ASU/SUMS/MTBI/SFI

Carlos Castillo-ChavezJoaquin Bustoz Jr. ProfessorArizona State University

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

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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

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Basic Epidemiological Models: SIR

Susceptible - Infected - Recovered

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S I R

€

μN

€

γI

€

β

€

μS

€

μI

€

μR

S(t): susceptible at time tI(t): infected assumed infectious at time tR(t): recovered, permanently immuneN: Total population size (S+I+R)

€

B(S,I) = βSI

N

€

β =contacts

time

⎛

⎝ ⎜

⎞

⎠ ⎟×

probability of transmission

contact

⎛

⎝ ⎜

⎞

⎠ ⎟

ASU/SUMS/MTBI/SFI

€

dS

dt= μN − βS

I

N− μS (1)

dI

dt= βS

I

N− μ + γ( )I (2)

dR

dt= γI − μR (3)

N = S + I + R (4)

dN

dt=

d

dtS + I + R( ) = 0 (5)

SIR - Equations

Per-capita death (or birth) rate

Per-capita recovery rate

Transmission coefficient

Parameters

€

μ

€

γ

€

β

€

β ≡contacts

unit time

⎛

⎝ ⎜

⎞

⎠ ⎟×

probability of transmission

contact

⎛

⎝ ⎜

⎞

⎠ ⎟

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SIR - Model (Invasion)

€

dS

dt= μN − βS

I

N− μS

dI

dt= βS

I

N− μ + γ( )I

S ≈ N

dI

dt= βI − μ + γ( )I = β − μ + γ( )( )I

or I(t) ≈ I(0)e β − μ +γ( )( ) t

I(t) ⇔ R0 =β

μ + γ>1

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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

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Establishment of a Critical Mass of Infectives!Ro >1 implies growth while Ro<1 extinction.

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Phase Portraits

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SIR Transcritical Bifurcation

unstable

€

I*(R0)

€

I*

€

R0

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Deliberate Release of Biological Agents

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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 Laboratory2University of Iowa3Cornell University

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The Model

Sn En

In R

V Q W

Sl El 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), Recovered (R), Dead (D)

S E I

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"An Epidemic Model with Virtual Mass Transportation"

Two neighborhood simulations

(NYC type city)1. There are 8 million long-term and 0.2 million

short-term (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

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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?

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Epidemics on Networks?

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

MMUR May 9, 2003/ 52 (18); 405-411

SARS propagation network in Singapore

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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 co-

habitation in the study of processes on networks

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Underlying Philosophy in Classical Epidemiology

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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

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Processes on Networks Temporal Scales

Single outbreak Long-term dynamics Evolutionary behavior

Nodes? Epidemiological Units Individuals

Cluster of individuals (friends)

Other aggregates? Farms?

Neighborhood

Cities, states, countries

Reach of the Network: Level of Aggregation

Highly trafficked locations in the city of Portland (EpiSims)

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Heterogeneity: Infection curves by routes of transmission (Ping Yang--Health Canada)

Feb .21Feb .25Mar .01Mar .05Mar .09Mar .13Mar .17Mar .21Mar .25Mar .29Apr .02Apr .06Apr .10Apr .14Apr .18Apr .22Apr .26Apr .30May 04May 08May 12May 16May 20May 24May 28

0

2

4

6

8

10

Health Care Settings

Health Care Settings

House Hold Transmission

Convenant

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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).

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Exponential Dynamics: Hong Kong and Singapore

Data

Model

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The Case of Toronto

Data

Model

Slow diagnosis and effective isolation

Fast diagnosis but imperfect isolation

Interventions

Fast diagnosis &effective isolation

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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)

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?

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Scale and topology

Feb 21, ‘03

Nov 05, ‘02

Prof. Liu

Mrs. Mok

Mrs. Siu-Chu

Johny Chen

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Modeling Challenges &Mathematical Approaches“Classical” Population Perspective

Deterministic Stochastic Computational Agent Based Models

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)

3

2

3

2

The network of actors in Hollywood

Julia Roberts Diane Lane

Eric Roberts

Mickey RourkeDenzel Washington

Richard Gere Kevin Bacon

(Watts and Strogatz, 1998)

The electric power grid of western US

(Watts and Strogatz, 1998)

The world wide web (www)

Home page

web page

web pageweb page

(Barabasi et al. (1999), Kumar (2000))

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 -γ

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

Directed, weighted network Data set contains a detailed description of the

movements of the individuals in the city of Portland.

Location 1

Location 4

Location 3Location 2

W i j

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

Out-degree

Nu

mb

er o

f lo

cati

o ns

γ

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

Out-traffic

Nu

mb

er o

f co

nn

ecti

ons

bet

wee

n lo

cati

ons

γ

Total traffic distribution

Total out-traffic

Nu

mb

er o

f lo

cati

ons

γ

Correlation in density

out-deg

ree

Nú

mer

o d

e lo

caci

ones

Tráfico total de salida por nodo

Semilog plot

out-degreeTotal out-

traffic

Log

(n

um

ber

of

loca

tion

s)

Hierarchical structures at different levels of aggregationC(k) k –β

a) b)

c) d)

a) Work activities

b) School activities

c) Social/rec activities

d) All activities

Cluster size distribution

Gerardo Chowell

t=4 hrs. t=5 hrs.

t=7 hrs.t=6 hrs.

Location-based networkSize 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

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Structure and Function of Complex Networks

•Introduction:

Strogatz, Nature (2001)

•Comprehensive study

Newman, SIAM Rev. (2003)QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

Results

Networktype

Clusteringcoefficient

Avg. characteristicpath length

Regular 0.5 12.8788Small World 0.36462 3.739Comp. Random 0.065967 2.4883

Random connections

Nearest neighbor, small world and random

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).

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LLYD Model

Scale-Free NetworksExponentially Distributed Networks

3)( −∝ kkP as

€

p → 0 z

k

ekP−

∝)( as

€

p →1

Barabasi-Albert (BA) Erdos-Renyi

Connectivity distribution

Out-degree

Nu

mb

er o

f lo

cati

o ns

γ

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Navigation in a Small-World (Kleinberg, 2000)

•Two dimensional lattice

•Long-range connection between node u and v, with probability , where

• 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.

where

€

r−α

€

r =| u1 − v1 | + | u2 − v2 |

€

T ≥ Ωnβ

€

Ω≡Ω(α ),β ≡ β (α )

a

b

c

d

v

u

Highly trafficked locations in the city of Portland

Building Epidemics

Transition Probabilities: P(S to I), P(I to R)

PSI = 1 – eβ t

PIR = 1 – eγ t

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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.

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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, β=4/7, γ=2/7.

The mean (red) of 50 realizations and the standard deviation are shown.

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SIR epidemics on Small worlds

Similar results are observed when initial infected nodes are chosen by highest degree

β=4/7, γ=2/7. The mean (red) of

50 realizations and the standard deviation are shown.

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SIR epidemics on Small worlds

Final epidemic size as a function of the transmission rate β.

5 initial infected nodes chosen at random

γ=2/7. The mean (solid) of

50 realizations and the standard deviation (bars) are shown.

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SIR epidemics on Scale-Free networks (Barabasi-Albert model)

Final epidemic size as a function of the transmission rate β.

5 initial infected nodes chosen at random

γ=2/7. The mean (red) of 50 realizations

and the standard deviation are shown.

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Small worlds: Epidemic duration(five sources placed at random)

• 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.

Conclusions

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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.