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Fractional nonlinear dynamics and forward bifurcation in a memory-based cholera model.

Zixuan Yang1, Jianwei Shen1

  • 1School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China.

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Fractional-order models reveal how memory effects in cholera transmission influence disease spread and control. Incorporating preventive behaviors and environmental factors, these models enhance epidemic resilience and reduce contamination.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Dynamical Systems

Background:

  • Cholera transmission exhibits complex memory-dependent and nonlinear dynamics.
  • Traditional integer-order models may not fully capture these intricate behaviors.
  • Understanding these dynamics is crucial for effective disease control.

Purpose of the Study:

  • To explore fractional-order epidemic models for cholera transmission.
  • To integrate preventive behaviors and environmental feedback into a fractional model.
  • To analyze the impact of memory effects on epidemic dynamics and control.

Main Methods:

  • Developed a fractional-order susceptible-infected-recovered-individuals adopting preventive measures-bacteria (SIR-IPM-B) model using the Caputo derivative.
  • Analyzed the existence, uniqueness, and boundedness of the model's solutions.
  • Derived the basic reproduction number (R0) and performed stability and bifurcation analyses.
  • Designed a fractional optimal control strategy for cholera intervention.

Main Results:

  • Fractional dynamics, influenced by memory, were shown to transition the system from a disease-free to an endemic state via forward bifurcation.
  • Numerical simulations demonstrated that fractional dynamics suppress infection peaks by extending transient memory effects.
  • The model indicated enhanced epidemic resilience and reduced environmental contamination due to fractional dynamics.
  • The study highlighted the significant impact of fractional-order memory and nonlinear coupling on epidemic thresholds and control effectiveness.

Conclusions:

  • Fractional-order models provide a more comprehensive framework for understanding cholera dynamics due to memory effects.
  • The integration of preventive behaviors and environmental factors in fractional models is effective for disease control.
  • Fractional dynamics offer a promising approach to enhance epidemic resilience and mitigate waterborne disease outbreaks.