Energy Landscapes and Non-Archimedean Pseudo- Differential Operators as Tools for Studying the Spreading of Infectious Diseases in a Situation of Extreme Social Isolation


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Authors: V. A. AGUILAR-ARTEAGA, I. S. GUTIERREZ AND A. TORRESBLANCA-BADILLO

DOI: 10.46793/KgJMat2406.827AA

Abstract:

In this article, we introduce a new type of pseudo-differential equations naturally connected with non-archimedean pseudo-differential operators and whose symbols are new classes of negative definite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation.



Keywords:

Energy landscapes, pseudo-differential operators, negative definite functions, evolution equations, strong Markov processes, p-adic analysis.



References:

[1]   V. A. Aguilar-Arteaga, M. Cruz-López and S. Estala-Arias, Non-archimedean analysis and a wave-type pseudodifferential equation on finite adéles., J. Pseudo-Differ. Oper. Appl. 11 (2020), 1139–1181. https://doi.org/10.1007/s11868-020-00343-1

[2]   V. A. Aguilar-Arteaga and S. Estala-Arias, Pseudodifferential operators and Markov processes on adéles, p-Adic Numbers Ultrametric Analysis and Applications 11 (2019), 89–113. https://doi.org/10.1134/S2070046619020018

[3]    S. Albeverio, A. Khrennikov and P. Kloeden, Memory retrieval as a p-adic dynamical system, Biosystems 49 (1999), 105–115. https://doi.org/10.1016/s0303-2647(98)00035-5

[4]   S. Albeverio, A. Y. Khrennikov and V. M. Shelkovich, Theory of p-Adic Distributions: Linear and Nonlinear Models, London Mathematical Society Lecture Note Series 370, Cambridge University Press, Cambridge, 2010. https://doi.org/10.1017/CBO9781139107167

[5]   A. V. Antoniouk, K. Oleschko, A. N. Kochubei and A. Y. Khrennikov, A stochastic p-adic model of the capillary flow in porous random medium, Physica A 505 (2018), 763–777. https://doi.org/10.1016/j.physa.2018.03.049

[6]   V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev and V. A. Osipov, p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes, J. Phys. A 35 (2002), 177–189. https://doi.org/10.1088/0305-4470/35/2/301

[7]   V. A. Avetisov, A. K. Bikulov and V. A. Osipov, p-Adic description of characteristic relaxation in complex systems, J. Phys. A 36 (2003), 4239–4246. https://doi.org/10.1088/0305-4470/36/15/301

[8]   O. M. Becker and M. Karplus, The topology of multidimensional protein energy surfaces: theory and application to peptide structure and kinetics, J. Chem. Phys. 106 (1997), 1495–1517. https://doi.org/10.1063/1.473299

[9]   L. F. Chacón-Cortes, I. G. García, A. Torresblanca-Badillo and A. Vargas, Finite time blow-up for a p-adic nonlocal semilinear ultradiffusion equation, J. Math. Anal. Appl. 494(2) (2021), Article ID 124599. https://doi.org/10.1016/j.jmaa.2020.124599

[10]   B. Christian and F. Gunnar, Potential Theory on Locally Compact Abelian Groups, Springer-Verlag, New York, Heidelberg, 1975. https://doi.org/10.1007/978-3-642-66128-0

[11]   H. Frauenfelder, B. H. McMahon and P. W. Fenimore, Myoglobin: the hydrogen atom of biology and paradigm of complexity, PNAS 100 (2003), 8615–8617. https://doi.org/10.1073/pnas.1633688100

[12]   A. V. Fuensanta, J. M. Mazón, J. D. Rossi and T. M. J. Julián, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs 165, Amer. Math. Soc., Real Sociedad Matemática Española, Providence, Madrid, 2010. http://dx.doi.org/10.1090/surv/165

[13]   I. G. García and A. Torresblanca-Badillo, Some classes of non-archimedean pseudo-differential operators related to Bessel potentials, J. Pseudo-Differ. Oper. Appl. 11 (2020), 1111–1137. https://doi.org/10.1007/s11868-020-00333-3

[14]   I. G. García and A. Torresblanca-Badillo, Strong Markov processes and negative definite functions associated with non-archimedean elliptic pseudo-differential operators, J. Pseudo-Differ. Oper. Appl. 11 (2020), 345–362. https://doi.org/10.1007/s11868-019-00293-3

[15]   I. G. García and A. Torresblanca-Badillo, Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators, J. Pseudo-Differ. Oper. Appl. 12 (2021), Article ID 12. https://doi.org/10.1007/s11868-021-00381-3

[16]   C. A. Gómez and J. D. Rossi, A nonlocal diffusion problem that approximates the heat equation with neumann boundary conditions, Journal of King Saud University 32 (2020), 17–20. https://doi.org/10.1016/j.jksus.2017.08.008

[17]    N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups, Imperial College Press, London, 2001. https://doi.org/10.1142/p245

[18]   T. Kazuaki, Boundary Value Problems and Markov Processes, Second Edition, Lecture Notes in Mathematics, Springer-Verlag, 2009. https://doi.org/10.1007/978-3-642-01677-6

[19]   A. Y. Khrennikov, Human subconscious as a p-adic dynamical system, Journal of Theoretical Biology 193 (1998), 179–196. https://doi.org/10.1006/jtbi.1997.0604

[20]   A. Y. Khrennikov, Toward an adequate mathematical model of mental space: Conscious/unconscious dynamics on m-adic trees, Biosystems 90 (2007), 656–675. https://doi.org/10.1016/j.biosystems.2007.02.004

[21]   A. Y. Khrennikov and A. N. Kochubei, p-Adic analogue of the porous medium equation, J. Fourier Anal. Appl. 24 (2018), 1401–1424. https://doi.org/10.1007/s00041-017-9556-4.

[22]   A. Y. Khrennikov, S. V. Kozyrev and W. A. Zúñiga-Galindo, Ultrametric pseudodifferential equations and applications, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2018. https://doi.org/10.1017/9781316986707

[23]   A. Y. Khrennikov and K. Oleschko, An ultrametric random walk model for disease spread taking into account social clustering of the population, Entropy 22(9) (202), Article ID 931. https://doi.org/10.3390/e22090931

[24]   A. Y. Khrennikov, K. Oleschko and M. C. López, Modeling fluid’s dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks., Entropy 18(7) (2016), Article ID 249. https://doi.org/10.3390/e18070249

[25]   S. V. Kozyrev, p-Adic pseudodifferential operators and p-adic wavelets, Theoret. and Math. Phys. 138 (2004), 322–332. https://doi.org/10.1023/B:TAMP.0000018449.72502.6f

[26]   S. V. Kozyrev, Dynamics on rugged landscapes of energy and ultrametric diffusion, p-Adic Numbers, Ultrametric Analysis, and Applications 2 (2010), 122–132. https://doi.org/10.1134/S2070046610020044

[27]   W. A. Zúñiga-Galindo, Pseudodifferential Equations over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174, Springer International Publishing, 2016. https://doi.org/10.1007/978-3-319-46738-2

[28]   K. Oleschko and A. Y. Khrennikov, Applications of p-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions, Theoret. and Math. Phys. 19 (2017), 154–163. https://doi.org/10.4213/tmf9142

[29]   E. Pourhadi, A. Y. Khrennikov, R. Saadati, K. Oleschko and M. C. López, Solvability of the p-adic analogue of navier-stokes equation via the wavelet theory, Entropy 21(11) (2019), Article ID 1129. https://doi.org/10.3390/e21111129

[30]   J. J. Rodríguez-Vega and W. A. Zúñiga-Galindo, Taibleson operators, p-adic parabolic equations and ultrametric diffusion, Pacific J. Math. 237 (2008), 327–347. https://doi.org/10.2140/PJM.2008.237.327

[31]   F. H. Stillinger and T. A. Weber, Hidden structure in liquids, Phys. Rev. A 25 (1982), 978–989. https://doi.org/10.1103/PhysRevA.25.978

[32]   F. H. Stillinger and T. A. Weber, Packing structures and transitions in liquids and solids, Science 225 (1984), 983–989. https://doi.org/10.1126/science.225.4666.983

[33]   M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, New Jersey, 1975. http://www.jstor.org/stable/j.ctt130hkh3

[34]   A. Torresblanca-Badillo, Non-archimedean generalized bessel potentials and their applications, J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2020.124874

[35]   A. Torresblanca-Badillo, Non-archimedean pseudo-differential operators on Sobolev spaces related to negative definite functions, J. Pseudo-Differ. Oper. Appl. 12 (2021), Article ID 7. https://doi.org/10.1007/s11868-021-00385-z

[36]   A. Torresblanca-Badillo and W. A. Zúñiga-Galindo, Non-archimedean pseudodifferential operators and feller semigroups, p-Adic Numbers, Ultrametric Analysis and Applications 10 (2018), 57–73. https://doi.org/10.1134/S2070046618010041

[37]   A. Torresblanca-Badillo and W. A. Zúñiga-Galindo, Ultrametric diffusion, exponential landscapes, and the first passage time problem, Acta Appl. Math. 157 (2018), 93–116. https://doi.org/10.1007/s10440-018-0165-2

[38]   V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics, World Scientific, 1994. https://doi.org/10.1142/1581