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Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment
1. | Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275 |
2. | College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035 |
References:
[1] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
doi: 10.1137/060672522. |
[2] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
[3] |
M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Mathematical Biosciences, 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[4] |
R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Wiley Online Library, 1992. |
[5] |
S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems, Nonlinear Analysis: Real World Applications, 3 (2002), 453-464.
doi: 10.1016/S1468-1218(01)00025-6. |
[6] |
C. Bain, Applied mathematical ecology, Journal of Epidemiology and Community Health, 44 (1990), p254.
doi: 10.1136/jech.44.3.254-b. |
[7] |
F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. |
[8] |
E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, 33 (1995), 250-260.
doi: 10.1007/BF00169563. |
[9] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, 2003.
doi: 10.1002/0470871296. |
[10] |
V. Capasso, Mathematical Structures of Epidemic Systems, Springer, 1993.
doi: 10.1007/978-3-540-70514-7. |
[11] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.
doi: 10.1016/0025-5564(78)90006-8. |
[12] |
C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution, Mathematical Biosciences, 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[13] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[14] |
D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites, The American Naturalist, 156 (2000), 459-477. |
[15] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM Journal on Mathematical Analysis, 33 (2001), 570-588.
doi: 10.1137/S0036141000371757. |
[16] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
[17] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Mathematical Biosciences, 206 (2007), 233-248.
doi: 10.1016/j.mbs.2005.07.005. |
[18] |
B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition), Springer, 2012.
doi: 10.1007/978-1-4614-1686-9. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag Berlin, 1981. |
[20] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[21] |
H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, In Applied Mathematical Ecology, Springer, (1989), 193-211.
doi: 10.1007/978-3-642-61317-3_8. |
[22] |
W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
[23] |
T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, Journal of Mathematical Biology, 46 (2003), 17-30.
doi: 10.1007/s00285-002-0165-7. |
[24] |
T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, 2 (2005), 743-751.
doi: 10.3934/mbe.2005.2.743. |
[25] |
Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89-130. |
[26] |
Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918.
doi: 10.3934/mbe.2014.11.877. |
[27] |
W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology, 53 (1991), 33-55. |
[28] |
K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[29] |
A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Mathematical Medicine and Biology, 22 (2005), 113-128.
doi: 10.1093/imammb/dqi001. |
[30] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation, 210 (2009), 141-150.
doi: 10.1016/j.amc.2008.12.085. |
[31] |
W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[32] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, 2010. |
[33] |
J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models, Mathematical Biosciences and Engineering, 3 (2006), 161-172.
doi: 10.3934/mbe.2006.3.161. |
[34] |
Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, 2009.
doi: 10.1142/7223. |
[35] |
C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. |
[36] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[37] |
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I, Journal of Differential Equations, 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
[38] |
R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods and Applications, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[39] |
R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[40] |
R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
doi: 10.1088/0951-7715/25/5/1451. |
[41] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[42] |
M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126.
doi: 10.1016/j.jtbi.2011.12.015. |
[43] |
R. Ross, The Prevention of Malaria(2nd ed.), Murray, London, 1911. |
[44] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97-122. |
[45] |
S. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[46] |
J. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
[47] |
H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, Journal of Mathematical Biology, 17 (1983), 163-177.
doi: 10.1007/BF00305757. |
[48] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[49] |
N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439.
doi: 10.1080/17513758.2011.614697. |
[50] |
V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of life reviews, 6 (2009), 267-310.
doi: 10.1016/j.plrev.2009.10.002. |
[51] |
W. D. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[52] |
W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[53] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
show all references
References:
[1] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
doi: 10.1137/060672522. |
[2] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
[3] |
M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Mathematical Biosciences, 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[4] |
R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Wiley Online Library, 1992. |
[5] |
S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems, Nonlinear Analysis: Real World Applications, 3 (2002), 453-464.
doi: 10.1016/S1468-1218(01)00025-6. |
[6] |
C. Bain, Applied mathematical ecology, Journal of Epidemiology and Community Health, 44 (1990), p254.
doi: 10.1136/jech.44.3.254-b. |
[7] |
F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. |
[8] |
E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, 33 (1995), 250-260.
doi: 10.1007/BF00169563. |
[9] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, 2003.
doi: 10.1002/0470871296. |
[10] |
V. Capasso, Mathematical Structures of Epidemic Systems, Springer, 1993.
doi: 10.1007/978-3-540-70514-7. |
[11] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.
doi: 10.1016/0025-5564(78)90006-8. |
[12] |
C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution, Mathematical Biosciences, 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[13] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[14] |
D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites, The American Naturalist, 156 (2000), 459-477. |
[15] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM Journal on Mathematical Analysis, 33 (2001), 570-588.
doi: 10.1137/S0036141000371757. |
[16] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
[17] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Mathematical Biosciences, 206 (2007), 233-248.
doi: 10.1016/j.mbs.2005.07.005. |
[18] |
B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition), Springer, 2012.
doi: 10.1007/978-1-4614-1686-9. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag Berlin, 1981. |
[20] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[21] |
H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, In Applied Mathematical Ecology, Springer, (1989), 193-211.
doi: 10.1007/978-3-642-61317-3_8. |
[22] |
W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
[23] |
T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, Journal of Mathematical Biology, 46 (2003), 17-30.
doi: 10.1007/s00285-002-0165-7. |
[24] |
T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, 2 (2005), 743-751.
doi: 10.3934/mbe.2005.2.743. |
[25] |
Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89-130. |
[26] |
Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918.
doi: 10.3934/mbe.2014.11.877. |
[27] |
W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology, 53 (1991), 33-55. |
[28] |
K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[29] |
A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Mathematical Medicine and Biology, 22 (2005), 113-128.
doi: 10.1093/imammb/dqi001. |
[30] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation, 210 (2009), 141-150.
doi: 10.1016/j.amc.2008.12.085. |
[31] |
W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[32] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, 2010. |
[33] |
J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models, Mathematical Biosciences and Engineering, 3 (2006), 161-172.
doi: 10.3934/mbe.2006.3.161. |
[34] |
Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, 2009.
doi: 10.1142/7223. |
[35] |
C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. |
[36] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[37] |
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I, Journal of Differential Equations, 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
[38] |
R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods and Applications, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[39] |
R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[40] |
R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
doi: 10.1088/0951-7715/25/5/1451. |
[41] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[42] |
M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126.
doi: 10.1016/j.jtbi.2011.12.015. |
[43] |
R. Ross, The Prevention of Malaria(2nd ed.), Murray, London, 1911. |
[44] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97-122. |
[45] |
S. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[46] |
J. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
[47] |
H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, Journal of Mathematical Biology, 17 (1983), 163-177.
doi: 10.1007/BF00305757. |
[48] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[49] |
N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439.
doi: 10.1080/17513758.2011.614697. |
[50] |
V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of life reviews, 6 (2009), 267-310.
doi: 10.1016/j.plrev.2009.10.002. |
[51] |
W. D. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[52] |
W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[53] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
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