import pymc as pm
import numpy as np

def model(data):
    time = np.array(data['Time'])
    infected_counts = np.array(data['Infected_Count'])
    
    with pm.Model() as model:
        # Priors for the logistic curve parameters
        # beta: growth rate, should be positive
        beta = pm.Gamma('beta', alpha=2, beta=1)
        
        # t_mid: midpoint of the logistic curve, centered around 1.0 based on data
        t_mid = pm.Normal('t_mid', mu=1.0, sigma=0.5)
        
        # Total population is fixed at 50
        K = 50
        
        # Deterministic logistic function
        # I(t) = K / (1 + exp(-beta * (t - t_mid)))
        expected_infected = pm.Deterministic(
            'expected_infected',
            K / (1 + pm.math.exp(-beta * (time - t_mid)))
        )
        
        # Likelihood
        # Using Gaussian likelihood for simplicity, as counts are relatively high
        # and the Gaussian approximation to Binomial is reasonable.
        # Alternatively, Binomial likelihood: Binomial(n=50, p=expected_infected/50)
        # Let's use Binomial for better statistical grounding.
        p_infected = expected_infected / K
        # Clip p to [0, 1] to avoid numerical issues
        p_infected = pm.math.clip(p_infected, 0, 1)
        
        likelihood = pm.Binomial(
            'likelihood',
            n=50,
            p=p_infected,
            observed=infected_counts
        )
        
    return model