Bulletin of Mathematical Biology

, Volume 77, Issue 11, pp 2004–2034 | Cite as

SIS and SIR Epidemic Models Under Virtual Dispersal

  • Derdei Bichara
  • Yun Kang
  • Carlos Castillo-Chavez
  • Richard Horan
  • Charles Perrings
Original Article

Abstract

We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number \({\mathcal {R}}_0 \) is computed as a function of a patch residence-time matrix \({\mathbb {P}}\). Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: There is a unique globally stable endemic equilibrium when \({\mathcal {R}}_0>1 \), while the disease-free equilibrium is globally stable when \({\mathcal {R}}_0\le 1 \). Our further analysis indicates that the dispersal behavior described by the residence-time matrix \({\mathbb {P}}\) has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence-time matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single-outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease-prevalence-driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state-dependent \({\mathbb {P}}\) on disease dynamics.

Keywords

Epidemiology SIS–SIR models Dispersal Residence times Global stability Adaptive behavior Final size relationship 

Mathematics Subject Classfication

Primary 34D23 92D25 60K35 

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Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Derdei Bichara
    • 1
  • Yun Kang
    • 2
  • Carlos Castillo-Chavez
    • 1
  • Richard Horan
    • 3
  • Charles Perrings
    • 4
  1. 1.SAL Mathematical, Computational and Modeling Science CenterArizona State UniversityTempeUSA
  2. 2.Sciences and Mathematics Faculty, College of Letters and SciencesArizona State UniversityMesaUSA
  3. 3.Department of Agricultural, Food and Resource EconomicsMichigan State UniversityEast LansingUSA
  4. 4.School of Life SciencesArizona State UniversityTempeUSA

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