Inspired by the recent work Sahin and Agha gave recursion formulas for  and  Horn hypergeometric functions (Sahin and Agha in Miskolc Math Notes 16(2):1153–1162, 2015). The object of work is to establish several new recursion relations, relevant differential recursion formulas, new integral operators, infinite summations and interesting results for Horn’s hypergeometric functions,  and .

We report the microstructure, optical properties and high photocatalytic performance, with correlations between them, of coprecipitated and post-annealed ZnO nanoparticles. The obtained ZnO single phase after post annealing at different temperatures (200 – 600 ^oC) was identified by X-ray diffraction (XRD) and its morphology was investigated by transmission electron microscopy (TEM). The TEM results indicate a gradual growth of the spherical-like nanoparticles to nanorod-like shapes by post annealing with common nanorods shape by annealing at 600 ^oC. XRD Rietveld refinements indicated that the global structure parameters such as lattice constants, bond lengths, and oxygen position are slightly affected by the post annealing. While the crystallite size and number of unit cells per a particle gradually increase, the dislocation density, micro-strain and residual stress gradually decrease by the post annealing with a further change at annealing temperature of 500 ^oC, indicating correlations to the particles shape. Based on optical absorbance measurements, it was found that the optical energy gap gradually decreases from 3.22 to 2.89 eV by increasing the annealing temperature up to 600 ^oC. After a period up to 90 min of irradiation, photo-catalytic efficiency of up to 95% was observed in the photo-degradation of methylene blue. In addition, the nanoparticles annealed at low temperature showed better performance in the photo-degradation of methylene blue. The observed high photocatalytic performance is discussed in correlation to the microstructure and optical properties of the investigated samples.

In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number R0 and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if R0< 1 then the uninfected steady state is globally asymptotically stable (GAS), and if R0> 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.

The purpose of this paper is to study the effect of the cytotoxic T lymphocytes (CTLs) on an HIV-1 dynamics. The model considers that the virus infects the macrophages in addition to the CD4+ T cells. The role of the CTLs is to kill the infected macrophages and CD4+ T cells. The time delay which accounts the time of infection and the time of producing new active HIV-1 is modeled. The HIV-1 dynamics is modeled as a 6-dimensional nonlinear delay differential equations. The incidence rate of infection and killer rate of infected cells are given by general nonlinear functions. We study the qualitative behavior of the system. The global stability analysis has been established using Lyapunov method and LaSalle invariance principle. We present an example and perform numerical simulations to emphasize our theoretical results.

The purpose of this paper is to investigate the qualitative behavior of HIV dynamics models with immune impairment. The HIV particles infect both CD4+ T cells and macrophages. Both latently and actively infected cells are incorporated into the models. The models consider multiple discrete or distributed delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The existence and global stability of the steady states are determined by the basic reproduction number. The global stability analysis of the steady states is performed using Lyapunov method. We solve the system of delay differential equations numerically to support the theoretical results.

The presence of cells latently infected with HIV is currently considered to be a major barrier to viral eradication within a patient. Here, we consider birth–death-immigration models for the latent cell population in a single patient, and present analytical results for the size of this population in the absence of treatment. We provide results both at steady state (viral set point), and during the non-equilibrium setting of early infection. We obtain semi-analytic results showing how latency-reversing drugs might be expected to affect the size of the latent pool over time. We also analyze the probability of rare mutant viral strains joining the latent cell population, allowing for steady-state and dynamic viral populations within the host.

The coronavirus disease 2019 (COVID-19) is a respiratory disease caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). In this paper, we analyze a within-host SARS-CoV-2/HIV coinfection model. The model is made up of eight ordinary differential equations. These equations describe the interactions between healthy epithelial cells, latently infected epithelial cells, productively infected epithelial cells, SARS-CoV-2 particles, healthy CD T cells, latently infected CD4+ T cells, productively infected CD4+ T cells, and HIV particles. We confirm that the solutions of the developed model are bounded and nonnegative. We calculate the different steady states of the model and derive their existence conditions. We choose appropriate Lyapunov functions to show the global stability of all steady states. We execute some numerical simulations to assist the theoretical contributions. Based on …

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The global features of a fractional-order general pathogen infection model with immune deficiency and two types of target cells are examined in this chapter. Both cell-to-cell and pathogen-to-cell transmissions are presented. We demonstrate that the system solutions are positivity and confined, Lyapunov function is considered to show global asymptotic stability for disease-free equilibrium point of the model. Analytical expression for basic reproduction number  and the required condition under which disease-free and chronic equilibrium points are asymptotically stable are determined. We carried out numerical simulations to utilize it to document the analytical results.

Pb (II) and Hg (II) ions were reacted in aqueous solution with two tridentate thiosemicarbazones, resulting in four complexes {[Pb(L¹)₂] 1, [Pb(L²)₂] 2, [Hg(L¹)₂] 3 and [Hg(L²)₂] 4; HL¹ = 4-(4-nitrophenyl)-1-((pyridin-2-yl)methylene)thiosemicarbazide and HL² = 4-(2-chlorophenyl)-1-((pyridin-2-yl)methylene)thiosemicarbazide}. The solid-state structures of complexes 2 and 3 were determined by X-ray crystallography, which also confirmed the monobasic tridentate NNS nature of the ligands, and the octahedral coordination geometry around the metal centers. Complexes 1-4 exhibited high antifungal activity against the pathogen Cladosporium sphaerospermum; all four complexes giving rise to a reduction in the fungal dry mass and sugar consumption. Complexes 2 and 4 that inhibited Cladosporium sphaerospermum growth at the concentration of 1.8 mM are more efficient inhibitors than the free ligand and metal ions. Further, the complexes gave varied values of the fungal total antioxidants (TA), superoxide dismutase (SOD) and catalase (CAT) activities dependent on their concentrations.