ECS 253 / MAE 253, Lecture 6 April 19, Percolation and Epidemiology on Networks

Transcription

1 ECS 253 / MAE 253, Lecture 6 April 19, 2018 Perclatin and Epidemilgy n Netwrks

2 Hmewrk 1 Assignment HW1.pdf: Due tday (at midnight) via Canvas. HW1b: Due tday (at midnight) via Canvas. HW1a: Prject pitch... Due tmrrw (at midnight) via Canvas. Wuld peple prefer that I make this deadline be Saturday night instead? HW2, HW2a, HW2b are all psted n the class lecture web page.

3 Prcesses n netwrks Spreading prcesses Search fr infrmatin Interplay f tplgy and functin (see, e.g.,) Barrat, et al, Dynamical prcesses n Cmplex Netwrks Prter and Gleesn, Dynamical systems n netwrks: tutrial A

4 Epidemilgy Understanding hw diseases/pinins spread n netwrks Human diseases Cmputer viruses (typically spread via netwrks) Typically attached t an executable prgram. Typically crrupt files n hst cmputer Cmputer wrms (spread directly frm cmputer t cmputer via netwrk cnnectins) Wrms are self-cntained. Generally harm the netwrk and cnsume bandwidth.

5 Starting simply Understand flw f ne virus n a static netwrk. SIR (Susceptible, Infected, Remved) SIS (Susceptible, Infected, Susceptible) S = dn t have the disease but can catch it if expsed. I = have the disease and can pass it n. R = recvered with permanent immunity (r remved ).

6 Traditinal mathematical epidemilgy β prbability f an S catching disease frm an I. γ prbability f an I recvering and becming an R. Neglect any spatial structure, and assume fully mixed (i.e., any individual is equally likely t cme int cntact with any ther). In graph thery terms, this wuld be the cmplete graph. Als called mean-field in physics.

7 The resulting rate equatins: The Kermack-McKendrick mdel: [Kermack and McKendrick, A Cntributin t the Mathematical Thery f Epidemics. Prc. Ry. Sc. Lnd. A 115, 1927] [Andersn and May, Ppulatin Bilgy f Infectius Diseases: Part I. Nature 280, 1979] Three cupled rdinary differential equatins: 1. ds dt = βis, 2. di dt = βis γi, 3. dr dt = γi.

8 Epidemilgical threshld T c = βs 0 γ Where S 0 is initial size f susceptible ppulatin. Fr T c < 1 disease dies ut, di/dt < 0. (An I infects less than ne S befre recvering r dying). Fr T c > 1 disease will spread until full ppulatin gets infected, di/dt > 0. (An I infects mre than ne S).

9 Disease spread n a netwrk (N lnger mean field)

10 Wave-like spreading in the distant past Frm Barabasi, Netwrk Science bk

11 Netwrk structure matters! GLEAMviz.rg see. e.g., Clizza, V., Barrat, A., Barthelemy, M., & Vespignani, A. The rle f the airline transprtatin netwrk in the predictin and predictability f glbal epidemics. PNAS, 103(7), (2006). Clizza, Vittria, Rmuald Pastr-Satrras, and Alessandr Vespignani. Reactin-diffusin prcesses and metappulatin mdels in hetergeneus netwrks. Nature Physics 3 (2007).

12 Incrprating netwrk structure in mathematical mdels: Simpler than SIS/SIR: Perclatin n a netwrk; an SI mdel SIS/SIR mdels

13 ER nset f the giant cmpnent: Perclatin

14 Bnd Perclatin (Cntact prcesses) [Grassberger, On the critical behavir f the general epidemic prcess and dynamical perclatin, Math. Bisci., 63, 1983.] Start ut with the cmplete graph as the underlying structure Assume randm distributin f initial carriers Prbability disease is transmitted crrespnds t the edge ccupancy prbability, p. The size f the largest cnnected cmpnent is the size f the largest utbreak (the number f Infecteds) (Recall the Erdös-Renyi randm graph)

15 Aside: Bnd perclatin versus site perclatin n a lattice Useful java applets: achter/

16 Incrprating netwrk structure: Bnd Perclatin (Cntact prcesses) [Grassberger, On the critical behavir f the general epidemic prcess and dynamical perclatin, Math. Bisci., 63, 1983.] Assume randmly chsen initial carrier. Prbability disease is transmitted crrespnds rughly t the edge ccupancy prbability. Remember the Erdös-Renyi randm graph, but here we are given an underlying netwrk and are activating selected edges.

17 Bnd perclatin, cnt. Lk at distributin f cluster sizes. These crrespnd t extent f disease spread. Nte all we get are the final I, S, R values. Says nthing abut the dynamics! Just the final state. The perclatin transitin crrespnds t the epidemic threshld. The size f the giant cmpnent crrespnds t the size f the epidemic. Hw d we chse the underlying graph? Almst every scial netwrk studied shws heavy-tailed distributin. The Internet has a highly right-skewed degree distributin. Pwer law randm graphs relatively easy t analyze.

18 Behavir n ER randm graphs: (λ is infectin rate) Slide curtesy f Amin Saberi

19 Perclatin/epidemic threshld n pwer law randm graphs, P (k) k γ fr 2 < γ 3 Netwrk rbustness and fragility: Perclatin n randm graphs, Callaway, Newman, Strgatz and Watts, Phys. Rev. Lett., 85 (2000). Epidemic Spreading in Scale-Free Netwrks, R. Pastr- Satrras and A. Vespignani Phys. Rev. Lett. 86 (2001). The Epidemic Threshld in Scale-Free Graphs, N. Berger, C. Brgs, J. Chayes, and A. Saberi, Sympsium n Discrete Algrithms (SODA), (2005). Find T c = 0, in ther wrds absence f epidemilgical threshld. Fr all β > 0 and S 0 > 0, the steady-state result is that sme nn-zer, fractin f the ppulatin has the disease.

20 Results frm Callaway et al Degree dist, p k k γ e k/c (pwer law with cutff). Let q k be prbability that a vertex f degree k is infected. Fr simplicity they analyze q k = q (independent f k). Then p k q is prbability f having degree k and being infected. Calculate s, the mean cluster size. functins... details mitted here) that Find (via generating s = q q 2 k 1 (q k 2 / k ) s when denminatr 1 q k 2 / k = 0, i.e., q c = k k 2

21 q c versus C, the cutff prbability P s threshld q c cutff 10 3 Critical infectin prbability fr epidemic utbreak q c 0 as the graph becmes a true pwer law (Epidemic if q > q c = k, which als means expected number f secnd k size 2 f cluster s neighbrs k 2 exceeds expected number f first k.) IG. 1. Prbability P s that a randmly chsen vertex belngs a cluster f s sites fr k 10, t 2.5, and q 0.65 frm

22 Frm Perclatin t SIS dynamics n a netwrk SIS Susceptible-Infected-Susceptible Epidemic Spreading in Scale-Free Netwrks, R. Pastr- Satrras and A. Vespignani Phys. Rev. Lett. 86 (2001). Rigrus prf: The Epidemic Threshld in Scale-Free Graphs, N. Berger, C. Brgs, J. Chayes, and A. Saberi, Sympsium n Discrete Algrithms (SODA), (2005). (Always enugh hubs that disease n a hub s neighbr and reinfects hub. (Recall this is SIS))

23 Fllwing Pastr-Satrras and A. Vespignani ρ k (t) is density f infected ndes f degree k at time t. (Hence [1 ρ k (t)] is prbability a nde f degree k is NOT infected.) λ = β/γ, the effective spreading rate. Set γ = 1. (Recall β is infectin rate, γ is recvery.) The time evlutin (a master equatin / rate equatin ): dρ k (t) dt = ρ k (t) λk [1 ρ k (t)] Θ(ρ(t)) First term: ndes recver with unit rate (γ = 1) Secnd term: Infectin rate λ, times number f neighbrs k, times prb nde f degree k is healthy, times prb f being cnnected t an infected nde Θ(ρ(t)).

24 Edge fllwing prbability Prb f fllwing an edge t a nde f degree k is k p k S prbability f neighbr being infected (ρ k (t) density f infected): Θ(ρ(t)) = k k p k ρ k (t) k k p k = 1 k k p k ρ k (t) k k edges reach nde f degree k:

25 Steady state f master eqn, dρ k dt = 0 implies: ρ k = λ k Θ 1 λ Θ Inserting int expressin fr Θ: Θ = 1 k k k p k λ k Θ 1 λ Θ (Nte Θ = 0 always satisfies, but is quite dull!... ρ k = 0)

26 Searching fr mre slutins t last equatin, in interval 0 < Θ 1 Taking derivative w.r.t. Θ f bth sides f last equatin: d dθ [ 1 k p k k k ] λ k Θ 1 λ Θ Θ=0 = 1, at λ = λ c slving this: 1 k k k p k λ c k = k 2 k λ c = 1 Critical spreading rate: λ c = k k 2 If k 2 but k finite, then λ c 0.

27 Last three slides, actually pieced frm three papers Epidemic Spreading in Scale-Free Netwrks, R. Pastr- Satrras and A. Vespignani Phys. Rev. Lett. 86 (2001). Epidemic dynamics in finite size scale-free netwrks, R Pastr- Satrras, A Vespignani Physical Review E (2002). Immunizatin f cmplex netwrks, R Pastr-Satrras, A Vespignani Physical Review E (2002).

28 Frm SIS t SIR PSV 01 and BBCS 05 cnsider SIS. R. M. May, A. L. Llyd Infectin dynamics n scale-free netwrks Phys. Rev. E, (2001). Shw similar results hld fr SIR. (Lrd Rbert May, funder f theretical eclgy/ppulatin bilgy/evlutinary game thery... great wiki entry) Immunizatin Many subsequent papers n immunizatin by kncking ut ndes. But the recvery rate depends n ther attributes f nde (age, medical histry...) and can verride netwrk structure. (i.e., less verall infected r less verall fatalities imprtant?)

29 Implicatins fr disease spread? Are human cntact netwrks and the Internet really like pwer law randm graphs? Yes, they have the pwer law degree disributin. But usually, als mre structure: Gegraphic crrelatin. Degree-degree crrelatins. High transitivity fr scial netwrks. Each f the three factrs alne can make T c > 0. Develping a mdel that accurately captures human cnnectivity still in the wrks.

30 Immunizatin: Cupling perclatin and netwrk resilience View vaccinatin as remving a particular set f vertices frm the netwrk. As we saw previusly, remving the high-degree ndes frm a pwer law randm graph, quickly destrys cnnectivity. Hw t find these hubs in a scial netwrk, fr instance a netwrk fr sexually transmitted diseases?

31 Identifying Hubs Want t sample edges rather than ndes. Chse nde at randm, prbability f chsing nde f degree k is p k. Chse an edge at randm, prbability f it leading t a nde f degree k prprtinal t kp k. Hw t chse an edge at randm?

32 Chse a persn at randm. Acquaintance vaccinatin Then chse a friend f that persn t vaccinate. Chen, ben-avraham, and Havlin, Efficient Immunizatin f Ppulatins and Cmputers, Phys. Rev. Lett. 91, (2003) Shw by cmputer simulatin and analytic calculatins that this is much mre effective than randm vaccinatin. This type f acquaintance vaccinatin actually used t cntrl small px and ft-and-muth ( ring vaccinatin )

33 Hw t mdel a real human ppulatin? (Using census data) [ Bansal, Purbhlul, Meyers, The Spread f Infectius Disease thrugh Cntact Netwrks, Talk given at MSRI, April 2005.] Nt published, but a vide can be viewed at: speakertalks?field pid= See further references at the end. Take actually census data frm the city f Vancuver.

34 Cnstructing cnnectivity via census data Husehlds Classrms Businesses Shpping

35 Wh t immunize?

36 Strategy ne: Immunize the hubs Receptinists Bus drivers Schl teachers This results in the least number f peple becming infected.

37 Elderly and children. Strategy tw: Immunize the mst frail Mre peple verall get infected, but less peple verall die as a result f the disease!

38 Further wrk n netwrk epidemilgy Bansal S, Purbhlul B, Meyers LA (2006) A Cmparative Analysis f Influenza Vaccinatin Prgrams. PLS Med 3(10) Brckman/Vespignani wrk n influenza (including where s Gerge mbility tracking, and transprtatin nets spreading disease), Effects f clustering n epidemic threshlds (Newman, Gleesn, Vlz alternate calculatins and implicatins)

39 Effectively breaking up different netwrks What ther types f ndes play key rles?

40 Other types f imprtant ndes A classic example frm Scial Netwrk Analysis (SNA) [ spap/water/netwrk/intr.htm] The Kite Netwrk Wh is imprtant and why?

41 The Kite Netwrk Degree Diane lks imprtant (a hub ). Betweenness Heather lks imprtant (a cnnectr / brker ). Clseness Fernand and Garth can access anyne via a shrt path. Bundary spanners as Fernand, Garth, and Heather are well-psitined t be innvatrs. Peripheral Players Ike and Jane may be an imprtant resurces fr fresh infrmatin.

42 A cntemprary scial netwrk (Taken frm

43 Betweenness Centrality [Freeman, L. C. A set f measures f centrality based n betweenness. Scimetry ] A measure f hw many shrtest paths between all ther vertices pass thrugh a given vertex.

44 Betweenness (frmal definitin) Fr a given vertex i: B(i) = s t i σ st (i) σ st Where σ st is the number f shrtest gedesic paths between s and t. And σ st (i) are the number f thse passing thrugh vertex i. (Calculating shrtest paths efficiently... s algrithm )

45 Betweenness and eigenvalues (bttlenecks) X Y R = τ [min] = 109 t X Y R = 4.61 τ [med] = 604 t X Y R = τ [max] = 5314 t Bttlenecks have large betweenness values. In scial netwrks betweenness is a measure f a ndes centrality and imprtance (culd be a prxy fr influence). In a rad netwrk, high betweenness culd indicate where alternate rutes are needed. Als a measure f the resilience f a netwrk (next page).

46 Targeted attack by different metrics Hlme P, Kim BJ, Yn CN, Han SK (2002) Attack vulnerability f cmplex netwrks. Phys. Rev. E 65: Degree centrality Betweeness centrality Typically (but nt always) high degree are high betweeness. High betweeness the mre effective strategy t break up a netwrk s cnnectivity.