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Let i ≥ 2, ∆ ≥ 0, 1 ≤ a ≤ b − ∆, n > (a+b)(ib+2m−2) a + n and δ(G) ≥ b 2 a + n + 2m, and let g, f be two integer-valued functions defined on V (G) such that a ≤ g(x) ≤ f (x) − ∆ ≤ b − ∆ for each x ∈ V (G). In this article, it is determined that G is a fractional (g, f, n , m)-critical deleted graph if max{d 1 , d 2 , • • • , d i } ≥ b(n+n) a+b for any independent subset {x 1 , x 2 ,. .. , x i } ⊆ V (G). The result is tight on independent set degree condition.

The intension of the present work is to present the stochastic numerical approach for solving human immunodeficiency virus (HIV) infection model of cluster of differentiation 4 of T-cells, i.e., CD4 + T cells. A reliable integrated intelligent computing framework using layered structure of neural network with different neurons and their optimization with efficacy of global search by genetic algorithms supported with rapid local search methodology of active-set method, i.e., hybrid of GA-ASM, is used for solving the HIV infection model of CD4 + T cells. A comparison between the present results for different neurons-based models and the numerical values of the Runge-Kutta method reveals that the present intelligent computing techniques is trustworthy, convergent and robust. Statistics-based observation on different performance indices further demonstrates the applicability, effectiveness and convergence of the present schemes.

The aim of the present paper is to state a simplified nonlinear mathematical model to describe the dynamics of the novel coronavirus (COVID-19). The design of the mathematical model is described in terms of four categories susceptible ([Formula: see text], infected ([Formula: see text], treatment ([Formula: see text] and recovered ([Formula: see text], i.e. SITR model with fractals parameters. These days there are big controversy on if is needed to apply confinement measure to the population of the word or if the infection must develop a natural stabilization sharing with it our normal life (like USA or Brazil administrations claim). The aim of our study is to present different scenarios where we draw the evolution of the model in four different cases depending on the contact rate between people. We show that if no confinement rules are applied the stabilization of the infection arrives around 300 days affecting a huge number of population. On the contrary with a contact rate small, due to confinement and social distancing rules, the stabilization of the infection is reached earlier.

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In the present investigation, a novel neuro-swarming intelligence-based numerical computing solver is developed for solving second order non-linear singular periodic (NSP) boundary value problems (BVPs), i.e., NSP-BVPs, using the modeling strength of artificial neural networks (ANN) optimized with global search efficacy of particle swarm optimization (PSO) supported with the methodology of rapid local search by interior-point scheme (IPS), i.e., ANN-PSO-IPS. In order to check the proficiency, robustness, and stability of the designed ANN-PSO-IPS, two numerical problems of the NSP-BVPs have been presented for different numbers of neurons. The outcomes of the proposed ANN-PSO-IPS are compared with the available exact solutions to establish the worth of the solver in terms of accuracy and convergence, which is further endorsed through results of statistical performance metrics based on multiple implementations.

This article is devoted to the study of invariant ε-scrambled sets. We show that every topologically mixing map with at least one fixed point contains at least one such set. Additionally we show that this condition can be weakened in the case of symbolic dynamics, e.g. mixing can be replaced by transitivity. Some relations between mixing and proximal relation are also studied.

In this study, the design of a novel model based on nonlinear third-order Emden–Fowler delay differential (EF-DD) equations is presented along with two types using the sense of delay differential and standard form of the second-order EF equation. The singularity at ξ = 0 at single or multiple points of each type of the designed EF-DD model are discussed. The detail of shape factors and delayed points is provided for both types of the designed third-order EF-DD model. For the verification and validation of the model, two numerical examples are presented of each case and numerical results have been performed using the artificial neural network along with the hybrid of global and local capabilities. The comparison of the obtained numerical results with the exact solutions shows the perfection and correctness of the designed third-order EF-DD model.