Towards Identifying and Predicting Spatial Epidemics of ... · Towards Identifying and Predicting...

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Towards Identifying and Predicting Spatial Epidemics of Complex Meta - population Networks 李翔 复旦大学 智慧网络与系统研究中心 2016-6-25 上海交通大学 Net-X 2016 学术论坛

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Page 1: Towards Identifying and Predicting Spatial Epidemics of ... · Towards Identifying and Predicting Spatial Epidemics of Complex Meta-population Networks 李翔 复旦大学 智慧网络与系统研究中心

Towards Identifying and Predicting Spatial Epidemics of Complex Meta-population

Networks

李 翔

复旦大学智慧网络与系统研究中心

2016-6-25

上海交通大学 Net-X 2016 学术论坛

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Outline

• Brief review and motivation

• Identification of spatial epidemic parameters

• Identification of invasion pathways

• From identification to prediction

Acknowledged contributionsPhd candidate: Jianbo WANG (FDU)

Dr. Cong LI (FDU)

Dr. Lin WANG (FDU - HKU)

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History repeats itself. ——Thucidides

• Smallpox, black-death,……, AIDS,……, SARS(2003), ……, A-H1N1(2009), ……E.Coli(2011),……Ebola/Zika,……?

• The modern medical technology can save more patients than before, while failed more effectively stopping a virus prevalence than before…

<Science>, June 20, 2003:Modeling the SARS Epidemic Transmission Dynamics and Control of SARS

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Daniel Bernoulli, on smallpox inoculation,1766

Prototype of compartment model

Epidemic models in a population

William Heaton Hamer

(1862-1936)

mass action principle

An incremental infection depends on the rate of

contact between susceptible and infectious individuals.

SIS

S I𝛽𝐼

𝜇

S I𝛽𝐼

R𝜇

SIR

∆𝐼 = 𝛽 ∙ 𝑆 ∙ 𝐼

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Epidemics on complex network

• Epidemic thresholdsR. Pastor-Satorras, A. Vespingnani (2001), Epidemic spreading in scale-free

networks, Physical Review Letters, 86, 3200.

C. Castellano, R. Pastor-Satorras (2010), Thresholds for epidemic spreading in networks, Physical Review Letters, 105, 218701.

P. Van Mieghem and R. van de Bovenkamp (2013), Non-Markovian infection spread dramatically alters the SIS epidemic threshold, Physical Review Letters, 110, 108701.

……

• Epidemic dynamicsT. Gross, C.J.D. D‘Limma, B. Blasius (2006). Epidemic dynamics on an adaptive

network. Physical Review Letters, 96: 208701.

X. Li, X.F. Wang (2006), Controlling the spreading in small-world evolving networks: stability, oscillation, and topology. IEEE Trans. Automatic Control, 51(3): 534-540.

P. Van Mieghem, J. Omic, R. E. Kooij (2009), Virus spread in networks, IEEE/ACM Transaction on Networking, 17(1), 1-14.

……

• Latest surveyR. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani (2015),

Epidemic processes in complex networks, Review of Modern Physics, 87(3), 925-979.

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Spatial epidemics on a meta-population

• Richard Levins (1969), Bull. Entomol. Soc. Am. 15, 237.

(Spatial ecology)

• Metapopulation: Divide the

whole population (country/world)

into several sub-populations

(cities), a subpopulation is

connected with others via public

transportation networks, e.g., the

air-line web, high-way web.

Colizza, et al. The role of the airline transportation

network in the prediction and predictability of global

epidemic. PNAS 103(2006): 2015-2020.

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Too many stories,

leave for a new book

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To identify/predict spatial epidemics

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Challenges

• Prediction/Forecast/Alarm, WHO/CDC

• Partial information available/observable

• Dynamical evolving situations

• One shot vs rounds of repeat

• Difficulty of stochastics to analyze and verify

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Q1: is it possible to identify the spatial

epidemic parameters on a meta-

population network?

Our trial:

Identification Algorithm on a minimal

Metapopulation Network

J.-B. Wang, L. Cao, and X. Li, “On estimating spatial epidemic parameters of a simplified

metapopulation model,” in Proc. 13th IFAC Symposium on Large Scale Complex Systems:

Theory and Applications, pp. 383–388, 2013.

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The SI Reaction-Diffusion Process on a

Metapopulation Network

[1] Hufnagel L, Brockmann D, Geisel T., PNAS, 101: 15124-15129, 2004.

[2] Colizza, V., Pastor-Satorras, R. & Vespignani A., Nature Phys.3, 276-282, 2007.

Reaction process 𝐼 + 𝑆՜𝛼2𝐼

Diffusion process 𝒳𝑖

𝑝𝑖𝑗𝒳𝑗 , 𝒳 represents 𝑆 𝑜𝑟 𝐼

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Problem Statement: a minimal case

SI dynamics:

Known:

• The time series of infected cases (number of infected individuals of each infected subpopulation 𝐼𝑖 𝑡 ) at time step 𝑡

Unknown (to identify):

• Infection rate 𝛼, diffusion rate 𝑝12

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Identification Method

ො𝛼 = argminln 𝐼 𝑡2 −ln(𝐼(𝑡1))

𝑡2−𝑡1− 𝛼∗

At the early stage of epidemic, 𝐼1(𝑡) ≪ 𝑁1(𝑡),𝑖 = 1,2.

at the whole population level, we have:

𝑖=1

2𝑑𝐼𝑖(𝑡)

𝑑𝑡≈ 𝛼

𝑖=1

2

𝐼𝑖(𝑡)

𝑑𝐼(𝑡)

𝑑𝑡≈ 𝛼𝐼(𝑡)

𝐼(𝑡) ≈ 𝑒𝛼(𝑡−𝑡0)𝐼(𝑡0)

Since 𝐼 𝑡 = σ𝑖=12 𝐼𝑖(𝑡),

Solve the equation:

1. Estimate infect rate 𝜶

Least square regression:

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Identification Method2. Identify diffusion rate 𝒑𝟏𝟐

At the first time of infected

subpopulation 1 invading susceptible

subpopulation 2:

Repeating 𝑚 rounds of 1

invading 2, we have the

likelihood function L(P):

Letting 𝑑𝐿(𝑝)

𝑑𝑝12= 0, we have:

Consistency and

Asymptotic normality:

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Simulation results

Estimation of the transmission rate 𝛼 at

the whole population level.

Estimation of diffusion rate 𝑝12 under

various runs of realizations.

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Main interest:

Identifying Epidemic

Spatial Invasion Pathways

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[1] A. Gautreau, A. Barrat, and M. Barthelemy. ”Global disease spread: statistics and estimation of arrival times,”Journal of theoretical

biology, vol. 251, no. 3, pp. 509–522, Apr. 2008.

[2] D. Balcan, V. Colizza, B. Gonc a̧lves, H. Hu, J. J. Ramasco and A. Vespignani, ”Multiscale mobility networks and the spatial spreading of

infectious diseases.” Proc. Natl. Acad. Sci., vol. 106 no. 51, pp. 21484–21489, Dec. 2009.

[3] D. Brockmann and D. Helbing, ”The Hidden Geometry of Complex, Network-Driven Contagion Phenomena,”Science, vol. 342, no. 6164,

pp. 1337–1342, Dec. 2013.

Exploring Epidemic Shortest PathsAverage-Arrival-Time (ARR)-Based Shortest Paths

Tree (SPT) [1] Monte Carlo-Maximum-Likelihood

(MCML)-Based Epidemic Invasion Tree [2]

Effective (EFF)-Distance-Based Most Probable Paths Tree [3]

𝑑𝑖𝑗 = 1 − 𝑇𝑖𝑗Distance between

subpopulation i, j

λ 𝑡𝑗 ≈ 𝜒(𝑗|𝑠)

≡ min𝑃𝑠,𝑗

𝑘,𝑙 ∈𝑃𝑠,𝑗

ln𝑁𝑘𝜆

𝑤𝑘𝑙− 𝛾

Shortest path between subpopulation s, j

𝑑𝑚𝑛 = 1 − log𝑃𝑚𝑛 ≥1Effective distance between

subpopulation m, n

Shortest paths between

subpopulation m, n𝐷𝑚𝑛 = min

Γ𝜆 Γ

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Q2: is it possible to retrospect the

stochastic pandemic spatial paths

among a networked metapopulation?

Our answer:

Invasion Pathways Identification Algorithm Based on

Dynamical Programming and Maximum Likelihood Estimation

J.-B. Wang, L. Wang, and X. Li, “Identifying spatial invasion of pandemics on meta-

population networks via anatomizing arrival history,” IEEE Trans. Cybernetics, 2016, Doi:

10.1109/TCYB.2015.2489702.

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Problem Statement with SI dynamics

Reaction process 𝐼 + 𝑆՜𝛽2𝐼

Diffusion process 𝒳𝑖

𝑝𝑖𝑗𝒳𝑗 , 𝒳 represents 𝑆 𝑜𝑟 𝐼

Known:

• Number of infected individuals of each infected subpopulation

𝐼𝑖 𝑡 at time step 𝑡, network topology(including diffusion rates)

Unknown (to identify):

• Spatial invasion pathways

[1] Hufnagel L, Brockmann D, Geisel T., PNAS, 101: 15124-15129, 2004.

[2] Colizza, V., Pastor-Satorras, R. & Vespignani A., Nature Phys.3, 276-282, 2007.

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Four Invasion Cases

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Algorithm steps:1) Invasion Partition (Dynamic Programming)

The whole invasion pathway T is anatomized into (at each epidemic arrival time (EAT))

four classes of invasion cases with number of Λ :

𝐼 ⟼ 𝑆, 𝐼 ⟼ 𝑛𝑆,𝑚𝐼 ⟼ 𝑆, 𝑚𝐼 ⟼ 𝑛𝑆

𝑇𝑤ℎ𝑜𝑙𝑒 𝑖𝑛𝑣𝑎𝑠𝑖𝑜𝑛 𝑝𝑎𝑡ℎ𝑤𝑎𝑦𝑠 = 𝑜𝑝𝑡

𝑖=1

Λ

ො𝑎𝑖

2) Identifying Each Invasion Case

Accurate identification + optimal identification (Maximum Likelihood Estimation)

ො𝑎𝑖 = arg max𝑎𝑖∈𝐺𝐼𝑁𝐶𝑖

𝑃(𝑎𝑖|𝐺𝐼𝑁𝐶𝑖)

Invasion Pathways Identification Algorithm

Invasion Cases

(INCs)

𝑚𝐼 ⟼ 𝑆 𝑚𝐼 ⟼ 𝑛𝑆

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Subpopulations’ observability

Illustration of neighbors subpopulation classification in terms of status transitions from 𝑡 − 1to 𝑡:(a) Observable subpopulation 𝑖. (b) Partially observable subpopulation 𝑖. (c) Unobservable

subpopulation 𝑖.

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Identification of 𝒎𝑰⟼ 𝑺Step 1 Accurate Identification:

Step 2 Optimal Identification:

Decompose the number of first

arrival infected individuals

𝑖=1

𝑚

ℋ𝑖1 = ℋ

Compute the likelihood of each potential solution:

Choose the maximal one

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Identification of 𝒎𝑰⟼ 𝒏𝑺Step 1 Accurate Identification:

Step 2 Optimal Identification:

Decompose the number of first

arrival infected individualsCompute the likelihood of each potential solution:

𝑖∈𝑌𝑘

𝑚

ℋ𝑖𝑘 = ℋ𝑘

Choose the maximal one

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Analysis Identifiability

Denote 𝜋 the probability

corresponding to the

most likely pathways

for a given INC. Thus

we have 𝜋 𝜎 =sup 𝑃(𝜎𝑖|ℰ) .

Property 1: Given an

INC 𝑚𝐼 ⟼ 𝑆 or 𝑚𝐼 ⟼𝑛𝑆, 𝑃 𝜎𝑖 ℰ =ς𝑘=1𝑚 Ω/σ𝑖=1

𝑀 ς𝑘=1𝑚 Ω .

there must exist 𝑃𝑚𝑖𝑛 and

𝑃𝑚𝑎𝑥satisfying

𝑃𝑚𝑖𝑛 ≤ 𝜋 𝜎 ≤ 𝑃𝑚𝑎𝑥

Define identifiability of invasion pathways to

characterize the difficulty level an INC can be identified

𝜫 = 𝝅 𝝈 (𝟏 − 𝓢).Theorem 3: Given an INC 𝑚𝐼 ⟼ 𝑆 or 𝑚𝐼 ⟼ 𝑛𝑆,

Π is the identifiability computed by the IPI algorithm.

There exist a lower boundary Π𝑚𝑖𝑛 = Τ1 𝑀 1 − 𝒮′ and

Π𝑚𝑎𝑥 = 𝜋 − 𝒮 𝜋 𝜎 that

Π𝑚𝑖𝑛 ≤ Π ≤ Π𝑚𝑎𝑥,

where 𝒮′ = −(1

log𝑀)(𝜋log 𝜋 + σ

1−𝜋

𝑀−1log(

1−𝜋

𝑀−1))).

𝒮 = −1

log𝑀

𝑖=1

𝑀

𝑃 𝜎𝑖 ℰ log𝑃(𝜎𝑖|ℰ)

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Example 1

The figure shows various identified accuracy for

the early stage and the whole invasion pathways on

the AAN.

American Airports Network (AAN)

N=404; total population=2.4 × 108 ,< k > = 16.

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Example 2

Large-scale BA metapopulation

network

N=3000; 𝑁𝑖=6× 105; total

population=1.8 × 109, < k > = 16.

The figure shows various identified

accuracy for the early stage and the whole

invasion pathways on 3000 subpopulations

of the BA networked metapopulation.

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Example 3

• L

Illustration of the actual invasion pathways

and the most likely identified invasion

pathways on the AAN. Subpopulation 1 is

the source.

Statistics analysis of the likelihoods

entropy and identifiability of wrongly

identified INCs on the AAN.

Source 1

Source 1

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Extension to Predict

Spatial Epidemics

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Q3: Is it possible to predict spatial

transmission of epidemics on a

meta-population network?

Our trial:

Prediction Algorithm Based on Maximum

Likelihood Estimation

J.-B. Wang, C. Li, and X. Li, “Predicting spatial transmission at the early stage of

epidemics on a networked metapopulation,” 12th IEEE International Conference on Control &

Automation (ICCA), pp. 116-121, Kathmandu, Nepal, June 1-3, 2016.

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The Reaction-Diffusion Process on a

Metapopulation Network

[1] Hufnagel L, Brockmann D, Geisel T., PNAS, 101: 15124-15129, 2004.

[2] Colizza, V., Pastor-Satorras, R. & Vespignani A., Nature Phys.3, 276-282, 2007.

Reaction process 𝐼 + 𝑆՜𝛽2𝐼

Diffusion process 𝒳𝑖

𝑝𝑖𝑗𝒳𝑗 , 𝒳 represents 𝑆 𝑜𝑟 𝐼

𝐼 represents a infected individual, 𝑆 represents a susceptible individual. 𝛽 is the infection

rate.

𝒳 is a placeholder. 𝑝𝑖𝑗 is the diffusion rate from subpopulation 𝑖 to subpopulation 𝑗.

unit time ∆𝑡

reaction ∆𝒕

𝟐diffusion

∆𝒕

𝟐

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Conditions to Predict

Known:

• The time series of infected cases (number of infected

individuals of each infected subpopulation 𝐼𝑖 𝑡 ) at time

step 𝒕

• The topology of the metapopulation networks (including

diffusion rates 𝑝𝑖𝑗 and demography)

Unknown:

• Infection rate 𝛽

To predict:

• The susceptible subpopulation which will be infected in

the next time step 𝒕 + 𝟏

Conditions to Predict

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A Prediction AlgorithmAlgorithm steps:1)Estimate Infection Rate 𝜷

2) Calculate the Infection Likelihood

𝐼𝑖 𝑡 + ∆𝑡 is number of infected individuals of subpopulation 𝑖, ∆𝑡 is the

unit time. 𝛽∗ is the possible value of actual 𝛽.

m is the number of infected neighbor subpopulations, ℒ𝑖 is the infection

likelihood of susceptible subpopulation 𝑖.

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A Prediction AlgorithmAlgorithm steps:3) Predict the Spatial Transmission

i) n = 1;

ii) n≥2

Algorithm analysis:

Entropy of likelihood vector:

𝒫𝑖 repsents the infection

likelihood of subpopulation 𝑖only 𝑖 will be infected at next

time step.

ො𝑣 is the label of predicted

subpopulation which will be

infected.

ො𝑣𝑛 is the labels of predicted

subpopulations which will be

infected at next time step.

$ is the set of all possible

infected subpopulations at next

time step.

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Simulation Results

The RankError results of our prediction algorithm and the corresponding RandError results of random

selected algorithm for beta=0.05 and beta=0.01.

The identification results for beta=0.05 and beta =0.1.

BA scale-free

metapopulation network

N = 404; 𝑁𝑖 = 6× 105;population = 2.424 × 108,< k > = 16.

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Simulation Results

The evolution of the number of possible

infected candidates and entropies of

likelihoods vector for beta=0.05.

The evolution of the number of possible

infected candidates and entropies of

likelihoods vector for beta=0.1.

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To be continued…

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Contributed publications

• J.B. Wang, L. Wang, and X. Li, “Identifying spatial invasion of pandemics on meta-population networks via anatomizing arrival history,” IEEE Trans. Cybernetics, 2016, Doi: 10.1109/TCYB.2015.2489702.

• X. Li, J.B. Wang, and C. Li, “Towards identifying epidemic processes with interplay between complex networks and human populations”, 2016 IEEE Conference on Norbert Wiener in the 21st Century.

• J.B. Wang, L. Cao, and X. Li, “On estimating spatial epidemic parameters of a simplified metapopulation model,” in Proc. 13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications, pp. 383–388, 2013.

• J.B. Wang, X. Li, and L. Wang, “Inferring spatial transmission of epidemics in networked metapopulations”, IEEE Int. Symposium on Circuits and Systems (ISCAS 2015), pp. 906-909, 2015.

• J.B. Wang, C. Li, and X. Li, “Predicting spatial transmission at the early stage of epidemics on a networked metapopulation,” 12th IEEE International Conference on Control & Automation (ICCA), pp. 116-121, 2016.

Page 39: Towards Identifying and Predicting Spatial Epidemics of ... · Towards Identifying and Predicting Spatial Epidemics of Complex Meta-population Networks 李翔 复旦大学 智慧网络与系统研究中心

Xiang LI

[email protected]

http://www.can.fudan.edu.cn