On Susceptible-Infected-Susceptible Epidemic Spreading: an Overview of Recent Study

Transcription

1 The 1 st Net-X (2017) On Susceptible-Infected-Susceptible Epidemic Spreading: an Overview of Recent Study Cong Li Adaptive Networks and Control Lab, Fudan University Collaborators: Xiaojie Li, Jianbo Wang, Jing Li, Xiang Li (Fudan Univ.) Bo Qu, Piet Van Mieghem, Huijuan Wang (TUD) Shanghai Jiao Tong University, 31st March,

2 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 2

3 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 3

4 Motivation for virus spread in networks virus Understanding the spread of a virus is the first step in preventing it. Infectious diseases enormous morbidity and mortality tremendous economic losses Computer viruses security threat to internet costly 4

5 Application of virus spread models Epidemic algorithms rumor gossip Error propagation in networks Gossiping Emotions as infectious diseases in social networks Public opinion spreading Adaptive Networks and Control Lab 5

6 Epidemics in Networks Homogeneous infection rate on all edges between infected and susceptible nodes Homogeneous curing rate t = b /d : effective spreading rate d d b for infected nodes 1 d Healthy 3 b 0 b Infected 2 Infected Adaptive Networks and Control Lab 6

7 Heterogeneous Mean-field (HMF) Approximation of the SIS Model Dynamical mean-field reaction rate equation is written as t r k (t) = -dr k (t)+ bk[1- r k (t)]q(r(t)) where r k (t) is the relative density of infected nodes with degree k, and Q(r(t)) is the probability that any given link points to an infected node. [1] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Physical review letters, vol. 86, no. 14, p. 3200, Adaptive Networks and Control Lab 7

8 N-intertwined Mean-field Approximation (NIMFA) of the SIS Model Each node j can be in either of the two states: 0 : healthy 1 : infected Markov continuous time: Infection rate β Curing rate δ Mathematically: X j is the state of node j Infinitesimal generator Q j (t) = é ê ê ë -q 1 j q 1 j q 2 j 0 -q 2 j b d N å ù é ú = ê ú û ê ë 1 q 1 j (t) = b a jk 1 {Xk (t )=1} k=1 -q 1 j q 1 j d -d ù ú ú û Adaptive Networks and Control Lab 8

9 N-intertwined Mean-field Approximation (NIMFA) of the SIS Model Markov theory requires that the infinitesimal generator in a matrix whose elements are NOT random variables However, this is not the case in our simple model Q j (t) = é ê ê ë -q 1 j q 1 j q 2 j -q 2 j ù ú ú û N å q 1 j (t) = b a jk 1 {Xk (t )=1} k=1 is replaced by its mean (the only approximation!) Q j (t) = é ê ëê -E[q 1 j ] E[q 1 j ] d -d ù ú ûú N å { ( ) = 1} E[q 1 j (t)] = b a jk Pr é ë X k t k=1 ù û [2] P. Van Mieghem, J. Omic, and R. Kooij, Virus spread in networks, IEEE/ACM Transactions on Networking, vol. 17, no. 1, pp. 1 14, 2009 Adaptive Networks and Control Lab 9

10 N-intertwined Mean-field Approximation (NIMFA) of the SIS Model N-intertwined model for virus spread where v k (t) = Pr[ X k (t) = 1] N Non-linear matrix equation: dv(t) dt = bav(t)- diag( v i ( t) )( bav(t) +du) [3] C. Li, R. van de Bovenkamp and P. Van Mieghem, Susceptible-Infected-Susceptible Model: A Comparison of N- intertwined and Heterogeneous Mean-field Approximations, Physical Review E, 86(2), , Adaptive Networks and Control Lab 10

11 Epidemic threshold of SIS Model β : infection rate per link δ : curing rate per node τ= β/ δ : effective spreading rate Epidemic threshold ( 1) 1 c 1 A [2] P. Van Mieghem, J. Omic, and R. Kooij, Virus spread in networks, IEEE/ACM Transactions on Networking, vol. 17, no. 1, pp. 1 14, 2009 Adaptive Networks and Control Lab 11

12 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 12

13 Immunization of epidemic in networks Metrics in Complex Networks Structural Metrics Average hopcount E[H] Global efficiency E[1/H] Clustering coefficient CG Degree diversity K Assortativity rd Spectral Metrics Spectral radius 1 Algebraic connectivity mn-1 Graph resistance RG Ratio m1/mn-1 Principal eigenvector x1 and many more Degree D Centrality Metrics Degree mass D (m) Closeness Cn Betweenness Bn K-shell index Ks Leverage Ln [4] C. Li, Q. Li, P. Van Mieghem, H. Eugene Stanley and H. Wang, Correlation Between Centrality Metrics and Their Application to the Opinion Model, European Physical Journal B, Vol. 88, No. 3, article 65, [5] M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse, Identication of influential spreaders in complex networks," Nature Physics, vol. 6, no. 11, pp ,

14 Q1: Does the ranking of nodal infection probability in SIS epidemic always the same? Our trial: Using NIMFA to calculate the nodal infection probability under different effective spreading rate

15 Counting of the nodal ranking changes Example: The red dish line changes dramatically from the medium vulnerable when τ=0.1 to the most vulnerable when τ=0.24 Figure 1. The meta-stable infection probability v k as a function of the effective infection rate τ for 10 random nodes in a real-world networks called Roget. [6] B. Qu, C. Li*(corresponding author), P. Van Mieghem, and H. Wang, Ranking of Nodal Infection Probability in Susceptible- Infected-Susceptible Epidemic, submitted to Scientific Reports,

16 Q2: How to estimate the change of the ranking of nodal infection probability in SIS epidemic? Our trial: calculating the number of crossings between any two trajectory v k τ 0 and v m τ 0

17 Counting the total number of crossing We define an N N matrix F with the elements fij f ij V τ 0, V τ 1 = v i τ 0 v j τ 0 v i τ 1 v j τ 1 to save whether there is a crossing between the trajectory v i τ and v j τ in the interval (τ 0, τ 1 ), and fij < 0 means there is a crossing between the two trajectories. The maximum number of crossing is N N 1 2 under the one-crossing assumption. The number of crossings in the interval (τ 0, τ 1 ) of infection rate equals to i 1 N χ(τ 0, τ 1 ) = 1 fij v τ 0,v τ 1 <0 j=1 j=1 Adaptive Networks and Control Lab 17

18 Recall:N-intertwined Mean-field Approximation (NIMFA) of the SIS Model N-intertwined model for virus spread where v k (t) = Pr[ X k (t) = 1] Infected probability at meta-stable state: N Non-linear matrix equation: v k τ = τ σ N j=1 a kj v j (τ) dv(t) dt = bav(t)- diag( v i ( t) )( bav(t) +du) [3] C. Li, R. van de Bovenkamp and P. Van Mieghem, Susceptible-Infected-Susceptible Model: A Comparison of N- intertwined and Heterogeneous Mean-field Approximations, Physical Review E, 86(2), ,

19 Lower bound of the total number of crossings From the infected probability at meta-stable state of NIMFA SIS approximation, 1 v k τ = τ σ N j=1 a kj v j (τ) We obtain τ is large enough τ > τ u if d k > d m then v k τ > v m τ On the other hand, when the effective spreading rate τ = τ c (1) + ε, the ranking of the infection probability v i (τ c (1) +ε) is the same as the ranking of the components of the principal eigenvector (x 1 ) i. Above discussion suggests that the total number of crossings of a graph can be lower bounded: 19 i 1 χ τ 1 N N c + ε, τ u = 1 1 j=1 fij v τ c +ε,v τ u <0 j=1 j=1 i 1 j=1 1 fij x 1,d <0 = χ l

20 Comparison Figure 2. The lower bound χ l versus the total number of crossings χ τ c 1 + ε, τ u in ER random graphs, BA random graphs and real-world networks. Adaptive Networks and Control Lab 20

21 Q3: what is the number of crossing in different intervals of τ? Our trial: Taylor expansion of the steady-state NIMFA infection probability

22 Number of crossings in different intervals of τ Theoretical conditions for the existing of a crossing: There is a crossing close to τ 0 and the corresponding infection probability vector V τ 0, there is a crossing close to τ 0 between the trajectory v k τ and the trajectory v m τ at τ 0 + ε km if ε km is positive and small enough. ε km = v k τ v m τ v k τ v m τ τ τ We compare the theoretical results and the numerical results from i 1 N χ(τ 0, τ 1 ) = 1 fij v τ 0,v τ 1 <0 j=1 j=1 Adaptive Networks and Control Lab 22

23 Where the crossings are more likely appear where α j = τ j τ c (1) τ Figure 3. the number χ α j 1, α j of crossings as a function of the normalized effective infection rate α j Theoretical results agree well with the numerical results. When α j is smaller, a smaller value of (α j α j 1 ) τ c (1) is needed for theoretical results to be accurate. The more crossings appear when the effective infection rate is smaller. Adaptive Networks and Control Lab 23

24 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 24

25 Q4: How to vaccinate the nodes when we consider the social cost? Our trial: with Zero-Determinant strategies An individual using Zero-determinant (ZD) strategy [7] can unilaterally set the expected cost of the opponent irrespective of his/her strategy. [7] W. H. Press and F. J. Dyson, Iterated Prisoners Dilemma contains strategies that dominate any evolutionary opponent, PNAS, 2012.

26 Vaccination game Players: an administrator user (AU) and other general users (GUs ) Strategies: AU the overall network security adopting ZD strategy invest in the antivirus protection/get vaccinated (Cooperator) minimize the social cost(all GUs total expected cost) do not invest in the antivirus protection, having a risk of being infected (Defector) Payoff: the cost of cooperation : W, the cost of defection : H In the stationary state of the epidemic process, the cost of getting infected: H v, ( m) i Adaptive Networks and Control Lab 26

27 Zero-determinant strategy in a complete graph the number of cooperators among RUs is n AU choose to cooperate the probability to cooperate for AU AU p Cn, Theorem 1. When the administrator (AU) takes an appropriate probability strategy vector, satisfying then we obtain the total expected cost of GUs, Previous round AU choose to defect AU p Dn, Current round p [ p,..., p,..., p p,..., p,..., p ] AU AU AU AU AU AU T AU C,0 C, n C, N 1, D,0 D, n D, N 1, where, E i and s i correspond to the expected cost and cost vector of node i. [8] X.-J. Li, C. Li, X. Li, Vaccinating SIS epidemics in networks with zero-determinant strategy, accepted by IEEE Symposium on circuits and systems (ISCAS), Adaptive Networks and Control Lab 27

28 In a complete graph with N nodes, the cost of immunization and infection are W and H, respectively. E is the economic incentive and τ is the effective spreading rate vi, ( m) ( m 1), if and m 2 m 1 0, otherwise Adaptive Networks and Control Lab 28

29 Simulation results W = 0.4, H = 0.5 effective spreading rate τ = 1.1 Figure 7 (a) shows that the social cost always converges to the same value regardless of the strategies of GUs when AU implements the given ZD strategy. (b) shows that the social cost corresponding to all feasible ZD strategies within the feasible region. When the two probabilities satisfy, the social cost is the minimum. Adaptive Networks and Control Lab 29

30 Comparison with other strategy Figure 8. the relative cost compared to that in [9] vs. (a) the size N of networks and (b) the effective spreading rate τ. ZD strategy has an advantage in the performance of optimizing the social cost. [9] S. Trajanovski, Y. Hayel, E. Altman, H. J. Wang and P. Van Mieghem, Decentralized protection strategies against SIS epidemics in networks, IEEE Transactions on Control of Network Systems, Adaptive Networks and Control Lab 30

31 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 31

32 Reaction-Diffusion Model on Metapopulation Networks unit time t reaction t 2 diffusion t 2 Reaction process I + S β 2I From t to t + t 2 p ijxj Diffusion process X i, X represents S or I to t + t I represents a infected individual, S represents a susceptible individual. β is the infection rate. X is a placeholder. p ij is the diffusion rate from subpopulation i to subpopulation j. From t + t 2 [10] Hufnagel L, Brockmann D, Geisel T., PNAS, 101: , [11] Colizza, V., Pastor-Satorras, R. & Vespignani A., Nature Phys.3, ,

33 Q5: Is it possible to predict spatial transmission of epidemics on a networked metapopulation? Our trial: Prediction Algorithm Based on Maximum Likelihood Estimation

34 Conditions to Predict Known: The time series of infected cases (number of infected individuals of each infected subpopulation I i t ) at time step t The topology of the metapopulation networks (including diffusion rates p ij and demography) To predict: The susceptible subpopulation(s) which will be infected in the next time step Adaptive Networks and Control Lab 34

35 A Prediction Algorithm Algorithm steps: 1)Estimate Infection Rate β I i t + t is number of infected individuals of subpopulation i, t is the unit time. β is the possible value of actual β. 2) Calculate the Infection Likelihood m is the number of infected neighbor subpopulations, L i is the infection likelihood of susceptible subpopulation i. Adaptive Networks and Control Lab 35

36 A Prediction Algorithm Algorithm steps: 3) Predict the Spatial Transmission i) n = 1; ii) n 2 P i repsents the infection likelihood of subpopulation i only i will be infected at next time step. v is the label of predicted subpopulation which will be infected. v n is the labels of predicted subpopulations which will be infected at next time step. [10] J.-B. Wang, C. Li* (corresponding author) and X. Li, Predicting spatial transmission at the early stage of epidemics on a networked metapopulation, Proceedings of the 12th IEEE International Conference on Control & Automation, Kathmandu, Nepal, 2016, Jun. 1-3, p Adaptive Networks and Control Lab 36

37 Simulation Results BA scale-free metapopulation network N = 404; N i = ; population = , < k > = 16. The identification results for beta=0.05. The identification results for beta=0.1. Adaptive Networks and Control Lab 37

38 Simulation Results The RankError results of our prediction algorithm and the corresponding RandError results of random selected algorithm for beta=0.05. The RankError results of our prediction algorithm and the corresponding RandError results of random selected algorithm for beta=0.1. Adaptive Networks and Control Lab 38

39 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 39

40 Edge Temporal networks 1 2 Temporally Switching Time Time time-order, temporal motif, bursts overlapping between temporal networks Adaptive Networks and Control Lab 40

41 Outline 1 Motivation, Introduction & Definitions 2 Ranking of Nodal Infection Probability 3 Vaccinating SIS Epidemics with Zero-Determinant Strategy 4 Predicting Spatial Transmission on Metapopulation 5 Epidemic in Temporal Networks 6 Take away message Adaptive Networks and Control Lab 41

42 Take Away Message Number of crossings is introduced to study the ranking of nodal infection probability and theoretical results are given. Zero-determinant strategy is utilized to vaccinate the SIS epidemics, in order to minimize the social cost. Maximum likelihood methods are applied to predict the metapopulation to be infected. Effect of the link overlap on epidemics is considered Adaptive Networks and Control Lab 42

43 Thanks! Cong Li Adaptive Networks and Control Lab Fudan University 43