American Institute of Mathematical Sciences

November  2015, 20(9): 2819-2858. doi: 10.3934/dcdsb.2015.20.2819

Mathematical analysis of population migration and its effects to spread of epidemics

 1 Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China 2 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received  June 2014 Revised  May 2015 Published  September 2015

In this paper we study some mathematical models describing evolution of population density and spread of epidemics in population systems in which spatial movement of individuals depends only on the departure and arrival locations and does not have apparent connection with the population density. We call such models as population migration models and migration epidemics models, respectively. We first apply the theories of positive operators and positive semigroups to make systematic investigation to asymptotic behavior of solutions of the population migration models as time goes to infinity, and next use such results to study asymptotic behavior of solutions of the migration epidemics models as time goes to infinity. Some interesting properties of solutions of these models are obtained.
Citation: Shangbin Cui, Meng Bai. Mathematical analysis of population migration and its effects to spread of epidemics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2819-2858. doi: 10.3934/dcdsb.2015.20.2819
References:
 [1] L. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profile of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309. doi: 10.1137/060672522. [2] L. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Disc. Cont. Dyn. Syst. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [3] W. Arendt, A. Grabosch, G. Greiner and et al, One-parameter Semigroups of Positive Operators (ed. R. Nagel), Springer-Verlag, Berlin, 1986. [4] J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multispecies epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60. doi: 10.1016/j.mbs.2005.09.002. [5] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Stud., 10 (2003), 175-193. doi: 10.1080/08898480306720. [6] J. Arino and P. van den Driessche, Disease spread in metapopulations, Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Inst. Comm. 48 (2006), 1-12. [7] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [8] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003. doi: 10.1002/0470871296. [9] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271. doi: 10.1080/17513750701450227. [10] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes, in Spatial Ecology (eds. R.S. Cantrell, C. Cosner and S. Ruan), Chapman Hall/CRC, Boca Raton, FL, (2009), 213-229. doi: 10.1201/9781420059861.ch11. [11] V. Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7. [12] C. Cosner, J. C. Beier, R. S. Cantrell and et al, The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol., 258 (2009), 550-560. doi: 10.1016/j.jtbi.2009.02.016. [13] C. Cosner, J. Dávila and S. Martinez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyna., 6 (2012), 395-405. doi: 10.1080/17513758.2011.588341. [14] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Diff. Equa., 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [15] J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Letters, 26 (2013), 831-835. doi: 10.1016/j.aml.2013.03.005. [16] J. Coville, J. Dávila and S. Martinez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equa., 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [17] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb., 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [18] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [19] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Disc. Cont. Dyna. Syst. Ser. B, 4 (2004), 893-910. doi: 10.3934/dcdsb.2004.4.893. [20] S. A. Gourley, R. Liu and J. Wu, Spatiotemporal patterns of disease spread: Interaction of physiological structure, spatial movements, disease progression and human intervention, in Structured Population Models in Biology and Epidemiology (Eds. P. Magal and S. Ruan), Lecture Notes in Math., Springer, Berlin, 1936 (2008), 165-208. doi: 10.1007/978-3-540-78273-5_4. [21] I. Gudelj and K. White, Spatial heterogeneity, social structure and disease dynamics of animal populations, Theor. Popul. Biol., 66 (2004), 139-149. doi: 10.1016/j.tpb.2004.04.003. [22] I. Gudelj, K. White and N. F. Britton, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004), 91-108. doi: 10.1016/S0092-8240(03)00075-2. [23] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189. [24] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [25] Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364. doi: 10.1016/j.jmaa.2005.01.034. [26] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. [27] A. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042. [28] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 13 (1926), 98-130. [29] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Chapter 10, 3rd ed., Springer-Verlag, New York, 2003. [30] J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3. [31] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. [32] R. Ross, The Prevention of Malaria, 2nd ed., John Murray, London, 1911. [33] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, 97-122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007. [34] M. Salmani and P. van den Driessche, A model for disease transmission in a patchy environment, Disc. Cont. Dyn. Sys. Ser. B, 6 (2006), 185-202. [35] H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York, 1974. [36] W. Wang and G. Mulone, Threshold of disease transmission in a patch environment, J. Math. Anal. Appl., 285 (2003), 321-335. doi: 10.1016/S0022-247X(03)00428-1. [37] W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [38] WHO, Global Alert and Response, Summary of probable SARS cases with onset of illness from 1 November 2002 to 31 July 2003. Available from: http://www.who.int/csr/sars/country/table 2004_04_21/en/. [39] Wikipedia, Timeline of the SARS outbreak,, Available from: , ().

show all references

References:
 [1] L. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profile of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309. doi: 10.1137/060672522. [2] L. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Disc. Cont. Dyn. Syst. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [3] W. Arendt, A. Grabosch, G. Greiner and et al, One-parameter Semigroups of Positive Operators (ed. R. Nagel), Springer-Verlag, Berlin, 1986. [4] J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multispecies epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60. doi: 10.1016/j.mbs.2005.09.002. [5] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Stud., 10 (2003), 175-193. doi: 10.1080/08898480306720. [6] J. Arino and P. van den Driessche, Disease spread in metapopulations, Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Inst. Comm. 48 (2006), 1-12. [7] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [8] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003. doi: 10.1002/0470871296. [9] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271. doi: 10.1080/17513750701450227. [10] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes, in Spatial Ecology (eds. R.S. Cantrell, C. Cosner and S. Ruan), Chapman Hall/CRC, Boca Raton, FL, (2009), 213-229. doi: 10.1201/9781420059861.ch11. [11] V. Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7. [12] C. Cosner, J. C. Beier, R. S. Cantrell and et al, The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol., 258 (2009), 550-560. doi: 10.1016/j.jtbi.2009.02.016. [13] C. Cosner, J. Dávila and S. Martinez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyna., 6 (2012), 395-405. doi: 10.1080/17513758.2011.588341. [14] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Diff. Equa., 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [15] J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Letters, 26 (2013), 831-835. doi: 10.1016/j.aml.2013.03.005. [16] J. Coville, J. Dávila and S. Martinez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equa., 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [17] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb., 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [18] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [19] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Disc. Cont. Dyna. Syst. Ser. B, 4 (2004), 893-910. doi: 10.3934/dcdsb.2004.4.893. [20] S. A. Gourley, R. Liu and J. Wu, Spatiotemporal patterns of disease spread: Interaction of physiological structure, spatial movements, disease progression and human intervention, in Structured Population Models in Biology and Epidemiology (Eds. P. Magal and S. Ruan), Lecture Notes in Math., Springer, Berlin, 1936 (2008), 165-208. doi: 10.1007/978-3-540-78273-5_4. [21] I. Gudelj and K. White, Spatial heterogeneity, social structure and disease dynamics of animal populations, Theor. Popul. Biol., 66 (2004), 139-149. doi: 10.1016/j.tpb.2004.04.003. [22] I. Gudelj, K. White and N. F. Britton, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004), 91-108. doi: 10.1016/S0092-8240(03)00075-2. [23] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189. [24] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [25] Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364. doi: 10.1016/j.jmaa.2005.01.034. [26] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. [27] A. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042. [28] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 13 (1926), 98-130. [29] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Chapter 10, 3rd ed., Springer-Verlag, New York, 2003. [30] J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3. [31] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. [32] R. Ross, The Prevention of Malaria, 2nd ed., John Murray, London, 1911. [33] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, 97-122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007. [34] M. Salmani and P. van den Driessche, A model for disease transmission in a patchy environment, Disc. Cont. Dyn. Sys. Ser. B, 6 (2006), 185-202. [35] H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York, 1974. [36] W. Wang and G. Mulone, Threshold of disease transmission in a patch environment, J. Math. Anal. Appl., 285 (2003), 321-335. doi: 10.1016/S0022-247X(03)00428-1. [37] W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [38] WHO, Global Alert and Response, Summary of probable SARS cases with onset of illness from 1 November 2002 to 31 July 2003. Available from: http://www.who.int/csr/sars/country/table 2004_04_21/en/. [39] Wikipedia, Timeline of the SARS outbreak,, Available from: , ().
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