Figure 2.11 (a and b) ------------------------- :download:Downloadable Source Code  .. image:: fig2p11a.png :width: 45 % .. image:: fig2p11b.png :width: 45 % :: import EoN import networkx as nx import matplotlib.pyplot as plt import scipy import random from scipy import integrate ''' Code to generate figure 2.11. This is a bit messy because we have to define the ODE models. Since python deals with ODEs by taking 1D arrays we have to set up all the variables into a single long vector. ''' def star(N): G = nx.Graph() G.add_node(0) for node_id in range(1,N): G.add_edge(0,node_id) return G def complete_graph_dX(X, t, tau, gamma, N): r'''This system is given in Proposition 2.3, taking Q=S, T=I f_{SI}(k) = f_{QT}= k*\tau f_{IS}(k) = f_{TQ} = \gamma \dot{Y}^0 = \gamma Y^1 - 0\\ \dot{Y}^1 = 2\gamma Y^2 + 0Y^0 - (\gamma + (N-1)\tau)Y^1 \dot{Y}^2 = 3\gamma Y^3 + (N-1)\tau Y^1 - (2\gamma+2(N-2))Y^2 ... \dot{Y}^N = (N-1)\tau Y^{N-1} - N\gamma Y^N Note that X has length N+1 ''' #X[k] is probability of k infections. dX = [] dX.append(gamma*X[1]) for k in range(1,N): dX.append((k+1)*gamma*X[k+1]+ (N-k+1)*(k-1)*tau*X[k-1] - ((N-k)*k*tau + k*gamma)*X[k]) dX.append((N-1)*tau*X[N-1] - N*gamma*X[N]) return scipy.array(dX) def complete_graph_lumped(N, tau, gamma, I0, tmin, tmax, tcount): times = scipy.linspace(tmin, tmax, tcount) X0 = scipy.zeros(N+1) #length N+1 of just 0 entries X0[I0]=1. #start with 100 infected. X = integrate.odeint(complete_graph_dX, X0, times, args = (tau, gamma, N)) #X[t] is array whose kth entry is p(k infected| time=t). I = scipy.array([sum(k*Pkt[k] for k in range(len(Pkt))) for Pkt in X]) S = N-I return times, S, I def star_graph_dX(X, t, tau, gamma, N): '''this system is given in Proposition 2.4, taking Q=S, T=I so f_{SI}(k) = f_{QT}(k) = k*tau f_{IS}(k) = f_{TQ}(k) = gamma X has length 2*(N-1)+2 = 2N''' # [[central node infected] + [central node susceptible]] #X = [Y_1^1, Y_1^2, ..., Y_1^{N}, Y_2^0, Y_2^1, ..., Y_2^{N-1}] #Note that in proposition Y^0 is same as Y_2^0 #and Y^N is same as Y_1^N #Y1[k]: central node infected, & k-1 peripheral nodes infected Y1vec = [0]+list(X[0:N]) #for Y_1^k, use Y1vec[k] #pad with 0 to make easier calculations Y_1^0=0 #the probability of -1 nodes infected is 0 #Y2[k]: central node susceptible & k peripheral nodes infected Y2vec = list(X[N:])+[0] #for Y_2^k use Y2vec[k] #padded with 0 to make easier calculations. Y_2^N=0 #the probability of N (of N-1) peripheral nodes infected is 0 dY1vec = [] dY2vec = [] for k in range(1, N): #k-1 peripheral nodes infected, central infected dY1vec.append((N-k+1)*tau*Y1vec[k-1] + (k-1)*tau*Y2vec[k-1] +k*gamma*Y1vec[k+1] - ((N-k)*tau + (k-1)*gamma+gamma)*Y1vec[k]) #now the Y^N equation dY1vec.append(tau*Y1vec[N-1] + (N-1)*tau*Y2vec[N-1] - N*gamma*Y1vec[N]) #now the Y^0 equation dY2vec.append(gamma*(N-1)*Y1vec[1] + gamma*Y2vec[1]-0) for k in range(1,N): #k peripheral nodes infected, central susceptible dY2vec.append(0 + gamma*Y1vec[k+1] + gamma*(k+1)*Y2vec[k+1] - (k*tau + 0 + k*gamma)*Y2vec[k]) return scipy.array(dY1vec + dY2vec) def star_graph_lumped(N, tau, gamma, I0, tmin, tmax, tcount): times = scipy.linspace(tmin, tmax, tcount) # [[central node infected] + [central node susceptible]] #X = [Y_1^1, Y_1^2, ..., Y_1^{N}, Y_2^0, Y_2^1, ..., Y_2^{N-1}] X0 = scipy.zeros(2*N) #length 2*N of just 0 entries X0[I0]=I0*1./N #central infected, + I0-1 periph infected prob X0[N+I0] = 1-I0*1./N #central suscept + I0 periph infected X = EoN.my_odeint(star_graph_dX, X0, times, args = (tau, gamma, N)) #X looks like [[central susceptible,k periph] [ central inf, k-1 periph]] x T central_inf = X[:,:N] central_susc = X[:,N:] I = scipy.array([ sum(k*central_susc[t][k] for k in range(N)) + sum((k+1)*central_inf[t][k] for k in range(N)) for t in range(len(X))]) S = N-I return times, S, I N=1000 I0=int(0.1*N) iterations = 100 #number of simulations to compare gamma = 1 tmin=0 tmax=5 tcount = 1001 report_times = scipy.linspace(tmin, tmax, 21) #for simulations plt.figure(0) tau = 0.005 G = nx.complete_graph(N) t, S, I = complete_graph_lumped(N, tau, gamma, I0, tmin, tmax, tcount) plt.plot(t, I/N, label = 'Prediction') #now check with simulation obs_I = 0*report_times print("done with complete graph ODE. Now simulating") for counter in range(iterations): IC = random.sample(range(N),I0) t, S, I = EoN.fast_SIS(G, tau, gamma, initial_infecteds = IC, tmax = tmax) obs_I += EoN.subsample(report_times, t, I) plt.plot(report_times, obs_I*1./(iterations*N), 'o', label='Simulation') plt.axis(ymin=0, ymax=1) plt.xlabel('$t$') plt.ylabel('$[I]$') plt.legend() plt.savefig('fig2p11a.png') print("done with complete graph. Now star --- warning, this may be slow") plt.clf() #for star, if 100 nodes randomly start infected, 1/10 cases have #central node infected and 99 peripheral. 9/10 have 100 peripheral. tau = 4. G = star(N) t, S, I = star_graph_lumped(N, tau, gamma, I0, tmin, tmax, tcount) plt.plot(t, I/N, label = 'Prediction') print("done with star ODE, now simulating") obs_I = 0*report_times for counter in range(iterations): IC = random.sample(range(N),I0) t, S, I = EoN.fast_SIS(G, tau, gamma, initial_infecteds = IC, tmax = tmax) obs_I += EoN.subsample(report_times, t, I) plt.plot(report_times, obs_I*1./(iterations*N), 'o', label='Simulation') plt.axis(ymin=0, ymax=1) plt.xlabel('$t$') plt.ylabel('$[I]$') plt.legend() plt.savefig('fig2p11b.png')