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. 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. 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. 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. 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. 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. 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. 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. 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.

[10]

S. M. Blower and H. B. Gershengorn, A tale of two futures: HIV and antiretroviral therapy in San Francisco,, Science, 287 (2000), 650. 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. 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. doi: 10.1216/rmjm/1181072101.

[13]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley, (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. 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. 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. doi: 10.1017/S0004972700029488.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (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. 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. 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. 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. 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.

[25]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species,, Journal of Differential Equations, 223 (2006), 400. 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. 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. doi: 10.1080/17513750802638712.

[28]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

[10]

S. M. Blower and H. B. Gershengorn, A tale of two futures: HIV and antiretroviral therapy in San Francisco,, Science, 287 (2000), 650. 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. 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. doi: 10.1216/rmjm/1181072101.

[13]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley, (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. 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. 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. doi: 10.1017/S0004972700029488.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (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. 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. 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. 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. 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.

[25]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species,, Journal of Differential Equations, 223 (2006), 400. 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. 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. doi: 10.1080/17513750802638712.

[28]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (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. 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. 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. 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. 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. doi: 10.1080/17513758.2011.614697.

[1]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489

[2]

Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057

[3]

Azmy S. Ackleh, Youssef M. Dib, S. R.-J. Jang. Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 683-698. doi: 10.3934/dcdsb.2007.7.683

[4]

Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545

[5]

Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1479-1494. doi: 10.3934/mbe.2018068

[6]

Hao Wang, Katherine Dunning, James J. Elser, Yang Kuang. Daphnia species invasion, competitive exclusion, and chaotic coexistence. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 481-493. doi: 10.3934/dcdsb.2009.12.481

[7]

Jing Ge, Ling Lin, Lai Zhang. A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2763-2776. doi: 10.3934/dcdsb.2017134

[8]

Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989

[9]

Chengxia Lei, Fujun Li, Jiang Liu. Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173

[10]

Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114

[11]

M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141

[12]

Ming Yang, Chulin Li. Valuing investment project in competitive environment. Conference Publications, 2003, 2003 (Special) : 945-950. doi: 10.3934/proc.2003.2003.945

[13]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[14]

Christopher DuBois, Jesse Farnham, Eric Aaron, Ami Radunskaya. A multiple time-scale computational model of a tumor and its micro environment. Mathematical Biosciences & Engineering, 2013, 10 (1) : 121-150. doi: 10.3934/mbe.2013.10.121

[15]

Maryam Esmaeili, Samane Sedehzade. Designing a hub location and pricing network in a competitive environment. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2018172

[16]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[17]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[18]

Mahin Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 185-202. doi: 10.3934/dcdsb.2006.6.185

[19]

Natalia Kudryashova. Strategic games in a competitive market: Feedback from the users' environment. Conference Publications, 2015, 2015 (special) : 745-753. doi: 10.3934/proc.2015.0745

[20]

Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]