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This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

The guiding principles for raising beekeeping products in developing countries provide support for commercializing honey production in order to generate income for beekeepers through the exploration of melliferous flora. The main objective of this study was to conduct a preliminary investigation into the pollen morphology of selected melliferous species that are frequently visited by honeybees in the vicinity of apiaries in the district of Mardan, Khyber Pakhtunkhwa. In this study, an updated overview was provided by reviewing the literature regarding palynological investigation of melliferous flora. The morpho-palynological features, dominant plant families visited by honeybees, and life forms of melliferous species were also discussed. The Acetolysis method was used to prepare the microscopic slides of pollen, measure and describe the qualitative and quantitative features using LM and SEM. Results showed maximum pollen polar and equatorial diameter in M. azadirachta (37.9 [micro]m) and (40.9 [micro]m) respectively. The polar to equatorial diameter ratio (P/E) was observed to be the highest in M. domestica (1.14). Colpi length was calculated to be the highest in the P. granatum (13.8 [micro]m). The highest percentage of pollen fertility was calculated in V. tricolor (92.5%), whereas the lowest was in O. rosea (64.28%). Studied characters under LM and SEM have appeared as a useful tool for the delimitation of melliferous flora, which may lead to the correct identification of the potential and valuable bee floral resources for beekeepers. The outcomes of this study aid in the assessment of melliferous flora and help in providing conservation measures for the sustainable development of beekeeping and the honeybee flora of the region. Les principes directeurs pour l'elevage de produits apicoles dans les pays en developpement soutiennent la commercialisation de la production de miel afm de generer des revenus pour les apiculteurs grace a l'exploration de la flore melifere. L'objectif principal de cette etude etait de mener une enquete preliminaire sur la morphologie pollinique de certaines especes meliferes frequemment visitees par les abeilles domestiques a proximite des ruchers dans le district de Mardan, Khyber Pakhtunkhwa. Dans cette etude, une vue d'ensemble mise a jour a ete fournie en passant en revue la litterature concemant la recherche palynologique de la flore melliferous. Les dispositifs morpho-palynologiques, les families dominantes de plantes visitees par des abeilles domestiques, et les formes de vie des especes melliferous ont ete egalement discutes. La methode d'acetolyse a ete utilisee pour preparer les lames microscopiques de pollen, mesurer et decrire les caracteristiques qualitatives et quantitatives a l'aide de LM et de SEM. Les resultats ont montre le diametre polaire et equatorial maximum du pollen chez M. azadirachta (37,9 [micro]m) et (40,9 [micro]m) respectivement. Le rapport diametre polaire/diametre equatorial (P/E) a ete observe comme le plus eleve chez M. domestica (1,14). La longueur de Colpi a ete calculee comme etant la plus elevee dans le granatum punica (13,8 [micro]m). Le pourcentage le plus eleve de fertilite du pollen a ete calcule chez V. tricolor (92,5%), tandis que le plus faible a ete chez O. rosea (64,28%). Les caracteres etudies sous LM et SEM sont apparus comme un outil utile pour la delimitation de la flore melifere, ce qui peut conduire a l'identification correcte des ressources florales potentielles et precieuses des abeilles pour les apiculteurs. Les resultats de cette etude aident a 1'evaluation de la flore melifere et aident a foumir des mesures de conservation pour le developpement durable de l'apiculture et de la flore des abeilles melliferes de la region. Keywords Apiaries * Honeybee * Melliferous flora * Palynology * Scanning electron microscopy, Introduction Beekeeping is very popular in South Asia including Pakistan, since much part of the country has auspicious habitats for honey production. Congenial climatic conditions and bee plants in the [...]