
(:html:)<font size=2><pre>
to:
(:source lang=python:)
Changed lines 68-69 from:
</pre></font>(:htmlend:)
to:
(:sourceend:)
Changed line 81 from:
Daniel P. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.
to:
Daniel P. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/UTTlZ6rVXxM" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

Changed lines 79-81 from:
!!! References

Daniel P. Word, George H
. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.
to:
!!! Reference

Daniel P
. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.

November 25, 2010, at 03:13 AM by 206.180.155.75 -
(:title Simulation of Infectious Disease Spread:)
(:keywords nonlinear, model, predictive control, APMonitor, differential, algebraic, modeling language, infectious disease spread, measles nonlinear dynamics:)
(:description Simulation of infectious disease spread (measles) with APMonitor:)

November 16, 2010, at 04:51 AM by 206.180.155.75 -
Changed line 9 from:
In this case, the starting population of 3,200,000 is composed of a group of Susceptible, Infectived, and Recovered individuals.  To account for underreporting of measles cases, a reporting factor is used in the model.  In this study, it is assumed that only 45% of the measles cases were reported.
to:
In this case, the starting population of 3,200,000 is composed of a group of Susceptible, Infected, and Recovered individuals.  To account for underreporting of measles cases, a reporting factor is used in the model.  In this study, it is assumed that only 45% of the measles cases were reported.
November 16, 2010, at 04:48 AM by 206.180.155.75 -
Changed lines 3-4 from:
Measles (sometimes known as English Measles) is spread through respiration (contact with fluids from an infected person's nose and mouth, either directly or through aerosol transmission), and is highly contagious—90% of people without immunity sharing living space with an infected person will catch it. The infection has an average incubation period of 14 days (range 6–19 days) and infectivity lasts from 2–4 days prior, until 2–5 days following the onset of the rash (i.e. 4–9 days infectivity in total).  Understanding the spread of the measles virus from historical data of major metropolitan areas will help researchers understand the spread of disease through a population.  In this case, the starting population of 3,200,000 is composed of a group of Susceptible, Infectived, and Recovered individuals.
to:
Measles (sometimes known as English Measles) is spread through respiration (contact with fluids from an infected person's nose and mouth, either directly or through aerosol transmission), and is highly contagious—90% of people without immunity sharing living space with an infected person will catch it. The infection has an average incubation period of 14 days (range 6–19 days) and infectivity lasts from 2–4 days prior, until 2–5 days following the onset of the rash (i.e. 4–9 days infectivity in total).

the measles virus from historical data of major metropolitan areas will help researchers understand the fundamentals of disease spread through a population.  This may help guide policy for school closure, travel restrictions, and other measures intended to slow disease spread until a suitable vaccine can be developed.
Changed lines 9-10 from:
To account for underreporting of measles cases, a reporting factor is used in the model.  In this case, it is assumed that only 45% of the measles cases were reported as new cases.
to:
In this case, the starting population of 3,200,000 is composed of a group of Susceptible, Infectived, and Recovered individuals.  To account for underreporting of measles cases, a reporting factor is used in the model.  In this study, it is assumed that only 45% of the measles cases were reported.
Changed line 17 from:
The seasonal transmission parameter can be used to estimate spread of other diseases.  The transmission parameter is the number of contact (on average) made with other individuals during a biweek period.
to:
The seasonal transmission parameter can also be used to estimate spread of other diseases.  The transmission parameter is the number of contact (on average) made with other individuals during a biweek period.  The transmission parameter can sometimes be decreased for brief periods of time to slow the spread of a disease.  Also, the number of susceptible individuals can be reduced through the use of vaccines.  In the case of limited vaccine supply, disease spread models can help identify the best use the resources to limit large outbreaks.
November 16, 2010, at 04:35 AM by 206.180.155.75 -
Measles (sometimes known as English Measles) is spread through respiration (contact with fluids from an infected person's nose and mouth, either directly or through aerosol transmission), and is highly contagious—90% of people without immunity sharing living space with an infected person will catch it. The infection has an average incubation period of 14 days (range 6–19 days) and infectivity lasts from 2–4 days prior, until 2–5 days following the onset of the rash (i.e. 4–9 days infectivity in total).  Understanding the spread of the measles virus from historical data of major metropolitan areas will help researchers understand the spread of disease through a population.  In this case, the starting population of 3,200,000 is composed of a group of Susceptible, Infectived, and Recovered individuals.

To account for underreporting of measles cases, a reporting factor is used in the model.  In this case, it is assumed that only 45% of the measles cases were reported as new cases.

Attach:measles_cases.png

The new case data shows a seasonal variation that is caused by an increased rate of contact between the susceptible population and the infective population.  Based on the new case data (cases), birth rate (mu), and recovery rate (gamma), the seasonal variation of the transmission parameter is estimated over a period of 20 years.

Attach:measles_transmission.png

The seasonal transmission parameter can be used to estimate spread of other diseases.  The transmission parameter is the number of contact (on average) made with other individuals during a biweek period.
November 16, 2010, at 04:07 AM by 206.180.155.75 -
Changed line 3 from:
Attach:sofc_overview.png
to:
Attach:measles_populations.png
November 16, 2010, at 03:54 AM by 206.180.155.75 -
Changed line 18 from:
! birth rate            (births/biweek/total population)
to:
! birth rate            (births/biweek/total population)
November 16, 2010, at 03:46 AM by 206.180.155.75 -

Attach:sofc_overview.png

----

(:html:)<font size=2><pre>
! APMonitor Modeling Language
!
! Data from New York for the years 1947-1965
! Data from Bangkok for the years 1975-1984
! Data from London is 1944-1966?
!
Model disease
Parameters
! population size        (individuals)
N          = 3.2e6
! birth rate            (births/biweek/total population)
mu          = 7.8e-4
! recovery rate          (recoveries/biweek/infectives)
gamma      = 0.07
! reporting fraction to account for underreporting
rep_frac    = 0.45
! cases reported        (new individuals infected per biweek)
cases = 180, >= 0
End Parameters

Variables
! transmission parameter (potentially infectious contacts/biweek)
beta = 10
! susceptibles          (individuals in the total population)
S = 0.06*N, >= 0, <= N
! infectives            (individuals infected)
I = 0.0001*N, >= 0, <= N
End Variables

Intermediates
! infection rate per biweek
R = beta * S * I / N
End Intermediates

Equations
$S = -R + mu * N$I =  R -  gamma * I
cases = rep_frac * R
End Equations
End Model
</pre></font>(:htmlend:)

----

!!! References

Daniel P. Word, George H. Abbott, Derek Cummings, and Carl D. Laird, "Estimating Seasonal Drivers in Childhood Infectious Diseases with Continuous Time and Discrete-Time Models", in Proceedings, 2010 American Control Conference, Baltimore, MD, June 29 - July 2, 2010, p. 5137-5142.