Epidemics on networks

Transcription

1 Epidemics on networks Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Network Science Leonid E. Zhukov (HSE) Lecture / 38

2 Lecture outline 1 Probabilistic model SI model SIS model SIR model 2 Simulations 3 Experimental results Leonid E. Zhukov (HSE) Lecture / 38

3 Probabilistic model network of potential contacts (adjacency matrix A) probabilistic model (state of a node): s i (t) - probability that at t node i is susceptible x i (t) - probability that at t node i is infected r i (t) - probability that at t node i is recovered β - infection rate (probably to get infected on a contact in time δt) γ - recovery rate (probability to recover in a unit time δt) from deterministic to probabilistic description connected component - all nodes reachable network is undirected (matrix A is symmetric) Leonid E. Zhukov (HSE) Lecture / 38

4 Probabilistic model Two processes: Node infection: P inf = s i (t) 1 (1 βx j (t)δt) βs i (t) x j (t)δt Node recovery: j N (i) j N (i) P rec = γx i (t)δt Leonid E. Zhukov (HSE) Lecture / 38

5 SI model SI Model S I Probabilities that node i: s i (t) - susceptible, x i (t) -infected at t x i (t) + s i (t) = 1 β - infection rate, probability to get infected in a unit time x i (t + δt) = x i (t) + βs i A ij x j δt infection equations j dx i (t) dt = βs i (t) j A ij x j (t) x i (t) + s i (t) = 1 Leonid E. Zhukov (HSE) Lecture / 38

6 SI model Differential equation dx i (t) dt = β(1 x i (t)) j A ij x j early time approximation, t 0, x i (t) 1 dx i (t) dt = β j A ij x j Solution in the basis dx(t) dt = βax(t) Av k = λ k v k x(t) = k a k (t)v k Leonid E. Zhukov (HSE) Lecture / 38

7 SI model k da k dt v k = β k da k (t) dt a k (t) = a k (0)e βλ kt, Aa k (t)v k = β k = βλ k a k (t) a k (t)λ k v k a k (0) = v T k x(0) Solution x(t) = k a k (0)e λ kβt v k t 0, λ max = λ 1 > λ k x(t) = v 1 e λ 1βt 1 growth rate of infections depends on λ 1 2 probability of infection of nodes depends on v 1, i.e v 1i Leonid E. Zhukov (HSE) Lecture / 38

8 SI model late- time approximation, t, x i (t) const dx i (t) dt = β(1 x i (t)) j A ij x j = 0 Ax 0 since λ min 0, 1 x i (t) 0 All nodes in connected component get infected t x i (t) 1 Connected component structure and distribution. Does initially infected node belong to GCC? Leonid E. Zhukov (HSE) Lecture / 38

9 SI model image from M. Newman, 2010 Leonid E. Zhukov (HSE) Lecture / 38

10 SI simulation 1 Every node at any time step is in one state {S, I } 2 Initialize c nodes in state I 3 On each time step each I node has a probability β to infect its nearest neighbors (NN), S I Model dynamics: I + S β 2I Leonid E. Zhukov (HSE) Lecture / 38

11 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

12 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

13 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

14 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

15 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

16 SI model simulation β = 0.5 Leonid E. Zhukov (HSE) Lecture / 38

17 SI model Leonid E. Zhukov (HSE) Lecture / 38

18 SIS model SIS Model S I S Probabilites that node i: s i (t) - susceptable, x i (t) -infected at t x i (t) + s i (t) = 1 β - infection rate, γ - recovery rate infection equations: dx i (t) dt = βs i (t) j A ij x j (t) γx i x i (t) + s i (t) = 1 Leonid E. Zhukov (HSE) Lecture / 38

19 SIS model Differential equation dx i (t) dt = β(1 x i (t)) j A ij x j γx i early time approximation, x i (t) 1 dx i (t) dt = β j A ij x j γx i dx i (t) dt = β j (A ij γ β δ ij)x j dx(t) dt dx(t) dt = β(a ( γ β )I)x(t) = βmx(t), M = A ( γ β )I Leonid E. Zhukov (HSE) Lecture / 38

20 SIS model Eigenvector basis Mv k = λ k v k, M = A ( γ β )I, Av k = λ k v k Solution v k = v k, λ k = λ k γ β x(t) = k a k (t)v k = k a k (0)v k eλ k βt = k a k (0)v k e (βλ k γ)t λ 1 λ k, critical: βλ 1 = γ -if βλ 1 > γ, x(t) v 1 e (βλ1 γ)t - growth -if βλ 1 < γ, x(t) 0 - decay Leonid E. Zhukov (HSE) Lecture / 38

21 SIS model Epidemic threshold R 0 : if β γ < R 0 - infection dies over time if β γ > R 0 - infection survives and becomes epidemic In SIS model: R 0 = 1 λ 1, Av 1 = λ 1 v 1 Leonid E. Zhukov (HSE) Lecture / 38

22 SIS model long time t : x i (t) const dx i (t) dt = β(1 x i ) j A ij x j γx i = 0 x i = j A ijx j γ β + j A ijx j Above the epidemic threshold (β/γ > R 0 ) - if β γ, x i (t) 1 - if β γ, x i γ β = j A ijx j, then λ 1 = γ β, x i(t) (v 1 ) i Leonid E. Zhukov (HSE) Lecture / 38

23 SIS simulation 1 Every node at any time step is in one state {S, I } 2 Initialize c nodes in state I 3 Each node stays infected τ γ = 0 τe τγ dτ = 1/γ time steps 4 On each time step each I node has a prabability β to infect its nearest neighbours (NN), S I 5 After τ γ time steps node recovers, I S Model dynamics: { I + S I β γ 2I S Leonid E. Zhukov (HSE) Lecture / 38

24 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

25 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

26 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

27 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

28 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

29 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

30 SIS model simulation β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

31 SIS model Leonid E. Zhukov (HSE) Lecture / 38

32 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

33 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

34 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

35 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

36 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

37 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

38 SIS model simulation β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

39 SIS model Leonid E. Zhukov (HSE) Lecture / 38

40 SIR model SIR Model S I R probabilities s i (t) -susceptable, x i (t) - infected, r i (t) - recovered s i (t) + x i (t) + r i (t) = 1 β - infection rate, γ - recovery rate Infection equation: dx i dt = βs i A ij x j γx i j dr i dt = γx i x i (t) + s i (t) + r i (t) = 1 Leonid E. Zhukov (HSE) Lecture / 38

41 SIR model Differential equation dx i (t) dt = β(1 r i x i ) j A ij x j γx i early time, t 0, r i 0, SIS = SIR dx i (t) dt = β(1 x i ) j A ij x j γx i Solution x(t) v 1 e (βλ 1 γ)t Leonid E. Zhukov (HSE) Lecture / 38

42 SIR model image from M. Newman, 2010 Leonid E. Zhukov (HSE) Lecture / 38

43 SIR simulation 1 Every node at any time step is in one state {S, I, R} 2 Initialize c nodes in state I 3 Each node stays infected τ γ = 1/γ time steps 4 On each time step each I node has a prabability β to infect its nearest neighbours (NN), S I 5 After τ γ time steps node recovers, I R 6 Nodes R do not participate in further infection propagation Model dynamics: { I + S I β γ 2I R Leonid E. Zhukov (HSE) Lecture / 38

44 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

45 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

46 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

47 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

48 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

49 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

50 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

51 SIR model Leonid E. Zhukov (HSE) Lecture / 38

52 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

53 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

54 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

55 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

56 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

57 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 38

58 SIR model Leonid E. Zhukov (HSE) Lecture / 38

59 SIR epidemics as percolation image from D. Easley and J. Kleinberg, 2010 Leonid E. Zhukov (HSE) Lecture / 38

60 SIR epidemics as percolation Bond percolation: only those nodes that are on a percolation path with infected node will get infected Leonid E. Zhukov (HSE) Lecture / 38

61 5 Networks, SIR Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free image from Keeling et al, 2005 Leonid E. Zhukov (HSE) Lecture / 38

62 5 Networks, SIR Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free Keeling et al, 2005 Leonid E. Zhukov (HSE) Lecture / 38

63 Effective distance J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013 Leonid E. Zhukov (HSE) Lecture / 38

64 Network synchronization, SIRS Small-world network at different values of disorder parameter p Kuperman et al, 2001 Leonid E. Zhukov (HSE) Lecture / 38

65 Flu contagion Infected - red, friends of infected - yellow N. Christakis, J. Fowler, 2010 Leonid E. Zhukov (HSE) Lecture / 38

66 References Epidemic outbreaks in complex heterogeneous networks. Y. Moreno, R. Pastor-Satorras, and A. Vespignani., Eur. Phys. J. B 26, 521?529, Networks and Epidemics Models. Matt. J. Keeling and Ken.T.D. Eames, J. R. Soc. Interfac, 2, , 2005 Simulations of infections diseases on networks. G. Witten and G. Poulter. Computers in Biology and Medicine, Vol 37, No. 2, pp , 2007 Small World Effect in an Epidemiological Model. M. Kuperman and G. Abramson, Phys. Rev. Lett., Vol 86, No 13, pp , 2001 Manitz J, Kneib T, Schlather M, Helbing D, Brockmann D. Origin Detection During Food-borne Disease Outbreaks - A Case Study of the 2011 EHEC/HUS Outbreak in Germany. PLoS Currents, 2014 Leonid E. Zhukov (HSE) Lecture / 38