Seite - 229 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

8.3 SpreadingofDisease:ASystemofFirstOrderODEs 229 forn= 0,1,.. .,Nt. This is an approximation since the differential equations are originallyvalid at all times t (usually in some finite interval [0,T]). Using forward finitedifferencesfor thederivatives results in anadditionalapproximation, Sn+1 −Sn Δt =−βSnIn, (8.26) In+1 −In Δt =βSnIn−γIn, (8.27) Rn+1 −Rn Δt =γIn . (8.28) As we can see, these equations are identical to the difference equations that naturally arise in the derivation of the model. However, other numerical methods than the Forward Euler scheme will result in slightly different difference equa- tions. 8.3.3 ProgrammingtheFEScheme;theSpecialCase The computation of (8.26)–(8.28) can be readily made in a computer program SIR1.py: import numpy as np import matplotlib.pyplot as plt # Time unit: 1 h beta = 10./(40*8*24) gamma = 3./(15*24) dt = 0.1 # 6 min D = 30 # Simulate for D days N_t = int(D*24/dt) # Corresponding no of time steps t = np.linspace(0, N_t*dt, N_t+1) S = np.zeros(N_t+1) I = np.zeros(N_t+1) R = np.zeros(N_t+1) # Initial condition S[0] = 50 I[0] = 1 R[0] = 0 # Step equations forward in time for n in range(N_t): S[n+1] = S[n] - dt*beta*S[n]*I[n] I[n+1] = I[n] + dt*beta*S[n]*I[n] - dt*gamma*I[n] R[n+1] = R[n] + dt*gamma*I[n] fig = plt.figure() l1, l2, l3 = plt.plot(t, S, t, I, t, R) fig.legend((l1, l2, l3), (’S’, ’I’, ’R’), ’center right’) plt.xlabel(’hours’)