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Competitive exclusion and coexistence in a two-strain pathogen model with diffusion
1. | Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010, United States |
2. | Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, United States |
References:
[1] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion principle for pathogens in an epidemic model with variable population size, Journal of Mathematical Biology, 47 (2003), 153-168.
doi: 10.1007/s00285-003-0207-9. |
[2] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175-188.
doi: 10.3934/dcdsb.2005.5.175. |
[3] |
A. S. Ackleh and P. Salceanu, Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality, Journal of Mathematical Biology, 68 (2014), 453-475.
doi: 10.1007/s00285-012-0636-4. |
[4] |
N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, Journal of Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[5] |
L. J. S. Allen, M. Langlais and C. J. Phillips, The dynamics of two viral infections in a single host population with applications to hantavirus, Mathematical Biosciences, 186 (2003), 191-217.
doi: 10.1016/j.mbs.2003.08.002. |
[6] |
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. |
[7] |
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. |
[8] |
V. Andreasen, J. Lin and S. A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, Journal Mathematical Biology, 35 (1997), 825-842.
doi: 10.1007/s002850050079. |
[9] |
V. Andreasen and A. Pugliese, Pathogen coexistence induced by density-dependent host mortality, Journal of Theoretical Biology, 177 (1995), 159-165. |
[10] |
S. M. Blower and H. B. Gershengorn, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654.
doi: 10.1126/science.287.5453.650. |
[11] |
H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, Journal of Mathematical Biology, 27 (1989), 179-190.
doi: 10.1007/BF00276102. |
[12] |
R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain Journal of Mathematics, 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[13] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003.
doi: 10.1002/0470871296. |
[14] |
C. Castillo-Chavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases, SIAM Journal on Applied Mathematics, 56 (1996), 494-508.
doi: 10.1137/S003613999325419X. |
[15] |
C. Castillo-Chavez, W. Huang and J. Li, The effects of females' susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, Journal of Mathematical Biology, 35 (1997), 503-522.
doi: 10.1007/s002850050063. |
[16] |
K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, submitted. |
[17] |
A. Ghoreishi and R. Logan, Positive solutions of a class of biological models in a heterogeneous environment, Bulletin of the Australian Mathematical Society, 44 (1991), 79-94.
doi: 10.1017/S0004972700029488. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. |
[20] |
S. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Transactions of the American Mathematical Society, 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[21] |
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. |
[22] |
V. Hutson, Y. Lou, K. Mischaikow and P. Polacik, Competing species near a degenerate limit, SIAM Journal on Mathematical Analysis, 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[23] |
V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, Journal of Differential Equations, 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[24] |
C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations under homogeneous Neumann boundary conditions, Funkcialaj Ekvacioj, 32 (1989), 191-213. |
[25] |
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426.
doi: 10.1016/j.jde.2005.05.010. |
[26] |
Y. Lou, S. Martínez and P. Polacik, Loops and branches of coexistence states in a Lotka-Volterra competition model, Journal of Differential Equations, 230 (2006), 720-742.
doi: 10.1016/j.jde.2006.04.005. |
[27] |
M. Martcheva, A non-autonomous multi-strain SIS epidemic model, Journal of Biological Dynamics, 3 (2009), 235-251.
doi: 10.1080/17513750802638712. |
[28] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[29] |
R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[30] |
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. |
[31] |
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. |
[32] |
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, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[33] |
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. |
show all references
References:
[1] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion principle for pathogens in an epidemic model with variable population size, Journal of Mathematical Biology, 47 (2003), 153-168.
doi: 10.1007/s00285-003-0207-9. |
[2] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175-188.
doi: 10.3934/dcdsb.2005.5.175. |
[3] |
A. S. Ackleh and P. Salceanu, Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality, Journal of Mathematical Biology, 68 (2014), 453-475.
doi: 10.1007/s00285-012-0636-4. |
[4] |
N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, Journal of Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[5] |
L. J. S. Allen, M. Langlais and C. J. Phillips, The dynamics of two viral infections in a single host population with applications to hantavirus, Mathematical Biosciences, 186 (2003), 191-217.
doi: 10.1016/j.mbs.2003.08.002. |
[6] |
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. |
[7] |
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. |
[8] |
V. Andreasen, J. Lin and S. A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, Journal Mathematical Biology, 35 (1997), 825-842.
doi: 10.1007/s002850050079. |
[9] |
V. Andreasen and A. Pugliese, Pathogen coexistence induced by density-dependent host mortality, Journal of Theoretical Biology, 177 (1995), 159-165. |
[10] |
S. M. Blower and H. B. Gershengorn, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654.
doi: 10.1126/science.287.5453.650. |
[11] |
H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, Journal of Mathematical Biology, 27 (1989), 179-190.
doi: 10.1007/BF00276102. |
[12] |
R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain Journal of Mathematics, 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[13] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003.
doi: 10.1002/0470871296. |
[14] |
C. Castillo-Chavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases, SIAM Journal on Applied Mathematics, 56 (1996), 494-508.
doi: 10.1137/S003613999325419X. |
[15] |
C. Castillo-Chavez, W. Huang and J. Li, The effects of females' susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, Journal of Mathematical Biology, 35 (1997), 503-522.
doi: 10.1007/s002850050063. |
[16] |
K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, submitted. |
[17] |
A. Ghoreishi and R. Logan, Positive solutions of a class of biological models in a heterogeneous environment, Bulletin of the Australian Mathematical Society, 44 (1991), 79-94.
doi: 10.1017/S0004972700029488. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. |
[20] |
S. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Transactions of the American Mathematical Society, 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[21] |
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. |
[22] |
V. Hutson, Y. Lou, K. Mischaikow and P. Polacik, Competing species near a degenerate limit, SIAM Journal on Mathematical Analysis, 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[23] |
V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, Journal of Differential Equations, 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[24] |
C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations under homogeneous Neumann boundary conditions, Funkcialaj Ekvacioj, 32 (1989), 191-213. |
[25] |
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426.
doi: 10.1016/j.jde.2005.05.010. |
[26] |
Y. Lou, S. Martínez and P. Polacik, Loops and branches of coexistence states in a Lotka-Volterra competition model, Journal of Differential Equations, 230 (2006), 720-742.
doi: 10.1016/j.jde.2006.04.005. |
[27] |
M. Martcheva, A non-autonomous multi-strain SIS epidemic model, Journal of Biological Dynamics, 3 (2009), 235-251.
doi: 10.1080/17513750802638712. |
[28] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[29] |
R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[30] |
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. |
[31] |
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. |
[32] |
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, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[33] |
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. |
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