Epidemics & Networks. Jesús Gómez Gardeñes Instituto de Biocomputación y Física de Sistemas Complejos (BIFI) Universidad de Zaragoza
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1 Epidemics & Networks Jesús Gómez Gardeñes Instituto de Biocomputación y Física de Sistemas Complejos (BIFI) Universidad de Zaragoza
2 DO WE NEED A MOTIVATION? Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
3 Infectious Disease Impact Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
4 Lets Model! Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
5 Compartmental models Aimed at capturing the global (population-level) dynamics from the microscopic contagion processes
6 Compartmental models Aimed at capturing the global (population-level) dynamics from the microscopic contagion processes Each individual can be in one of n states at time t S - Susceptible (Healty) I - Infected (and infectious) R - Recovered (immune/dead) (From Petter Holme s blog)
7 Compartmental models The transitions (e.g. S! I) is mediated by some rates ( ) and the encounters between individuals. µ!! S - Susceptible (Healty) I - Infected (and infectious) R - Recovered (immune/dead) (From Petter Holme s blog)
8 Compartmental models The transitions (e.g. S! I) is mediated by some rates ( ) and the encounters between individuals. The final impact of an SIR epidemic is given by the fraction of affected (Recovered) individuals µ!! S - Susceptible (Healty) I - Infected (and infectious) R - Recovered (immune/dead) (From Petter Holme s blog)
9 Some examples SIS S μ I SIR S I μ R SEIR S E α I μ R Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
10 Some examples SIS S μ I μ SIR S I R α μ SEIR S E I R QUESTION: What is the minimum value of epidemic outbreak to take place? for the Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
11 Part I Simple Models
12 Approaches to Epidemic Modeling From ideal (simple) to real (complex) interactions Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
13 Approaches to Epidemic Modeling From ideal (simple) to real (complex) interactions Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
14 Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
15 SIR model SIR basic processes µ Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
16 SIR model SIR basic processes µ Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
17 SIS versus SIR Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
18 SIS versus SIR Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
19 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
20 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
21 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
22 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
23 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
24 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
25 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
26 SIR model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
27 (SIS) (SIR) Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
28 Part II Beyond Mean field
29 Approaches to Epidemic Modeling From ideal (simple) to real (complex) interactions Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
30 Approaches to Epidemic Modeling From ideal (simple) to real (complex) interactions Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
31 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
32 Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
33 heterogeneous mean field Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
34 Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
35 Contagion Processes & Networks Probably the most prolific area of Network Science: more than 5000 papers since 2001 more than 3300 citations in Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
36 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
37 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
38 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
39 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
40 k k Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
41 k k k k Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
42 k k k k Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
43 k k k k k k k µ µ 2 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
44 k µ µ 2 µ hk 2 i hki =1 c = µhki hk 2 i 6= µ hki Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
45 Epidemics on general Networks c hki hk 2 i in Homogenous networks: hk 2 i = hki 2 c 1 hki i Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
46 Epidemics on general Networks c hki hk 2 i in Homogenous networks: hk 2 i = hki 2 c 1 hki in Scale-Free networks P (k) k if 2 < < 3 then hk 2 i!1 i c! 0 Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
47 Epidemics on general Networks c hki hk 2 i in Homogenous networks: hk 2 i = hki 2 c 1 hki in Scale-Free networks P (k) k if 2 < < 3 then hk 2 i!1 i c c! 0 Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
48 Alternative Analysis Microscopic Markov Chain Approach Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
49 SIS model p i (t): Probability that agent (node) i is infected at time t Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
50 SIS model p i (t): Probability that agent (node) i is infected at time t Prob. it remains inf. if inf. p i (t + 1) = p i (t)(1 µ)+(1 p i (t)) q i (t) Prob. it gets infected if healthy Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
51 SIS model p i (t): Probability that agent (node) i is infected at time t Prob. it remains inf. if inf. p i (t + 1) = p i (t)(1 µ)+(1 p i (t)) q i (t) Prob. it gets infected if healthy q i (t) =1 NY j=1 [1 A ij p j (t)] Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
52 SIS model p i (t): Probability that agent (node) i is infected at time t Prob. it remains inf. if inf. p i (t + 1) = p i (t)(1 µ)+(1 p i (t)) q i (t) Prob. it gets infected if healthy q i (t) =1 NY j=1 [1 A ij p j (t)] <I>(t) = NX p i (t) i=1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
53 SIS model A ij =1 lim t!1 p i(t)! p i Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
54 SIS model A ij =1 lim t!1 p i(t)! p i p i (t + 1) = p i (t) =p i Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
55 SIS model A ij =1 lim t!1 p i(t)! p i p i (t + 1) = p i (t) =p i µp i =(1 p i )q i Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
56 SIS model A ij =1 lim t!1 p i(t)! p i p i (t + 1) = p i (t) =p i µp i =(1 p i )q i q i =1 NY j=1 1 Aij p j Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
57 SIS model A ij =1 lim t!1 p i(t)! p i p i (t + 1) = p i (t) =p i µp i =(1 p i )q i q i =1 NY j=1 1 Aij p j Epidemic Onset p? i =? i 1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
58 SIS model A ij =1 lim t!1 p i(t)! p i p i (t + 1) = p i (t) =p i µp i =(1 p i )q i q i =1 NY j=1 1 Aij p j Epidemic Onset p? i =? i 1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
59 SIS model Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
60 SIS model p? i =? i 1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
61 SIS model p? i =? i 1 µ? i =(1? i ) NX A ij j j=1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
62 SIS model p? i =? i 1 µ? i =(1? i ) NX NX µ? i = A ij j A ij? i j j=1 j=1 NX A ij j j=1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
63 SIS model p? i =? i 1 µ? i =(1? i ) NX NX µ? i = A ij j A ij? i j j=1 j=1 NX A ij j j=1 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
64 SIS model p? i =? i 1 µ? i =(1? i ) NX NX µ? i = A ij j A ij? i j j=1 j=1 NX A ij j j=1 µ should be an eigenvalue of the Adjacency matrix Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
65 SIS model p? i =? i 1 µ? i =(1? i ) NX NX µ? i = A ij j A ij? i j j=1 j=1 NX A ij j j=1 µ should be an eigenvalue of the Adjacency matrix c = µ max (A) Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
66 SIS model Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
67 SIS model c = µ max (A) Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
68 SIS model c = µ max (A) Chung F., et al. Spectra of random graphs with given expected degrees, PNAS 100, 6313 (2003) Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
69 SIS model c = µ max (A) Chung F., et al. Spectra of random graphs with given expected degrees, PNAS 100, 6313 (2003) Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
70 Part III Metapopulation Models
71 Beyond contact networks: Facing a global challenge The black death (~ 700 years ago) HIV (2008) SARS (2009) H1N1 (2011) Wikipedia Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
72 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
73 Lecture II: Stochas4c Epidemics Processes Lecture III: in & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
74 Computational Epidemiology Goal Add Spatial Resolution & Design Localized Immunization policies PNAS 103, 5935 (2006) Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
75 Theory Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
76 Applications Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
77 Basic Model Different populations connected as a network. Each individual moves with probability p from one population to a neighboring one, according to the empirical observations*. j Inside each subpopulation, at each time step, epidemic dynamics takes place ( and ). µ wij wji Then, each individual comes back to its original subpopulation (node). i Epidemics Lecture III: & Networks, Walks, Conges4on Jesús Gómez & Epidemics, Gardeñes, Aberdeen Universidad January de Zaragoza 15, 2014
78 Lets construct the Markov Equations!
Module 5: Introduction to Stochastic Epidemic Models with Inference
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