def test_quadratic_processes(self):
S, E, I, R = sympy.symbols("S E I R")
epi = SymbolicEpiModel([S, E, I, R])
quadratic_rates = [I * S]
quadratic_events = [Matrix([[-1, +1, 0, 0]])]
epi.add_transmission_processes([
(S, I, 1, I, E),
])
for r0, r1 in zip(quadratic_rates, epi.quadratic_rate_functions):
assert (r0 == r1)
for e0, e1 in zip(quadratic_events, epi.quadratic_event_updates):
assert (all([_e0 == _e1 for _e0, _e1 in zip(e0, e1)]))
def symbolic_simulation_temporally_forced():
from epipack import SymbolicEpiModel
import sympy as sy
from epipack.plottools import plot
import numpy as np
S, I, R, eta, rho, omega, t, T = \
sy.symbols("S I R eta rho omega t T")
N = 1000
SIRS = SymbolicEpiModel([S,I,R],N)\
.set_processes([
(S, I, 3+sy.cos(2*sy.pi*t/T), I, I),
(I, rho, R),
(R, omega, S),
])
SIRS.set_parameter_values({
rho: 1,
omega: 1 / 14,
T: 100,
})
SIRS.set_initial_conditions({S: N - 100, I: 100})
_t = np.linspace(0, 150, 1000)
result = SIRS.integrate(_t)
t_sim, result_sim = SIRS.simulate(max(_t))
ax = plot(_t, result)
plot(t_sim, result_sim, ax=ax)
ax.get_figure().savefig('symbolic_model_time_varying_rate.png', dpi=300)
def test_stochastic_well_mixed(self):
S, E, I, R = sympy.symbols("S E I R")
N = 75000
tmax = 100
model = SymbolicEpiModel([S, E, I, R], N)
model.set_processes([
(S, I, 2, E, I),
(I, 1, R),
(E, 1, I),
])
model.set_initial_conditions({S: N - 100, I: 100})
tt = np.linspace(0, tmax, 10)
result_int = model.integrate(tt)
t, result_sim = model.simulate(tmax,
sampling_dt=1,
return_compartments=[S, R])
for c, res in result_sim.items():
#print(c, np.abs(1-res[-1]/result_int[c][-1]))
#print(c, np.abs(1-res[-1]/result_sim[c][-1]))
assert (np.abs(1 - res[-1] / result_int[c][-1]) < 0.05)
def test_linear_rates(self):
S, E, I, R = sympy.symbols("S E I R")
epi = SymbolicEpiModel([S, E, I, R])
epi.add_transition_processes([
(E, 1, I),
(I, 1, R),
])
linear_rates = [E, I]
linear_events = [Matrix([[0, -1, +1, 0]]), Matrix([[0, 0, -1, +1]])]
for r0, r1 in zip(linear_rates, epi.linear_rate_functions):
assert (r0 == r1)
for e0, e1 in zip(linear_events, epi.linear_event_updates):
assert (all([_e0 == _e1 for _e0, _e1 in zip(e0, e1)]))
def test_time_dependent_rates(self):
B, t = sympy.symbols("B t")
epi = SymbolicEpiModel([B])
epi.add_fission_processes([
(B, t, B, B),
])
epi.set_initial_conditions({B: 1})
result = epi.integrate([0, 3], adopt_final_state=True)
assert (np.isclose(epi.y0[0], np.exp(3**2 / 2)))
epi.set_initial_conditions({B: 1})
result = epi.integrate(np.linspace(0, 3, 10000),
integrator='euler',
adopt_final_state=True)
eul = epi.y0[0]
real = np.exp(3**2 / 2)
assert (np.abs(1 - eul / real) < 1e-2)
def test_interactive_integrator(self):
S, I, R, R0, tau, omega = sympy.symbols("S I R R_0 tau omega")
I0 = 0.01
model = SymbolicEpiModel([S,I,R])\
.set_processes([
(S, I, R0/tau, I, I),
(I, 1/tau, R),
(R, omega, S),
])\
.set_initial_conditions({S:1-I0, I:I0})
parameters = {
R0: LogRange(min=0.1, max=10, step_count=1000),
tau: Range(min=0.1, max=10, value=8.0),
omega: 1 / 14
}
t = np.linspace(0.01, 200, 1000)
integrator = InteractiveIntegrator(model,
parameters,
t,
figsize=(3, 4),
return_compartments=[S, I])
keys = sorted([str(k) for k in integrator.sliders.keys()])
assert (all([a == b for a, b in zip(sorted(['R_0', 'tau']), keys)]))
integrator.update_parameters()
keys = sorted([str(k) for k in integrator.lines.keys()])
assert (all([a == b for a, b in zip(sorted(['I', 'S']), keys)]))
class change():
new = True
integrator.update_xscale(change())
assert (integrator.ax.get_xscale() == 'log')
assert (integrator.ax.get_yscale() == 'linear')
integrator.update_yscale(change())
assert (integrator.ax.get_xscale() == 'log')
assert (integrator.ax.get_yscale() == 'log')
integrator = InteractiveIntegrator(model,
parameters,
t,
figsize=(3, 4))
keys = sorted([str(k) for k in integrator.lines.keys()])
assert (all([
a == b
for a, b in zip(sorted([str(C) for C in model.compartments]), keys)
]))
def test_basic_analytics(self):
S, I, R, eta, rho, omega, t = symbols("S I R eta rho omega, t")
SIRS = SymbolicEpiModel([S, I, R])
SIRS.set_processes([
#### transmission process ####
# S + I (eta)-> I + I
(S, I, eta, I, I),
#### transition processes ####
# I (rho)-> R
# R (omega)-> S
(I, rho, R),
(R, omega, S),
])
odes = SIRS.ODEs()
expected = [
Eq(Derivative(S, t), -I * S * eta + R * omega),
Eq(Derivative(I, t), I * (S * eta - rho)),
Eq(Derivative(R, t), I * rho - R * omega)
]
assert (all([got == exp for got, exp in zip(odes, expected)]))
fixed_points = SIRS.find_fixed_points()
expected = FiniteSet((S, 0, 0), (rho / eta, R * omega / rho, R))
assert (all([got == exp for got, exp in zip(fixed_points, expected)]))
J = SIRS.jacobian()
expected = Matrix([[-I * eta, -S * eta, omega],
[I * eta, S * eta - rho, 0], [0, rho, -omega]])
N = SIRS.N_comp
assert (all(
[J[i, j] == expected[i, j] for i in range(N) for j in range(N)]))
eig = SIRS.get_eigenvalues_at_disease_free_state()
expected = {-omega: 1, eta - rho: 1, 0: 1}
assert (all([v == expected[k] for k, v in eig.items()]))
def test_exceptions(self):
B, mu, t = sympy.symbols("B mu t")
epi = SymbolicEpiModel([B])
epi.add_fission_processes([
(B, mu, B, B),
])
epi.set_initial_conditions({B: 1})
self.assertRaises(ValueError, epi.integrate, [0, 1])
self.assertRaises(ValueError, SymbolicEpiModel, [t])
self.assertRaises(ValueError,
epi.get_eigenvalues_at_disease_free_state)
def test_adding_quadratic_processes(self):
S, E, I, A, R, rho = sympy.symbols("S E I A R rho")
epi = SymbolicEpiModel([S, E, I, A, R])
epi.set_processes([
(S, I, rho, I, E),
])
epi.add_transmission_processes([
(S, A, rho, A, E),
])
quadratic_rates = [I * S * rho, S * A * rho]
quadratic_events = [
Matrix([[-1, +1, 0, 0, 0]]),
Matrix([[-1, +1, 0, 0, 0]])
]
for r0, r1 in zip(quadratic_rates, epi.quadratic_rate_functions):
assert (r0 == r1)
for e0, e1 in zip(quadratic_events, epi.quadratic_event_updates):
assert (all([_e0 == _e1 for _e0, _e1 in zip(e0, e1)]))
def test_changing_population_size(self):
A, B, C, t = sympy.symbols("A B C t")
epi = SymbolicEpiModel([A, B, C],
10,
correct_for_dynamical_population_size=True)
epi.set_initial_conditions({A: 5, B: 5})
epi.set_processes([
(A, B, 1, C),
], allow_nonzero_column_sums=True)
dydt = epi.dydt()
assert (dydt[0] == -1 * A * B / (A + B + C))
_, res = epi.simulate(1e9)
assert (res[C][-1] == 5)
epi.set_processes([
(None, 1 + sympy.log(1 + t), A),
(A, 1 + sympy.log(1 + t), B),
(B, 1 + sympy.log(1 + t), None),
],
allow_nonzero_column_sums=True)
rates, comp_changes = epi.get_numerical_event_and_rate_functions()
_, res = epi.simulate(200, sampling_dt=0.05)
vals = np.concatenate([res[A][_ > 10], res[B][_ > 10]])
rv = poisson(vals.mean())
measured, bins = np.histogram(vals,
bins=np.arange(10) - 0.5,
density=True)
theory = [
rv.pmf(i) for i in range(0,
len(bins) - 1) if measured[i] > 0
]
experi = [
measured[i] for i in range(0,
len(bins) - 1) if measured[i] > 0
]
# make sure the kullback-leibler divergence is below some threshold
assert (entropy(theory, experi) < 2e-3)
assert (np.median(res[A]) == 1)
def test_compartments(self):
comps = sympy.symbols("S E I R")
epi = SymbolicEpiModel(comps)
assert (all(
[i == epi.get_compartment_id(C) for i, C in enumerate(comps)]))
print()
print(epi.ODEs())
print(epi.find_fixed_points())
omega = sympy.symbols("omega")
epi = SymbolicSIRSModel(eta, rho, omega)
print()
print(epi.ODEs())
print(epi.find_fixed_points())
import sympy
from epipack import SymbolicEpiModel
S, I, eta, rho = sympy.symbols("S I eta rho")
SIS = SymbolicEpiModel([S, I])
SIS.add_transmission_processes([
(I, S, eta, I, I),
])
SIS.add_transition_processes([
(I, rho, S),
])
print(SIS.find_fixed_points())
print(SIS.get_eigenvalues_at_fixed_point({S: 1}))
print("==========")
SIS = SymbolicEpiModel([S, I])
SIS.set_processes([
(I, S, eta / (1 - I), I, I),