Transcript DIMACS Workshop in Epidemiology June 24

Tutorials 2: Epidemiological Mathematical Modeling Applications
of Networks.
Mathematical Modeling of Infectious Diseases: Dynamics and Control (15
Aug - 9 Oct 2005)
Jointly organized by Institute for Mathematical Sciences, National University of
Singapore and Regional Emerging Diseases Intervention (REDI) Centre,
Singapore
http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm
Singapore, 08-24-2005
Carlos Castillo-Chavez
Joaquin Bustoz Jr. Professor
Arizona State University
ASU/SUMS/MTBI/SFI
Bioterrorism
The possibility of bioterrorist acts
stresses the need for the development
of theoretical and practical
mathematical frameworks to
systemically test our efforts to
anticipate, prevent and respond to acts
of destabilization in a global
community
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Epidemic Models
ASU/SUMS/MTBI/SFI
Basic Epidemiological Models: SIR
Susceptible - Infected - Recovered
ASU/SUMS/MTBI/SFI
B ( S, I )   S
contacts   probability of transmission
  
 
 time  
contact
I
N
S(t):
susceptible at time t

I(t): infected assumed infectious at time t
R(t): recovered, permanently immune
N: Total population size (S+I+R)
N

S
I
S



I
I


ASU/SUMS/MTBI/SFI

R
R



SIR - Equations
dS
dt
dI
 N  S
 S
Parameters
(1)
N
 S
dt
dR
I
I
N
    I
 I   R
(2 )
( 3)
dt
dN
dt

d
dt
S 
(4 )
I  R  0
Per-capita death (or birth) rate

Per-capita recovery rate

Transmission coefficient


N  S I R


 contacts   probability of transmission
  
 
unit
time
contact

 
(5
 )
ASU/SUMS/MTBI/SFI



SIR - Model (Invasion)
dS
dt
dI
 N  S
I
 S
N
 S
dt
I
N
     I
S  N
dI
dt
  I      I       I
or
I(t)
I ( t )  I (0 )e

        t
R0 


ASU/SUMS/MTBI/SFI
1
Ro
“Number of secondary infections
generated by a “typical” infectious individual in a
population of mostly susceptibles
at a demographic steady state
Ro<1
No epidemic
Ro>1
Epidemic
ASU/SUMS/MTBI/SFI
Establishment of a Critical Mass of Infectives!
Ro >1 implies growth while Ro<1 extinction.
ASU/SUMS/MTBI/SFI
Phase Portraits
ASU/SUMS/MTBI/SFI

SIR
Transcritical Bifurcation
I
*
*
I (R0 )

R0
unstable

ASU/SUMS/MTBI/SFI
Deliberate Release of
Biological Agents
ASU/SUMS/MTBI/SFI
Effects of Behavioral Changes in a Smallpox Attack
Model
Impact of behavioral changes on response logistics and
public policy (appeared in Mathematical Biosciences, 05)
Sara Del Valle1,2
Herbert Hethcote2, Carlos Castillo-Chavez1,3, Mac Hyman1
1Los Alamos National Laboratory
2University of Iowa
3Cornell University
ASU/SUMS/MTBI/SFI
The Model
Sn
V
En
In
Q
S
Sl
R
W
E
El
I
Il
D
The subscript refers to normally active (n) or less active (l):
Susceptibles (S), Exposed (E), Infectious (I), Vaccinated (V),
Quarantined (Q), Isolated (W), ASU/SUMS/MTBI/SFI Recovered (R), Dead (D)
"An Epidemic Model with
Virtual Mass Transportation"
ASU/SUMS/MTBI/SFI
Two neighborhood simulations
(NYC type city)
1.
There are 8 million long-term and 0.2 million shortterm (tourists) residents in NYC.
2.
Time span of simulation is 30 days +.
3.
Control parameters in the model are: q1 and q2
(vaccination rates)
4.
We use two ``neighborhoods”, one for NYC residents
and the second for tourists.
Curve R0 (q1, q2)
=1
Plot R0 (q1, q2) vs q1 and
q2
Conclusions
•Integrated control policies are most effective:
behavioral changes and vaccination have a huge
impact.
•Policies must include “transient” populations
•Delays are bad.
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Worst Case Scenarios?
ASU/SUMS/MTBI/SFI
Epidemics on Networks?
ASU/SUMS/MTBI/SFI
SARS propagation network in Singapore
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see thi s picture.
MMUR May 9, 2003/ 52 (18); 405-411
ASU/SUMS/MTBI/SFI
Epidemics on Networks? Some caveats
Appeal: Networks look like the real world
 Typical: To study “fixed” graphs (small
world, scale-free)
 Network/Graph structure not fixed over
time
 A connected to B requires temporal cohabitation in the study of processes on
networks

ASU/SUMS/MTBI/SFI
Underlying Philosophy in Classical
Epidemiology
ASU/SUMS/MTBI/SFI
Ecological view point






Invasion (Networks useful at this level).
Short temporal scales--single outbreaks (Networks useful at this
level).
Persistence
Co-existence
Evolution
Co-evolution
ASU/SUMS/MTBI/SFI
Processes on Networks Temporal Scales
 Single
outbreak
 Long-term dynamics
 Evolutionary behavior
ASU/SUMS/MTBI/SFI
Nodes? Epidemiological Units

Individuals

Cluster of individuals (friends)

Other aggregates? Farms?
Reach of the Network: Level of Aggregation

Neighborhood

Cities, states, countries
Highly trafficked locations in the city of
Portland (EpiSims)
Convenant
ASU/SUMS/MTBI/SFI
M a y 28
M a y 24
M a y 20
M a y 16
M a y 12

M a y 08
M a y 04
A pr .30
A pr .26
A pr .22
A pr .18
A pr .14
A pr .10
A pr .06
A pr .02
M a r .29
M a r .25
M a r .21
M a r .17
M a r .13

M a r .09
M a r .05
M a r .01
F e b .25
F e b .21
Heterogeneity: Infection curves by routes of
transmission
(Ping Yang--Health Canada)

Health Care Settings
Health Care Settings

House Hold
Transm ission


Toronto. SARS was introduced to Toronto by a couple (Guests F and G) at Hotel M
in Hong Kong. On February 23, 2003 they returned to Toronto. Two days later,
the woman developed SARS, infected 5 out of her 6 adult family members and
caused the first outbreak in Toronto. In mid-May, an undiagnosed case at North
York General Hospital led to a second outbreak among other patients, family
members and healthcare workers (from Glen Webb’s presentation).
ASU/SUMS/MTBI/SFI
Exponential Dynamics: Hong Kong and Singapore
Data
Model
ASU/SUMS/MTBI/SFI
The Case of Toronto
Fast
diagnosis
&
effective
isolation
Fast
diagnosis
but
imperfect
isolation
Data
Model
Interventions
ASU/SUMS/MTBI/SFI
Slow
diagnosis
and
effective
isolation
Predicting the Final Size of the
Epidemic in Toronto
Model prediction = 396 cases
(J. Theor. Biol 224, 1-8, 2003)
 Actual number as of June 23, 2003 = 377 (Health
Canada website)

ASU/SUMS/MTBI/SFI
Complexity and Networks: Population’s
Characteristics

Gender, ethnicity, race

Social, age, economic structure

Cultural and Communication structures
Connectedness?

Local small isolated populations

Large multi-connected populations

Who mixes with whom?
Scale and topology
Nov 05, ‘02
Feb 21, ‘03
Mrs. Siu-Chu
Mrs. Mok
ASU/SUMS/MTBI/SFI
Prof. Liu
Johny Chen
Modeling Challenges &Mathematical Approaches
“Classical” Population Perspective
Deterministic
 Stochastic
 Computational
 Agent Based Models

ASU/SUMS/MTBI/SFI
Scaling Laws for the Movement of People
Between Locations in a Large City
Gerardo Chowell et al.
Scaling Laws for the Movement of People between locations in a
large city, Physical Review E, 68, 066102 (2003), Chowell, Hyman,
Eubank and Castillo-Chavez
LA-UR-02-6658
Outline

Statistical properties of real world networks
–
–
–
–
–
Network of actors in Hollywood
www
Internet
Scientific collaboration network
Power generator network of western US
Outline

Analysis of a Real World Network: The city of
Portland
–
–
–
–
–
–
Location-based network
Topological properties
Traffic distribution
Total traffic distribution per location
Correlation between connectivity and traffic
distributions
Time evolution of the network
Statistical properties of networks



Connectivity distribution (degree distribution)
Clustering (C)
Characteristic path length (L)
2
3
3
2
The network of actors in Hollywood
(Watts and Strogatz, 1998)
Julia Roberts
Richard Gere
Diane Lane
Kevin Bacon
Denzel Washington
Eric Roberts
Mickey Rourke
The electric power grid of
western US
(Watts and Strogatz, 1998)
The world wide web (www)
(Barabasi et al. (1999), Kumar (2000))
Home page
web page
web page
web page
Internet
(Faloutsos et al., 1998)
Scientific collaboration networks

M. J. Newman
Statistical properties of real
world networks

Small-world effect
–
–

High levels of clustering
Short Characteristic path length
Connectivity distribution has a tail that decays as a
power law of the form:

P(k) ~ k -
Connectivity distribution for two real
world networks
Random Graph Models of Networks. M. E. J. Newman, 2002
City of Portland:
A Social Network
Location-based network
W ij
Location 1
Location 2
Location 3
Location 4


Directed, weighted network
Data set contains a detailed description of the
movements of the individuals in the city of Portland.
Highly trafficked locations in Portland
Statistical properties
The clustering coefficient for our location-based
network is C = 0.058 (roughly 350 times larger
than the expected value for an equivalent random
graph).
 The same situation arises for the electric power
grid of western US where C=0.08.
 Average distance between nodes = 3.38 (diameter
= 9).

Connectivity distribution
.7
Out-degree
Strong or weak connections ?
Very little is known about the distribution of
the strength of the connections in real world
networks.
 Only their structural properties have been
analyzed. The main reason being the lack of
data to quantify the strength of the
connections.

Traffic distribution
3.7
Out-traffic
Number of locations
Total traffic distribution
.7
Total out-traffic
Número de locaciones
Correlation in density
Tráfico total de salida por nodo
Log (number of locations)
Semilog plot
Total outtraffic
Hierarchical structures at different
levels of aggregation
C(k) a k –
a)
Work activities
b)
School activities
c)
Social/rec activities
d)
All activities
a)
b)
c)
d)
Cluster size distribution
t=4 hrs.
t=6 hrs.
Gerardo Chowell
t=5 hrs.
t=7 hrs.
Location-based network
Size of the largest cluster
Gerardo Chowell

This is joint work with J.M. Hyman, S. Eubank and
G Chowell.

Scaling Laws for the Movement of People between
locations in a large city, Physical Review E, 68,
066102 (2003), Chowell, Hyman, Eubank and
Castillo-Chavez
Structure and Function of Complex Networks
•Introduction:
Strogatz, Nature (2001)
•Comprehensive study
Newman, SIAM Rev. (2003)
ASU/SUMS/MTBI/SFI
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Results
Random connections
Nearest neighbor, small
world and random
N e tw o rk
ty p e
C lu s te rin g
c o e ffic ie n t
A v g . c h a ra c te r is tic
p a th le n g th
R e g u la r
S m a ll W o rld
0 .5
0 .3 6 4 6 2
1 2 .8 7 8 8
3 .7 3 9
Com p. Random
0 .0 6 5 9 6 7
2 .4 8 8 3
Questions
Compare disease
spread on a




Nearest Neighbor
Network
Random Network
Small World Network
p = 1 when random and
p about 0 when nearest neighbor dominates
Small-world and Scale-free networks
Small world network of size 70
with probability of random
connections p = 0.1
Scale-free network of size 70
illustrating the presence of a
small number of highly nodes
connected (hubs).
LLYD Model
Exponentially Distributed
Networks
Scale-Free Networks
P (k )  k
3
as
p 0
P (k )  e
Barabasi-Albert (BA)

k
z
as p  1
Erdos-Renyi


ASU/SUMS/MTBI/SFI
Connectivity distribution
.7
Out-degree
Navigation in a Small-World
(Kleinberg, 2000)
•Two dimensional lattice
•Long-range connection between
node u and v, with probability r  a ,
where r | u1  v 1 |  | u 2  v 2 |
v
b
• Greedy heuristic algorithm: each
message holder forwards
the

 message across a connection that
brings it as close as possible to the
target in the lattice distance
•T: Expected delivery time. T   n
where    (a ),    (a )
ASU/SUMS/MTBI/SFI
a
u
d

c
Highly trafficked locations in the city of
Portland
Building Epidemics

Transition Probabilities: P(S to I), P(I to
PSI = 1 – eIdt
PIR = 1 – edt
R)
ASU/SUMS/MTBI/SFI
Epidemics on small-world networks
The rate of growth of SIR
epidemics increases in a
nonlinear fashion as
disorder in the network
increases.
The rate of growth rd for the
analogous deterministic
homogeneous mixing model
is shown.
The role of the network structure in
epidemics
The dotted graph shows
the rate of growth of SIR
epidemics when the initial
infectious source has the
highest number of faraway
connections (train stations,
airports, etc) while the
continuous line is the
result of placing the source
at random.
Rate of growth of epidemics in small-world networks
Growth in the number of infected in an SIR model where individuals live in a ring.
Curves give the average number of infected (50 simulations) in a population of
1000 while the growth is exponential. p = 0, disorder parameter, corresponds to no
long term connections and p =1 implies that everybody is connected to each other.
Graph on the left from a single source (idea, virus, rumor). Top curve is when the
spread begin at a pressure point. Lower spread begins at a random point. Graph on
the right, three randomly placed sources of infection (ideas, whatever) versus one.
SIR epidemics on Small worlds



For small worlds, a
sharp transition occurs
at small values of the
disorder parameter p.
5 initial infected nodes
chosen at random,
=4/7, =2/7.
The mean (red) of 50
realizations and the
standard deviation are
shown.
ASU/SUMS/MTBI/SFI
SIR epidemics on Small worlds



Similar results are
observed when
initial infected
nodes are chosen
by highest degree
=4/7, =2/7.
The mean (red) of
50 realizations and
the standard
deviation are
shown.
ASU/SUMS/MTBI/SFI
SIR epidemics on Small worlds




Final epidemic size
as a function of the
transmission rate .
5 initial infected
nodes chosen at
random
=2/7.
The mean (solid) of
50 realizations and
the standard
deviation (bars) are
shown.
ASU/SUMS/MTBI/SFI
SIR epidemics on Scale-Free networks
(Barabasi-Albert model)
Final epidemic
size as a function
of the
transmission rate
.
 5 initial infected
nodes chosen at
random
 =2/7.
 The mean (red)
of 50 realizations
and the standard
deviation are
shown.

ASU/SUMS/MTBI/SFI
Small worlds: Epidemic duration
(five sources placed at random)
ASU/SUMS/MTBI/SFI
Conclusions
•
•
•
•
•
The impact of alternative agents of disease transmission
and evolution—like transportation systems seems
critically important.
The study of epidemics on different topologies (networks)
is essential (mobile individuals cause a lot of
``problems”).
Worst case scenarios may occur in random networks but
the focus should be on worst case plausible scenarios-one
cannot ignore behavioral changes.
Worst case scenarios depend on topology
Bigger outbreaks are sometimes caused by releases at
pressure points in the network.
Bioterrorism: Mathematical Modeling
Applications in Homeland Security.
H. T. Banks and C. Castillo-Chavez, Editors
Frontiers in Applied Mathematics 28
Globalization and the possibility of bioterrorist acts have highlighted
the pressing need for the development of theoretical and practical
mathematical frameworks that may be useful in our systemic efforts to
anticipate, prevent, and respond to acts of destabilization.
Bioterrorism: Mathematical Modeling Applications in Homeland
Security collects the detailed contributions of selected groups of experts
from the fields of biostatistics, control theory, epidemiology, and
mathematical biology who have engaged in the development of
frameworks, models, and mathematical methods needed to address
some of the pressing challenges posed by acts of terror. The ten chapters
of this volume touch on a large range of issues in the subfields of
biosurveillance, agroterrorism, bioterror response logistics, deliberate
release of biological agents, impact assessment, and the spread of
fanatic behaviors.
ASU/SUMS/MTBI/SFI