American Institute of Mathematical Sciences

Advanced
January  2008, 21(1): 1-20. doi: 10.3934/dcds.2008.21.1

Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model

Linda J. S. Allen 1, , B. M. Bolker 2, , Yuan Lou 3, and  A. L. Nevai 4,

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States
2. 

Department of Zoology, University of Florida, Gainesville, FL 32611-8525, United States
3. 

Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210
4. 

Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, United States

Received  January 2007 Revised  October 2007 Published  February 2008

To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} < 1$, the unique disease-free equilibrium is globally asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.
Keywords: basic reproduction number, disease-free equilibrium, dispersal, endemic equilibrium., Spatial heterogeneity.
Mathematics Subject Classification: Primary: 92D30, 92D40, 91D25; Secondary: 34C60, 37N25, 92D5.
Citation: Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[1]

Roger M. Nisbet, Kurt E. Anderson, Edward McCauley, Mark A. Lewis. Response of equilibrium states to spatial environmental heterogeneity in advective systems. Mathematical Biosciences & Engineering, 2007, 4 (1) : 1-13. doi: 10.3934/mbe.2007.4.1
[2]

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
[3]

Yuan-Hang Su, Wan-Tong Li, Fei-Ying Yang. Effects of nonlocal dispersal and spatial heterogeneity on total biomass. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4929-4936. doi: 10.3934/dcdsb.2019038
[4]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501
[5]

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
[6]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020217
[7]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[8]

L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183
[9]

Wendi Wang. Population dispersal and disease spread. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 797-804. doi: 10.3934/dcdsb.2004.4.797
[10]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[11]

Alain Chenciner, Jacques Féjoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 421-438. doi: 10.3934/dcdsb.2008.10.421
[12]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020002
[13]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35
[14]

Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002
[15]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[16]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[17]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[18]

Alain Chenciner. The angular momentum of a relative equilibrium. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1033-1047. doi: 10.3934/dcds.2013.33.1033
[19]

Robert Stephen Cantrell, Chris Cosner, Yuan Lou. Evolution of dispersal and the ideal free distribution. Mathematical Biosciences & Engineering, 2010, 7 (1) : 17-36. doi: 10.3934/mbe.2010.7.17
[20]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (323)
  • HTML views (0)
  • Cited by (73)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences