Epidemics on networks Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural Analysis and Visualization of Networks Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 1 / 28
Lecture outline 1 Probabilistic model SI model SIS model SIR model 2 Stochastic modeling 3 Experimental results Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 2 / 28
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 13 07.04.2015 3 / 28
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 13 07.04.2015 4 / 28
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) = βs i (t) j A ij x j (t) x i (t) + s i (t) = 1 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 5 / 28
SI model Differential equation dx i (t) = β(1 x i (t)) j A ij x j early time approximation, t 0, x i (t) 1 dx i (t) = β j A ij x j Solution in the basis dx(t) = βax(t) Av k = λ k v k x(t) = k a k (t)v k Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 6 / 28
SI model k da k v k = β k da k (t) 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 13 07.04.2015 7 / 28
SI model late- time approximation, t, x i (t) const dx i (t) = 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 13 07.04.2015 8 / 28
SI model image from M. Newman, 2010 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 9 / 28
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) = βs i (t) j A ij x j (t) γx i x i (t) + s i (t) = 1 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 10 / 28
SIS model Differential equation dx i (t) = β(1 x i (t)) j A ij x j γx i early time approximation, x i (t) 1 dx i (t) = β j A ij x j γx i dx i (t) = β j (A ij γ β δ ij)x j dx(t) dx(t) = β(a ( γ β )I)x(t) = βmx(t), M = A ( γ β )I Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 11 / 28
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 13 07.04.2015 12 / 28
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 13 07.04.2015 13 / 28
SIS model long time t : x i (t) const dx i (t) = 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 13 07.04.2015 14 / 28
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 = βs i A ij x j γx i j dr i = γx i x i (t) + s i (t) + r i (t) = 1 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 15 / 28
SIR model Differential equation dx i (t) = 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) = 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 13 07.04.2015 16 / 28
SIR model image from M. Newman, 2010 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 17 / 28
SIS simulaiton 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 image from D. Easley and J. Kleinberg, 2010 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 18 / 28
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 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 19 / 28
SIR simulation image from D. Easley and J. Kleinberg, 2010 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 20 / 28
Stochastic modeling Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 21 / 28
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 13 07.04.2015 22 / 28
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 13 07.04.2015 23 / 28
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 13 07.04.2015 24 / 28
Effective distance J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 25 / 28
Network synchronization, SIRS Small-world network at different values of disorder parameter p Kuperman et al, 2001 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 26 / 28
Flu contagion N. Christakis, J. Fowler, 2010 Leonid E. Zhukov (HSE) Lecture 13 07.04.2015 27 / 28
References Networks and Epidemics Models. Matt. J. Keeling and Ken.T.D. Eames, J. R. Soc. Interfac, 2, 295-307, 2005 Simulations of infections diseases on networks. G. Witten and G. Poulter. Computers in Biology and Medicine, Vol 37, No. 2, pp 195-205, 2007 Small World Effect in an Epidemiological Model. M. Kuperman and G. Abramson, Phys. Rev. Lett., Vol 86, No 13, pp 2909-2912, 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 13 07.04.2015 28 / 28