# Epidemic

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Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks.Phys. Rev. Lett. 86, 3200 (2001).[1]

Tizzoni M, Sun K, Benusiglio D, et al. The Scaling of Human Contacts and Epidemic Processes in Metapopulation Networks[J]. Scientific reports, 2015, 5.[2]

```import numpy as np
import networkx as nx
import matplotlib.pyplot as plt
%pylab inline

def SIS_net(G,p,mu,status):
# we make the process syncronous
# at each time step all infected indivuals can infect their peers
# status is a vector that takes two values for each node
# 0 -> S
# 1 -> I
temp_inf=set()
temp_rec=set()
for i in G.nodes():
# if the node is infected
if status[i]==1:
# we go through her neighbors and we try to infect the S
for j in G.neighbors(i):
if status[j]==0 and uniform(0,1)<p:
temp_inf.add(j) # this node will be infected next

# recovery process
if uniform(0,1)<mu:

# update the status
for i in temp_inf:
status[i]=1
for i in temp_rec:
status[i]=0

we need to select the initial infected population
let us create a function to select a fraction p of nodes
as initial seeds

def set_seeds(status,p):
# we get the number of infected given a fraction p
n_of_infected=int(p*status.size)
# we get the list of nodes and shuffle it
list_nodes=[]
for i in range(status.size):
list_nodes.append(i)
shuffle(list_nodes)

# the first n_of_infected will be the seeds
for i in range(n_of_infected):
status[list_nodes[i]]=1

def infecting(G, p):
N=G.number_of_nodes()
status=np.zeros(N,int)
# for each node i, status[i]=0 indicates that the node is susceptible
# for each node i, status[i]=1 indicates that the node is infected
# let us set the seed
set_seeds(status,0.1)
stop=50
mu=0.5
it=np.zeros(stop,float)
for t in range(stop):
SIS_net(G,p,mu,status)
# number of infected nodes
# the sum is the number of infected!
inf=float(status.sum())
it[t]=inf/N
return it

plot_num = 5
cmap = cm.get_cmap('Accent_r', plot_num)

def plotDegreeDistribution(G):
plt.plot(nx.degree_histogram(G) , 'ro', markersize = 8)      #返回图中所有节点的度分布序列
plt.legend(['Degree'])
plt.xlabel('Degree')
plt.ylabel('Number of nodes')
plt.title('Degree Distribution')
plt.xscale('log')
plt.yscale('log')
plt.show()
G = nx.barabasi_albert_graph(500, 1, seed=None)
plotDegreeDistribution(G)

for k, p in enumerate(np.linspace(0.8,0.9, plot_num)):
it = infecting(G, p)
plt.plot(it,c= cmap(k),linewidth=4,alpha=0.5, label = 'P = '+str(np.round(p, 2)) )
plt.xlabel('t')
plt.ylabel('i(t)')
lg = plt.legend(loc = 0 )#bbox_to_anchor=(1.3, 0.965))
lg.draw_frame(False)
plt.show()

```

The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.

# 参考文献

- Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001). pdf - Tizzoni M, Sun K, Benusiglio D, et al. The Scaling of Human Contacts and Epidemic Processes in Metapopulation Networks[J]. Scientific reports, 2015, 5. pdf