import pymc as pm
import numpy as np

def model(data):
    """
    Construct a PyMC model for the death_process task.
    
    Parameters:
    data (dict): A dictionary containing 'Time' and 'Infected_Count' arrays.
    
    Returns:
    pm.Model: The PyMC model.
    """
    times = np.array(data['Time'])
    infected_counts = np.array(data['Infected_Count'])
    N = 50  # Total population size
    
    with pm.Model() as pmodel:
        # Define parameters for the model
        # We assume a linear relationship between time and the logit of the probability of infection
        # p(t) = sigmoid(alpha + beta * t)
        # Then the number of infected follows a Binomial(N, p(t))
        
        # Priors for the linear coefficients
        alpha = pm.Normal('alpha', mu=0, sigma=10)
        beta = pm.Normal('beta', mu=0, sigma=10)
        
        # Calculate the probability of infection at each time point
        logit_p = alpha + beta * times
        p = pm.math.sigmoid(logit_p)
        
        # Observed likelihood
        infected_observed = pm.Binomial('infected_observed', n=N, p=p, observed=infected_counts)
        
    return pmodel