Towards Computational Epidemiology Using Stochastic Cellular Automata in Modeling Spread of Diseases Sangeeta Venkatachalam, Armin R. Mikler Computational Epidemiology Research Laboratory University of North Texas Email: {venkatac, mikler}@cs.unt.edu This research is in part supported by the National Science Foundation award: NSF-0350200

Overview Mathematical Epidemiology Cellular Automata and Epidemiology Stochastic Cellular Automata - A Global Model Composition Model Experiments Summary

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.

Susceptibles Infectives Removals (SIR) model Homogeneous mixing of people Every individual makes same contacts No demographics considered Geographical distances not considered

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

Parameters considered Latent period Infectious period Contact Infectivity Population Index case Multiple index cases Location of index case Illustrates time-line for infection (influenza)

Stochastic Cellular Automata A Global Model ΥC 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. Definition of a Fuzzy Set Neighborhood of cell Ci,j is global SCA Gi,j := {(Ck,l, ΥC i ,j, C k ,l) |for all Ck,l Є C, 0 ≤ Υ Ci,j, Ck,l ≤ 1} C is a set of all cells in the CA.

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

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

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. Infection spread is slower if behavioral change is considered.

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

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

Composition Model -Experiment The population distribution over the region is non-uniform. Contacts made between cells depends on the population of the cell. Assumption : Regions with high population make more contacts than regions with low population. Simulation parameters: Disease Simulated : Influenza like disease Incubation period : 3 days Infectious period: 3 days Recovery period: 5 days Infectivity : 0.020 Contact rate/person : 11

Composition Model -Experiment Population distribution over the north Denton region. Infected Population distribution over the north Denton region. Total Population infected at the end of simulation: 48000 Total Population of 110000 distributed over a grid size of 50 * 100.

Contact Rate Contact rate defines the number of contacts an individual is involved in a day. May vary depending on the age or occupation of the individual. Contact is considered as any situation which may lead to a successful disease transmission. The graph illustrates the epidemic curves for the same disease parameters with varying contact rates. Can be thought of as for different demographics such as age groups and occupation. Evidently incidence is lower for lower rates of contact. d

Experiment-- Infectivity The probability of a contact resulting in a successful disease transmission depends on the infectivity/virulence parameter Experiment was conducted to analyze the prevalence of influenza for varied levels of infectivity. Incidence is lower for lower levels of infectivity. Epidemic curves for varied levels of infectivity

Experiment-- Immunity The probability of a contact with an infectious person resulting in a successful disease transmission depends on the immunity of the individual. Experiment was conducted considering that people residing in a particular region were immune to the particular virus as means of either vaccination or previous infection. The results show lower level of prevalence of disease in that region compared to other regions. Region Immunized

Composition Model – Modeling Distance Dependence Dichotomy introduced between local and global interactions. Global interactions are between any two cells in the grid Local interactions are within a Manhattan block distance of given distance k. Incidence of disease prevalence decreases with higher proportions of local mixing. Disease prevails over a longer period of time with higher proportions of local mixing. Epidemic curves for the different rates of global and local mixing.

Spatial Spread of Influenza simulated over Northern Denton County with local contacts

Index case

Spatial Spread of Influenza simulated over Northern Denton County with local and global contacts

Summary Designing tools for investigating local disease clusters through simulation. What’s 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.