Dynamical bifurcation of a sewage treatment model with general higher-order perturbation

Epidemic threshold of a COVID-19 model with gaussian white noise and semi-Markov switching

“…There are two general approaches to studying the effect of stochastic perturbations on the dynamics of the model: one is to study stochastic differential equations with Gaussian white noise [15] , [16] , [17] , [18] , [19] , and the other potential technique is to study stochastic differential equations driven by continuous-time Markov chains [20] , [21] . However, since the dwell times of Markov chains in each environmental regime are assumed to follow an exponential distribution, these models driven by continuous-time Markov chains have some limitations in practical applications.…”

The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis

“…With the fast evolution of applied sciences, shorthand description and verbal discussion are no longer adequate to depict and clarify real-world problems in physics, chemistry, biology, economics, etc [1][2][3][4][5][6][7]. To this end, the mathematical formulation has become requisite to comprehend our environment and probe phenomena at scales of time and space that are difficult to attain empirically [8].…”

“…Generally, the amalgamation of time delays with fractional derivatives has several advantages and provides excellent insight into the infection dynamics [48][49][50][51]. In accordance with our knowledge, system (3) has not yet been studied due to its complexity. In this research we try to analyze it by investigating some of its long-term characteristics.…”

“…The remaining of this article is structured as follows: in section 2, we provide certain fundamental and preliminary results concerning fractional calculus. In section 3, we establish the well-posedness of our enhanced model (3) in the sense that it has a unique positive and bounded solution, and we present the steady points •  ,   1 ,   2 of system (3) and their related critical values •  and  1 . These two thresholds make it possible to classify the dynamic behavior of our system.…”

“…In addition, since m > 0, Assume that  > and the condition(19) holds. If Q > ,5 , then   2 is locally asymptotically stable for all  t 0 Hopf bifurcation analysis Now, by considering the time delay τ 2 as a bifurcation critical sill, we probe the Hopf bifurcation phenomenon for model(3). If Θ 0 < 0, i.e., has at least one positive real root.…”