Simple Dynamical Model in PyMC3

Question

I'm trying to put together a model of a dynamical system in PyMC3, to infer two parameters. The model is the basic SIR, commonly used in epidemiology :

dS/dt = - r0 * g * S * I

dI/dt = g * I ( r * S - 1 )

where r0 and g are parameters to be inferred. So far, I'm unable to get very far at all. The only examples I've seen of putting together a Markov chain like this yields errors about recursion being too deep. Here's my example code.

# Time
t = np.linspace(0, 8, 200)

# Simulated observation
def SIR(y, t, r0, gamma) :
    S = - r0 * gamma * y[0] * y[1]
    I = r0 * gamma * y[0] * y[1] - gamma * y[1]
    return [S, I]

# Currently no noise, we just want to infer params r0 = 16 and g = 0.5
solution = odeint(SIR, [0.99, 0.01, 0], t, args=(16., 0.5))


with pymc.Model() as model :
    r0 = pymc.Normal("r0", 15, sd=10)
    gamma = pymc.Uniform("gamma", 0.3, 1.)

    # Use forward Euler to solve
    dt = t[1] - t[0]

    # Initial conditions
    S = [0.99]
    I = [0.01]

    for i in range(1, len(t)) :
        S.append(pymc.Normal("S%i" % i, \
                         mu = S[-1] + dt * (-r0 * gamma * S[-1] * I[-1]), \
                         sd = solution[:, 0].std()))
        I.append(pymc.Normal("I%i" % i, \
                         mu = I[-1] + dt * ( r0 * gamma * S[-1] * I[-1] - gamma * I[-1]), \
                         sd = solution[:, 1].std()))

    Imcmc = pymc.Normal("Imcmc", mu = I, sd = solution[:, 1].std(), observed = solution[:, 1])

    #start = pymc.find_MAP()
    trace = pymc.sample(2000, pymc.NUTS())

Any help would be much appreciated. Thanks !

Answer 1

I would try defining a new distribution. Something like the following. However, this is not quite working, and I'm not quite sure what I did wrong.

class SIR(Distribution): 
def __init__(self, gamma, r0,dt, std): 
    self.gamma = gamma
    self.r0 = r0
    self.std = std
    self.dt = dt

def logp(self, SI):
    r0 = self.r0 
    std = self.std 
    gamma = self.gamma 
    dt = self.dt

    S=SI[:,0]
    I=SI[:,1]

    Si = S[1:]
    Si_m1 = S[:-1]
    Ii = I[1:]
    Ii_m1 = I[:-1]

    Sdelta = (Si - Si_m1)
    Idelta = (Ii - Ii_m1)

    Sexpected_delta = dt* (-r0 * gamma * Si_m1 * Ii_m1)
    Iexpected_delta = dt * gamma * Ii_m1 *( r0 * Si_m1 - 1 )


    return (Normal.dist(Sexpected_delta, sd=std).logp(Sdelta) +
            Normal.dist(Iexpected_delta, sd=std).logp(Idelta))


with Model() as model: 
    r0 = pymc.Normal("r0", 15, sd=10)
    gamma = pymc.Normal("gamma", 0.3, 1.)
    std = .5
    dt = t[1]-t[0]


    SI = SIR('SI', gamma, r0, std,dt, observed=solution[:,:2])

    #start = pymc.find_MAP(start={'gamma' : .45, 'r0' : 17})
    trace = pymc.sample(2000, pymc.NUTS())