In this paper, a Covid-19 dynamical transmission model of a coupled non-linear fractional differential equation in the Atangana-Baleanu Caputo sense is proposed. The basic dynamical transmission features of the proposed system are briefly discussed. The qualitative as well as quantitative results on the existence and uniqueness of the solutions are evaluated through the fixed point theorem. The Ulam-Hyers stability analysis of the suggested system is established. The two-step Adams-Bashforth-Moulton (ABM) numerical method is employed to find its numerical solution. The numerical simulation is performed to accesses the impact of various biological parameters on the dynamics of Covid-19 disease.

In this study, a mathematical model for the differentiation of stem cells is proposed to understand the dynamics of cell differentiation in a complex network. For this, myeloid cells, which are differentiated from stem cells, are introduced in this study. We introduce the threshold quantity $ \mathcal{R}_{0} $ to understand the population dynamics of stem cells. The local stability analysis of three equilibria, namely $ (i) $ free equilibrium points, $ (ii) $ absence of stem and progenitor cells, and $ (iii) $ endemic equilibrium points are investigated in this study. The model is first formulated in non-fractional order and after that converted into a fractional sense by utilizing the Atangana-Baleanu derivative in Caputo (ABC) sense in the form of a non-singular kernel. The model is solved by using numerical techniques. It is seen that the myeloid cell population significantly affects the stem cell population.

In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established.

In this research, we have introduced compartments for asymptomatic and symptomatic individuals, along with reduced susceptibility, as key factors defining our investigation. The study is carried out in diverse scenarios, considering them as crucial for the essential generation number of the model, set at 3.18( r 0 > 1 ). The persistent reproduction differential method was used to explore the impact of continuous adrenocorticotropic hormone (ACTH) administration on the global gene expression in primary cultures of both fetal and adult adrenal cells. The study also investigates ACTH’s genetic effects on both adult and fetal human adrenal cells. The conclusion of this study is demonstrated through relevant and correct medical applications.