### Abstract

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.

Original language | English |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 21 |

Issue number | 1 |

State | Published - May 2008 |

### Keywords

- Basic reproduction number
- Disease-free equilibrium
- Dispersal
- Endemic equilibrium
- Spatial heterogeneity

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## Cite this

*Discrete and Continuous Dynamical Systems*,

*21*(1), 1-20.