Competitive exclusion and coexistence in a two-strain pathogen model with diffusion

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2016, 13(1): 1-18. doi: 10.3934/mbe.2016.13.1

Competitive exclusion and coexistence in a two-strain pathogen model with diffusion

Azmy S. Ackleh ^1, , Keng Deng ^1, and Yixiang Wu ^2,

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

Received December 2014 Revised August 2015 Published October 2015

Abstract
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We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number $R_0$ and show that when the model parameters are constant (spatially homogeneous), if $R_0 >1$ then one strain will outcompete the other strain and drive it to extinction, but if $R_0 \le 1$ then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition $R_0 >1$: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.

Keywords: coexistence, homogeneous environment, heterogeneous environment., competitive exclusion, Multiple-strain pathogen model, basic reproduction number.

Mathematics Subject Classification: Primary: 92D30, 91D25; Secondary: 35K57, 37N25, 35B4.

Citation: Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 1-18. doi: 10.3934/mbe.2016.13.1

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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