Stochastic Modeling and Combined Spatial Pattern Analysis of Epidemic Spreading
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
[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.
[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.
@article{"International Journal of Information, Control and Computer Sciences:53704", author = "S. Chadsuthi and W. Triampo and C. Modchang and P. Kanthang and D. Triampo and N. Nuttavut", title = "Stochastic Modeling and Combined Spatial Pattern Analysis of Epidemic Spreading", 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", volume = "5", number = "3", pages = "252-8", }
