Towards Computational Epidemiology Using Stochastic Cellular Automata in Modeling Spread of Diseases

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Towards Computational Epidemiology Using
Stochastic Cellular Automata in Modeling Spread
of Diseases

• Sangeeta Venkatachalam, Armin R. Mikler
• Computational Epidemiology Research Laboratory
  (cerl.unt.edu)
• Department of Computer Science and Engineering
• University of North Texas
• Email venkatac, mikler_at_cs.unt.edu

This research is in part supported by the
National Science Foundation award NSF-0350200
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Overview

• Computational Epidemiology
• Mathematical Epidemiology
• Introduction to Cellular Automata
• Cellular Automata and Epidemiology
• Neighborhood Restriction
• Stochastic Cellular Automata - A Global Model

• Experiments
• Composition Model
• Summary

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Towards Computational Epidemiology

• Address broader aspects of Epidemiology
• Disease Tracking, Analysis, and Surveillance
• High Performance Computing (HPC)
• Simulation
• Data visualization.

• Investigating disease outbreaks and risk
  assessment in spatially delineated environments
• Investigating vaccination strategies to control
  the spread of a disease
• Investigating spread of disease over large
  distances

Contribute towards establishing computational
epidemiology as a new research domain!!
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Mathematical Epidemiology

• Susceptibles Infectives Removals (SIR) model

SIR state diagram

• A SIR model simulation of a disease spread
• The graph shows the transient curves for the
  susceptibles , infectives and removals during the
  course of a disease epidemic in a given
  population.

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The Model
Vaccination
Disease Parameters
Population
Demographics
Interaction factors
Distances
Data Sets
Visualization
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Cellular Automata

• State of each cell Ci,j depends on the
  neighbors Ci,j1, Ci,j-1, Ci1,j, Ci-1,j for the
  von Neumann neighborhood
• State of each cell Ci,j depends on Ci,j1,
  Ci,j-1, Ci1,j, Ci-1,j, Ci1,j-1, Ci1,j1,
  Ci-1,j-1, Ci-1,j1 for the Moore neighborhood

Von Neumann and Moore Neighborhood
The color of a cell changes based on the majority
color of its neighbors
Cellular Automata Update from time step t-1 to t
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Cellular Automata
Deterministic
Stochastic

• State of the cells after a time period.
• Both types of population are grouped in large
  groups (patches).
• At this stage, both populations seem to be
  stable.
• State (color) of cell does not change anymore.

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Infection Time-line
Illustrates time-line for infection (influenza)
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Cellular Automata and Epidemiology
Disease Parameters Latent period 2
days Infectious period 3 days Recovery period 2
days Infectivity of 7, 10,12,15
Cellular Automata with a neighborhood of 8
cells. Comparison of growth rate for different
infectivities is shown
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Analysis of Vaccination
Comparison of random vaccination (5 of the
population vaccinated) and no vaccination
Comparison of Random vaccination and Ring
vaccination
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Neighborhood Restriction

• Cell layers with respect to central cell on layer
  1.
• Total neighbors of a layers is summation of its
  outer-line and inner-line neighborhoods.
• Effective neighbors per cell is the ration of
  neighboring cells to the cells in the current
  layer.

Cell Layers
Li 1 i1 (2i-1)2 (2i-3)2 igt1
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Neighborhood Restriction

• Graph illustrates the effective inner-line and
  outer-line neighbors from layer1 up to layer50.
• Effective outer-line neighbors converges to 1 for
  higher layers.
• Effective inner-line neighbors increases to 1 for
  higher layers.

Effective Neighborhood
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Stochastic Cellular Automata A Global Model

• Definition of a Fuzzy Set
• Neighborhood of cell Ci,j is global SCA
• Gi,j (Ck,l, ?C i ,j, C k ,l) for all Ck,l ?
  C, 0 ? Ci,j, Ck,l 1
• C is a set of all cells in the CA.

?C i ,j, C k ,l represents an interaction
coefficient that controls all possible
interactions between a cell Ci,j and its global
neighborhood Gi,j. A function of inter-cell
distance and cell population density.
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Interaction Metrics

• Interaction Coefficient defined as
• 1/Euclidean distance between the cells
• Interaction coefficient based on distance
• Interaction coefficient based on distance and
  population
• Global Interaction Coefficient
• Infection factor is calculated as the ratio of
  interaction coefficients between the cells and
  the global interaction coefficient

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Distance dependence of disease spread and
Neighborhood restriction

• Assumption Individual is more likely to make
  contact with some one closer than some one
  farther.
• Spread of disease is slower when the assumption
  is considered.
• Spread of disease is distance dependent

Comparison of spread of disease considering and
not considering distance dependence for contacts

• Traditional cellular automata with a 8
  neighborhood restricts the spread of infection
  due to neighborhood saturation.
• The graph compares the infection in traditional
  CA and the global neighborhood model.

Comparison of spread of disease in restricted
neighborhood of 8 and global neighborhood
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Distance dependence of disease spread

• Assumption Individual is more likely to make
  contact with some one closer than some one
  farther.
• Spread of disease is slower when the assumption
  is considered.
• Spread of disease is distance dependent

Comparison of spread of disease considering and
not considering distance dependence for contacts
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Neighborhood Restriction

• Traditional cellular automata with a 8
  neighborhood restricts the spread of infection
  due to neighborhood saturation.
• The graph compares the infection in traditional
  CA and the global neighborhood model.

Comparison of spread of disease in restricted
neighborhood of 8 and global neighborhood
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Experiments Behavior change

• Assumption Sick or infected individuals are
  less likely to make contacts during the
  infectious period.
• Model adjusts the contact rate of individuals
  based on the number of days infected.

• The graph compares the infection spread for the
  model with the behavior change and without
  behavior changes. Spread of disease is distance
  dependent.
• Infection spread is slower if behavioral change
  is considered.

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Experiments

• Spread of a disease for different contact rates.
• Disease parameters
• Contact rates of 8, 15, 25
• Infectivity of 0.005
• As the contact rate decrease spread of disease
  is slower and prolonged.

Spread of a disease for different contact rates.

• Spread of different diseases on a specific
  population with fixed contact rate.
• Disease parameters such as latency, infectious
  period, infectivity and recovery different with
  respect to a disease.
• The graph illustrates different diseases spread
  differently in a given population set.

Spread of different diseases in a given population
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Composition Model
Assumption Sub-regions (or cells) with a
larger proportion of a certain demographic may
display increased or decrease prevalence of a
certain disease as compared to a sub-region with
a larger proportion of a different demographic
Composition model reflects the spread of the
infection in each sub-region.

• Cell interaction is controlled by age proportions
  and population densities.

Observed Cumulative Epidemic caused by Temporally
and Spatially Distributed Local Outbreaks
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Summary

• Designing tools for investigating local disease
  clusters through simulation.
• Whats New?
• Utilizing GIS and EPI information for modeling
• Combining different simulation paradigms
• Designing of a Global Stochastic CA
• The goal Contribute to establish computational
  epidemiology as a new research domain.