Impact of Travel Between Patches for Spatial Spread of Disease

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, ℜ0. Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of ℜ0 on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches.

[1]  Julien Arino,et al.  A multi-species epidemic model with spatial dynamics. , 2005, Mathematical medicine and biology : a journal of the IMA.

[2]  Xiao-Qiang Zhao,et al.  An Epidemic Model with Population Dispersal and Infection Period , 2006, SIAM J. Appl. Math..

[3]  Xiao-Qiang Zhao,et al.  An epidemic model in a patchy environment. , 2004, Mathematical biosciences.

[4]  Alan Hastings,et al.  Can spatial variation alone lead to selection for dispersal , 1983 .

[5]  Lisa Sattenspiel,et al.  Simulating the effect of quarantine on the spread of the 1918–19 flu in Central Canada , 2003, Bulletin of mathematical biology.

[6]  J. Arino,et al.  A multi-city epidemic model , 2003 .

[7]  D. Butler.  Doubts hang over source of bird flu spread , 2006, Nature.

[8]  C. Macken,et al.  Mitigation strategies for pandemic influenza in the United States. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[9]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[10]  D. Cummings,et al.  Strategies for mitigating an influenza pandemic , 2006, Nature.

[11]  G. Rota.  Non-negative matrices in the mathematical sciences: A. Berman and R. J. Plemmons, Academic Press, 1979, 316 pp. , 1983 .

[12]  Wendi Wang,et al.  Threshold of disease transmission in a patch environment , 2003 .

[13]  E O Powell,et al.  Theory of the chemostat. , 1965, Laboratory practice.

[14]  J. Hyman,et al.  The role of spatial mixing in the spread of foot-and-mouth disease. , 2006, Preventive veterinary medicine.

[15]  J. Arino,et al.  Disease Spread in Metapopulations , 2005 .

[16]  J A P Heesterbeek,et al.  The metapopulation dynamics of an infectious disease: tuberculosis in possums. , 2002, Theoretical population biology.

[17]  P. van den Driessche,et al.  A model for disease transmission in a patchy environment , 2005 .

[18]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[19]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[20]  J. R. Scotti,et al.  Available From , 1973 .

[21]  Julien Arino,et al.  Quarantine in a multi-species epidemic model with spatial dynamics. , 2007, Mathematical biosciences.

[22]  S. Ruan.  The effect of global travel on the spread of SARS. , 2005, Mathematical biosciences and engineering : MBE.

[23]  D. Cummings,et al.  Strategies for containing an emerging influenza pandemic in Southeast Asia , 2005, Nature.

[24]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[25]  M. McPeek,et al.  The Evolution of Dispersal in Spatially and Temporally Varying Environments , 1992, The American Naturalist.

[26]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[27]  J. Hyman,et al.  Modeling the Spread of Influenza Among Cities , 2003 .