Abstract—We present analysis of spatial patterns of generic disease spread simulated by a stochastic long-range correlation SIR model, where individuals can be infected at long distance in a power law distribution. We integrated various tools, namely perimeter, circularity, fractal dimension, and aggregation index to characterize and investigate spatial pattern formations. Our primary goal was to understand for a given model of interest which tool has an advantage over the other and to what extent. We found that perimeter and circularity give information only for a case of strong correlation— while the fractal dimension and aggregation index exhibit the growth rule of pattern formation, depending on the degree of the correlation exponent (β). The aggregation index method used as an alternative method to describe the degree of pathogenic ratio (α). This study may provide a useful approach to characterize and analyze the pattern formation of epidemic spreading Keywords—spatial pattern epidemics, aggregation index, fractal dimension, stochastic, long-rang epidemics I. INTRODUCTION ATTERN formation phenomena, occurring via the aggregation process or clustering of particles, has been the subject of increased interest [1]. Spatial pattern analysis plays an important role in many fields of research, ranging from the microscopic to macroscopic scale, including bacteria colonies [2], epidemiology [3], forests, and ecology [4]. Spatial technology enables epidemiologists to create detailed maps and employ spatial cluster statistics to garner insights about patterns of disease [3]. There has been significant development in creating predictive models to better understand the pattern formation of epidemics; see reviews [5- 7]. The mathematical epidemiological model usually takes the form of a deterministic model, which consists of a system of ordinary differential equation (ODE) models describing changes in the number of susceptible, infected, and recovered individuals in a given population [8]. Typically, the ODE S. Chadsuthi, C. Modchang and N. Nuttavut are with Biophysics Group, Department of Physics, Faculty of Science, Mahidol University, Rama VI, Ratchathewi District, Bangkok Thailand 10400 W. Triampo is with Biophysics Group, Department of Physics, Faculty of Science, Mahidol University, Rama VI, Ratchathewi District, Bangkok Thailand 10400, ThEP Center, CHE, 328 Si Ayutthaya Road, Bangkok, Thailand 10400, Institute for Innovative Learning, Mahidol University, 999, Phuttamonthon 4 Road, Salaya, Nakorn Pathom 73170, Thailand (corresponding author to provide phone: +662-441-9816 ext. 1131; fax +662- 441-9322; e-mail wtriampo@gmail.com, scwtr@mahidol.ac.th) P. Kanthang is with Department of Physics, Facutly of Science and Technology, Rajamangala University of Technology, Phra Nakhon, Bangkok Thailand 10800 D. Triampo is with Department of Chemistry, Center of Excellence for Innovation in Chemistry, Faculty of Science, Mahidol University, Rama VI, Ratchathewi District, Bangkok, Thailand 10400 deterministic model neglects spatial correlation by assuming that the system is spatially homogeneous; this is also termed the mean field approximation approach [9]. In addition, the ODE deterministic model can exactly determine the transient evolution of the system once the initial condition is given. However, for epidemical phenomena where there is a large degree of spatial organization or pattern formation of spreading, the ODE deterministic model may be unrealistic. It is important to account for spatial variation and to study the landscape as well as the pattern formations of epidemic phenomena. With regard to spatial pattern driven forces such as diffusion, deterministic partial differential equation (PDE) models are well known conventional tools for analyzing dynamical aspects [10], but such models do not take into account the noises or stochastic fluctuations associated with spatiotemporal dynamics. Consequently, stochastic partial differential equation (SPDE) models, such as the Langevin equation, are needed [11]. However, it usually is difficult to obtain analytical solutions to compare with experimental data. Therefore, computer simulations can be of great help in investigating spatial patterns that are typically due to the effect of noise. One of the most efficient is the Monte Carlo based spatial cellular automata model. In our current work, we were interested in studying the spatial pattern formation of epidemic spreading using the Monte Carlo simulation approach. Even though the model of interest is generic; we believe that our findings could be useful in understanding how diseases spread, and how to prevent epidemic spreading. To characterize the spatial pattern of epidemics, there are many parameters that can be used to analyze the patterns—for example, area, perimeter [12], circularity [12, 13], fractal dimension [14], and aggregation index [15]. These parameters provide different information at least to some extent. Area and perimeter measurements are very familiar and straight forward to understand. More interesting is circularity, which is a numerical quantity representing the degree to which a shape is compact. It is calculated from the perimeter, which is defined as a path surrounding an area. This measurement of a region is a common technique used to characterize pattern patchiness and compactness [12]. Topologically, the circularity value should be invariant under similarity transformations of the shape, such as scaling, rotation and translation. However this measurement can be used to describe the interaction between individuals. Fractal dimension (D f ) can be defined as a measure of structural complexity [1]. It has attracted considerable attention from many mathematicians because its fractional quality is in sharp contrast to the integer dimensions (zero, one, two, and three) of Euclidean geometry. It is an index used to indicate how completely a fractal appears to fill spaces, as S. Chadsuthi, W. Triampo, C. Modchang, P. Kanthang , D. Triampo, N. Nuttavut Stochastic Modeling and Combined Spatial Pattern Analysis of Epidemic Spreading P World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:5, No:3, 2011 292 International Scholarly and Scientific Research & Innovation 5(3) 2011 ISNI:0000000091950263 Open Science Index, Computer and Information Engineering Vol:5, No:3, 2011 publications.waset.org/4594/pdf