import pymc as pm
import numpy as np

def model(data):
    """
    Defines a PyMC model for predicting infected counts in a population of 50.
    
    Args:
    data (dict): A dictionary containing 'Time' and 'Infected_Count' arrays.
    
    Returns:
    pm.Model: The defined PyMC model.
    """
    time = np.array(data['Time'])
    infected_counts = np.array(data['Infected_Count'])
    N = 50  # Total population size
    
    with pm.Model() as m:
        # Priors for the logistic regression coefficients
        # alpha: intercept, beta: slope
        alpha = pm.Normal('alpha', mu=0, sigma=5)
        beta = pm.Normal('beta', mu=0, sigma=5)
        
        # Compute the probability of infection at each time point
        # Using the logistic function: p = 1 / (1 + exp(-(alpha + beta * t)))
        logit_p = alpha + beta * time
        p = pm.math.invlogit(logit_p)
        
        # Observed likelihood: Binomial distribution
        # The number of infected individuals is binomially distributed with N trials and probability p
        infected_observed = pm.Binomial(
            'infected_observed',
            n=N,
            p=p,
            observed=infected_counts
        )
        
    return m