To help the unfamiliar to understand the application and relevance of this algorithm, consider the following dynamical system derived from an epidemiological problem:
Suppose an infected individual arrives in his home town after his/her vacation carrying an infectious disease entirely new to his fellow citizens. This means no one in town has antibodies for that disease. Let this set of people be called the "susceptibles" and let S be the number of people in this group. Our Infected individual will initially be the only member of a group called the "infectious" which size is denoted by I. The rate of encounters between susceptibles and infectious is proportional to their numbers (S*I). Everytime an infectious meet a susceptible there is a chance b that the disease is transmitted, so the probability of a transmission event is b*S*I. This is called an individual state transition, since the individual went from a susceptible state to an infectious one.
The dynamical system used as an example here contains this and other state transitions described in the code below:
As can be seen there are five different states in this model with the last one R, being an absorbing state, since all individuals converge to that state with time.
"""
Example of an SEIR model with two Infectious classes: subclinical(Is) and clinical(Ic)
Is
/ \
S -> E R
\ /
Ic
States:
S: Susceptible
E: Exposed
Is: Infectious subclinical
Ic: Infectious clinical
R: Recovered
Transition rates:
b,ks,kc,rs,rc = (0.001, 0.1, 0.1, 0.01, .01)
Transitions:
S -> E : b*S*(Is+Ic)
E -> Is : ks*E
E -> Ic : kc*E
Is -> R : rs*Is
Ic -> R : rc*Ic
"""
from gillespie import Model
import time
from numpy import array
vars = ['S','E','Is','Ic','R']
#rates: b,ks,kc,rs,rc
r = (0.001, 0.1, 0.1, 0.01, .01)
ini = (490,0,10,0,0)
prop = (lambda r,ini:r[0]*ini[0]*(ini[2]+ini[3]),
lambda r,ini:r[1]*ini[1],
lambda r,ini:r[2]*ini[1],
lambda r,ini:r[3]*ini[2],
lambda r,ini:r[4]*ini[3]
)
tmat = array([[-1,0,0,0,0],
[1,-1,-1,0,0],
[0,1,0,-1,0],
[0,0,1,0,-1],
[0,0,0,1,1]
])
#for e in prop:
# print e()
M=Model(vnames=vars,rates = r,inits=ini,tmat=tmat,propensity=prop)
t0 = time.time()
M.run(tmax=80,reps=100)
print 'total time: ',time.time()-t0
t,series,steps = M.getStats()
print steps,'steps'
from pylab import plot , show, legend, errorbar
plot(t,series.mean(axis=2),'-o')
legend(vars,loc=0)
show()
the code that simulates this stochastic dynamic model is in the gillespie module:
from numpy.random import uniform, multinomial, exponential,random
from numpy import arange, array, empty,zeros,log
#from math import log
import time
import multiprocessing
import psyco
psyco.full()
#global ini
#ini=[500,1,0]
class Model:
def __init__(self,vnames,rates,inits, tmat,propensity):
'''
* vnames: list of strings
* rates: list of fixed rate parameters
* inits: list of initial values of variables
* propensity: list of lambda functions of the form:
lambda r,ini: some function of rates ans inits.
'''
self.vn = vnames
self.rates = rates
self.inits = inits
self.tm = tmat
self.pv = propensity#[compile(eq,'errmsg','eval') for eq in propensity]
self.pvl = len(self.pv) #length of propensity vector
self.nvars = len(self.inits) #number of variables
self.time=None
self.series=None
self.steps=0
def getStats(self):
return self.time,self.series,self.steps
def run(self,method='SSA', tmax=10, reps=1):
self.res = zeros((tmax,self.nvars,reps),dtype=float)
tvec = arange(tmax, dtype=int)
if method =='SSA':
for i in xrange(reps):
steps = self.GSSA(tmax,i)
print steps,' steps'
elif method == 'SSAct':
pass
self.time=tvec
self.series=self.res
self.steps=steps
def GSSA(self, tmax=50,round=0):
'''
Gillespie Direct algorithm
'''
ini = self.inits
r = self.rates
pvi = self.pv
l=self.pvl
pv = zeros(l,dtype=float)
tm = self.tm
tc = 0
steps = 0
self.res[0,:,round]= ini
a0=1
for tim in xrange(1,tmax):
while tc < tim:
for i in xrange(l):
pv[i] = pvi[i](r,ini)
#pv = abs(array([eq() for eq in pvi]))# #propensity vector
a0 = pv.sum() #sum of all transition probabilities
# print tim, pv, a0
tau = (-1/a0)*log(random())
event = multinomial(1,pv/a0) # event which will happen on this iteration
ini += tm[:,event.nonzero()[0][0]]
#print tc, ini
tc += tau
steps +=1
if a0 == 0: break
self.res[tim,:,round] = ini
if a0 == 0: break
# tvec = tvec[:tim]
# self.res = res[:tim,:,round]
return steps
def CR(self,pv):
"""
Composition reaction algorithm
"""
pass
def main():
vars = ['s','i','r']
ini= [500,1,0]
rates = [.001,.1]
tm = array([[-1,0],[1,-1],[0,1]])
prop=[lambda r, ini:r[0]*ini[0]*ini[1],lambda r,ini:r[0]*ini[1]]
M = Model(vnames = vars,rates = rates,inits=ini, tmat=tm,propensity=prop)
t0=time.time()
M.run(tmax=80,reps=1000)
print 'total time: ',time.time()-t0
#print res
# from pylab import plot , show, legend
# plot(t,res,'-.')
# legend(M.vn,loc=0)
# show()
if __name__=="__main__":
import cProfile
cProfile.run('main()',sort=1)
main()
The Gillespie SSA is a very simple algorithm, so I thought it would be nice to blog about it and have other Pythonistas take a look at my code and possibly find a way to optimize it further. Having the core code run as fast as possible is very important for this application, because you normally have to run thousands of replicates of a every run due to its stochastic nature. This "need for speed", made me try Cython, but due to my inexperience with it, I couldn't squeeze much performance out of it:
That's it. I hope this code will serve to inspire other scientists which need to use this type of simulations. In the next few days I will organize it in a package and make it available somewhere.
from numpy.random import uniform, multinomial, exponential
#from numpy import arange, array, empty,zeros
import numpy as np
cimport numpy as np
import time
DTYPE = np.double
ctypedef np.double_t DTYPE_t
ctypedef np.int_t INT_t
cdef class Model:
cdef object vn,rates,inits,pv
cdef np.ndarray tm,res,time,series
cdef int pvl,nvars,steps
cdef object ts
def __init__(self,vnames,rates,inits, tmat,propensity):
'''
* vnames: list of strings
* rates: list of fixed rate parameters
* inits: list of initial values of variables
* propensity: list of lambda functions of the form:
lambda r,ini: some function of rates ans inits.
'''
self.vn = vnames
self.rates = rates
self.inits = inits
self.tm = tmat
self.pv = propensity#[compile(eq,'errmsg','eval') for eq in propensity]
self.pvl = len(self.pv) #length of propensity vector
self.nvars = len(self.inits) #number of variables
self.time = np.zeros(1)
self.series = np.zeros(1)
self.steps = 0
def run(self, method='SSA', int tmax=10, int reps=1):
cdef np.ndarray[DTYPE_t,ndim=3] res = np.zeros((tmax,self.nvars,reps),dtype=float)
tvec = np.arange(tmax)
self.res = res
cdef int i, steps
if method =='SSA':
for i from 0 <= i<reps:
steps = self.GSSA(tmax,i)
print steps,' steps'
elif method == 'SSAct':
pass
self.time=tvec
self.series=self.res
self.steps=steps
def getStats(self):
return self.time,self.series,self.steps
cpdef int GSSA(self, int tmax=50,int round=0):
'''
Gillespie Direct algorithm
'''
ini = self.inits
r = self.rates
pvi = self.pv
cdef int l,steps,i,tim
cdef double a0,tc, tau
#cdef np.ndarray[INT_t] tvec
cdef np.ndarray[DTYPE_t] pv
l=self.pvl
pv = np.zeros(l, dtype=float)
tm = self.tm
#tvec = np.arange(tmax,dtype=int)
tc = 0
steps = 0
self.res[0,:,round]= ini
a0=1.
for tim from 1<= tim <tmax:
while tc < tim:
for i from 0 <= i <l:
pv[i] = pvi[i](r,ini)
#pv = abs(array([eq() for eq in pvi]))# #propensity vector
a0 = a_sum(pv,l) #sum of all transition probabilities
#print ini#,tim, pv, a0
tau = (-1/a0)*np.log(np.random.random())
event = multinomial(1,(pv/a0)) # event which will happen on this iteration
ini += tm[:,event.nonzero()[0][0]]
#print tc, ini
tc += tau
steps +=1
if a0 == 0: break
self.res[tim,:,round] = ini
if a0 == 0: break
# tvec = tvec[:tim]
# self.res = res[:tim,:,round]
return steps
def CR(self,pv):
"""
Composition reaction algorithm
"""
pass
def main():
vars = ['s','i','r']
cdef np.ndarray ini= np.array([500,1,0],dtype = int)
cdef np.ndarray rates = np.array([.001,.1],dtype=float)
cdef np.ndarray tm = np.array([[-1,0],[1,-1],[0,1]])
prop = [l1,l2]
M = Model(vnames = vars,rates = rates,inits=ini, tmat=tm,propensity=prop)
t0=time.time()
M.run(tmax=80,reps=1000)
print 'total time: ',time.time()-t0
cdef double a_sum(np.ndarray a, int len):
cdef double s
cdef int i
s=0
for i from 0 <=i <len:
s+=a[i]
return s
def l1(np.ndarray r,np.ndarray ini):
return r[0]*ini[0]*ini[1]
def l2(np.ndarray r,np.ndarray ini):
return r[1]*ini[1]
I would appreciate any suggestions to speed up the gillespie module, both the Python and the Cython versions.
