# American Institute of Mathematical Sciences

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

## 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

Received  December 2014 Revised  August 2015 Published  October 2015

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.
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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] V. Andreasen and A. Pugliese, Pathogen coexistence induced by density-dependent host mortality, Journal of Theoretical Biology, 177 (1995), 159-165. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003. doi: 10.1002/0470871296.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model,, submitted., ().   Google Scholar [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.  Google Scholar [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations under homogeneous Neumann boundary conditions, Funkcialaj Ekvacioj, 32 (1989), 191-213.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] M. Martcheva, A non-autonomous multi-strain SIS epidemic model, Journal of Biological Dynamics, 3 (2009), 235-251. doi: 10.1080/17513750802638712.  Google Scholar [28] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] V. Andreasen and A. Pugliese, Pathogen coexistence induced by density-dependent host mortality, Journal of Theoretical Biology, 177 (1995), 159-165. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003. doi: 10.1002/0470871296.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model,, submitted., ().   Google Scholar [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.  Google Scholar [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations under homogeneous Neumann boundary conditions, Funkcialaj Ekvacioj, 32 (1989), 191-213.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] M. Martcheva, A non-autonomous multi-strain SIS epidemic model, Journal of Biological Dynamics, 3 (2009), 235-251. doi: 10.1080/17513750802638712.  Google Scholar [28] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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