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Analysis of a heterogeneous SEIRS patch model with asymmetric mobility kernel.

Shuangshuang Yin1, Jianhong Wu2, Pengfei Song1,2

  • 1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, China.

Mathematical Biosciences and Engineering : MBE
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This study introduces a spatial SEIRS model, revealing that the mobility of exposed and recovered individuals significantly impacts disease dynamics, unlike previous models focusing only on susceptible and symptomatic groups.

Keywords:
SEIRS epidemic modelasymmetric mobility kernelbasic reproduction numberspatial heterogeneity

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Spatial Dynamics

Background:

  • Spatial epidemic models are crucial for understanding disease spread.
  • Previous models often simplified mobility patterns.
  • Heterogeneity in population movement influences disease transmission.

Purpose of the Study:

  • To develop a spatial heterogeneous SEIRS patch model with an asymmetric mobility kernel.
  • To analyze the global dynamics of disease spread in relation to the basic reproduction ratio ($ \mathcal{R}_{0} $).
  • To investigate the impact of mobility for all disease compartments (Susceptible, Exposed, Infected, Recovered).

Main Methods:

  • Establishment of a spatial heterogeneous SEIRS patch model.
  • Definition and analysis of the basic reproduction ratio ($ \mathcal{R}_{0} $).
  • Investigation of global dynamics and stability of equilibria.
  • Analysis of monotonicity of $ \mathcal{R}_{0} $ with respect to diffusion coefficients.
  • Exploration of long-term behavior as susceptible diffusion approaches zero.

Main Results:

  • The basic reproduction ratio ($ \mathcal{R}_{0} $) determines global dynamics.
  • Monotonicity of $ \mathcal{R}_{0} $ with diffusion coefficients is case-dependent.
  • The mobility of exposed and recovered individuals is critical for disease dynamics.
  • Long-term behavior of the endemic equilibrium was characterized under specific conditions.

Conclusions:

  • The mobility of all individuals, including exposed and recovered, is essential for accurate disease modeling.
  • Spatial heterogeneity and asymmetric mobility kernels significantly affect epidemic spread.
  • The study provides a more comprehensive understanding of disease dynamics in spatially structured populations.