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Abstract This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay [b(ℓ)(y(n−1)(ℓ))γ]′+∫cdq(ℓ,ξ)yγ(g(ℓ,ξ))d(ξ)=0,ℓ≥ℓ0, $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition ∫ℓ0∞1b1γ(ℓ)dℓ

In this work, the catalytic activity of three different sizes of gold nano particles (AuNPs) (12, 30, and 45 nm) synthesized by the citrate reduction process studied in the conventional redox reaction of K 3 Fe (CN 6 ) -3 to K 4 Fe (CN 6 ) -4 using NaBH 4 (reductant) at four different temperatures (5 °C, 10 °C, 15 °C, and 20 °C) and measured by UV-visible spectrophotometry. Comparative kinetic analysis of different sizes of AuNPs including rate constant, activation energy, Entropy values and Frequency of collisions are reported for the first time. Transmission electron microscopy analysis is employed to investigate morphology and particle size. Spherical nanoparticles of size 12, 30, and 45 nm were observed. The UV-visible spectra were recorded at regular intervals, and it was seen that the peak of K 3 Fe (CN 6 ) -3 decreased gradually with time, at the same time surface plasmon resonance of AuNPs remained constant. As reaction catalysts, AuNPs maintain their optical density which shows their stability during the course of reaction. The kinetic parameters i.e., rate constant, and activation energy (k, t 1/2 , E a ) were determined for three distinct sizes of gold nanoparticles (AuNPs) using the reductant at various concentrations. The value of k increases by increasing reductant concentration. This rise was significant for the small AuNPs. Increasing gold nanoparticle size (12, 30, 45 nm) decreased rate constant. As the size of AuNPs decreased the E a reduced as well, i.e. 17.325 k cal mol -1 for 12 nm, 19 k cal mol -1 for 30 nm and 21 k cal mol -1 for 45 nm AuNPs. For 50 mM of NaBH 4 , k for 45 nm AuNPs is 0.10728 s -1 , but for 12 nm AuNPs, the value of k is 0.145 s -1 , indicating that the 12 nm AuNPs have the greatest rate constant values. The rate of reaction rises with an increase in reductant concentration and temperature, but this increase is significant in the case of small-sized nanoparticles, i.e., 12 nm, which have a high surface area and low E a . Besides being a model redox reaction, the reduction of K 3 Fe (CN 6 ) -3 to K 4 Fe (CN 6 ) -4 has industrial use in making fertilizers and paint industry, anti-coating agent in colour photography, in dot etching and in amperometric biosensors., (© 2023. The Author(s).)

Game theory is an excellent mathematical tool to describe the interaction between the immune system and cancerous leukocytes ( c . leu ). The feature of cancerous leukocytes to differentiate and mutate to give rise to leukemia is in the domain of ecological models as well. In this work, the dynamic of leukemia is described and compared by two models: firstly by a simple probabilistic mathematical model using the zero-sum two player game of Hawk and Dove, and secondly by Leslie Predator Prey model of ecology. The main goal of this study is to compare the results of both models and then discuss the treatment of leukemia i.e., Hematopoietic Stem cell transplant with the best model among them. Hawk and Dove model also describes the cell to cell interaction of cancerous leukocytes and healthy leukocytes ( leu ) after diagnoses and the condition of the patient before and after treatments. In this work, Hematopoietic Stem cell transplant is discussed by using concepts of a zero-sum three player game. Also, both models will be characterized by determining the stability properties, identifying basins of attraction, and locating the equilibrium points to see, at what extent the patient's survival is possible with leukemia in its body. Results for both models will be presented graphically., Competing Interests: The authors declare that they have no conflicts of interest., (Copyright © 2022 Mariam Sultana et al.)

During the past few years, freelancing has grown exponentially due to the pandemic and subsequent economical changes in the world. In fact, in the last ten years, a drastic increase in freelancing has been observed; people quit their jobs to be their own boss. There are various reasons for this: downsizing of employees, not having fun in their jobs, unemployment, part time work to earn more, etc. Observing this vast change, many individuals on Facebook/YouTube, NGOs, and government departments started teaching freelancing as a course; to date, thousands of youngsters have been trained to start their careers as freelancers. It has been observed that the ratio of informed freelancers is more successful than those who start their careers independently. We construct a compartmental model to explore the influence of information on the expansion of freelancing in this article, which was motivated by this surge in freelancing. Following that, the model is subjected to dynamical analysis utilizing dynamical systems and differential equation theory. To validate our analytical conclusions, we used numerical simulation.

Motivation . Currently, the COVID-19 pandemic represents a critical issue all over the world. On May 11, 2020, at 05 : 41 GMT, approximately 0.28 million individuals had perished because of the COVID-19 pandemic, and the figure is continuously growing rapidly. Unfortunately, millions of people have died due to this pandemic. As a result, this issue forced governments and other corresponding organizations to take significant action, such as the lockdown and vaccinations. Furthermore, scientists have developed several vaccinations, and the World Health Organization (WHO) has urged governments and people to get vaccinated to eradicate this pandemic. Consequently, the findings of any scientific research into this phenomenon are highly interesting. Problem Statement . To enhance individual protection, it is now critical to analyze and compare the percentage of people fully vaccinated against COVID-19. It is constantly of interest in the field of big data science and other related disciplines to provide the best analysis and modeling of COVID-19 data. Methodology . Through this paper, we aimed to compare individuals who have been completely vaccinated against COVID-19 in two locations: North American countries and Arabian Peninsula countries. Simple techniques for comparing individuals who have been completely vaccinated against COVID-19 have been applied, which may be used to generate the foundation for conclusions. Most significantly, a modern statistical model was created to present the best assessment of individuals completely vaccinated against COVID-19 data in nations in North America and the Arabian Peninsula. Some of the suggested statistical model features were proposed. Furthermore, the estimate of the model parameters was driven using the maximum likelihood estimation method. Results . The flexibility provided by the proposed statistical model is useful for describing the percentage of the individuals completely vaccinated against COVID-19, which provides a close fit with the COVID-19 data. Implications . The proposed statistical model can be used for statistics and generate new statistical distributions that can be used to compare and predict the process of people's willingness to vaccinate and take the vaccine to try to eliminate COVID-19., Competing Interests: There are no conflicts of interest in this paper's publishing., (Copyright © 2022 Abdullah Ali H. Ahmadini et al.)

This report examines the heat and mass transfer in three-dimensional second grade non-Newtonian fluid in the presence of a variable magnetic field. Heat transfer is presented with the involvement of thermal relaxation time and variable thermal conductivity. The generalized theory for mass flux with variable mass diffusion coefficient is considered in the transport of species. The conservation laws are modeled in simplified form via boundary layer theory which results as a system of coupled non-linear partial differential equations. Group similarity analysis is engaged for the conversion of derived conservation laws in the form of highly non-linear ordinary differential equations. The solution is obtained vial optimal homotopy procedure (OHP). The convergence of the scheme is shown through error analysis. The obtained solution is displayed through graphs and tables for different influential parameters.

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.