Figure 9.25:

Envelope versus time for hepatitis B virus model; initial guess and estimated parameters fit to data.

Code for Figure 9.25

Text of the GNU GPL.

main.py

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# [makes] hbv_det.dat

from paresto import Paresto
from misc import octave_save
import numpy as np
import matplotlib.pyplot as plt
import os
from pathlib import Path
plt.ion()

# Define the model
pe = Paresto()
pe.print_level = 1
pe.nlp_solver_options = {'ipopt': {'mumps_scaling': 0}}

pe.x = ['ca', 'cb', 'cc']
pe.p = ['k1', 'k2', 'k3', 'k4', 'k5', 'k6']
pe.d = ['ma', 'mb', 'mc']

def hbv_rxs(t, y, p):
    kr = [10**p.k1,
          10**p.k2,
          10**p.k3,
          10**p.k4,
          10**p.k5,
          10**p.k6]
    rhs = [
        kr[1]*y.cb - kr[3]*y.ca,
        kr[0]*y.ca - kr[1]*y.cb - kr[5]*y.cb*y.cc,
        kr[2]*y.ca - kr[4]*y.cc - kr[5]*y.cb*y.cc
    ]
    return rhs

pe.ode = hbv_rxs

# Measurement weights
mweight = np.sqrt([1, 1e-2, 1e-4])
w = {'a': mweight[0], 'b': mweight[1], 'c': mweight[2]}

def lsq_weighted(t, y, p, w):
    retval = [
        w['a'] * (y.ca - y.ma),
        w['b'] * (y.cb - y.mb),
        w['c'] * (y.cc - y.mc)
    ]
    return retval

pe.lsq = lambda t, y, p: lsq_weighted(t, y, p, w)

# Reaction rate constants
krac = np.array([2, 0.025, 1000, 0.25, 1.9985, 7.5e-6])
thetaac = np.log10(krac)

p_names = pe.p
p_true = {fn: thetaac[i] for i, fn in enumerate(p_names)}

# Output times
tfinal = 100
ndata = 51
tdata = np.linspace(0, tfinal, ndata)
pe.tout = tdata

# Create a paresto instance
pe.initialize()

# Initial condition
small = 0
x0_ac = [1, small, small]

# Simulate using the paresto instance
pin_ac = dict(p_true)
pin_ac['ca'] = x0_ac[0]; pin_ac['cb'] = x0_ac[1]; pin_ac['cc'] = x0_ac[2]
y_ac, zsim = pe.simulate(pin_ac)
y_ac = y_ac.reshape(3, -1)
# Add noise — load from octave-generated file if available (for comparison)
_meas_file = Path('hbv_det_meas.dat')
if _meas_file.exists():
    y_noisy = np.loadtxt(_meas_file, comments='#')  # shape (3, ndata)
    y_noisy = np.maximum(y_noisy, 0)
else:
    np.random.seed(1)
    R = np.diag([0.1, 0.1, 0.1])**2
    noise = np.sqrt(R) @ np.random.normal(0,1,y_ac.shape)
    y_noisy = y_ac * (1 + noise)
    y_noisy = np.maximum(y_noisy, 0)

# Initial guess for rate constants
logkrinit = np.array([0.80, -1.13, 3.15, -0.77, -0.16, -5.46])
p_init = logkrinit.copy()

# Simulate outputs for the initial guess
pin_init = {fn: p_init[i] for i, fn in enumerate(p_names)}
pin_init['ca'] = x0_ac[0]; pin_init['cb'] = x0_ac[1]; pin_init['cc'] = x0_ac[2]
y_init, zsim = pe.simulate(pin_init)

# Initialize parameters
del_value = 1

pe.ic.k1 = logkrinit[0]
pe.ic.k2 = logkrinit[1]
pe.ic.k3 = logkrinit[2]
pe.ic.k4 = logkrinit[3]
pe.ic.k5 = logkrinit[4]
pe.ic.k6 = logkrinit[5]
pe.ic.ca = x0_ac[0]
pe.ic.cb = x0_ac[1]
pe.ic.cc = x0_ac[2]

pe.lb_ub()

for fn in p_names:
    setattr(pe.lb, fn, getattr(pe.ic, fn) - del_value)
    setattr(pe.ub, fn, getattr(pe.ic, fn) + del_value)
# State initial conditions are fixed (lb == ub == ic)

# Estimate parameters
est, y, p = pe.optimize(y_noisy.reshape(3,-1,1))

# Calculate confidence intervals with 95% confidence
conf = pe.confidence(est, 0.95)

print('Initial guess')
print({fn: logkrinit[i] for i, fn in enumerate(p_names)})

print('Estimated parameters')
print('\n'.join(f'{k}: {float(v)}' for k, v in est.theta.items()))

print('Bounding box intervals')
print('\n'.join(f'{k}: {float(v)}' for k, v in conf.bbox.items()))

print('True parameters')
print(thetaac)

# Eigenvalues/eigenvectors of the symmetric Hessian; eigh returns them
# sorted ascending, so column 0 is the weak (poorly-determined) direction.
lam, u = np.linalg.eigh(conf.H)

# Perturb parameters in the weak direction
logkrp = np.array(list(est.theta.values())) + 0.5 * u[:, 0]

# Simulate with perturbed parameters
pp = logkrp
pin_pp = {fn: pp[i] for i, fn in enumerate(p_names)}
pin_pp['ca'] = x0_ac[0]; pin_pp['cb'] = x0_ac[1]; pin_pp['cc'] = x0_ac[2]
yp, zsim = pe.simulate(pin_pp)

# Prepare data for saving
store1 = np.column_stack([pe.tout, y_ac.T, y_init.reshape(3,-1).T, yp.reshape(3,-1).T])
store2 = np.column_stack([pe.tout, y_noisy.T])
# Save store2 first then store1 (matches Octave: save hbv_det.dat store2 store1)
octave_save('hbv_det.dat',
    ('store2', store2),
    ('store1', store1),
)

# Plotting
if not os.getenv('OMIT_PLOTS') == 'true':
    fig, axs = plt.subplots(3, 2, figsize=(12, 10))

    # Plot 1
    axs[0, 0].plot(pe.tout, y_noisy[0, :], '+', label='Noisy Data')
    axs[0, 0].plot(pe.tout, y_ac[0], label='Original Data')
    axs[0, 0].plot(pe.tout, yp[0], label='Simulated Data')
    axs[0, 0].set_xlim(0, 100)
    axs[0, 0].set_ylim(0, 60)
    axs[0, 0].set_title('hbv_det_cccdna')


    # Plot 2
    axs[1, 0].plot(pe.tout, y_noisy[1, :], '+', label='Noisy Data')
    axs[1, 0].plot(pe.tout, y_ac[1], label='Original Data')
    axs[1, 0].plot(pe.tout, yp[1], label='Simulated Data')
    axs[1, 0].set_xlim(0, 100)
    axs[1, 0].set_ylim(0, 600)
    axs[1, 0].set_title('hbv_det_rcdna')

    # Plot 3
    axs[2, 0].plot(pe.tout, y_noisy[2, :], '+', label='Noisy Data')
    axs[2, 0].plot(pe.tout, y_ac[2], label='Original Data')
    axs[2, 0].plot(pe.tout, yp[2], label='Simulated Data')
    axs[2, 0].set_xlim(0, 100)
    axs[2, 0].set_ylim(0, 30000)
    axs[2, 0].set_title('hbv_det_env')

    # Plot 4
    axs[0, 1].plot(pe.tout, y_noisy[0, :], '+', label='Noisy Data')
    axs[0, 1].plot(pe.tout, y_ac[0], label='Original Data')
    axs[0, 1].plot(pe.tout, yp[0], label='Simulated Data')
    axs[0, 1].set_xlim(0, 100)
    axs[0, 1].set_ylim(0, 60)
    axs[0, 1].set_title('hbv_det_cccdnap')

    # Plot 5
    axs[1, 1].plot(pe.tout, y_noisy[1, :], '+', label='Noisy Data')
    axs[1, 1].plot(pe.tout, y_ac[1], label='Original Data')
    axs[1, 1].plot(pe.tout, yp[1], label='Simulated Data')
    axs[1, 1].set_xlim(0, 100)
    axs[1, 1].set_ylim(0, 600)
    axs[1, 1].set_title('hbv_det_rcndap')

    # Plot 6
    axs[2, 1].plot(pe.tout, y_noisy[2, :], '+', label='Noisy Data')
    axs[2, 1].plot(pe.tout, y_ac[2], label='Original Data')
    axs[2, 1].plot(pe.tout, yp[2], label='Simulated Data')
    axs[2, 1].set_xlim(0, 100)
    axs[2, 1].set_ylim(0, 30000)
    axs[2, 1].set_title('hbv_det_envp')

    plt.tight_layout()
    if not os.getenv('OMIT_PLOTS') == 'true':
        plt.show()