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Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University

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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

Carlos Castillo-Chavez

Joaquin Bustoz Jr. Professor

Arizona State University

ASU/SUMS/MTBI/SFI

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

Homeland Security

Telecom

Pharmaceuticals

Ports &

Airports

Buildings

Response

Attribution

Food

Water Supply

Urban

Treatment and

Consequence Management

Roads & Transport

Electric Power

Detection

Interdiction

Warning

ASU/SUMS/MTBI/SFI

From defense threat reduction agency

Food Safety

Medical Surveillance

Animal/Plant Health

Other Public Health

Choke Points

Urban Monitoring

Characterization

Metros

Data Mining,

Fusion, and

Management

State and

Local

Governments

Emergency

Management

Tools

Federal

Response

Plan

ASU/SUMS/MTBI/SFI

Toxic Industrials

Ricardo Oliva:

Ricardo Oliva:

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

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

- 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?

ASU/SUMS/MTBI/SFI

- Invasion
- Persistence
- Co-existence
- Evolution
- Co-evolution
- Control

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- Cell
- Individuals
- Houses/Farms
- Generalized households
- Communities
- Cities/countries

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- Single outbreaks
- Long-term dynamics
- Evolutionary behavior

ASU/SUMS/MTBI/SFI

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

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

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

ASU/SUMS/MTBI/SFI

Critical Response Time in FMD epidemics

A. L. Rivas, S. Tennenbaum, C. Castillo-Chávez et al.{American Journal of Veterinary Research}(Canadian Journal of Veterinary Research)

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

: 1-5 cases

(1- 7 days

post-onset)

1-5 cases

(8-14 days

post-onset)

3

2

1

The context--Foot and Mouth Disease

BRAZIL

ARGENT .INA

ATLANTIC OCEAN

“exponential”growth

Number of daily cases

The Basic Reproductive Number R0

R0is 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?

ASU/SUMS/MTBI/SFI

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

3.0 days

2.6 days

1.4 days

ASU/SUMS/MTBI/SFI

Basic Epidemiological Models: SIR

Susceptible - Infected - Recovered

ASU/SUMS/MTBI/SFI

R

S

I

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)

ASU/SUMS/MTBI/SFI

SIR - Equations

Parameters

ASU/SUMS/MTBI/SFI

ASU/SUMS/MTBI/SFI

“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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ASU/SUMS/MTBI/SFI

Phase Portraits

ASU/SUMS/MTBI/SFI

unstable

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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

MODEL

- 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)

ASU/SUMS/MTBI/SFI

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)

Sn

En

In

R

Q

W

V

S

E

I

Sl

El

Il

D

ASU/SUMS/MTBI/SFI

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

with

ASU/SUMS/MTBI/SFI

ASU/SUMS/MTBI/SFI

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.

ASU/SUMS/MTBI/SFI

Mass Transportation and Epidemics

ASU/SUMS/MTBI/SFI

ASU/SUMS/MTBI/SFI

NSU

NSU

SU

SU

Subway

SU

SU

NSU

NSU

Subway Transportation Model

ASU/SUMS/MTBI/SFI

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

- The population is divided into N neighborhoods;
- Epidemiologically each individual is in one of four status: susceptible, exposed, infectious, and recovered;
- A person is either a subway user or not
- 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).

Definitions

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.

Formulae of Mixing Probabilities

(depends on activity level and allocated time)

Identities of Mixing Probabilities

- 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 i

Si

Ei

Ii

Ai

Ri

Model Equations for neighborhood i

Nonsubway users Subway users

Infection Rates

Rate of infection for NSU

Rate of infection for SU

R0 for Two Neighborhoods

(a special case)

- There are 8 million long-term and 0.2 million short-term (tourists) residents in NYC.
- Time span of simulation is 30 days +.
- Control parameters in the model are: q1 and q2(vaccination rates)
- 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)

Conclusions

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

ASU/SUMS/MTBI/SFI