Archive for the ‘Diffusion’ Category

Understanding Basic Epidemic Models with Python

Published by chengjun on March 14th, 2013

Modeling is one way for understanding analytic models. Python supplies many tools to do that.

SIR

First, the python script for SIR

# -*- coding: utf-8 -*-

###################################
### Written by Ilias Soumpasis    #
### ilias.soumpasis@ucd.ie (work) #
### ilias.soumpasis@gmail.com	  #
###################################

import scipy.integrate as spi
import numpy as np
import pylab as pl

beta=1.4247
gamma=0.14286
TS=1.0
ND=70.0
S0=1-1e-6
I0=1e-6
INPUT = (S0, I0, 0.0)

def diff_eqs(INP,t):
	'''The main set of equations'''
	Y=np.zeros((3))
	V = INP
	Y[0] = - beta * V[0] * V[1]
	Y[1] = beta * V[0] * V[1] - gamma * V[1]
	Y[2] = gamma * V[1]
	return Y   # For odeint

t_start = 0.0; t_end = ND; t_inc = TS
t_range = np.arange(t_start, t_end+t_inc, t_inc)
RES = spi.odeint(diff_eqs,INPUT,t_range)

print RES

#Ploting
pl.plot(RES[:,0], '-bs', label='Susceptibles')  # I change -g to g--  # RES[:,0], '-g',
pl.plot(RES[:,2], '-g^', label='Recovereds')  # RES[:,2], '-k',
pl.plot(RES[:,1], '-ro', label='Infectious')
pl.legend(loc=0)
pl.title('SIR epidemic without births or deaths')
pl.xlabel('Time')
pl.ylabel('Susceptibles, Recovereds, and Infectious')
pl.savefig('2.1-SIR-high.png', dpi=900) # This does, too
pl.show()

p2.1-SIS
Second, the python script for SIS

# -*- coding: utf-8 -*-

import scipy.integrate as spi
import numpy as np
import pylab as pl

beta=1.4247
gamma=0.14286
I0=1e-6
ND=70
TS=1.0
INPUT = (1.0-I0, I0)

def diff_eqs(INP,t):
	'''The main set of equations'''
	Y=np.zeros((2))
	V = INP
	Y[0] = - beta * V[0] * V[1] + gamma * V[1]
	Y[1] = beta * V[0] * V[1] - gamma * V[1]
	return Y   # For odeint

t_start = 0.0; t_end = ND; t_inc = TS
t_range = np.arange(t_start, t_end+t_inc, t_inc)
RES = spi.odeint(diff_eqs,INPUT,t_range)

print RES

#Ploting
pl.plot(RES[:,0], '-bs', label='Susceptibles')
pl.plot(RES[:,1], '-ro', label='Infectious')
pl.legend(loc=0)
pl.title('SIS epidemic without births or deaths')
pl.xlabel('Time')
pl.ylabel('Susceptibles and Infectious')
pl.savefig('2.5-SIS-high.png', dpi=900) # This does increase the resolution.
pl.show()
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Understanding dyad in ERGM

Published by chengjun on October 20th, 2012

It’s a bit difficult to understand the terms used by ERGM.

I am working on predicting how friendships influence the information diffusion using weibo landscape data with ERGM.

According to the statnet library:

With the parameter of “dyadcov” term, we add three statistics to the model, each equal to the sum of the covariate values for all dyads occupying one of the three possible non-empty dyad states (mutual, upper-triangular asymmetric, and lower-triangular asymmetric dyads, respectively).

Obviously, there are three kinds of dyads.

Check Wasserman and Faust’ book of Social Network Analysis, and I find the figure above.

 However, most of us don’t know the difference between upper-triangular asymmetric and lower-triangular asymmetric dyads.

Based on my understanding, it’s related to the direction of ties, see the figure below (am i right?):

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Posted in Data, Diffusion, Network
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Slides of Bass Diffusion Model

Published by chengjun on April 26th, 2012

http://weblab.com.cityu.edu.hk/blog/chengjun/files/2012/04/Bass-Diffusion-Model.pdf

Introduction

I had made slides to understand the bass diffusion model which is proposed by Frank M. Bass in 1969. The model proposes that diffusion is motivated both by inovativeness and imitation: first, the innovative early adopters adopted the products, after which the followers will imitate them and adopt the products.

•The Bass model coefficient (parameter) of innovation is p.
•The Bass model coefficient (parameter) of imitation is q.

The slides will show you how to deprive the equation of Bass diffusion model by both discrete-time model and continuous model (using Hazard rate).
“The probability of adopting by those who have not yet adopted is a linear function of those who had previously adopted.”
By solving the differential equation of bass diffusion model using Mathematica and Simulating the model using R code.
# basss diffusion model
# chengjun, 20120424@canberra

# refer to http://en.wikipedia.org/wiki/Bass_diffusion_model
# and http://book.douban.com/subject/4175572/discussion/45634092/

# BASS Diffusion Model
# three parameters:
# the total number of people who eventually buy the product, m;
# the coefficient of innovation, p;
# and the coefficient of imitation, q

# example
T79 <- 1:10
Tdelt <- (1:100) / 10
Sales <- c(840,1470,2110,4000, 7590, 10950, 10530, 9470, 7790, 5890)
Cusales <- cumsum(Sales)
Bass.nls <- nls(Sales ~ M * ( ((P+Q)^2 / P) * exp(-(P+Q) * T79) ) /(1+(Q/P)*exp(-(P+Q)*T79))^2,
                start = list(M=60630, P=0.03, Q=0.38))
summary(Bass.nls)

# get coefficient
Bcoef <- coef(Bass.nls)
m <- Bcoef[1]
p <- Bcoef[2]
q <- Bcoef[3]
# setting the starting value for M to the recorded total sales.
ngete <- exp(-(p+q) * Tdelt)

# plot pdf
Bpdf <- m * ( (p+q)^2 / p ) * ngete / (1 + (q/p) * ngete)^2
plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l')
points(T79, Sales)

# plot cdf
Bcdf <- m * (1 - ngete)/(1 + (q/p)*ngete)
plot(Tdelt, Bcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l')
points(T79, Cusales)

# when q=0, only Innovator without immitators.
Ipdf<- m * ( (p+0)^2 / p ) * exp(-(p+0) * Tdelt) / (1 + (0/p) * exp(-(p+0) * Tdelt))^2
# plot(Tdelt, Ipdf, xlab = "Year from 1979",ylab = "Isales per year", type='l')
Impdf<-Bpdf-Ipdf
plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l', col="red")
lines(Tdelt,Impdf,col="green")
lines(Tdelt,Ipdf,col="blue")

# when q=0, only Innovator without immitators.
Icdf<-m * (1 - exp(-(p+0) * Tdelt))/(1 + (0/p)*exp(-(p+0) * Tdelt))
# plot(Tdelt, Icdf, xlab = "Year from 1979",ylab = "ICumulative sales", type='l')
Imcdf<-m * (1 - ngete)/(1 + (q/p)*ngete)-Icdf
plot(Tdelt, Imcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l', col="red")
lines(Tdelt,Bcdf,col="green")
lines(Tdelt,Icdf,col="blue")
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