Sangeeta Venkatachalam, Armin R. Mikler Computational Epidemiology Research Laboratory

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

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

3. 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. Mathematical Epidemiology Susceptibles Infectives Removals (SIR) model

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

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

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

7. 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. Stochastic Cellular Automata A Global Model 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.

8. 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

9. 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

10. 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. Experiments – Behavior change

11. 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 Distance dependence of disease spread

12. 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

13. 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 • 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.

14. 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. Composition Model -Experiment

15. 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. 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 asany situation which may lead to a successful disease transmission. d

16. 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-- Infectivity

17. 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. Experiment-- Immunity Region Immunized

18. 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. Composition Model – Modeling Distance Dependence

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

24. Index case

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