Bulletin of Mathematical Biology

, Volume 80, Issue 12, pp 3106–3126 | Cite as

Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One

  • S. SugayaEmail author
  • V. M. Kenkre
Original Article


The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction–diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016–3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction–diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.


Smoluchowski Interacting random walks Diffusion Infection transmission Epidemics Hantavirus 


  1. Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover Publications, TorontozbMATHGoogle Scholar
  2. Abramson G, Giuggioli L, Kenkre VM, Dragoo J, Parmenter R, Parmenter C, Yates TL (2006) Diffusion and home range parameters of rodents: peromyscus maniculatus in New Mexico. Ecol Complex 3:64CrossRefGoogle Scholar
  3. Abramson G, Kenkre VM (2002) Spatiotemporal patterns in the Hantavirus infection. Phys Rev E 66:011912CrossRefGoogle Scholar
  4. Abramson G, Wio HS (1995) Time behavior for diffusion in the presence of static imperfect traps. Chaos Solitons Fractals 6:1CrossRefGoogle Scholar
  5. Aguirre MA, Abramson G, Bishop AR, Kenkre VM (2002) Simulations in the mathematical modeling of the spread of the Hantavirus. Phys Rev E 66:041908CrossRefGoogle Scholar
  6. Anderson RM, May RM (1991) Infectious diseases of humans. Oxford University Press Inc., New YorkGoogle Scholar
  7. Berg HC (1983) Random walks in biology. Princeton University Press, PrincetonGoogle Scholar
  8. Brauer F, Castillo-Chávez C (2001) Mathematical models in population biology and epidemiology. Springer, New YorkCrossRefGoogle Scholar
  9. Cantrell RS, Cosner C (2003) Spatial ecology via reaction–diffusion equations. Wiley, HobokenzbMATHGoogle Scholar
  10. Carslaw HS, Jaeger CJ (1959) Condition of heats in solids. Oxford University Press, OxfordGoogle Scholar
  11. Dickmann U, Law R, Metz JAJ (2000) The geometry of ecological interactions. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  12. Giuggioli L, Abramson G, Kenkre VM, Parmenter C, Yates TL (2006) Theory of home range estimation from displacement measurements of animal populations. J Theor Biol 240:126MathSciNetCrossRefGoogle Scholar
  13. Giuggioli L, Abramson G, Kenkre VM, Suzán E, Marcé G, Yates TL (2005) Diffusion and home range parameters from rodent population measurements in Panama Bull. Math Biol 67(5):1135CrossRefGoogle Scholar
  14. Hemenger RP, Lakatos-Lindenberg K, Pearlstein RM (1974) Impurity quenching of molecular excitons. III. Partially coherent excitons in linear chains. J Chem Phys 60:3271CrossRefGoogle Scholar
  15. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599MathSciNetCrossRefGoogle Scholar
  16. Kenkre VM (1980) Theory of exciton annihilation in molecular crystals. Phys Rev B 22:2089CrossRefGoogle Scholar
  17. Kenkre VM (1982) Exciton dynamics in molecular crystals and aggregates. In: Springer tracts in modern physics, vol 94, Springer, Berlin (and references therein)Google Scholar
  18. Kenkre VM (1982) A theoretical approach to exciton trapping in systems with arbitrary trap concentration. Chem Phys Lett 93:260CrossRefGoogle Scholar
  19. Kenkre VM, Parris PE (1983) Exciton trapping and sensitized luminescence: a generalized theory for all trap concentrations. Phys Rev B 27:3221CrossRefGoogle Scholar
  20. Kenkre VM (2003) Memory formalism, nonlinear techniques, and kinetic equation approaches. In: Proceedings of the PASI on modern challenges in statistical mechanics: patterns, noise, and the interplay of nonlinearity and complexity, AIPGoogle Scholar
  21. Kenkre VM, Giuggioli L, Abramson G, Camelo-Neto G (2007) Theory of hantavirus infection spread incorporating localized adult and itinerant juvenile mice. Eur Phys J B 55:461CrossRefGoogle Scholar
  22. Kenkre VM (2004) Results from variants of the fisher equation in the study of epidemics and bacteria. Physica A 342:242CrossRefGoogle Scholar
  23. Kenkre VM (2005) Statistical mechanical considerations in the theory of the spread of the Hantavirus. Physica A 356:121CrossRefGoogle Scholar
  24. Kenkre VM, Parmenter RR, Peixoto ID, Sadasiv L (2005) A theoretical framework for the analysis of the west nile virus epidemic. Math Comput Model 42:313MathSciNetCrossRefGoogle Scholar
  25. Kenkre VM, Sugaya S (2014) Theory of the transmission of infection in the spread of epidemics: interacting random walkers with and without confinement. Bull Math Biol 76:3016–3027MathSciNetCrossRefGoogle Scholar
  26. McKane AJ, Newman TJ (2004) Stochastic models in population biology and their deterministic analogs. Phys Rev E 70:041902MathSciNetCrossRefGoogle Scholar
  27. MacInnis D, Abramson G, Kenkre VM (2008) University of New Mexico preprint; see also D. MacInnis, Ph. D. thesis, unpublished, University of New MexicoGoogle Scholar
  28. Montroll EW, West BJ (1979) On an enriched collection of stochastic process. Fluctuation phenomena, North-Holland, AmsterdamGoogle Scholar
  29. Nasci RS, Savage HM, White DJ, Miller JR, Cropp BC, Godsey MS, Kerst AJ, Bennet P, Gottfried K, Lanciotti RS (2001) West Nile virus in overwintering Culex mosquitoes, New York City, 2000. Emerg Infect Dis 7:4Google Scholar
  30. Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. Springer, New YorkCrossRefGoogle Scholar
  31. Redner S (2001) A guide to first-passage processes. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  32. Reichl LE (2009) A modern course in statistical physics. WILEY-VCH Verlag, WeinheimzbMATHGoogle Scholar
  33. Risken H (1989) The Fokker–Planck equation: Methods of solution and applications. Springer, BerlinCrossRefGoogle Scholar
  34. Roberts GE, Kaufman H (1966) Table of laplace transforms. W. B. Saunders Company, PhiladelphiazbMATHGoogle Scholar
  35. Redner S, ben-Avraham D (1990) Nearest-neighbor distances of diffusing particles from a single trap. J Phys A: Math Gen 23:L1169CrossRefGoogle Scholar
  36. Szabo A, Lamm G, Weiss GH (1984) Localized partial traps in diffusion processes and random walks. J Stat Phys 34:225CrossRefGoogle Scholar
  37. Spendier K, Kenkre VM (2013) Solutions for some reaction–diffusion scenarios. J Phys Chem B 117:15639CrossRefGoogle Scholar
  38. Spendier K, Sugaya S, Kenkre VM (2013) Reaction–diffusion theory in the presence of an attractive harmonic potential. Phys Rev E 88:062142CrossRefGoogle Scholar
  39. Sugaya S (2016) Ph.D. Thesis, University of New Mexico (unpublished)Google Scholar
  40. Strausbaugh LJ, Martin AA, Gubler DJ (2001) West Nile encephalitis: an emerging disease in the United States. Clin Infect Dis 33:1713–1719CrossRefGoogle Scholar
  41. Wax N (1954) Selected papers on noise and stochastic processes. Dover Publications INC., New YorkzbMATHGoogle Scholar
  42. Yates TL, Mills JN, Parmenter CA, Ksiazek TG, Parmenter RR, Vande Castle JR, Calisher CH, Nichol ST, Abbott KD, Young JC, Morrison ML, Beaty BJ, Dunnum JL, Baker RJ, Salazar-Bravo J, Peters CJ (2002) The ecology and evolutionary history of an emergent disease: hantavirus pulmonary syndrome. Bioscience 52:989CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.MSC01 1100, 1University of New MexicoAlbuquerqueUSA
  2. 2.MSC01 1130, 1University of New MexicoAlbuquerqueUSA

Personalised recommendations