ASU/SUMS/MTBI/SFI

Carlos Castillo-ChavezJoaquin Bustoz Jr. ProfessorArizona State University

Tutorials 1: Epidemiological Mathematical Modeling Applications in Homeland Security.

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, 08-23-2005

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

ASU/SUMS/MTBI/SFI

From defense threat reduction agency

Buildings

Urban

Ports &Airports

Food Water Supply

Roads

&

Transp

ortElectric

Power

Warning

Interdiction

Detection

Treatment andConsequence Management

Attribution

PharmaceuticalsTelecom

Response

ASU/SUMS/MTBI/SFI

From defense threat reduction agency

Food SafetyMedical SurveillanceAnimal/Plant HealthOther Public Health

Urban Monitoring

CharacterizationMetros

Toxic Industrials

Choke Points

Federal

Response

Plan

Data Mining,Fusion, and

Management

EmergencyManagement

Tools

State andLocal

Governments

From defense threat reduction agency

ASU/SUMS/MTBI/SFI

Research Areas

•Biosurveillance;•Agroterrorism; •Bioterror response logistics; •Deliberate release of biological agents; •Impact assessment at all levels;•Causes: spread of fanatic behaviors.

Ricardo Oliva:Ricardo Oliva:Ricardo Oliva:Ricardo Oliva:

ASU/SUMS/MTBI/SFI

Modeling Challenges &Mathematical ApproachesFrom a “classical” perspective to a global scale

Deterministic Stochastic Computational Agent Based Models

ASU/SUMS/MTBI/SFI

Some theoretical/modeling challenges

•Individual and Agent Based Models--what can they do?

•Mean Field or Deterministic Approaches--how do we average?

•Space? Physical or sociological?

•Classical approaches (PDEs, meta-population models) or network/graph theoretic approaches

•Large scale simulations--how much detail?

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Ecological/Epidemiological view point

Invasion Persistence Co-existence Evolution Co-evolution Control

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Epidemiological/Control Units

Cell Individuals Houses/Farms Generalized households Communities Cities/countries

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Temporal Scales

Single outbreaks Long-term dynamics Evolutionary behavior

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Social Complexity

Spatial distribution Population structure Social Dynamics Population Mobility Demography--Immigration Social hierarchies Economic systems/structures

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Links/Topology/Networks

Local transportation network Global transportation network Migration Topology (social and physical) Geography--borders.

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Control/Economics/Logistics

Vaccination/Education Alternative public health approaches Cost, cost & cost Public health infrastructure Response time

Critical Response Time in Critical Response Time in FMD epidemicsFMD epidemics

A. L. Rivas, A. L. Rivas, S. Tennenbaum, S. Tennenbaum,

C. Castillo-Chávez et al.C. Castillo-Chávez et al.{American Journal of Veterinary Research}(Canadian Journal of Veterinary Research)

It is critical to determine the time It is critical to determine the time needed and available to implement a needed and available to implement a

successful intervention.successful intervention.

11 22 33

BRAZILBRAZIL

AA RR GG EE NN T

T

.. II NN AA

ATLANTIC OCEANATLANTIC OCEAN

: 1-5 cases

(1- 7 days

post-onset)

1-5 cases

(8-14 days

post-onset)

The context--Foot and Mouth DiseaseThe context--Foot and Mouth Disease

0

5

10

15

20

25

30

35

40

Day 1 (April 23, 01) Day 2 (April 24, 01)Day 3 (April 25, 01)Day 4 (April 26, 01)Day 5 (April 27, 01)Day 6 (April 28, 01)Day 7 (April 29, 01)Day 8 (April 30, 01)Day 9 (May 1, 01)Day 10 (May 2, 01)Day 11 (May 3, 01)Day 12 (May 4, 01)Day 13 (May 5, 01)Day 14 (May 6, 01)Day 15 (May 7, 01)Day 16 (May 8, 01)Day 17 (May 9, 01)Day 18 (May 10, 01)Day 19 (May 11, 01)Day 20 (May 12, 01)Day 21 (May 13, 01)Day 22 (May 14, 01)Day 23 (May 15, 01)Day 24 (May 16, 01)Day 25 (May 17, 01)Day 26 (May 18, 01)Day 27 (May 19, 01)Day 28 (May 20, 01)Day 29 (May 21, 01)Day 30 (May 22, 01)

Region 1

Region 2

Region 3

“exponential”growth

Daily cases in the first month of the epidemicDaily cases in the first month of the epidemicN

um

be

r o

f d

aily

cas

es

ASU/SUMS/MTBI/SFI

The Basic Reproductive Number R0

R0 is the average number of secondary cases generated by an

infectious unit when it is introduced into a susceptible population (at demographic steady state) of the same units.

If R0 >1 then an epidemic is expected to occur--number of infected units increases

If R0 < 1 then the number of secondary infections is not enough to sustain an apidemic.

The goal of public health interventions is to reduce R0 to a number below 1.

However, timing is an issue! How fast do we need to respond?

1.4 days

2.6 days

3.0 days

Estimated CRTs for implementing intervention(s) resulting in R_o <= 1 (successful intervention)

ASU/SUMS/MTBI/SFI

Epidemic Models

ASU/SUMS/MTBI/SFI

Basic Epidemiological Models: SIR

Susceptible - Infected - Recovered

ASU/SUMS/MTBI/SFI

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

⎛

⎝ ⎜

⎞

⎠ ⎟

ASU/SUMS/MTBI/SFI

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

ASU/SUMS/MTBI/SFI

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

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SIR Transcritical Bifurcation

unstable

€

I*(R0)

€

I*

€

R0

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 Laboratory2University of Iowa3Cornell University

ASU/SUMS/MTBI/SFI

•All individuals are susceptible

•The population is divided into two groups: normally active and less active

•No vital dynamics included (single outbreak)

•Disease progression: Exposed (latent) and Infectious

•News of a smallpox outbreak leads to the implementation of the following interventions:

–Quarantine–Isolation–Vaccination (ring and mass vaccination)–Behavioral changes (3 levels: high, medium & low)

MODEL

ASU/SUMS/MTBI/SFI

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

ASU/SUMS/MTBI/SFI

The Model

The behavioral change rates are modeled by a non-negative, bounded, monotone increasing function i (for i = S, E, I) given by

€

ϕ i =ai(In + Il )

1+ bi(In + Il )

1

day with

€

ϕ S < ϕ E < ϕ I

ASU/SUMS/MTBI/SFI

Numerical Simulations

ASU/SUMS/MTBI/SFI

Numerical Simulations

ASU/SUMS/MTBI/SFI

Conclusions•Behavioral changes play a key role.

• Integrated control policies are most effective: behavioral changes and vaccination have a huge impact.

•Delays are bad.

Mass Transportation and Epidemics

ASU/SUMS/MTBI/SFI

"An Epidemic Model with Virtual Mass Transportation"

ASU/SUMS/MTBI/SFI

Mass Transportation Systems/HUBS

Baojun SongJuan Zhang

Carlos Castillo-Chavez

ASU/SUMS/MTBI/SFI

Subway Transportation ModelSubway Transportation Model

Subway

NSU

SU SU

NSU

SU

NSU

SU

NSU

Vaccination Strategies

• Vaccinate civilian health-care and public health workers• Ring vaccination (Trace vaccination)• Mass vaccination• Mass vaccination if ring vaccination fails•Integrated approaches likely to be most effective

Assumptions

1.The population is divided into N neighborhoods;

2.Epidemiologically each individual is in one of four status: susceptible, exposed, infectious, and recovered;

3.A person is either a subway user or not4.A ``vaccinated” class is included--

everybody who is successfully vaccinated is sent to the recovered class

Proportionate mixing

K subpopulations with densities N1(t), N2(t), …, Nk(t) at time t.

 cl : the average number of contacts per individual, per unit time

among members of the lth subgroup.  

Pij : the probability that an i-group individual has a contact with a

j-group individual given that it had a contact with somebody.

Proportionate mixing(Mixing Axioms)

(1) Pij >0

(2)

(3) ci Ni Pij = cj Nj Pji

 

Then

is the only separable solution satisfying (1) , (2), and (3). 

€

Pijj=1

k∑ =1

€

Pij =P j =c

jN

j

clN

ll =1

K

∑

ASU/SUMS/MTBI/SFI

the mixing probability between non-subway users from neighborhood i given that they mixed.

the mixing probability of non-subway and subway users from neighborhood i, given that they mixed.

the mixing probability of subway and non-subway users from neighborhood i, given that they mixed.

the mixing probability between subway users from neighborhood i, given that they mixed.

the mixing probability between subway users from neighborhoods i and j, given that they mixed.

the mixing probability between non-subway users from neighborhoods i and j, given that they mixed.

the mixing probability between non-subway user from neighborhood i and subway users from neighborhood j, given that they mixed.

iaibP

jbiaP

jaiaP

jbibP

ibiaP

iaiaP

ibibP

Definitions

Formulae of Mixing Probabilities(depends on activity level and allocated time)

Identities of Mixing Probabilities

State Variables i index of neighborhood Wi number of individuals of susceptibles of SU in

neighborhood i Xi number of individuals of exposed of SU in

neighborhood i Yi number of individuals of infectious of SU in

neighborhood i Zi number of individuals of recovered of SU in

neighborhood i Si number of individuals of susceptibles of NSU in

neighborhood i Ei number of individuals of exposed of NSU in

neighborhood i Ii number of individuals of infectious of NSU in

neighborhood i Ri number of individuals of recovered of NSU in

neighborhood i

Smallpox Model for NSU in neighborhood iSmallpox Model for NSU in neighborhood i

Ei Ii

Ri

Si

iEql2

iSql1

iE

iEμ iId )( +μ

iIα

iRμ

Ai)(tBi

iSμ

Model Equations for neighborhood i Model Equations for neighborhood i

Nonsubway users Subway usersNonsubway users Subway users

)()()()()(

)(

)()(

)()(

21

2

1

tRtItEtStQ

EqlSqlRIdt

dR

IdEdtdI

EqlEEtBdt

dE

SqlStBAdt

dS

iiiii

iii

iii

iiiii

iiiii

+++=

++−=

+++=

++−=

+−−=

μα

αμ

μ

μ

)()()()()(

)(

)()(

)()(

21

2

1

tZtYtXtWtT

XqlWqlZYdt

dZ

YdXdt

dY

XqlXXtVdt

dX

WqlWtVdt

dW

iiiii

iiiii

iii

iiiii

iiiii

+++=

++−=

+++=

++−=

+−−=

μα

αμ

μ

μΛ

Infection Rates Infection Rates

Rate of infection for NSU

Rate of infection for SU

€

Vi(t) = β ibiW i P a i

Ii

Ti

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟+ Qi

+ P bi

Yi

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟

Ti

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟+ Qi

+ P b j

Y j

ρ j

ρ j + σ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Tj

ρ j

ρ j + σ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

j=1

N

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

€

Bi(t) = β iaiSi˜ P a i

Ii

Ti

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟+ Qi

+ ˜ P bi

Yi

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟

Ti

σ i

ρ i + σ i

⎛

⎝ ⎜

⎞

⎠ ⎟+ Qi

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

RR00 for Two Neighborhoods for Two Neighborhoods(a special case)(a special case)

1 ,0 ,0 === iiq σρ

},max{ 2,01,00 RRR =

€

R0, i

= βia

i

φ

μ + φ

⎛

⎝ ⎜

⎞

⎠ ⎟

1

μ + α + d

⎛

⎝ ⎜

⎞

⎠ ⎟

ai(A

i/μ )

(aiA

i+ b

iΛ

i) /μ

⎛

⎝ ⎜

⎞

⎠ ⎟

Ai/μ

(Ai

+ Λi) /μ

⎛

⎝ ⎜

⎞

⎠ ⎟

+βib

i

φ

μ + φ

⎛

⎝ ⎜

⎞

⎠ ⎟

1

μ + α + d

⎛

⎝ ⎜

⎞

⎠ ⎟

bi(Λ

i/μ )

(aiA

i+ b

iΛ

i) /μ

⎛

⎝ ⎜

⎞

⎠ ⎟

Λi/μ

(Ai

+ Λi) /μ

⎛

⎝ ⎜

⎞

⎠ ⎟

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

Cumulative deaths: One day delay (q1 = q2=0.5)

Cases: One day delay (q1 = q2=0.5)

Cumulative deaths: One day delay (q1 = q2=0.8)

Cases: One day delay (q1 = q2=0.5)

ASU/SUMS/MTBI/SFI

Conclusions•Integrated control policies are most effective: behavioral changes and vaccination have a huge impact.

•Delays are bad.