def Legrand_Ebola_SEIHFR(param=None):
'''
The Legrand Ebola model with 6 compartments that includes the
H = hospitalization and F = funeral state. Note that because this
is an non-autonomous system, there are in fact a total of 7 states
after conversion. The set of equations that describes the model are
.. math::
\\frac{dS}{dt} &= -(\\beta_{I}SI + \\beta_{H}SH + \\beta_{F}SF) \\\\
\\frac{dE}{dt} &= (\\beta_{I}SI + \\beta_{H}SH + \\beta_{F}SF) - \\alpha E \\\\
\\frac{dI}{dt} &= \\alpha E - (\\gamma_{H} \\theta_{1} + \\gamma_{I}(1-\\theta_{1})(1-\\delta_{1}) + \gamma_{D}(1-\\theta_{1})\\delta_{1})I \\\\
\\frac{dH}{dt} &= \\gamma_{H}\\theta_{1}I - (\\gamma_{DH}\\delta_{2} + \\gamma_{IH}(1-\\delta_{2}))H \\\\
\\frac{dF}{dt} &= \\gamma_{D}(1-\\theta_{1})\\delta_{1}I + \\gamma_{DH}\\delta_{2}H - \\gamma_{F}F \\\\
\\frac{dR}{dt} &= \\gamma_{I}(1-\\theta_{1})(1-\\delta_{1})I + \\gamma_{IH}(1-\\delta_{2})H + \\gamma_{F}F.
References
----------
.. [1] Understanding the dynamics of Ebola epidemics,
Legrand J. et al. Epidemiology and Infection,
Volume 135, Issue 4, pg 610-621, 2007
Examples
--------
>>> x0 = [1.0, 3.0/200000.0, 0.0, 0.0, 0.0, 0.0, 0.0]
>>> t = numpy.linspace(0, 25, 100)
>>> ode = common_models.Legrand_Ebola_SEIHFR([('beta_I',0.588),('beta_H',0.794),('beta_F',7.653),('omega_I',10.0/7.0),('omegaD',9.6/7.0),('omega_H',5.0/7.0),('omega_F',2.0/7.0),('alphaInv',7.0/7.0),('delta',0.81),('theta',0.80),('kappa',300.0),('interventionTime',7.0)]).setInitialValue(x0,t[0])
>>> solution = ode.integrate(t[1::])
>>> ode.plot()
'''
# define our states
state = ['S', 'E', 'I', 'H', 'F', 'R', 'tau']
# and initial parameters
params = ['beta_I', 'beta_H', 'beta_F',
'omega_I', 'omega_D', 'omega_H', 'omega_F',
'alphaInv', 'delta', 'theta',
'kappa', 'interventionTime']
# we now construct a list of the derived parameters
# which has 2 item
# name
# equation
derivedParamList = [
('gamma_I', '1/omega_I'),
('gamma_D', '1/omega_D'),
('gamma_H', '1/omega_H'),
('gamma_F', '1/omega_F'),
('alpha', '1/alphaInv'),
('gamma_IH', '1/((1/gamma_I) - (1/gamma_H))'),
('gamma_DH', '1/((1/gamma_D) - (1/gamma_H))'),
('delta_1', 'delta * gamma_I / (delta * gamma_I + (1 - delta) * gamma_D)'),
('delta_2', 'delta * gamma_IH / (delta * gamma_IH + (1 - delta) * gamma_DH)'),
('theta_A', 'theta * (gamma_I * (1 - delta_1) + gamma_D * delta_1)'),
('theta_1', 'theta_A/ (theta_A + (1 - theta) * gamma_H)'),
('beta_H_Time', 'beta_H * (1 - (1/ (1+exp(-kappa*(tau-interventionTime)))))'),
('beta_F_Time', 'beta_F * (1 - (1/ (1+exp(-kappa*(tau-interventionTime)))))')
]
# alternatively, we can do it on the operate ode model
ode = OperateOdeModel(state, params)
# add the derived parameter
ode.setDerivedParamList(derivedParamList)
# define the set of transitions
# name of origin state
# name of target state
# equation
# type of equation, which is a transition between two state in this case
transitionList = [
Transition(origState='S', destState='E',
equation='(beta_I * S * I + beta_H_Time * S * H + beta_F_Time * S * F)',
transitionType=TransitionType.T),
Transition(origState='E', destState='I',
equation='alpha * E',
transitionType=TransitionType.T),
Transition(origState='I', destState='H',
equation='gamma_H * theta_1 * I',
transitionType=TransitionType.T),
Transition(origState='I', destState='F',
equation='gamma_D * (1 - theta_1) * delta_1 * I',
transitionType=TransitionType.T),
Transition(origState='I', destState='R',
equation='gamma_I * (1 - theta_1) * (1 - delta_1) * I',
transitionType=TransitionType.T),
Transition(origState='H', destState='F',
equation='gamma_DH * delta_2 * H',
transitionType=TransitionType.T),
Transition(origState='H', destState='R',
equation='gamma_IH * (1 - delta_2) * H',
transitionType=TransitionType.T),
Transition(origState='F', destState='R',
equation='gamma_F * F',
transitionType=TransitionType.T)
]
#print transitionList
bdList = [Transition(origState='tau', equation='1',
transitionType=TransitionType.B)]
# see how we can insert the transitions later, after initializing the ode object
# this is not the preferred choice though
ode.setTransitionList(transitionList)
ode.setBirthDeathList(bdList)
# set return, depending on whether we have input the parameters
if param is None:
return ode
else:
ode.setParameters(param)
return ode
def Legrand_Ebola_SEIHFR(param=None):
'''
The Legrand Ebola model with 6 compartments that includes the
H = hospitalization and F = funeral state. Note that because this
is an non-autonomous system, there are in fact a total of 7 states
after conversion. The set of equations that describes the model are
.. math::
\\frac{dS}{dt} &= -(\\beta_{I}SI + \\beta_{H}SH + \\beta_{F}SF) \\\\
\\frac{dE}{dt} &= (\\beta_{I}SI + \\beta_{H}SH + \\beta_{F}SF) - \\alpha E \\\\
\\frac{dI}{dt} &= \\alpha E - (\\gamma_{H} \\theta_{1} + \\gamma_{I}(1-\\theta_{1})(1-\\delta_{1}) + \gamma_{D}(1-\\theta_{1})\\delta_{1})I \\\\
\\frac{dH}{dt} &= \\gamma_{H}\\theta_{1}I - (\\gamma_{DH}\\delta_{2} + \\gamma_{IH}(1-\\delta_{2}))H \\\\
\\frac{dF}{dt} &= \\gamma_{D}(1-\\theta_{1})\\delta_{1}I + \\gamma_{DH}\\delta_{2}H - \\gamma_{F}F \\\\
\\frac{dR}{dt} &= \\gamma_{I}(1-\\theta_{1})(1-\\delta_{1})I + \\gamma_{IH}(1-\\delta_{2})H + \\gamma_{F}F.
References
----------
.. [1] Understanding the dynamics of Ebola epidemics,
Legrand J. et al. Epidemiology and Infection,
Volume 135, Issue 4, pg 610-621, 2007
Examples
--------
>>> x0 = [1.0, 3.0/200000.0, 0.0, 0.0, 0.0, 0.0, 0.0]
>>> t = numpy.linspace(0, 25, 100)
>>> ode = common_models.Legrand_Ebola_SEIHFR([('beta_I',0.588),('beta_H',0.794),('beta_F',7.653),('omega_I',10.0/7.0),('omega_D',9.6/7.0),('omega_H',5.0/7.0),('omega_F',2.0/7.0),('alphaInv',7.0/7.0),('delta',0.81),('theta',0.80),('kappa',300.0),('interventionTime',7.0)]).setInitialValue(x0,t[0])
>>> solution = ode.integrate(t[1::])
>>> ode.plot()
'''
# define our states
state = ['S', 'E', 'I', 'H', 'F', 'R', 'tau']
# and initial parameters
params = ['beta_I', 'beta_H', 'beta_F',
'omega_I', 'omega_D', 'omega_H', 'omega_F',
'alphaInv', 'delta', 'theta',
'kappa', 'interventionTime']
# we now construct a list of the derived parameters
# which has 2 item
# name
# equation
derivedParamList = [
('gamma_I', '1/omega_I'),
('gamma_D', '1/omega_D'),
('gamma_H', '1/omega_H'),
('gamma_F', '1/omega_F'),
('alpha', '1/alphaInv'),
('gamma_IH', '1/((1/gamma_I) - (1/gamma_H))'),
('gamma_DH', '1/((1/gamma_D) - (1/gamma_H))'),
('delta_1', 'delta * gamma_I / (delta * gamma_I + (1 - delta) * gamma_D)'),
('delta_2', 'delta * gamma_IH / (delta * gamma_IH + (1 - delta) * gamma_DH)'),
('theta_A', 'theta * (gamma_I * (1 - delta_1) + gamma_D * delta_1)'),
('theta_1', 'theta_A/ (theta_A + (1 - theta) * gamma_H)'),
('beta_H_Time', 'beta_H * (1 - (1/ (1+exp(-kappa*(tau-interventionTime)))))'),
('beta_F_Time', 'beta_F * (1 - (1/ (1+exp(-kappa*(tau-interventionTime)))))')
]
# alternatively, we can do it on the operate ode model
ode = OperateOdeModel(state, params)
# add the derived parameter
ode.setDerivedParamList(derivedParamList)
# define the set of transitions
# name of origin state
# name of target state
# equation
# type of equation, which is a transition between two state in this case
transitionList = [
Transition(origState='S', destState='E',
equation='(beta_I * S * I + beta_H_Time * S * H + beta_F_Time * S * F)',
transitionType=TransitionType.T),
Transition(origState='E', destState='I',
equation='alpha * E',
transitionType=TransitionType.T),
Transition(origState='I', destState='H',
equation='gamma_H * theta_1 * I',
transitionType=TransitionType.T),
Transition(origState='I', destState='F',
equation='gamma_D * (1 - theta_1) * delta_1 * I',
transitionType=TransitionType.T),
Transition(origState='I', destState='R',
equation='gamma_I * (1 - theta_1) * (1 - delta_1) * I',
transitionType=TransitionType.T),
Transition(origState='H', destState='F',
equation='gamma_DH * delta_2 * H',
transitionType=TransitionType.T),
Transition(origState='H', destState='R',
equation='gamma_IH * (1 - delta_2) * H',
transitionType=TransitionType.T),
Transition(origState='F', destState='R',
equation='gamma_F * F',
transitionType=TransitionType.T)
]
#print transitionList
bdList = [Transition(origState='tau', equation='1',
transitionType=TransitionType.B)]
# see how we can insert the transitions later, after initializing the ode object
# this is not the preferred choice though
ode.setTransitionList(transitionList)
ode.setBirthDeathList(bdList)
# set return, depending on whether we have input the parameters
if param is None:
return ode
else:
ode.setParameters(param)
return ode
