Coronavirus infection (COVID-19) is a considerably dangerous disease with a high demise rate around the world. There is no known vaccination or medicine until our time because the unknown aspects of the virus are more significant than our theoretical and experimental knowledge. One of the most effective strategies for comprehending and controlling the spread of this epidemic is to model it using a powerful mathematical model. However, mathematical modeling with a fractional operator can provide explanations for the disease's possibility and severity. Accordingly, basic information will be provided to identify the kind of measure and intrusion that will be required to control the disease's progress. In this study, we propose using a fractional-order SEIARPQ model with the Caputo sense to model the coronavirus (COVID-19) pandemic, which has never been done before in the literature. The stability analysis, existence, uniqueness theorems, and numerical solutions of such a model are displayed. All results were numerically simulated using MATLAB programming. The current study supports the applicability and influence of fractional operators on real-world problems.

In the present investigations, we construct a new mathematical for the transmission dynamics of corona virus (COVID-19) using the cases reported in Kingdom of Saudi Arabia for March 02 till July 31, 2020. We investigate the parameters values of the model using the least square curve fitting and the basic reproduction number is suggested for the given data is ℛ ≈ 1.2937. The stability results of the model are shown when the basic reproduction number is ℛ < 1. The model is locally asymptotically stable when ℛ < 1. Further, we show some important parameters that are more sensitive to the basic reproduction number ℛ using the PRCC method. The sensitive parameters that act as a control parameters that can reduce and control the infection in the population are shown graphically. The suggested control parameters can reduce dramatically the infection in the Kingdom of Saudi Arabia if the proper attention is paid to the suggested controls.

In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown-like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS-CoV-2 virus into a system of equations to examine the existence and uniqueness results with the Atangana-Baleanu derivative by using a fixed-point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India.

An implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross-diffusion system possesses a formal gradient-flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite-volume scheme is based on two-point flux approximations with "double" upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin-Lions lemma of "degenerate" type. Numerical simulations of a calcium-selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.

The construction of projection operators, which commute with the exterior derivative and at the same time are bounded in the proper Sobolev spaces, represents a key tool in the recent stability analysis of finite element exterior calculus. These so-called bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projections defined by the degrees of freedom. However, an undesired property of these bounded projections is that, in contrast to the canonical projections, they are nonlocal. The purpose of this article is to discuss a recent alternative construction of bounded cochain projections, which also are local. A key tool for the new construction is the structure of a double complex, resembling the Čech-de Rham double complex of algebraic topology.

Elevating the temperature of cancerous cells is known to increase their susceptibility to subsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher intensity heat sources may be used to ablate the tissue. These facts are the basis for hyperthermia based cancer treatments. Of the many available modalities for delivering the heat source, the application of a laser heat source under the guidance of real-time treatment data has the potential to provide unprecedented control over the outcome of the treatment process [7, 18]. The goals of this work are to provide a precise mathematical framework for the real-time finite element solution of the problems of calibration, optimal heat source control, and goal-oriented error estimation applied to the equations of bioheat transfer and demonstrate that current finite element technology, parallel computer architecture, data transfer infrastructure, and thermal imaging modalities are capable of inducing a precise computer controlled temperature field within the biological domain.