Symbolic Models¶
Provides an API to define epidemiological models in terms of sympy symbolic expressions.

class
epipack.symbolic_epi_models.
SymbolicEpiModel
(compartments, initial_population_size=1, correct_for_dynamical_population_size=False)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicMixin
,epipack.numeric_epi_models.EpiModel
Define a model based on the analytical framework offered by Sympy.
This class uses the eventbased framework where statechange vectors are associated with event rates.
 Parameters
compartments (list)  A list of
sympy.Symbol
instances that symbolize compartments.initial_population_size (float, default = 1.0)  The population size at \(t = 0\).
correct_for_dynamical_population_size (bool, default = False)  If
True
, the quadratic coupling terms will be divided by the sum of all compartments, otherwise they will be divided by the initial population size.

correct_for_dynamical_population_size
¶ If
True
, the quadratic coupling terms will be divided by the sum of all compartments, otherwise they will be divided by the initial population size. Type

birth_rate_functions
¶ A list of functions that return rate values based on time
t
and state vectory
. Each entry corresponds to an event update inself.birth_event_updates
. Type
list of symbolic expressions

birth_event_updates
¶ A list of vectors. Each entry corresponds to a rate in
birth_rate_functions
and quantifies the change in individual counts in the compartments. Type
list of sympy.Matrix

linear_rate_functions
¶ A list of functions that return rate values based on time
t
and state vectory
. Each entry corresponds to an event update inself.linear_event_updates
. Type
list of symbolic expressions

linear_event_updates
¶ A list of vectors. Each entry corresponds to a rate in
linear_rate_functions
and quantifies the change in individual counts in the compartments. Type
list of sympy.Matrix

quadratic_rate_functions
¶ A list of functions that return rate values based on time
t
and state vectory
. Each entry corresponds to an event update inself.quadratic_event_updates
. Type
list of symbolic expressions

quadratic_event_updates
¶ A list of vectors. Each entry corresponds to a rate in
quadratic_rate_functions
and quantifies the change in individual counts in the compartments. Type
list of sympy.Matrix

y0
¶ The initial conditions.
 Type
numpy.ndarray

rates_have_explicit_time_dependence
¶ Internal switch that's flipped when a nonconstant rate is passed to the model.
 Type

dydt
()[source]¶ Compute the momenta of the epidemiological model as symbolic expressions.
 Parameters
t (
float
)  Current timey (numpy.ndarray)  The entries correspond to the compartment frequencies (or counts, depending on population size).

set_linear_events
(event_list, allow_nonzero_column_sums=False, reset_events=True)[source]¶ Define the linear transition events between compartments.
 Parameters
A list of tuples that contains transition events in the following format:
[ ( ("affected_compartment_0",), rate, [ ("affected_compartment_0", dN0), ("affected_compartment_1", dN1), ... ], ), ... ]
allow_nonzero_column_sums (
bool
, default : False)  Traditionally, epidemiological models preserve the total population size. If that's not the case, switch off testing for this.reset_events (bool, default : True)  Whether to reset all linear events to zero before converting those.
Example
For an SEIR model with infectious period
tau
and incubation periodtheta
.epi.set_linear_events([ ( ("E",), 1/theta, [ ("E", 1), ("I", +1) ] ), ( ("I",), 1/tau, [ ("I", 1), ("R", +1) ] ), ])
Read as "compartment E reacts with rate \(1/\theta\) which leads to the decay of one E particle to one I particle."

set_quadratic_events
(event_list, allow_nonzero_column_sums=False, reset_events=True)[source]¶ Define the quadratic transition events between compartments.
 Parameters
A list of tuples that contains transmission events in the following format:
[ ( ("coupling_compartment_0", "coupling_compartment_1"), rate, [ ("affected_compartment_0", dN0), ("affected_compartment_1", dN1), ... ], ), ... ]
allow_nonzero_column_sums (
bool
, default : False)  Traditionally, epidemiological models preserve the total population size. If that's not the case, switch off testing for this.reset_events (bool, default : True)  Whether to reset all linear events to zero before converting those.
Example
For an SEIR model with infection rate
eta
.epi.set_quadratic_events([ ( ("S", "I"), eta, [ ("S", 1), ("E", +1) ] ), ])
Read as
"Coupling of S and I leads to the decay of one S particle to one E particle with rate \(\eta\).".

class
epipack.symbolic_epi_models.
SymbolicMixin
[source]¶ Bases:
object
Provides methods that are useful to both
epipack.symbolic_epi_models.SymbolicEpiModel
andepipack.symbolic_matrix_epi_models.SymbolicMatrixEpiModel

get_eigenvalues_at_disease_free_state
(disease_free_state=None)[source]¶ Obtain the Jacobian's eigenvalues at the disease free state.
 Parameters
disease_free_state (dict, default = None) 
A dictionary where a compartment symbol maps to an expression (the value of this compartment in the fixed point). If compartments are missing, it is implicitly assumed that this compartment has a value of zero.
If
None
, the disease_free_state is assumed to be atdisease_free_state = { S: 1 }
. Returns
eigenvalues  Each entry maps an eigenvalue expression to its multiplicity.
 Return type

get_eigenvalues_at_fixed_point
(fixed_point_dict)[source]¶ Obtain the Jacobian's eigenvalues at a given fixed point.
 Parameters
fixed_point_dict (dict)  A dictionary where a compartment symbol maps to an expression (the value of this compartment in the fixed point). If compartments are missing, it is implicitly assumed that this compartment has a value of zero.
 Returns
eigenvalues  Each entry maps an eigenvalue expression to its multiplicity.
 Return type

get_jacobian_at_fixed_point
(fixed_point_dict, simplify=True)[source]¶ Obtain the Jacobian at a given fixed point.
 Parameters
fixed_point_dict (dict)  A dictionary where a compartment symbol maps to an expression (the value of this compartment in the fixed point). If compartments are missing, it is implicitly assumed that this compartment has a value of zero.
simplify (bool)  whether or not to let sympy try to simplify the expressions
 Returns
J  The Jacobian matrix at the given fixed point.
 Return type
sympy.Matrix

get_numerical_dydt
(lambdify_modules='numpy')[source]¶ Returns values of the given compartments at the demanded time points (as a numpy.ndarray of shape
(return_compartments), len(time_points)
. Parameters
time_points (np.ndarray)  An array of time points at which the compartment values should be evaluated and returned.
return_compartments (list, default = None)  A list of compartments for which the result should be returned. If
return_compartments
is None, all compartments will be returned.integrator (str, default = 'dopri5')  Which method to use for integration. Currently supported are
'euler'
and'dopri5'
. If'euler'
is chosen, \(\delta t\) will be determined by the difference of consecutive entries intime_points
.adopt_final_state (bool, default = False)  Whether or not to adopt the final state of the integration
 Returns
dydt  A function
dydt(t, y, *args, **kwargs)
that returns the numerical momenta of this system at timet
and state vectory
. Return type
func

get_numerical_event_and_rate_functions
()[source]¶ Converts the symbolic event lists and corresponding symbolic rates to functions that return numeric event lists and numeric rates based on the current time and state vector.
This function is needed in the
epipack.numeric_epi_models.EpiModel
base class for stochastic simulations. Returns
get_event_rates (func)  A function that takes the current time
t
and state vectory
and returns numerical event rate lists.get_compartment_changes (funx)  A function that takes a numerical list of event
rates
and returns a random event state change vector with probability proportional to its entry inrates
.


class
epipack.symbolic_epi_models.
SymbolicODEModel
(ODEs)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicEpiModel
Define a model purely based on a list of ODEs.
 Parameters
ODEs (list) 
A list of symbolic ODEs in format
sympy.Eq(sympy.Derivative(Y, t), expr)

add_fission_processes
(*args, **kwargs)[source]¶ Define linear fission processes between compartments.
 Parameters
process_list (
list
oftuple
) A list of tuples that contains fission rates in the following format:
[ ("source_compartment", rate, "target_compartment_0", "target_compartment_1" ), ... ]
Example
For pure exponential growth of compartment B.
epi.add_fission_processes([ ("B", growth_event, "B", "B" ), ])

add_fusion_processes
(*args, **kwargs)[source]¶ Define fusion processes between compartments.
 Parameters
process_list (
list
oftuple
) A list of tuples that contains fission rates in the following format:
[ ("coupling_compartment_0", "coupling_compartment_1", rate, "target_compartment_0" ), ... ]
Example
Fusion of reactants "A", and "B" to form "C".
epi.add_fusion_processes([ ("A", "B", reaction_rate, "C" ), ])

add_linear_events
(*args, **kwargs)[source]¶ Add linear events without resetting the existing event terms. See
epipack.numeric_epi_models.EpiModel.set_linear_events()
for docstring.

add_quadratic_events
(*args, **kwargs)[source]¶ Add quadratic events without resetting the existing event terms. See
epipack.numeric_epi_models.EpiModel.set_quadratic_events()
for docstring.

add_transition_processes
(*args, **kwargs)[source]¶ Define the linear transition processes between compartments.
 Parameters
process_list (
list
oftuple
) A list of tuples that contains transitions events in the following format:
[ ( source_compartment, rate, target_compartment ), ... ]
Example
For an SEIR model.
epi.add_transition_processes([ ("E", symptomatic_rate, "I" ), ("I", recovery_rate, "R" ), ])

add_transmission_processes
(*args, **kwargs)[source]¶ A wrapper to define quadratic process rates through transmission reaction equations. Note that in stochastic network/agent simulations, the transmission rate is equal to a rate per link. For the meanfield ODEs, the rates provided to this function will just be equal to the prefactor of the respective quadratic terms.
For instance, if you analyze an SIR system and simulate on a network of mean degree \(k_0\), a basic reproduction number \(R_0\), and a recovery rate \(\mu\), you would define the single link transmission process as
("I", "S", R_0/k_0 * mu, "I", "I")
For the meanfield system here, the corresponding reaction equation would read
("I", "S", R_0 * mu, "I", "I")
 Parameters
process_list (
list
oftuple
) A list of tuples that contains transitions rates in the following format:
[ ("source_compartment", "target_compartment_initial", rate "source_compartment", "target_compartment_final", ), ... ]
Example
For an SEIR model.
epi.add_transmission_processes([ ("I", "S", +1, "I", "E" ), ])

set_linear_events
(*args, **kwargs)[source]¶ Define the linear transition events between compartments.
 Parameters
A list of tuples that contains transition events in the following format:
[ ( ("affected_compartment_0",), rate, [ ("affected_compartment_0", dN0), ("affected_compartment_1", dN1), ... ], ), ... ]
allow_nonzero_column_sums (
bool
, default : False)  Traditionally, epidemiological models preserve the total population size. If that's not the case, switch off testing for this.reset_events (bool, default : True)  Whether to reset all linear events to zero before converting those.
Example
For an SEIR model with infectious period
tau
and incubation periodtheta
.epi.set_linear_events([ ( ("E",), 1/theta, [ ("E", 1), ("I", +1) ] ), ( ("I",), 1/tau, [ ("I", 1), ("R", +1) ] ), ])
Read as "compartment E reacts with rate \(1/\theta\) which leads to the decay of one E particle to one I particle."

set_processes
(*args, **kwargs)[source]¶ Converts a list of reaction process tuples to event tuples and sets the rates for this model.
 Parameters
process_list (
list
oftuple
) A list containing reaction processes in terms of tuples.
[ # transition process ( source_compartment, rate, target_compartment), # transmission process ( coupling_compartment_0, coupling_compartment_1, rate, target_compartment_0, target_ccompartment_1), # fission process ( source_compartment, rate, target_compartment_0, target_ccompartment_1), # fusion process ( source_compartment_0, source_compartment_1, rate, target_compartment), # death process ( source_compartment, rate, None), # birth process ( None, rate, target_compartment), ]
allow_nonzero_column_sums (bool, default : False)  Traditionally, epidemiological models preserve the total population size. If that's not the case, switch off testing for this.
reset_events (bool, default : True)  If this is True, reset all events to zero before setting the new ones.
ignore_rate_position_checks (bool, default = False)  This function usually checks whether the rate of a reaction is positioned correctly. You can turn this behavior off for transition, birth, death, and transmission processes. (Useful if you want to define symbolic transmission processes that are compartmentdependent).

set_quadratic_events
(*args, **kwargs)[source]¶ Define the quadratic transition events between compartments.
 Parameters
A list of tuples that contains transmission events in the following format:
[ ( ("coupling_compartment_0", "coupling_compartment_1"), rate, [ ("affected_compartment_0", dN0), ("affected_compartment_1", dN1), ... ], ), ... ]
allow_nonzero_column_sums (
bool
, default : False)  Traditionally, epidemiological models preserve the total population size. If that's not the case, switch off testing for this.reset_events (bool, default : True)  Whether to reset all linear events to zero before converting those.
Example
For an SEIR model with infection rate
eta
.epi.set_quadratic_events([ ( ("S", "I"), eta, [ ("S", 1), ("E", +1) ] ), ])
Read as
"Coupling of S and I leads to the decay of one S particle to one E particle with rate \(\eta\).".

simulate
(*args, **kwargs)[source]¶ Returns values of the given compartments at the demanded time points (as a numpy.ndarray of shape
(return_compartments), len(time_points)
.If
return_compartments
is None, all compartments will be returned. Parameters
tmax (float)  maximum length of the simulation
return_compartments (list of compartments, default = None:)  The compartments for which to return time series. If
None
, all compartments will be returned.sampling_dt (float, default = None)  Temporal distance between samples of the compartment counts. If
None
, every change will be returned.sampling_callback (funtion, default = None)  A function that's called when a sample is taken
 Returns
t (numpy.ndarray)  times at which compartment counts have been sampled
result (dict)  Dictionary mapping a compartment to a time series of its count.

class
epipack.symbolic_epi_models.
SymbolicSIModel
(infection_rate, initial_population_size=1)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicEpiModel
An SI model derived from
epipack.symbolic_epi_models.SymbolicEpiModel
.

class
epipack.symbolic_epi_models.
SymbolicSIRModel
(infection_rate, recovery_rate, initial_population_size=1)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicEpiModel
An SIR model derived from
epipack.symbolic_epi_models.SymbolicEpiModel
.

class
epipack.symbolic_epi_models.
SymbolicSIRSModel
(infection_rate, recovery_rate, waning_immunity_rate, initial_population_size=1)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicEpiModel
An SIRS model derived from
epipack.symbolic_epi_models.SymbolicEpiModel
.

class
epipack.symbolic_epi_models.
SymbolicSISModel
(infection_rate, recovery_rate, initial_population_size=1)[source]¶ Bases:
epipack.symbolic_epi_models.SymbolicEpiModel
An SIS model derived from
epipack.symbolic_epi_models.SymbolicEpiModel
.

epipack.symbolic_epi_models.
get_temporal_interpolation
(time_data, value_data, interpolation_degree=1)[source]¶ Obtain a symbolic piecewise function that interpolates between values given in
value_data
for the intervals defined intime_data
, based on a spline interpolation of degreeinterpolation_degree
. Ifinterpolation_degree == 0
, the function changes according to step functions. In this casetime_data
needs to have one value more thanvalue_data
.The values in
time_data
andvalue_data
can be symbols or numeric values.