Bulletin of Mathematical Biology

, Volume 76, Issue 12, pp 3016–3027 | Cite as

Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement

  • V. M. KenkreEmail author
  • S. Sugaya
Original Article


A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in \(n\) dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in \(2n\)-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated, and it is shown how to use them in complex systems consisting of dense populations.


Smoluchowski Interacting random walks Diffusion   Infection transmission Epidemics 



It is a pleasure for us to acknowledge helpful conversations with Professor Kathrin Spendier of the University of Colorado and Matthew Chase in our own research group. This research was supported by the Consortium of the Americas for Interdisciplinary Science and by the Program in Interdisciplinary Biological and Biomedical Sciences of the University of New Mexico.


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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.Consortium of the Americas for Interdisciplinary Science, Department of Physics and AstronomyUniversity of New MexicoAlbuquerqueUSA

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