I am trying to fit a Susceptible-Infectious-Recovered (SIR) ODE model to infection data in python using scipy.optimize, but the infection curve of the ODE model leads to a delayed rise in infections and an overestimated peak height (see the orange line below).
The data comes from an agent-based simulation with homogeneous mixing and three states (S, I, and R). In theory, this should be in accordance with the assumptions of ODE models and the model should be able to fit perfectly. I am not sure whether my fitting method needs improvement or whether the agent-based simulation creates a shape that cannot be reproduced with a simple ODE model.
Below is the data and my attempt.
import numpy as npimport matplotlib.pyplot as pltfrom scipy import integrate, optimizeinf_data = [0.005,0.007,0.009,0.012,0.015,0.019,0.024,0.03,0.036,0.044,0.053,0.063,0.074,0.086,0.098,0.112,0.126,0.141,0.157,0.172,0.188,0.203,0.218,0.232,0.247,0.259,0.272,0.282,0.292,0.302,0.308,0.315,0.318,0.321,0.322,0.321,0.319,0.317,0.312,0.307,0.301,0.294,0.286,0.279,0.27,0.262,0.253,0.244,0.236,0.226,0.217,0.207,0.197,0.188,0.18,0.171,0.163,0.155,0.147,0.14,0.132,0.125,0.119,0.112,0.106,0.1,0.095,0.089,0.085,0.08,0.076,0.071,0.067,0.064,0.06,0.056,0.053,0.05,0.047,0.044,0.041,0.039,0.037,0.034,0.032,0.03,0.028,0.026,0.025,0.023,0.022,0.021,0.019,0.018,0.017,0.016,0.015,0.014,0.013,0.012,0.011] #percentage of infectedtime_points = np.arange(len(inf_data))def sir_model(y, x, beta, gamma): S, I, R = y dS = -beta * S * I / N dI = beta * S * I / N - gamma * I dR = gamma * I return dS, dI, dRdef fit_odeint(time_points, beta, gamma): return integrate.odeint(sir_model, (S0, I0, R0), time_points, args=(beta, gamma))[:,1]# Initial valuesN = 1.0I0 = inf_data[0]S0 = N - I0R0 = 0.0popt, pcov = optimize.curve_fit(fit_odeint, time_points, inf_data)fitted = fit_odeint(time_points, *popt)plt.plot(time_points, inf_data, 'o')plt.plot(time_points, fitted)plt.show()