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Social and Technological Network Analysis Lecture 8: Epidemics Spreading Dr. Cecilia Mascolo

Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

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Page 1: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Social  and  Technological    Network  Analysis  

 Lecture  8:  Epidemics  Spreading  

 Dr.  Cecilia  Mascolo  

 

Page 2: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

In  This  Lecture  

•  In  this  lecture  we  introduce  the  process  of  spreading  epidemics  in  networks.  

– This  has  been  studied  widely  in  biology.  – But  it  also  has  important  parallels  in  informaEon/idea  diffusion  in  networks.  

Page 3: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Epidemics  vs  Cascade  Spreading  

•  In  cascade  spreading  nodes  make  decisions  based  on  pay-­‐off  benefits  of  adopEng  one  strategy  or  the  other.  

•  In  epidemic  spreading    – Lack  of  decision  making.  – Process  of  contagion  is  complex  and  unobservable  •  In  some  cases  it  involves  (or  can  be  modeled  as  randomness).  

Page 4: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Branching  Process  

•  Simple  model.  •  First  wave:    A  person  carrying  a  disease  enters  the  populaEon  and  transmit  to  all  he  meets  with  probability  p.  He  meets  k  people:  a  porEon  of  which  will  be  infected.  

•  Second  wave:  each  of  the  k  people  goes  and  meet  k  different  people.  So  we  have  a  second  wave  of  kxk=k2  people.  

•  Subsequent  waves:  same  process.  

Page 5: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Example  with  k=3  

High  contagion  probability:  The  disease  spreads  

Low  contagion  probability:  The  disease  dies  out  

Page 6: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Basic  ReproducEve  Number  

•  Basic  ReproducEve  Number  R0=p*k  –  It  determines  it  the  disease  will  spread  or  die  out.  

•  In  the  branching  process  model,  if  R0<1  the  disease  will  die  out  aYer  a  finite  number  of  waves.  If  R0>1,  with  probability  >0,  the  disease  will  persist  by  infecEng  at  least  one  person  in  each  wave.  

Page 7: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Measures  to  limit  the  spreading  

•  When  R0  is  close  1,  slightly  changing  p  or  k  can  result  in  epidemics  dying  out  or  happening.  – QuaranEning  people/nodes  reduces  k.  – Encouraging  be]er  sanitary  pracEces  reduces  germs  spreading  [reducing  p].  

•  LimitaEons  of  this  model:  – No  realisEc  contact  networks:  no  triangles!  – Nodes  can  infect  only  once.  – No  nodes  recover.  

 

Page 8: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Formal  Epidemics  Models    The  SI  Model  

•  S:  suscepEble  individuals.    •  X:  infected  individuals,  when  infected  they  can  infect  others  conEnuously.  

•  n:  total  populaEon.  •  <k>  average  contacts  per  individual  •  β=λ<k>  is  the  infecEon  rate  per  individual  (0≤λ≤1)  •  SuscepEble  contacts  per  unit  of  Eme  βS/n.  •  Overall  rate  of  infecEon  XβS/n.    

SuscepEble(S)  

Infected    (X)  

Page 9: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SI  Model  

dXdt

= !SXn

dSdt

= !!SXn

s = Sn

!

x =Xn

s =1! xdxdt= !x(1! x)

       

Logis6c  Growth  Equa6on  

Eme  

infected

 

x(t) = x0e!t

1! x0 + x0e!t

Page 10: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SIR  Model  

•  Infected  nodes  recover  at  a  rate  γ.  •  A  node  stays  infected  for  τ  Eme.  •  Branching  process  is  SIR  with  τ=1.  

SuscepEble(S)  

Infected    (I)  

Removed  (R)  

!

dsdt

= "#sx

dxdt

= #sx " $x

drdt

= $x

s+ x + r =1

Page 11: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Example  

 •  Numerical  examples  of  soluEon:  •  β=1,  γ=0.4,  s(at  start)=0.99,  x(at  start)=0.01,  r(at  start)=0  

FracEo

n  of  nod

es  

Infected  

recovered  suscepEble  

Eme  

Page 12: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SIS  Model  

SuscepEble(S)  

Infected    (X)  

dsdt= ! x !"sx

dxdt= "sx !! x

s+ x =1

• If    β  >  γ  growth  curve  like  in  SI  but  never  reaching  all  populaEon  infected.  The  fracEon  of  infected-­‐>0  as  β  approaches  γ.  • If  β<  γ  the  infecEon  will  die  out  exponenEally.  • SIS  has  the  same  R0  as  SIR.  

Page 13: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Epidemic  Threshold    

•  When  would  the  epidemic  develop  and  when  would  it  die  out?  

•  It  depends  on  the  relaEonship  of  β  and  γ:  – Basic  ReproducEve  Number  R0=β/γ  –  If  the  infecEon  rate  [per  unit  of  Eme]  is  higher  than  the  removal  rate  the  infecEon  will  survive  otherwise  it  will  die  out.  

–  In  SI,  γ=0  so  the  epidemics  always  happen.  

Page 14: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

LimitaEons  of  SIR  

•  Contagion  probability  is  uniform  and  “on-­‐off”  

•  Extensions  

–  Probability  q  of  recovering  in  each  step.  –  Infected  state  divided  into  intermediate  states  (early,  middle  and  final  infecEon  Emes)  with  varying  probability  during  each.  

– We  have  assumed  homogenous  mixing  :  assumes  all  nodes  encounter  each  others  with  same  probability:  we  could  assume  different  probability  per  encounter.  

Page 15: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Relaxing  AssumpEons  

•  Homogeneous  Mixing:  a  node  connects  to  the  same  average  number  of  other  nodes  as  any  other.  

•  Most  real  networks  are  not  Erdos-­‐Renyi  random  networks  (for  which  the  homogeneous  mixing  assumpEon  holds).  

•  Most  networks  have  heterogeneous  degree  distribuEons.  – Scale  free  networks!  

Page 16: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Would  the  model  apply  to  SF?  

•  Pastor-­‐Satorras  and  Vespignani  [2001]  have  considered  the  life  of  computer  viruses  over  Eme  on  the  Internet:  

Surviving  probability  of  virus   Virus  survived  on  average  6-­‐9/14  months  depending  on  type  

Page 17: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

How  to  jusEfy  this  survival  Eme?  

•  The  virus  survival  Eme  is  considerably  high  with  respect  to  the  results  of  epidemic  models  of  spreading/recovering:  – Something  wrong  with  the  epidemic  threshold!  

•  Experiment:  SIS  over  a  generated  Scale  Free  network  (exponent  -­‐3).  

Page 18: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

No  Epidemic  Threshold  for  SF!  

Random  SF    

threshold  

Random  Network  

Scale  Free  Network  InfecEons  proliferate  in  SF  networks  independently    of  their  spreading  rates!  

Page 19: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Following  result  on  ImmunizaEon  

•  Random  network  can  be  immunized  with  some  sort  of  uniform  immunizaEon  process  [oblivious  of  the  characterisEcs  of  nodes].  

•  This  does  not  work  in  SF  networks  no  ma]er  how  many  nodes  are  immunized  [unless  it  is  all  of  them].  

•  Targeted  immunizaEon  needs  to  be  applied  – Keeping  into  account  degree!  

Page 20: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

ImmunizaEon  on  SF  Networks  

•  Red=SF  •  Black=  Random  

Uniform  ImmunizaEon  

Targeted  ImmunizaEon   Uniform  and  Targeted  ImmunizaEon  

FracEon  of  immunized  nodes  

Density  of  Infected  

Page 21: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Local  ImmunizaEon  

•  Global  knowledge  on  the  network  structure  is  rarely  (or  never)  available  

•  Local  immunizaEon  strategy:  – Select  g  nodes  at  random  – Ask  to  each  of  them  to  pass  over  the  vaccine  to  one  of  their  neighbors  

– As  a  result,  a  node  with  degree  k  is  immunized  with  a  probability  kP(k)  (hubs  are  immunized  with  higher  probability!)  

Page 22: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SIRS  Model  

•  SIR  but  aYer  some  Eme  an  R  node  can  become  suscepEble  again.  

•  A  number  of  epidemics  spread  in  this  manner  (remaining  latent  for  a  while  and  having  bursts).  

SuscepEble(S)  

Infected    (I)  

Refractory  (R)  

Page 23: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

ApplicaEon  of  SIRS  to    Small  World  Models  

Page 24: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Numerical  Results  

•  c  is  the  rewiring  probability  

Page 25: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SARS  PredicEon  

•  SARS:  severe  acute  respiratory  syndrome  •  SIR  like  model  with  more  parameters  and  homogenous  mixing  

•  Travel  data  and  census  data  •  WHO  data  about  SARS  spreading  to  evaluate  the  model  

•  Outbreak  starts  in  Hong  Kong  

Page 26: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Epidemics  Flow  

Page 27: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

The  Parameters  of  the  Model  

•  Parameters  used:  

Page 28: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Adding  Travel  and  Geography  

•  A  model  per  city  •  ProbabiliEes  of  an  individual  of  moving  from  one  city  to  the  next  follows  proporEons  of  traffic  observed  in  the  air  travel  data  

Page 29: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Predicted  Outbreak  Likelihood  

Page 30: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Comparison  with  Data  

Page 31: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SI  and  Weak  Ties    (Onnela  2007)  

•  What  is  the  role  of  Ee  strength  in  informaEon  diffusion/epidemic  spreading?  

•  Let  us  remember  the  findings  in  Lecture  2…  

Page 32: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SI  over  CommunicaEon  Network  

Page 33: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

SI  and  Ee  strength  Medium  Ees  are  the  most  used:  Weak  Ees:  not  enough  Eme  to  communicate  Strong  Ees:  stuck  inside  a  community  

Page 34: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

Summary    

•  Epidemics  are  very  complex  processes.  •  ExisEng  models  have  been  increasingly  capable  of  capturing  their  essence.  

•  However  there  are  sEll  a  number  of  open  issues  related  to  the  modelling  of  real  disease  spreading  or  informaEon  disseminaEon.  

•  Mixing  geographic  factors  with  epidemics  model  is  very  relevant  and  important.  

•  Tie  strength  add  another  perspecEve.  

Page 35: Social’and’Technological’’ Network’Analysis’ Lecture’8 ......References’ • Chapter’21’ • PastorISatorras,’R.’and’Vespignani,A.’Epidemic Spreading’in’ScaleIFree’Networks.’Phys.’Rev.’Le].(86),’

References  •  Chapter  21  •  Pastor-­‐Satorras,  R.  and  Vespignani,  A.  Epidemic  Spreading  in  Scale-­‐Free  Networks.  Phys.  Rev.  Le].(86),  n.14.  Pages  =  3200-­‐-­‐3203.  2001.  

•  Pastor-­‐Satorras,  R.  and  Vespignani,  A.  ImmunizaEon  of  Complex  Networks.  Physical  Review  E  65.  2002.  

•  Marcelo  Kuperman  and  Guillermo  Abramson.  Small  world  effect  in  an  epidemiological  model.  Physical  Review  Le]ers,  86(13):2909–2912,  March  2001.  

•  V.  Colizza,  A.  Barrat,  M.  Barthélemy  and  A.  Vespignani.  Predictability  and  epidemic  pathways  in  global  outbreaks  of  infecEous  diseases:  the  SARS  case  study.  BMC  Medicine  2007,  5:34  


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