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Analysis of Stochastic SIRC Model with Cross Immunity Based on Ornstein-Uhlenbeck Process.

Zhiming Ni1, Daqing Jiang1, Zhongwei Cao2

  • 1College of Science, China University of Petroleum (East China), Qingdao, 266580 People's Republic of China.

Qualitative Theory of Dynamical Systems
|May 1, 2023
PubMed
Summary

This study analyzes a stochastic SIRC model with an Ornstein-Uhlenbeck process, proving the existence of unique solutions and stationary distributions. It establishes conditions for disease extinction, validated by numerical simulations.

Keywords:
Cross immunityOrnstein–Uhlenbeck processStationary distributionStochastic SIRC epidemic model

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Area of Science:

  • Epidemiology
  • Stochastic Processes
  • Mathematical Biology

Background:

  • Deterministic models like the SIRC model are foundational in epidemiology.
  • Stochastic processes introduce randomness, offering a more realistic representation of disease dynamics.
  • The Ornstein-Uhlenbeck process is a key tool for modeling random fluctuations.

Purpose of the Study:

  • To analyze a stochastic SIRC model incorporating the Ornstein-Uhlenbeck process.
  • To establish the existence and uniqueness of global solutions for the stochastic SIRC model.
  • To investigate the conditions for disease extinction and the properties of equilibrium points.

Main Methods:

  • Existence and uniqueness proofs for global solutions.
  • Lyapunov function construction to demonstrate ergodic stationary distributions.
  • Definition of a quasi-endemic equilibrium point.
  • Derivation of the probability density function for the linearized system.
  • Numerical simulations for validation.

Main Results:

  • The existence and uniqueness of a global solution for the stochastic SIRC model are proven.
  • Ergodic stationary distributions for the stochastic SIRC system are demonstrated.
  • A quasi-endemic equilibrium is defined, and its properties analyzed.
  • The probability density function near the equilibrium is obtained.
  • Sufficient conditions for disease extinction are established and proven.

Conclusions:

  • The stochastic SIRC model with an Ornstein-Uhlenbeck process exhibits well-defined solutions and stable distributions.
  • The study provides a theoretical framework for understanding disease dynamics under randomness.
  • Conditions for disease extinction are mathematically derived and numerically verified.