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May  2017, 22(3): 1167-1187. doi: 10.3934/dcdsb.2017057

## Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion

 1 Department of Mathematics, University of Louisiana at Lafayette, 105 E. University Circle, Lafayette, LA 70503, USA 2 Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA 3 Department of Mathematics, University of Florida, 1400 Stadium Rd, Gainesville, FL 32611, USA

* Corresponding author: yixiang.wu@vanderbilt.edu. Current address: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

** The authors are partially supported by NSF grant DMS-1220342 and DMS-1515661/DMS-1515442

Received  August 2015 Revised  November 2015 Published  December 2016

This paper investigates a two strain SIS model with diffusion, spatially heterogeneous coefficients of the reaction part and distinct diffusion rates of the separate epidemiological classes. First, it is shown that the model has bounded classical solutions. Next, it is established that the model with spatially homogeneous coefficients leads to competitive exclusion and no coexistence is possible in this case. Furthermore, it is proved that if the invasion number of strain $j$ is larger than one, then the equilibrium of strain $i$ is unstable; if, on the other hand, the invasion number of strain $j$ is smaller than one, then the equilibrium of strain $i$ is neutrally stable. In the case when all diffusion rates are equal, global results on competitive exclusion and coexistence of the strains are established. Finally, evolution of dispersal scenario is considered and it is shown that the equilibrium of the strain with the larger diffusion rate is unstable. Simulations suggest that in this case the equilibrium of the strain with the smaller diffusion rate is stable.

Citation: Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057
##### References:
 [1] A. S. Ackleh, K. Deng and Y. Wu, Competitive exclusion and coexistence in a two-strain pathogen model with diffusion, Math. Biosci. Eng., 13 (2016), 1-18.  doi: 10.3934/mbe.2016.13.1. [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, Series A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [3] H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190.  doi: 10.1007/BF00276102. [4] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations Wiley, Chichester, West Sussex, UK, 2003. [5] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete and Continuous Dynamical Systems, 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701. [6] Dockery, Hutson, Mischaikow and Polácik, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120. [7] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Communications in Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269. [8] Hsu, Smith and 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. [9] Hutson, Mischaikow and Polácik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106. [10] K. I. Kim, Z. Lin and L. Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Analysis: Real World Applications, 11 (2010), 313-322.  doi: 10.1016/j.nonrwa.2008.11.015. [11] K. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous completion-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481. [12] J. Lopez-Gomez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems Nonlinear Analysis: Theory, Methods & Applications 31 (1998), 687-699. [13] K. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336. [14] Lou and Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004. [15] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2. [16] M. Martcheva, B. M. Bolker and R. D. Holt, Vaccine-induced pathogen strain replacement: What are the mechanisms?, J. Royal Sco. Interface, 5 (2008), 3-13.  doi: 10.1098/rsif.2007.0236. [17] M. Mimura, Coexistence in competition-diffusion systems, differential equations models in biology, epidemiology and ecology, Lecture Notes in Biomathematics, 92 (1991), 235-246.  doi: 10.1007/978-3-642-45692-3_17. [18] Z. Qiu, Q. Kong, X. Li and M. Martcheva, The vector-host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [19] 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. [20] P. Waltman, Coexistence in chemostat-like models, Rocky Mountain J. Math., 20 (1990), 777-807.  doi: 10.1216/rmjm/1181073042.

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##### References:
 [1] A. S. Ackleh, K. Deng and Y. Wu, Competitive exclusion and coexistence in a two-strain pathogen model with diffusion, Math. Biosci. Eng., 13 (2016), 1-18.  doi: 10.3934/mbe.2016.13.1. [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, Series A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [3] H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190.  doi: 10.1007/BF00276102. [4] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations Wiley, Chichester, West Sussex, UK, 2003. [5] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete and Continuous Dynamical Systems, 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701. [6] Dockery, Hutson, Mischaikow and Polácik, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120. [7] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Communications in Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269. [8] Hsu, Smith and 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. [9] Hutson, Mischaikow and Polácik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106. [10] K. I. Kim, Z. Lin and L. Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Analysis: Real World Applications, 11 (2010), 313-322.  doi: 10.1016/j.nonrwa.2008.11.015. [11] K. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous completion-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481. [12] J. Lopez-Gomez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems Nonlinear Analysis: Theory, Methods & Applications 31 (1998), 687-699. [13] K. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336. [14] Lou and Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004. [15] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2. [16] M. Martcheva, B. M. Bolker and R. D. Holt, Vaccine-induced pathogen strain replacement: What are the mechanisms?, J. Royal Sco. Interface, 5 (2008), 3-13.  doi: 10.1098/rsif.2007.0236. [17] M. Mimura, Coexistence in competition-diffusion systems, differential equations models in biology, epidemiology and ecology, Lecture Notes in Biomathematics, 92 (1991), 235-246.  doi: 10.1007/978-3-642-45692-3_17. [18] Z. Qiu, Q. Kong, X. Li and M. Martcheva, The vector-host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [19] 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. [20] P. Waltman, Coexistence in chemostat-like models, Rocky Mountain J. Math., 20 (1990), 777-807.  doi: 10.1216/rmjm/1181073042.
Experiment 1 Total population in the homogeneous coefficient case
Experiment 1 The two pathogen strains at final time $t=150$
Experiment 2 Total population with large diffusion rate
Experiment 3 Total population with small diffusion rate
Experiment 4 Susceptible and infected individuals at time $t=100$. The top figure is showing the final state of susceptible individuals. The bottom figure is showing the coexistence of strains $I_1$ and $I_2$ at time $t=100$
Total population in the case $d_1>d_2$
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