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

Yixiang Wu; Necibe Tuncer; Maia Martcheva

doi:10.3934/dcdsb.2017057

Article Contents

2017, Volume 22, Issue 3: 1167-1187. Doi: 10.3934/dcdsb.2017057

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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
* 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: July 31, 2015

Revised: October 31, 2015

Published: September 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35K57, 35B35;Secondary:92D25.

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Figure 1. Experiment 1 Total population in the homogeneous coefficient case

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Figure 2. Experiment 1 The two pathogen strains at final time $t=150$

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Figure 3. Experiment 2 Total population with large diffusion rate

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Figure 4. Experiment 3 Total population with small diffusion rate

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Figure 5. 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 $t=100$

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Figure 6. Total population in the case $d_1>d_2$

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