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Pattern selection mechanism from the equilibrium point and limit cycle.

Qianqian Zheng1, Jianwei Shen2, Vikas Pandey3

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This study explores how infectious disease outbreaks form patterns using a mathematical SIR model. It reveals that diffusion-driven instability modes can lead to stable spot patterns, crucial for epidemic control strategies.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Dynamical Systems

Background:

  • Infectious disease outbreaks often show periodic behavior, representable as limit cycles.
  • The role of periodic behavior in diffusion-induced infection clustering within SIR models is understudied.
  • Turing instability is a key mechanism for pattern formation in biological systems.

Purpose of the Study:

  • To investigate the emergence of Turing instability from stable equilibrium and limit cycles in a spatiotemporal SIR model.
  • To illustrate the dynamical and biological mechanisms underlying infectious disease pattern formation.
  • To identify how different instability modes influence pattern selection and epidemic control.

Main Methods:

  • Utilized a spatiotemporal diffusion-driven SIR model.
  • Identified Hopf bifurcation to confirm stable limit cycles using the First Lyapunov coefficient.
  • Analyzed the competition between different instability modes to understand pattern emergence.

Main Results:

  • Demonstrated the emergence of Turing instability from both stable equilibrium and limit cycles.
  • Confirmed the existence of stable limit cycles through Hopf bifurcation analysis.
  • Observed that the competition between instability modes leads to various patterns, with spot patterns emerging as stable formations.
  • Showcased the significant impact of susceptible, infected, and recovered individuals on pattern types.

Conclusions:

  • Instability modes are critical in selecting pattern formations, directly correlating with the number of observed spot patterns.
  • The study elucidates the dynamical and biological mechanisms driving spot pattern formation in infectious diseases.
  • Findings provide a basis for developing effective epidemic prevention strategies by understanding pattern dynamics.