Carlos castillo chavez joaquin bustoz jr professor arizona state university
This presentation is the property of its rightful owner.
Sponsored Links
1 / 61

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


  • 44 Views
  • Uploaded on
  • Presentation posted in: General

Tutorials 1: Epidemiological Mathematical Modeling Applications in Homeland Security. Mathematical Modeling of Infectious Diseases: Dynamics and Control (15 Aug - 9 Oct 2005)

Download Presentation

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

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Carlos castillo chavez joaquin bustoz jr professor arizona 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

Carlos Castillo-Chavez

Joaquin Bustoz Jr. Professor

Arizona State University

ASU/SUMS/MTBI/SFI


Bioterrorism

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

From defense threat reduction agency

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 agency1

From defense threat reduction agency

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


Research areas

Ricardo Oliva:

Ricardo Oliva:

Research Areas

  • Biosurveillance;

  • Agroterrorism;

  • Bioterror response logistics;

  • Deliberate release of biological agents;

  • Impact assessment at all levels;

  • Causes: spread of fanatic behaviors.

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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

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?

ASU/SUMS/MTBI/SFI


Ecological epidemiological view point

Ecological/Epidemiological view point

  • Invasion

  • Persistence

  • Co-existence

  • Evolution

  • Co-evolution

  • Control

ASU/SUMS/MTBI/SFI


Epidemiological control units

Epidemiological/Control Units

  • Cell

  • Individuals

  • Houses/Farms

  • Generalized households

  • Communities

  • Cities/countries

ASU/SUMS/MTBI/SFI


Temporal scales

Temporal Scales

  • Single outbreaks

  • Long-term dynamics

  • Evolutionary behavior

ASU/SUMS/MTBI/SFI


Social complexity

Social Complexity

  • Spatial distribution

  • Population structure

  • Social Dynamics

  • Population Mobility

  • Demography--Immigration

  • Social hierarchies

  • Economic systems/structures

ASU/SUMS/MTBI/SFI


Links topology networks

Links/Topology/Networks

  • Local transportation network

  • Global transportation network

  • Migration

  • Topology (social and physical)

  • Geography--borders.

ASU/SUMS/MTBI/SFI


Control economics logistics

Control/Economics/Logistics

  • Vaccination/Education

  • Alternative public health approaches

  • Cost, cost & cost

  • Public health infrastructure

  • Response time

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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)


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

: 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


Daily cases in the first month of the epidemic

“exponential”growth

Daily cases in the first month of the epidemic

Number of daily cases


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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

3.0 days

2.6 days

1.4 days


Epidemic models

Epidemic Models

ASU/SUMS/MTBI/SFI


Basic epidemiological models sir

Basic Epidemiological Models: SIR

Susceptible - Infected - Recovered

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

SIR - Equations

Parameters

ASU/SUMS/MTBI/SFI


Sir model invasion

SIR - Model (Invasion)

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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

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

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Phase Portraits

ASU/SUMS/MTBI/SFI


Sir transcritical bifurcation

SIR Transcritical Bifurcation

unstable

ASU/SUMS/MTBI/SFI


Deliberate release of biological agents

Deliberate Release of Biological Agents

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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 model

The Model

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 model1

The Model

  • 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


Numerical simulations

Numerical Simulations

ASU/SUMS/MTBI/SFI


Numerical simulations1

Numerical Simulations

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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

Mass Transportation and Epidemics


An epidemic model with virtual mass transportation

"An Epidemic Model with Virtual Mass Transportation"

ASU/SUMS/MTBI/SFI


Mass transportation systems hubs baojun song juan zhang carlos castillo chavez

Mass Transportation Systems/HUBSBaojun SongJuan ZhangCarlos Castillo-Chavez

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

NSU

NSU

SU

SU

Subway

SU

SU

NSU

NSU

Subway Transportation Model

ASU/SUMS/MTBI/SFI


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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

Assumptions

  • 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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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.


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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).


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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.


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Formulae of Mixing Probabilities

(depends on activity level and allocated time)


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Identities of Mixing Probabilities


State variables

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Smallpox Model for NSU in neighborhood i

Si

Ei

Ii

Ai

Ri


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Model Equations for neighborhood i

Nonsubway users Subway users


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Infection Rates

Rate of infection for NSU

Rate of infection for SU


Carlos castillo chavez joaquin bustoz jr professor arizona state university

R0 for Two Neighborhoods

(a special case)


Two neighborhood simulations nyc type city

Two neighborhood simulations(NYC type city)

  • 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.


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Curve R0 (q1, q2) =1


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Plot R0 (q1, q2) vs q1 and q2


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

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


Carlos castillo chavez joaquin bustoz jr professor arizona state university

Conclusions

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

  • Delays are bad.

ASU/SUMS/MTBI/SFI


  • Login