/
Euler_Newton_method.py
287 lines (259 loc) · 8.88 KB
/
Euler_Newton_method.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
import numba
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
from model import f_Wu, Df_Wu, model_Wu_init
from model import f_Rong, Df_Rong, model_Rong_init
from matplotlib import rc
rc('font',**{'family':'sans-serif','serif':['Computer Modern Roman'],'sans-serif':['Helvetica']})
rc('text', usetex=True)
matplotlib.rcParams['text.usetex'] = True
'''Defining simulation parameters'''
tol = 1e-8
t_0 = 0
t_end = 800 # first two years
h = 0.001
Nt = int(1 + (t_end - t_0) / h)
t = np.linspace(t_0, t_end, Nt)
assert t[-1] == t_end
'''model and parameters'''
u = np.zeros((5,Nt))
# para, u[:,0] = model_Wu_init()
para, u[:,0] = model_Rong_init()
@numba.jit(nopython=True)
def g(u_old, u_new, h, f_new):
return u_new - u_old - h * f_new
@numba.jit(nopython=True)
def Dg(h, Df):
return np.identity(5) - h * Df
@numba.jit(nopython=True)
def Newton(x_old, u_old, p = para):
J_old = Dg(h, Df_Rong(x_old, p=para))
g_old = g(u_old, x_old, h, f_Rong(x_old, p=para))
return x_old - np.dot(np.linalg.pinv(J_old), g_old)
def NDF2(x_old, u_old, x_init, h):
kappa = -1/9
gamma = 1.5
A = np.identity(5) - 0.6*h*Df_Wu(x_old)
invA = np.linalg.inv(A)
psi = 3 / 2 * u_old[:, -1] - 12/5 * u_old[:, -2] - 9/10 * u_old[:, -3]
rhs = 0.6*h*f_Wu(x_old) - psi - (x_old - x_init)
x_new = x_old + np.matmul(invA, rhs)
return x_new
solver = 'Euler'
if solver == 'Euler':
for i in range(1, Nt):
guess = u[:, i - 1] + h * f_Rong(u[:, i - 1])
x_old = guess
x_new = Newton(x_old, u[:, i-1])
k = 0
dev = np.abs(x_old - x_new)
while dev[0] > tol and dev[1] > tol and dev[2] > tol:
x_old = x_new
x_new = Newton(x_old, u[:, i-1])
dev = np.abs(x_old - x_new)
k += 1
u[:, i] = x_new
if not (i % 10000):
print('Solved at time ', str(i), ' after ', str(k), ' Newton iterations.')
elif solver == 'Wolfbrandt':
for i in range(1, Nt):
print(str(i))
f1 = f_Wu(u[:, i - 1], p=para)
Jf = Df_Wu(u[:, i - 1], p=para)
d = 1 / (2 + np.sqrt(2))
W = np.identity(5) - h * d * Jf
W_inv = np.linalg.inv(W)
k1 = np.matmul(W_inv, f1)
f2 = f_Wu(u[:, i - 1] + 2 / 3 * h * k1, p=para)
k2 = np.matmul(W_inv, f2 - 4 / 3 * h * d * np.matmul(Jf, k1))
u[:, i] = u[:, i - 1] + 0.25 * h * (k1 + 3 * k2)
elif solver == 'NDF2':
for i in range(1,3):
u[:, i] = u[:, 0]
# guess = u[:, i - 1] + h * f_Wu(u[:, i-1])
#
# x_old = guess
# x_new = Newton(x_old, u[:, i - 1])
# dev = np.abs(x_old - x_new)
# while dev[0] > tol and dev[1] > tol and dev[2] > tol:
# x_old = x_new
# x_new = Newton(x_old, u[:, i - 1])
# dev = np.abs(x_old - x_new)
# u[:, i] = x_new
for i in range(3, Nt):
print(str(i))
if i == 100:
print(str(i))
guess = 3*u[:,i-1] - 3*u[:,i-2] - u[:,i-3]
x_new = NDF2(guess,u[:,i-3:i],guess,h)
# try:
# if x_new == 0:
# print(str(i))
# except ValueError:
# pass
# dev = np.abs(x_old - x_new)
# while dev[0] > tol and dev[1] > tol and dev[2] > tol:
for m in range(0,2):
x_old = x_new
x_new = NDF2(x_old,u[:,i-3:i],guess,h)
# dev = np.abs(x_old - x_new)
u[:, i] = x_new
# if not (i % 100):
# print('Solved at time ', str(i), ' after ', str(k), ' Newton iterations.')
np.save('data/eRT_04_alpha_02/u.npy', u)
fig1 = plt.figure(1)
plt.plot(t, u[0,:]*10e-3, 'black', label = r'$\displaystyle uninfected \; T \; cells$')
plt.grid()
plt.xlim((0, 400))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [cells/\mu l]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.tight_layout()
plt.show()
plt.savefig('images/eRT_0_alpha_0/treated_T.pdf', dpi = 300, pad_inches=0.25)
# ttl1 = plt.title(r'$\displaystyle uninfected \;\; T \;\; cells$', fontsize = 15)
# ttl1.set_position([.5, 1.25])
# plt.subplot(3,1,2)
fig2 = plt.figure(2)
plt.yscale('log')
plt.plot(t, u[1,:], 'black', label = 'Ts')
plt.grid()
plt.xlim((0, 400))
# plt.legend()
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.show()
plt.tick_params(labelsize=16)
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_Ts.pdf', dpi = 300, pad_inches=0.25)
fig3 = plt.figure(3)
plt.yscale('log')
plt.plot(t, u[3,:], 'black', label = 'Vs')
plt.grid()
plt.xlim((0, 400))
# plt.ylim((-10, 10))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.show()
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_Vs.pdf', dpi = 300, pad_inches=0.25)
# plt.subplot(3,1,3)
fig4 = plt.figure(4)
plt.yscale('log')
plt.plot(t, u[2,:], 'black', label = 'Tr')
plt.grid()
# plt.legend()
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.show()
plt.tight_layout()
plt.tick_params(labelsize=16)
plt.savefig('images/eRT_0_alpha_0/treated_Tr.pdf', dpi = 300, pad_inches=0.25)
fig5 = plt.figure(5)
plt.yscale('log')
plt.plot(t, u[4,:], 'black', label = 'Vr')
plt.grid()
plt.xlim((0, 400))
# plt.ylim((-15, 5))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.show()
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_Vr.pdf', dpi = 300, pad_inches=0.25)
fig6 = plt.figure(6)
plt.yscale('log')
plt.plot(t, u[4,:] + u[3,:], 'black', label = 'Vr')
plt.grid()
plt.xlim((0, 400))
# plt.ylim((-10, 10))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.show()
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_total_V.pdf', dpi = 300, pad_inches=0.25)
fig7 = plt.figure(7)
plt.yscale('log')
plt.plot(t, u[4,:] + u[3,:], '0.65', label = '$\displaystyle V_r + V_s$')
plt.plot(t, u[3,:], 'black', Linestyle = ':', label = '$\displaystyle V_s$')
plt.plot(t, u[4,:], 'black', Linestyle = '--', label = '$\displaystyle V_r$')
plt.grid()
plt.legend()
plt.xlim((0, 400))
# plt.ylim((-10, 10))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.show()
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_overview_V.pdf', dpi = 300, pad_inches=0.25)
fig8 = plt.figure(8)
plt.yscale('log')
plt.plot(t, u[2,:] + u[1,:], '0.65', label = '$\displaystyle T_r + T_s$')
plt.plot(t, u[1,:], 'black', Linestyle = ':', label = '$\displaystyle T_s$')
plt.plot(t, u[2,:], 'black', Linestyle = '--', label = '$\displaystyle T_r$')
plt.grid()
plt.legend()
plt.xlim((0, 400))
# plt.ylim((-10, 10))
plt.xlabel(r'$\displaystyle time \; [days]$', labelpad=10, fontsize = 18)
plt.ylabel(r'$\displaystyle cell \; concentration \; [log_{10}(cells/ml^3)]$', labelpad=10, fontsize=18)
plt.tick_params(labelsize=16)
plt.show()
plt.tight_layout()
plt.savefig('images/eRT_0_alpha_0/treated_overview_infected_T.pdf', dpi = 300, pad_inches=0.25)
# plt.subplots_adjust(wspace = 0.4)
# plt.show()
#
# fig2 = plt.figure(2)
# plt.subplot(2,2,1)
# plt.plot(t, u[0,:], 'green', label = 'T')
# plt.grid()
# plt.legend()
# # plt.xlabel('time (days)')
# # plt.ylabel('cell count (cells/mm^3)')
#
# # plt.subplot(2,2,2)
# # plt.yscale('log')
# # plt.plot(t, u[1,:], 'yellow', label = 'Ts (wild Type)')
# # plt.grid()
# # plt.legend()
# # # plt.xlabel('time (days)')
# # plt.ylabel('cell count (cells/mm^3)')
#
# plt.subplot(2,2,2)
# plt.yscale('log')
# plt.plot(t, u[3,:], 'yellow', Linestyle = '--', label = 'Vs (wild type)')
# plt.grid()
# plt.legend()
# # plt.xlabel('time (days)')
# plt.ylabel('cell count (cells/mm^3)')
#
# # plt.subplot(2,2,3)
# # plt.yscale('log')
# # plt.plot(t, u[2,:], 'red', label = 'Tr')
# # plt.grid()
# # plt.legend()
# # plt.xlabel('time (days)')
#
# plt.subplot(2,2,3)
# plt.yscale('log')
# plt.plot(t, u[4,:], 'red', Linestyle = '--', label = 'Vr (resistant)')
# plt.grid()
# plt.legend()
# plt.xlabel('time (days)')
# # plt.ylabel('cell count (cells/mm^3)')
#
# plt.subplot(2,2,4)
# plt.yscale('log')
# plt.plot(t, u[4,:] + u[3,:], 'red', Linestyle = '--', label = 'Vr + Vs')
# plt.grid()
# plt.legend()
# plt.xlabel('time (days)')
#
# plt.subplots_adjust(wspace = 0.4)
# plt.show()
print('Finish')