Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

Abstract

In the one-dimensional reaction-diffusion domain of this study, semi-analytical solutions are used for a delayed viral infection system with logistic growth. Through an ordinary differential equations system, the Galerkin technique is believed to estimate the prevailing partial differential equations. In addition, Hopf bifurcation maps are constructed. The effect of diffusion coefficient stricture and delay on the model is comprehensively investigated, and the outcomes demonstrate that diffusion and delay can stabilize or destabilize the system. We found that, as the delay parameter values rise, the values of the Hopf bifurcations for growth and the rates of viral death are augmented, whereas the rate of production is decreased. For the growth, production, and death rates strictures, there is determination of an asymptotically unstable region and a stable region. Illustrations of the unstable and stable limit cycles, as well as the Hopf bifurcation points, are found to prove the formerly revealed outcomes in the Hopf bifurcation map. The results of the semi-analytical solutions and numerical assessments revealed that the semi-analytical solutions are highly effective.

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

“…The method can be thought of most usefully as a temporalspatial separation. This technique considers a spatial form of the profile concentration, which is described in [5,7,23,24]. Galerkin's method indicates an analytical technique, which utilizes the orthogonality of rudimentary roles set, so to consider the delay ODEs model from the PDE equation.…”

“…Note that the parameter u a = 0.5 is used in all figures and examples in this paper. The fourth-order Runge-Kutta technique [8,24] is utilized to compute the solution of a delay ODEs system, while the finite-difference approximation [5,7] is used for the numerical results of a single PDE equation. The spatial and temporal discretizations solved in this paper will be t = 5×10 -3 and x = 0.01.…”

“…Reaction-diffusion phenomenon with time delays has been incorporated into many fields of biological applications. These applications have explained a number of practical applications in our everyday life by using partial differential equations (PDEs), for instance, in population ecology [15,17,20,30,31], animals [2,4,26], cell [5,19,22,25,33], chemicals [1,3,10], and heat and mass transfer [13,27]. This model can introduce instability, via a Hopf bifurcation, with the subsequent development of limit cycles.…”

“…The oscillatory phenomena (periodic solutions) have been shown by utilizing a continuous well-stirred tank reactor (CSTR). CSTR has a significant importance to show good outcomes with theoretical and experimental investigations: for more details, see [4,5,8] and the references therein.…”

“…Semi-analytical methods have been used to discuss many delay systems with reactiondiffusion phenomenon, such as delay logistic equations [6,7], predator-prey model [2], viral infection system [5], pellet systems [24], the Belousov-Zhabotinsky (BZ) reaction [9], the Brusselator model [3], the reversible Selkov model [1], the equation of Nicholsons blowflies [8], and the limited food model [4]. The outcomes for all papers that used this method revealed an excellent agreement between semi-analytical ODEs outcomes and the numerical solutions pertain to PDEs equations.…”