Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain)

This paper proposes a stochastic model to study the evolution of normal and excess weight population between 24 - 65 years old in the region of Valencia (Spain). An approximate solution process of the random model is obtained by taking advantage of Wiener-Hermite expansion together with a perturbation method (WHEP). The random model takes as starting point a classical deterministic SIS—type epidemiological model in order to improve it in several ways. Firstly, the stochastic model enhances the deterministic one because it considers uncertainty in its formulation, what it is considered more realistic in dealing with a complex problem as obesity is. Secondly, WHEP approach provides valuable information such as average and variance functions of the approximate solution stochastic process to random model. This fact is remarkable because other techniques only provide predictions in some a priori chosen points. As a consequence, we can compute and predict the expectation and the variance of normal and excess weight population in the region of Valencia for any time. This information is of paramount value to both doctors and health authorities to set optimal investment policies and strategies.

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J. Cortés, J. Romero, M. Roselló and R. Villanueva, "Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain)," American Journal of Computational Mathematics, Vol. 2 No. 4, 2012, pp. 274-281. doi: 10.4236/ajcm.2012.24037.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] M. Joelson and A. Ramamonjiarisoa, “Random Fields of Water Surface Waves Using Wiener-Hermite Functional Series Expansions,” Journal of Fluid of Mechanics, Vol. 496, 2003, pp. 313-334. doi:10.1017/S002211200300644X [2] Y. Kayanuma and D. Nelson, “Wiener-Hermite Expansion Formalism for the Stochastic Model of a Driven Quantum System,” Chemical Physics, Vol. 268, No. 1-3, 2001, pp. 177-188. doi:10.1016/S0301-0104(01)00305-6 [3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062 [4] C. Braumann, “Variable Effort Harvesting Models in Random Environments: Generalization to Density-Dependent Noise Intensities,” Mathematical Biosciences, Vol. 177-178, 2002, pp. 229-245. doi:10.1016/S0025-5564(01)00110-9 [5] E. A. Gawad, M. A. El-Tawil and M. A. Nassar, “Nonlinear Oscillatory Systems with Random Excitation,” Modelling, Simulation & Control B, Vol. 23, No. 1, 1989, pp. 55-63. [6] M. A. El-Tawil and G. Mahmoud, “The Solvability of Parametrically Forced Oscillators Using WHEP Technique,” Mechanics Mechanical Engineering, Vol. 3, No. 2, 1999, pp. 181-188. [7] M. A. El-Tawil and N. A. Al-Mulla, “Using Homotopy WHEP Technique for Solving a Stochastic Nonlinear Diffusion Equation,” Mathematical and Computer Modelling, Vol. 51, No. 9-10, 2010, pp. 1277-1284. doi:10.1016/j.mcm.2010.01.013 [8] M. A. El-Tawil and N. A. Al-Mulla, “Solving Nonlinear Diffusion Equations without Stochastic Homogeneity Using Homotopy Perturbation Method,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, No. 5, 2009, pp. 687-698. [9] M. A. El-Tawil and A. S. Al-Johani, “Approximate Solution of a Mixed Nonlinear Stochastic Oscillator,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2236-2259. doi:10.1016/j.camwa.2009.03.057 [10] A. S. Al-Jihany, “Comparisons between WHEP and Homotopy Perturbation Techniques in Solving Stochastic Cubic Oscillatory Problems,” The Open Applied Mathematics Journal, Vol. 4, 2010, pp. 24-30. [11] A. M. Evangelista, A. R. Ortiz, K. R. Rios-Soto and A. Urdapilleta. “USA the Fast Food Nation: Obesity as an Epidemic.” T-7, MS B284, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, 2004. http://math.lanl.gov/SummerPrograms/Reports2004/ev_or_ri_ur.pdf [12] L. Jódar, F. Santonja and G. Gonzalez-Parra, “Modeling Dynamics of Infant Obesity in the Region of Valencia, Spain,” Computers & Mathematics with Applications, Vol. 56, No. 3, 2008, pp. 679-689. doi:10.1016/j.camwa.2008.01.011 [13] N. A. Christakis and J. H. Fowler, “The Spread of Obesity in a Large Social Network over 32 Years,” The New England Journal of Medicine, Vol. 357, No. 4, 2007, pp. 370-379. doi:10.1056/NEJMsa066082 [14] J. D. Murray, “Mathematical Biology,” Springer, Berlin, 2002. [15] G. Gonzalez-Parra, L. Jódar, F. Santonja and R. J. Villanueva, “Age-Structured Model for Childhood Obesity,” Mathematical Population Studies, Vol. 171, No. 1, 2010, pp. 1-17. doi:10.1080/07481180903467218 [16] F. Santonja, R. J. Villanueva, L. Jódar and G. Gonzalez-Parra, “Mathematical Modeling of Social Obesity Epidemic in the Region of Valencia, Spain,” Mathematical and Computer Modelling of Dynamical Systems, Vol. 16, No. 1, 2010, pp. 23-34. doi:10.1080/13873951003590149 [17] Valencian Department of Health, “Health Survey, Year 2000,” 2010. http://www.san.gva.es/val/prof/homeprof.html [18] Valencian Department of Health, “Health Survey, Year 2005, 2010. http://www.san.gva.es/val/prof/homeprof.html [19] J. J. Arrizabalaga, L. Masmiquel, J. Vidal, A. Calaas, M. J. Díaz, P. P. García, S. Monereo, J. Moreiro, B. Moreno, W. Ricart and F. Cordido, “Recomendaciones y Algoritmo de Tratamiento Del Sobrepeso y la Obesidad en Personas Adultas (in Spanish),” Medicina Clínica, Vol. 122, No. 3, 2004, pp. 104-110. doi:10.1157/13056816 [20] E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Springer, Berlin, 1992. [21] A. J. Arenas, G. Gonzalez-Parra and J. A. Morano, “Stochastic Modelling of the Transmission of Respiratory Synctytial Virus (RSV) in the Region of Valencia (Spain),” Byosystems, Vol. 96, No. 3, 2009, pp. 206-212. doi:10.1016/j.biosystems.2009.01.007 [22] E. A. Gawad and M. A. El-Tawil, “General Stochastic Oscillatory Systems,” Applied Mathematical Modelling, Vol. 17, No. 6, 1993, pp. 329-335. doi:10.1016/0307-904X(93)90058-O [23] N. Wiener, “Nonlinear Problems in Random Theory,” MIT Press, New York, 1958. [24] R. H. Cameron and W. T. Martin, “The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals,” Annals of Mathematics, Vol. 48, No. 2, 1947, pp. 385-392. doi:10.2307/1969178 [25] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, “A Methodology for Performing Global Uncertainty and Sensititivity Analysis in Systems Biology,” Journal of Theoretical Biology, Vol. 254, 2008, pp. 178-196.