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<p>To investigate how time delays and random environmental fluctuations influence HIV transmission, we develop a novel stochastic reaction-diffusion model that incorporates distributed delays, colored noise, and the Beddington-DeAngelis functional response. For this stochastic partial differential equation system, we establish the existence of a unique invariant measure by using the generalized coupling method and prove the exponential convergence of the corresponding Markov semigroup toward this measure, thereby characterizing the long-term statistical behavior of the system in a probabilistic sense. Building on this foundation, drug therapy is introduced as a control mechanism, leading to an optimal time-delay control problem aimed at minimizing treatment cost while suppressing infected cells and viral loads. Since exact optimal controls may be unattainable in realistic scenarios, we further construct a near-optimal time-delay control framework and derive sufficient and necessary conditions for the existence of such controls by using the Hamiltonian function. Finally, numerical simulations further illustrate the effects of time delays and random noise on geometric ergodicity and control, which provides theoretical basis and numerical support for understanding the HIV transmission mechanism and designing effective intervention strategies.</p>

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Geometric ergodicity and near-optimal time-delay control of a stochastic HIV model with colored noise and diffusion

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Dan Shi,
  • Qimin Zhang

摘要

To investigate how time delays and random environmental fluctuations influence HIV transmission, we develop a novel stochastic reaction-diffusion model that incorporates distributed delays, colored noise, and the Beddington-DeAngelis functional response. For this stochastic partial differential equation system, we establish the existence of a unique invariant measure by using the generalized coupling method and prove the exponential convergence of the corresponding Markov semigroup toward this measure, thereby characterizing the long-term statistical behavior of the system in a probabilistic sense. Building on this foundation, drug therapy is introduced as a control mechanism, leading to an optimal time-delay control problem aimed at minimizing treatment cost while suppressing infected cells and viral loads. Since exact optimal controls may be unattainable in realistic scenarios, we further construct a near-optimal time-delay control framework and derive sufficient and necessary conditions for the existence of such controls by using the Hamiltonian function. Finally, numerical simulations further illustrate the effects of time delays and random noise on geometric ergodicity and control, which provides theoretical basis and numerical support for understanding the HIV transmission mechanism and designing effective intervention strategies.