Stochastic Modeling and Combined Spatial Pattern Analysis of Epidemic Spreading

Authors: S. Chadsuthi, W. Triampo, C. Modchang, P. Kanthang, D. Triampo, N. Nuttavut

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, fractaldimension, stochastic, long-rang epidemics

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1059483

References:

[1] A.-L. Barabási, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995.
[2] J. Northcott, M.C. Andersen, G.W. Roemer, E.L. Fredrickson, M. Demers, J. Truett, P.L. Ford, Spatial Analysis of Effects of Mowing and Burning on Colony Expansion in Reintroduced Black-Tailed Prairie Dog, Restor. Ecol. 16, 495-502, 2008.
[3] J. Meliker, G. Jacquez, P. Goovaerts, G. Copeland, M. Yassine, Spatial cluster analysis of early stage breast cancer: a method for public health practice using cancer registry data, Cancer Cause Control 20, 1061- 1069, 2009.
[4] R. Schlicht, Y. Iwasa, Spatial pattern analysis in forest dynamics: deviation from power law and direction of regeneration waves. Ecol. Res. 22, 197-203, 2007.
[5] G. Sun, Z. Jin, Q. Liu, L. Li, Pattern formation in a spatial SI model with non-linear incidence rates, J. Stat. Mech. 11, 11011, 2007.
[6] Q. Liu, Z. Jin, Formation of spatial patterns in an epidemic model with constant removal rate of the infectives, J. Stat. Mech. 5002, 2007.
[7] D. Eisinger, H. Thulke, Spatial pattern formation facilitates eradication of infectious diseases, J. Appl. Ecol. 45, 415-423, 2008.
[8] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl. 56, 3118-3129, 2008.
[9] F. Santos, J. Rodrigues, J. Pacheco, Epidemic spreading and cooperation dynamics on homogeneous small-world networks, Phys. Rev. E 72, 56128, 2005.
[10] U. Kitron, Landscape ecology and epidemiology of vector-borne diseases: tools for spatial analysis, J. Med. Entomol. 35, 435-445, 1998.
[11] L. Cobb, Stochastic differential equations for the social sciences, Mathematical frontiers of the social and policy sciences, 37-68, 1981.
[12] J. Ma, Q.-M. Zhao, Circular pattern extraction in wafer fault mining, International Conference on Wavelet Analysis and Pattern Recognition, Hong Kong, 123-127, 2008.
[13] J. Žunić, K. Hirota, Measuring Shape Circularity, in: J. Ruiz-Shulcloper, W.G. Kropatsch (Eds.), Progress in Pattern Recognition, Image Analysis and Applications, Springer Berlin, Heidelberg, 94-101, 2008.
[14] U. Purintrapiban, V. Kachitvichyanukul, Detecting patterns in process data with fractal dimension, Comput. Ind. Eng. 45, 653-667, 2003.
[15] H.S. He, B.E. DeZonia, D.J. Mladenoff, An aggregation index (AI) to quantify spatial patterns of landscapes, Landscape Ecol. 15, 591-601, 2000.
[16] C.M. Hagerhall, T. Purcell, R. Taylor, Fractal dimension of landscape silhouette outlines as a predictor of landscape preference, J. Environ. Psychol. 24, 247-255, 2004.
[17] O. Biham, O. Malcai, D.A. Lidar, D. Avnir, Pattern formation and a clustering transition in power-law sequential adsorption, Phys. Rev. E 59, R4713, 1999.
[18] Z.-J. Tan, X.-W. Zou, Z.-Z. Jin, Percolation with long-range correlations for epidemic spreading, Phys. Rev. E 62, 8409, 2000.
[19] Z.-J. Tan, C. Long, X.-W. Zou, W. Zhang, Z.-Z. Jin, Epidemic spreading in percolation worlds, Phys Lett. A 300, 317-323, 2002.
[20] J. Adamek, M. Keller, A. Senftleben, H. Hinrichsen, Epidemic spreading with long-range infections and incubation times, J. Stat. Mech-Theory E., 09002, 2005.
[21] J.A. van der Goot, G. Koch, M.C. de Jong, M. van Boven, Quantification of the effect of vaccination on transmission of avian influenza (H7N7) in chickens, P. Natl. Acad. Sci. USA 102, 18141- 18146, 2005.
[22] F. Wang, Z. Ma, Y. Shag, A competition model of HIV with recombination effect. Math. Comput. Model. 38, 1051-1065, 2003.
[23] F. Stagnitti, A model of the effects of nonuniform soil-water distribution on the subsurface migration of bacteria: Implications for land disposal of sewage, Math. Comput. Model. 29, 41-52, 1999.
[24] R.M. Weseloh, Short and Long Range Dispersal in the Gypsy moth (Lepidoptera: Lymantriidae) Fungal Pathogen, Entomophaga maimaiga (Zygomycetes: Entomophthorales), Environ. Entomol. 32, 111-122, 2003.
[25] M.B. Gravenor, N. Stallard, R. Curnow, A.R. McLean, Repeated challenge with prion disease: the risk of infection and impact on incubation period, P. Natl. Acad. Sci. USA 100, 10960-10965, 2003.
[26] J. Rabinovich, N. Schweigmann, V. Yohai, C. Wisnivesky-Colli, Probability of Trypanosoma cruzi transmission by Triatoma infestans (Hemiptera: Reduviidae) to the opossum Didelphis albiventris (Marsupialia: Didelphidae), Am. J. Trop. Med. Hyg. 65, 125-130, 2001.
[27] F. Ginelli, H. Hinrichsen, R. Livi, D. Mukamel, A. Torcini, Contact processes with long range interactions, J. Stat. Mech., 08008, 2006.
[28] J. Kilday, F. Palmieri, M.D. Fox, Classifying mammographic lesions using computerized image analysis, IEEE Trans. Med. Imaging 12, 664- 669, 1993.
[29] D. Hamburger, O. Biham, D. Avnir, Apparent fractality emerging from models of random distributions, Phys. Rev. E 53, 3342-3358, 1996.
[30] E. Bribiesca, Measuring 2-D shape compactness using the contact perimeter, Comput. Math. Appl. 33, 1-9, 1997.
[31] K. Shanmugan, A. Breipohl, Random signals: detection, estimation, and data analysis, John Wiley & Sons Inc, 1988.
[32] J. Lessler, J.H. Kaufman, D.A. Ford, J.V. Douglas, The cost of simplifying air travel when modeling disease spread, PLoS ONE 4, 4403, 2009.
[33] K.D. Reed, J.K. Meece, J.S. Henkel, S.K. Shukla, Birds, migration and emerging zoonoses: west nile virus, lyme disease, influenza A and enteropathogens, Clin Med. Res. 1, 5-12, 2003.
[34] M. Bohm, K.L. Palphramand, G. Newton-Cross, M.R. Hutchings, P.C.L. White, Dynamic interactions among badgers: implications for sociality and disease transmission, J. Anim. Ecol. 77, 735-745, 2008.
[35] M.F.M. Lima, J.A. Tenreiro Machado, M. Crisostomo, Filtering method in backlash phenomena analysis, Math. Comput. Model. 49, 1494-1503, 2009.
[36] D. Stauffer, A. Aharony, Introduction to percolation theory, Taylor & Francis, London, 2003.
[37] P.L. Leath, Cluster size and boundary distribution near percolation threshold, Phys. Rev. B 14, 5046-5055, 1976.
[38] P. Meakin, Fractals, Scaling and Growth Far From Equilibrium, Cambridge University Press, Cambridge, 1998.
[39] B. Gompertz, On the nature of the function expressive of the law of human mortality and on a new mode of determining life contin-gencies, Philos. Trans. R. Soc. Lond. 115, 513-585, 1825.