A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations

Abstract

Many turbidostat models are affected by environmental noise due to various complicated and uncertain factors, and Ornstein-Uhlenbeck process is a more effective and precise way. We formulate a stochastic turbidostat system incorporating Ornstein-Uhlenbeck process in this paper, develop dynamical behavior for the stochastic model, which include the existence and uniqueness of globally positive equilibrium, su cient conditions of the extinction, the existence of a unique stationary distribution and an expression of density function of quasi-stationary distribution around the positive solution of the deterministic model. The results indicate that the weaker volatility intensity can ensure the existence and uniqueness of stationary distribution, and the stronger reversion speed can lead to the extinction of microorganism. The validity of analytical results is veri ed through numerical simulation, which assess the in uence of the reversion speed and the volatility intensity on the long-term behavior of microorganism.

Dynamics and optimal therapy of a stochastic HTLV‐1 model incorporating Ornstein–Uhlenbeck process

“…This phenomenon can be characterized by an Ornstein-Uhlenbeck (OU) process. The topic has also lately drawn quite a bit of researchers [20][21][22][23]. For example, Mu et al [20] incorporated the mean-reverting OU process into a turbidostat system and obtained the threshold parameter with respect to extinction, ergodic stationary distribution (SD), and density function.…”

“…The topic has also lately drawn quite a bit of researchers [20][21][22][23]. For example, Mu et al [20] incorporated the mean-reverting OU process into a turbidostat system and obtained the threshold parameter with respect to extinction, ergodic stationary distribution (SD), and density function. Liu [22] formulated a HLIV model with OU process and established sufficient conditions for the existence of a SD and the extinction of infected cells.…”

Dynamics and density function of a stochastic differential infectivity epidemic model with Ornstein–Uhlenbeck process

“…In the current literature, there are two common approaches to incorporate random effects into deterministic models, including Gaussian linear white noise [13][14][15][16][17] and mean-reverting Ornstein-Uhlenbeck process (OU process). [18][19][20][21][22][23][24] For instance, Chang et al 15 proposed a stochastic SIR model with two different diseases cross-infection and immunization and analyzed its dynamical behavior. Lan et al 16 developed a stochastic SIRS epidemic model, which integrates the impact of available resources for public health.…”

“…Combined with the actual transmission dynamics of the epidemic and the papers, 10–12 the transmission rate $$$ {\beta}_i $$$ should be assumed to be a significant variable involved in the stochasticity. In the current literature, there are two common approaches to incorporate random effects into deterministic models, including Gaussian linear white noise 13–17 and mean‐reverting Ornstein–Uhlenbeck process (OU process) 18–24 . For instance, Chang et al 15 proposed a stochastic SIR model with two different diseases cross‐infection and immunization and analyzed its dynamical behavior.…”