https://github.com/blab/antibody-response-pulse
In [9]:
'''
author: Alvason Zhenhua Li
date: 04/09/2015
'''
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import os
from matplotlib.ticker import FuncFormatter
import alva_machinery_event_bcell as alva
AlvaFontSize = 23
AlvaFigSize = (15, 5)
numberingFig = 0
# plotting
dir_path = '/Users/al/Desktop/GitHub/antibody-response-pulse/bcell-array/figure'
file_name = 'Virus-Bcell-IgM-IgG'
figure_name = '-equation'
file_suffix = '.png'
save_figure = os.path.join(dir_path, file_name + figure_name + file_suffix)
numberingFig = numberingFig + 1
plt.figure(numberingFig, figsize=(12, 6))
plt.axis('off')
plt.title(r'$ Virus-Bcell-IgM-IgG \ equations \ (antibody-response \ for \ sequential-infection) $'
, fontsize = AlvaFontSize)
plt.text(0, 7.0/9, r'$ \frac{\partial V_n(t)}{\partial t} = \
+ \xi_{v}V_{n}(t)(1 - \frac{V_n(t)}{V_{max}}) \
- \phi_{m} V_{n}(t) M_{n}(t) \
- \phi_{g} V_{n}(t) \sum_{j = 1}^{N} (1 - \frac{|j - n|}{r + |j - n|})G_{j}(t) $'
, fontsize = 1.2*AlvaFontSize)
plt.text(0, 5.0/9, r'$ \frac{\partial B_n(t)}{\partial t} = \
+ \xi_{b} \
+ \beta_{m} B_{n}(t) V_{n}(t) \
+ \beta_{g} B_{n}(t)\sum_{j = 1}^{N} (1 - \frac{|j - n|}{r + |j - n|})V_{j}(t) \
- \mu_{b} B_{n}(t) \
+ m_b V_{m}\frac{B_{n-1}(t) - 2B_n(t) + B_{n+1}(t)}{(\Delta n)^2} $'
, fontsize = 1.2*AlvaFontSize)
plt.text(0, 3.0/9,r'$ \frac{\partial M_n(t)}{\partial t} = \
+ \xi_{m} B_{n}(t) \
- \phi_{m} M_{n}(t) V_{n}(t) \
- \mu_{m} M_{n}(t) $'
, fontsize = 1.2*AlvaFontSize)
plt.text(0, 1.0/9,r'$ \frac{\partial G_n(t)}{\partial t} = \
+ \xi_{g} B_{n}(t) \
- \phi_{g} G_{n}(t) \sum_{j = 1}^{N} (1 - \frac{|j - n|}{r + |j - n|})V_{j}(t) \
- \mu_{g} G_{n}(t) $'
, fontsize = 1.2*AlvaFontSize)
plt.savefig(save_figure, dpi = 100, bbox_inches='tight')
plt.show()
# define the V-B-M-G partial differential equations
# define the V-B-M-G partial differential equations
'''define the V-B-M-G partial differential equations'''
# inverted-monod equation
def monodInvert(half_radius, i):
if half_radius == 0:
gOut = i*0
# numpy.reshape will not change the structure of i,
# so that the first element of i(unkonwn-size-array) can be setted by array_to_list[0]
array_to_list = np.reshape(i,[i.size,1])
array_to_list[0] = 1
else: gOut = 1 - np.absolute(i)/(half_radius + np.absolute(i))
return (gOut)
# cross immunity
def crossI_neighborSum_X(gI, half_radius, gX):
total_neighbor_X = gX.shape[0]
I_neighborSum = np.zeros(total_neighbor_X)
# all I[xn] with neighbor-sum
ratioM = np.zeros([total_neighbor_X, total_neighbor_X])
gXX = np.tile(gX, [total_neighbor_X, 1])
gII = np.tile(gI, [total_neighbor_X, 1])
ratioM[:, :] = monodInvert(half_radius, gXX[:, :] - gXX[:, :].T)
I_neighborSum[:] = np.sum(ratioM[:, :] * gII[:, :].T, axis = 0)
if half_radius == 0:
I_neighborSum = np.copy(gI)
return (I_neighborSum)
def dVdt_array(VBMGxt = [], *args):
# naming
V = VBMGxt[0]
B = VBMGxt[1]
M = VBMGxt[2]
G = VBMGxt[3]
x_totalPoint = VBMGxt.shape[1]
# there are n dSdt
dV_dt_array = np.zeros(x_totalPoint)
# each dSdt with the same equation form
dV_dt_array[:] = +inRateV*V[:]*(1 - V[:]/maxV) \
- killRateVm*M[:]*V[:] \
- killRateVg*V[:]*crossI_neighborSum_X(G, cross_radius, gX)[:]
return(dV_dt_array)
def dBdt_array(VBMGxt = [], *args):
# naming
V = VBMGxt[0]
B = VBMGxt[1]
M = VBMGxt[2]
G = VBMGxt[3]
x_totalPoint = VBMGxt.shape[1]
# there are n dSdt
dB_dt_array = np.zeros(x_totalPoint)
# each dSdt with the same equation form
Bcopy = np.copy(B)
centerX = Bcopy[:]
leftX = np.roll(Bcopy[:], 1)
rightX = np.roll(Bcopy[:], -1)
leftX[0] = centerX[0]
rightX[-1] = centerX[-1]
dB_dt_array[:] = +inRateB \
+ actRateBm*V[:]*B[:] \
+ (actRateBg \
+ alva.event_recovered \
+ alva.event_OAS_boost)*B[:]*crossI_neighborSum_X(V, cross_radius, gX)[:] \
- (outRateB)*B[:] \
+ mutatRateB*V[:]*(leftX[:] - 2*centerX[:] + rightX[:])/(dx**2)
return(dB_dt_array)
def dMdt_array(VBMGxt = [], *args):
# naming
V = VBMGxt[0]
B = VBMGxt[1]
M = VBMGxt[2]
G = VBMGxt[3]
x_totalPoint = VBMGxt.shape[1]
# there are n dSdt
dM_dt_array = np.zeros(x_totalPoint)
# each dSdt with the same equation form
dM_dt_array[:] = +inRateM*B[:] - consumeRateM*M[:]*V[:] - outRateM*M[:]
return(dM_dt_array)
def dGdt_array(VBMGxt = [], *args):
# naming
V = VBMGxt[0]
B = VBMGxt[1]
M = VBMGxt[2]
G = VBMGxt[3]
x_totalPoint = VBMGxt.shape[1]
# there are n dSdt
dG_dt_array = np.zeros(x_totalPoint)
# each dSdt with the same equation form
Gcopy = np.copy(G)
centerX = Gcopy[:]
leftX = np.roll(Gcopy[:], 1)
rightX = np.roll(Gcopy[:], -1)
leftX[0] = centerX[0]
rightX[-1] = centerX[-1]
dG_dt_array[:] = +inRateG*B[:] \
- consumeRateG*G[:]*crossI_neighborSum_X(V, cross_radius, gX)[:] \
- outRateG*G[:]
return(dG_dt_array)
In [10]:
# setting parameter
timeUnit = 'year'
if timeUnit == 'hour':
hour = float(1)
day = float(24)
elif timeUnit == 'day':
day = float(1)
hour = float(1)/24
elif timeUnit == 'year':
year = float(1)
day = float(1)/365
hour = float(1)/24/365
maxV = float(60) # max virus/micro-liter
inRateV = 0.2/hour # in-rate of virus
killRateVm = 0.0003/hour # kill-rate of virus by antibody-IgM
killRateVg = killRateVm # kill-rate of virus by antibody-IgG
inRateB = 0.0075/hour # in-rate of B-cell
outRateB = inRateB/1 # out-rate of B-cell
actRateBm = killRateVm # activation rate of naive B-cell
actRateBg = killRateVg # activation rate of naive B-cell
inRateM = 0.16/hour # in-rate of antibody-IgM from naive B-cell
outRateM = inRateM/1 # out-rate of antibody-IgM from naive B-cell
consumeRateM = killRateVm # consume-rate of antibody-IgM by cleaning virus
inRateG = inRateM/10 # in-rate of antibody-IgG from memory B-cell
outRateG = outRateM/250 # out-rate of antibody-IgG from memory B-cell
consumeRateG = killRateVg # consume-rate of antibody-IgG by cleaning virus
mutatRateB = 0.00004/hour # Virus mutation rate
cross_radius = float(0.4) # radius of cross-immunity (the distance of half-of-value in the Monod equation)
# time boundary and griding condition
minT = float(0)
maxT = float(10*12*28*day)
totalPoint_T = int(6*10**3 + 1)
gT = np.linspace(minT, maxT, totalPoint_T)
spacingT = np.linspace(minT, maxT, num = totalPoint_T, retstep = True)
gT = spacingT[0]
dt = spacingT[1]
# space boundary and griding condition
minX = float(0)
maxX = float(9)
totalPoint_X = int(maxX - minX + 1)
gX = np.linspace(minX, maxX, totalPoint_X)
gridingX = np.linspace(minX, maxX, num = totalPoint_X, retstep = True)
gX = gridingX[0]
dx = gridingX[1]
gV_array = np.zeros([totalPoint_X, totalPoint_T])
gB_array = np.zeros([totalPoint_X, totalPoint_T])
gM_array = np.zeros([totalPoint_X, totalPoint_T])
gG_array = np.zeros([totalPoint_X, totalPoint_T])
# initial output condition
#gV_array[1, 0] = float(2)
# [viral population, starting time] ---first
origin_virus = int(2)
current_virus = int(6)
infection_period = 12*28*day
viral_population = np.zeros(int(maxX + 1))
viral_population[origin_virus:current_virus + 1] = 4
infection_starting_time = np.arange(int(maxX + 1))*infection_period
event_infect = np.zeros([int(maxX + 1), 2])
event_infect[:, 0] = viral_population
event_infect[:, 1] = infection_starting_time
event_infect[0, 1] = 0
print ('event_infect = {:}'.format(event_infect))
# [viral population, starting time] ---repeated
viral_population = np.zeros(int(maxX + 1))
viral_population[origin_virus:current_virus + 1] = 0
infection_starting_time = np.arange(int(maxX + 1))*0
event_repeated = np.zeros([int(maxX + 1), 2])
event_repeated[:, 0] = viral_population
event_repeated[:, 1] = infection_starting_time
print ('event_repeated = {:}'.format(event_repeated))
# [vaccine population, starting time] ---first
origin_vaccine = int(1)
current_vaccine = int(2)
vaccine_period = 12*28*day
vaccine_population = np.zeros(int(maxX + 1))
vaccine_population[origin_vaccine:current_vaccine + 1] = 0
vaccine_starting_time = np.arange(int(maxX + 1))*vaccine_period
event_vaccine = np.zeros([int(maxX + 1), 2])
event_vaccine[:, 0] = vaccine_population
event_vaccine[:, 1] = vaccine_starting_time
event_vaccine[0, 1] = 0
print ('event_vaccine = {:}'.format(event_vaccine))
#[origin-virus, current-virus, recovered-day, repeated-parameter, OAS+, OSA-]
min_cell = 1 # minimum cell
recovered_time = 14*day # recovered time of 1st-time infection
actRateBg_recovered = actRateBg*10 # activation rate of memory B-cell for repeated-infection (same virus)
inRateG_OAS_boost = actRateBg*15 # boosting in-rate of antibody-IgG from memory B-cell for origin-virus
actRateBg_OAS_press = 0.00/hour # depress act-rate from memory B-cell for non-origin-virus
outRateB_OAS_slow = 0.0 # not applied in infection
event_infection_parameter = np.array([origin_virus,
current_virus,
min_cell,
recovered_time,
actRateBg_recovered,
inRateG_OAS_boost,
actRateBg_OAS_press,
outRateB_OAS_slow])
# vaccination_parameter
# vaccination_parameter
# vaccination_parameter
min_cell_v = 1 # minimum cell
recovered_time_v = 14*day # recovered time of 1st-time infection
actRateBg_recovered_v = actRateBg*9 # activation rate of memory B-cell for repeated-infection (same virus)
inRateG_OAS_boost_v = 0.0 # boosting in-rate of antibody-IgG from memory B-cell for origin-virus
actRateBg_OAS_press_v = 0.00 # depress act-rate from memory B-cell for non-origin-virus
outRateB_OAS_slow_v = 0.0
event_vaccination_parameter = np.array([origin_vaccine,
current_vaccine,
min_cell_v,
recovered_time_v,
actRateBg_recovered_v,
inRateG_OAS_boost_v,
actRateBg_OAS_press_v,
outRateB_OAS_slow_v])
event_parameter = np.array([event_infection_parameter, event_vaccination_parameter])
event_table = np.array([event_parameter, event_infect, event_repeated, event_vaccine])
# Runge Kutta numerical solution
pde_array = np.array([dVdt_array, dBdt_array, dMdt_array, dGdt_array])
initial_Out = np.array([gV_array, gB_array, gM_array, gG_array])
gOut_array = alva.AlvaRungeKutta4XT(pde_array, initial_Out, minX, maxX, totalPoint_X, minT, maxT, totalPoint_T, event_table)
# plotting
gV = gOut_array[0]
gB = gOut_array[1]
gM = gOut_array[2]
gG = gOut_array[3]
numberingFig = numberingFig + 1
for i in range(totalPoint_X):
figure_name = '-response-%i'%(i)
figure_suffix = '.png'
save_figure = os.path.join(dir_path, file_name + figure_name + file_suffix)
plt.figure(numberingFig, figsize = AlvaFigSize)
plt.plot(gT, gV[i], color = 'red', label = r'$ V_{%i}(t) $'%(i), linewidth = 3.0, alpha = 0.5)
plt.plot(gT, gM[i], color = 'blue', label = r'$ IgM_{%i}(t) $'%(i), linewidth = 3.0, alpha = 0.5)
plt.plot(gT, gG[i], color = 'green', label = r'$ IgG_{%i}(t) $'%(i), linewidth = 3.0, alpha = 0.5)
plt.plot(gT, gM[i] + gG[i], color = 'gray', linewidth = 5.0, alpha = 0.5, linestyle = 'dashed'
, label = r'$ IgM_{%i}(t) + IgG_{%i}(t) $'%(i, i))
plt.grid(True, which = 'both')
plt.title(r'$ Antibody \ responses \ to \ Virus-{%i} $'%(i), fontsize = AlvaFontSize)
plt.xlabel(r'$time \ (%s)$'%(timeUnit), fontsize = AlvaFontSize)
plt.ylabel(r'$ Neutralization \ \ titer $', fontsize = AlvaFontSize)
plt.xlim([minT, maxT])
plt.xticks(fontsize = AlvaFontSize*0.6)
plt.yticks(fontsize = AlvaFontSize*0.6)
plt.ylim([2**0, 2**14])
plt.yscale('log', basey = 2)
plt.legend(loc = (1,0), fontsize = AlvaFontSize)
plt.savefig(save_figure, dpi = 100, bbox_inches='tight')
plt.show()
In [11]:
# Normalization stacked graph
numberingFig = numberingFig + 1
plt.figure(numberingFig, figsize = AlvaFigSize)
plt.stackplot(gT, gM + gG, alpha = 0.3)
plt.title(r'$ Stacked-graph \ of \ Antibody $', fontsize = AlvaFontSize)
plt.xlabel(r'$time \ (%s)$'%(timeUnit), fontsize = AlvaFontSize)
plt.ylabel(r'$ Neutralization \ \ titer $', fontsize = AlvaFontSize)
plt.xticks(fontsize = AlvaFontSize*0.6)
plt.yticks(fontsize = AlvaFontSize*0.6)
plt.ylim([2**0, 2**14])
plt.yscale('log', basey = 2)
plt.grid(True)
plt.show()
In [12]:
# expected peak of the antibody response
totalColor = current_virus - origin_virus + 1
AlvaColor = [plt.get_cmap('rainbow')(float(i)/(totalColor)) for i in range(1, totalColor + 1)]
sample_time = 60*day
# plotting
figure_name = '-landscape'
figure_suffix = '.png'
save_figure = os.path.join(dir_path, file_name + figure_name + file_suffix)
numberingFig = numberingFig + 1
plt.figure(numberingFig, figsize = (12, 9))
for i in range(origin_virus, current_virus + 1):
detect_xn = current_virus + 2 - i
if detect_xn == origin_virus:
virus_label = '$ origin-virus $'
elif detect_xn == current_virus:
virus_label = '$ current-virus $'
else: virus_label = '$ {:}th-virus $'.format(detect_xn - origin_virus + 1)
detect_time = int(totalPoint_T/(maxT - minT)*(detect_xn*infection_period + sample_time))
plt.plot(gX, gM[:, detect_time] + gG[:, detect_time], marker = 'o', markersize = 20
, color = AlvaColor[detect_xn - origin_virus], label = virus_label)
plt.fill_between(gX, gM[:, detect_time] + gG[:, detect_time], facecolor = AlvaColor[detect_xn - origin_virus]
, alpha = 0.5)
plt.grid(True, which = 'both')
plt.title(r'$ Antibody \ Landscape \ (Sequential-Infection)$', fontsize = AlvaFontSize)
plt.xlabel(r'$ Virus \ space \ (Antigenic-distance) $', fontsize = AlvaFontSize)
plt.ylabel(r'$ Neutralization \ \ titer $', fontsize = AlvaFontSize)
plt.xlim([minX, maxX])
plt.xticks(fontsize = AlvaFontSize)
plt.yticks(fontsize = AlvaFontSize)
plt.ylim([2**0, 2**14])
plt.yscale('log', basey = 2)
plt.legend(loc = (1,0), fontsize = AlvaFontSize)
plt.savefig(save_figure, dpi = 100, bbox_inches='tight')
plt.show()
# each frame
numberingFig = numberingFig + 1
for i in range(origin_virus, current_virus + 1):
plt.figure(numberingFig, figsize = (9,3))
detect_xn = current_virus + 2 - i
if detect_xn == origin_virus:
virus_label = '$ origin-virus $'
elif detect_xn == current_virus:
virus_label = '$ current-virus $'
else: virus_label = '$ {:}th-virus $'.format(detect_xn - origin_virus + 1)
detect_time = int(totalPoint_T/(maxT - minT)*(detect_xn*infection_period + sample_time))
plt.plot(gX, gM[:, detect_time] + gG[:, detect_time], marker = 'o', markersize = 20
, color = AlvaColor[detect_xn - origin_virus], label = virus_label)
plt.fill_between(gX, gM[:, detect_time] + gG[:, detect_time], facecolor = AlvaColor[detect_xn - origin_virus]
, alpha = 0.5)
plt.ylim([2**0, 2**14])
plt.yscale('log', basey = 2)
plt.show()
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