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

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.
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, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[1]

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, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217

[2]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[3]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[4]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[5]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

[6]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021045

[7]

Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066

[8]

Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2203-2215. doi: 10.3934/jimo.2020065

[9]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[10]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[11]

Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021035

[12]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[13]

Yu Jin, Xiao-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1991-2010. doi: 10.3934/dcdsb.2020362

[14]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[15]

Kun Hu, Yuanshi Wang. Dynamics of consumer-resource systems with consumer's dispersal between patches. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021077

[16]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092

[17]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[18]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044

[19]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[20]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (584)
  • HTML views (0)
  • Cited by (147)

[Back to Top]