用Python在地图上模拟疫情扩散

用Python在地图上模拟疫情扩散

受杰森的《Almost Looks Like Work》启发，我来展示一些病毒传播模型。需要注意的是这个模型并不反映现实情况，因此不要误以为是西非可怕的传染病。相反，它更应该被看做是某种虚构的僵尸爆发现象。那么，让我们进入主题。

这就是SIR模型，其中字母S、I和R反映的是在僵尸疫情中，个体可能处于的不同状态。

• S 代表易感群体，即健康个体中潜在的可能转变的数量。
• I 代表染病群体，即僵尸数量。
• R 代表移除量，即因死亡而退出游戏的僵尸数量，或者感染后又转回人类的数量。但对与僵尸不存在治愈者，所以我们就不要自我愚弄了（如果要把SIR模型应用到流感传染中，还是有治愈者的）。

至于β(beta)和γ(gamma):

• β(beta)表示疾病的传染性程度，只要被咬就会感染。
• γ(gamma)表示从僵尸走向死亡的速率，取决于僵尸猎人的平均工作速率，当然，这不是一个完美的模型，请对我保持耐心。

S′=−βIS告诉我们健康者变成僵尸的速率，S′是对时间的导数。

I′=βIS−γI告诉我们感染者是如何增加的，以及行尸进入移除态速率（双关语）。

R′=γI只是加上(gamma I)，这一项在前面的等式中是负的。

上面的模型没有考虑S/I/R的空间分布，下面来修正一下！

一种方法是把瑞典和北欧国家分割成网格，每个单元可以感染邻近单元，描述如下：

实验完整代码如下：

Main.py

# -*- coding: utf-8 -*-
import numpy as np
import math
import matplotlib.pyplot as plt    
from matplotlib import rcParams
import matplotlib.image as mpimg
from PIL import Image

rcParams['font.family'] = 'serif'
rcParams['font.size'] = 16
rcParams['figure.figsize'] = 12, 8


beta = 0.010
gamma = 1

def euler_step(u, f, dt):
    return u + dt * f(u)

def f(u):
    S = u[0]
    I = u[1]
    R = u[2]

    new = np.array([-beta*(S[1:-1, 1:-1]*I[1:-1, 1:-1] + \
                            S[0:-2, 1:-1]*I[0:-2, 1:-1] + \
                            S[2:, 1:-1]*I[2:, 1:-1] + \
                            S[1:-1, 0:-2]*I[1:-1, 0:-2] + \
                            S[1:-1, 2:]*I[1:-1, 2:]),
                     beta*(S[1:-1, 1:-1]*I[1:-1, 1:-1] + \
                            S[0:-2, 1:-1]*I[0:-2, 1:-1] + \
                            S[2:, 1:-1]*I[2:, 1:-1] + \
                            S[1:-1, 0:-2]*I[1:-1, 0:-2] + \
                            S[1:-1, 2:]*I[1:-1, 2:]) - gamma*I[1:-1, 1:-1],
                     gamma*I[1:-1, 1:-1]
                    ])

    padding = np.zeros_like(u)
    padding[:,1:-1,1:-1] = new
    padding[0][padding[0] < 0] = 0
    padding[0][padding[0] > 255] = 255
    padding[1][padding[1] < 0] = 0
    padding[1][padding[1] > 255] = 255
    padding[2][padding[2] < 0] = 0
    padding[2][padding[2] > 255] = 255

    return padding

img = Image.open('popdens2.png')
img = img.resize((img.size[0]/2,img.size[1]/2)) 
img = 255 - np.asarray(img)
imgplot = plt.imshow(img)
imgplot.set_interpolation('nearest')

S_0 = img[:,:,1]
I_0 = np.zeros_like(S_0)
I_0[309,170] = 1 # patient zero

R_0 = np.zeros_like(S_0)

T = 900                         # final time
dt = 1                          # time increment
N = int(T/dt) + 1               # number of time-steps
t = np.linspace(0.0, T, N)      # time discretization

# initialize the array containing the solution for each time-step
u = np.empty((N, 3, S_0.shape[0], S_0.shape[1]))
u[0][0] = S_0
u[0][1] = I_0
u[0][2] = R_0

import matplotlib.cm as cm
theCM = cm.get_cmap("Reds")
theCM._init()
alphas = np.abs(np.linspace(0, 1, theCM.N))
theCM._lut[:-3,-1] = alphas

for n in range(N-1):
    u[n+1] = euler_step(u[n], f, dt)

from images2gif import writeGif

keyFrames = []
frames = 60.0

for i in range(0, N-1, int(N/frames)):
    imgplot = plt.imshow(img, vmin=0, vmax=255)
    imgplot.set_interpolation("nearest")
    imgplot = plt.imshow(u[i][1], vmin=0, cmap=theCM)
    imgplot.set_interpolation("nearest")
    filename = "outbreak" + str(i) + ".png"
    plt.savefig(filename)
    keyFrames.append(filename)

images = [Image.open(fn) for fn in keyFrames]
gifFilename = "outbreak.gif"
writeGif(gifFilename, images, duration=0.3)
plt.clf()

image2gif.py

""" MODULE images2gif

Provides a function (writeGif) to write animated gif from a series
of PIL images or numpy arrays.

This code is provided as is, and is free to use for all.

Almar Klein (June 2009)

- based on gifmaker (in the scripts folder of the source distribution of PIL)
- based on gif file structure as provided by wikipedia

"""

try:
    import PIL
    from PIL import Image, ImageChops
    from PIL.GifImagePlugin import getheader, getdata
except ImportError:
    PIL = None

try:
    import numpy as np
except ImportError:
    np = None    

# getheader gives a 87a header and a color palette (two elements in a list).
# getdata()[0] gives the Image Descriptor up to (including) "LZW min code size".
# getdatas()[1:] is the image data itself in chuncks of 256 bytes (well
# technically the first byte says how many bytes follow, after which that
# amount (max 255) follows).


def intToBin(i):
    """ Integer to two bytes """
    # devide in two parts (bytes)
    i1 = i % 256
    i2 = int( i/256)
    # make string (little endian)
    return chr(i1) + chr(i2)


def getheaderAnim(im):
    """ Animation header. To replace the getheader()[0] """
    bb = "GIF89a"
    bb += intToBin(im.size[0])
    bb += intToBin(im.size[1])
    bb += "\x87\x00\x00"
    return bb


def getAppExt(loops=0):
    """ Application extention. Part that secifies amount of loops. 
    if loops is 0, if goes on infinitely.
    """
    bb = "\x21\xFF\x0B"  # application extension
    bb += "NETSCAPE2.0"
    bb += "\x03\x01"
    if loops == 0:
        loops = 2**16-1
    bb += intToBin(loops)
    bb += '\x00'  # end
    return bb


def getGraphicsControlExt(duration=0.1):
    """ Graphics Control Extension. A sort of header at the start of
    each image. Specifies transparancy and duration. """
    bb = '\x21\xF9\x04'
    bb += '\x08'  # no transparancy
    bb += intToBin( int(duration*100) ) # in 100th of seconds
    bb += '\x00'  # no transparant color
    bb += '\x00'  # end
    return bb


def _writeGifToFile(fp, images, durations, loops):
    """ Given a set of images writes the bytes to the specified stream.
    """

    # init
    frames = 0
    previous = None

    for im in images:

        if not previous:
            # first image

            # gather data
            palette = getheader(im)[1]
            data = getdata(im)
            imdes, data = data[0], data[1:]            
            header = getheaderAnim(im)
            appext = getAppExt(loops)
            graphext = getGraphicsControlExt(durations[0])

            # write global header
            fp.write(header)
            fp.write(palette)
            fp.write(appext)

            # write image
            fp.write(graphext)
            fp.write(imdes)
            for d in data:
                fp.write(d)

        else:
            # gather info (compress difference)              
            data = getdata(im) 
            imdes, data = data[0], data[1:]       
            graphext = getGraphicsControlExt(durations[frames])

            # write image
            fp.write(graphext)
            fp.write(imdes)
            for d in data:
                fp.write(d)

#             # delta frame - does not seem to work
#             delta = ImageChops.subtract_modulo(im, previous)            
#             bbox = delta.getbbox()
#             
#             if bbox:
#                 
#                 # gather info (compress difference)              
#                 data = getdata(im.crop(bbox), offset = bbox[:2]) 
#                 imdes, data = data[0], data[1:]       
#                 graphext = getGraphicsControlExt(durations[frames])
#                 
#                 # write image
#                 fp.write(graphext)
#                 fp.write(imdes)
#                 for d in data:
#                     fp.write(d)
#                 
#             else:
#                 # FIXME: what should we do in this case?
#                 pass

        # prepare for next round
        previous = im.copy()        
        frames = frames + 1

    fp.write(";")  # end gif
    return frames


def writeGif(filename, images, duration=0.1, loops=0, dither=1):
    """ writeGif(filename, images, duration=0.1, loops=0, dither=1)
    Write an animated gif from the specified images. 
    images should be a list of numpy arrays of PIL images.
    Numpy images of type float should have pixels between 0 and 1.
    Numpy images of other types are expected to have values between 0 and 255.
    """

    if PIL is None:
        raise RuntimeError("Need PIL to write animated gif files.")

    images2 = []

    # convert to PIL
    for im in images:

        if isinstance(im,Image.Image):
            images2.append( im.convert('P',dither=dither) )

        elif np and isinstance(im, np.ndarray):
            if im.dtype == np.uint8:
                pass
            elif im.dtype in [np.float32, np.float64]:
                im = (im*255).astype(np.uint8)
            else:
                im = im.astype(np.uint8)
            # convert
            if len(im.shape)==3 and im.shape[2]==3:
                im = Image.fromarray(im,'RGB').convert('P',dither=dither)
            elif len(im.shape)==2:
                im = Image.fromarray(im,'L').convert('P',dither=dither)
            else:
                raise ValueError("Array has invalid shape to be an image.")
            images2.append(im)

        else:
            raise ValueError("Unknown image type.")

    # check duration
    if hasattr(duration, '__len__'):
        if len(duration) == len(images2):
            durations = [d for d in duration]
        else:
            raise ValueError("len(duration) doesn't match amount of images.")
    else:
        durations = [duration for im in images2]


    # open file
    fp = open(filename, 'wb')

    # write
    try:
        n = _writeGifToFile(fp, images2, durations, loops)
        print n, 'frames written'
    finally:
        fp.close()


if __name__ == '__main__':
    im = np.zeros((200,200), dtype=np.uint8)
    im[10:30,:] = 100
    im[:,80:120] = 255
    im[-50:-40,:] = 50

    images = [im*1.0, im*0.8, im*0.6, im*0.4, im*0]
    writeGif('lala3.gif',images, duration=0.5, dither=0)

实验原始图像与实验后图像如下：