COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained $\mathcal{R}_{0} \approx 1.50$R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

The Internet of Medical Things (IoMT) heralds a transformative era in healthcare, with the potential to revolutionize patient care, healthcare services, and medical research. As with all technological progressions, IoMT introduces a suite of complex challenges, predominantly centered on security. In particular, ensuring the integrity, confidentiality, and availability of health data in real-time communication stands paramount, given the sensitivity of the information and the ramifications of potential breaches or misuse. In light of these challenges, existing security frameworks, while commendable, exhibit limitations. Specifically, they often grapple with comprehensive anomaly detection, effective resistance to replay attacks, and robust protection against threats like man-in-the-middle attacks, eavesdropping, data tampering, and identity spoofing. The proposed framework integrates state-of-the-art encryption techniques, cutting-edge pattern recognition modules, and adaptive learning mechanisms. These components collaboratively ensure data integrity during transmission, provide robust resistance against conventional and novel attack vectors, and adapt to evolving threats through continuous learning. Moreover, the framework incorporates sophisticated checksum techniques and advanced behavioral analysis, further enhancing its protective capabilities. Our system demonstrated significant improvements in anomaly detection and attack resistance metrics, consistently outperforming benchmark solutions like MRMS and BACKM-EHA.

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator $D_{q,\lambda }^m\left(\rho ,\sigma \right)$ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc $\mathcal{U}$. Furthermore, we analyze novel findings with respect to the inverse function $\left({\mu }^{-1}\right)$ within the class of q-starlike functions in $\mathcal{U}$. The findings in this paper are easy to understand and show a connection between present and past studies.

Introduction: This paper aims to measure the performance of early detection methods, which are usually used for infectious diseases. Methodology: By using real data of confirmed Coronavirus cases from the Kingdom of Saudi Arabia and Italy, the moving epidemic method (MEM) and the moving average cumulative sums (Mov. Avg Cusum) methods are used in our simulation study. Results: Our results suggested that the CUSUM method outperforms the MEM in detecting the start of the Coronavirus outbreak.

In this study, for $0< q<1$ , we first introduce a function $\varphi _{q}\left ( \tau \right ) = \frac{e^{q\tau }-1}{q\left ( 1-q\tau \right ) }$ and prove that $\varphi _{q}\left ( \tau \right ) $ is convex univalent in the open unit disk $\mathcal{U}$ for all $q\in \left ( 0,0.32\right ) $ . Further, using the subordination technique and considering q-difference operator, we define a new subclass $S_{\varphi }^{\ast }(q)$ of q-starlike functions associated with a function $1+\varphi _{q}\left ( \tau \right ) $ , which is in the class $\mathcal{P}$ . Some new geometric properties, such as coefficient bounds, Feketo–Szego inequalities, and the upper bound of the second-order Hankel determinant, are investigated for the function h belonging to the class $S_{\varphi }^{\ast }(q)$ .

This paper looks at adapting a recent approach found in the literature for pricing short-term American options to price American straddle options with two free boundaries. We provide a series solution in which explicit formulas for the coefficients are given. Hence, no complicated, recursive systems or nonlinear integral equations need to be solved, and the method efficiently provides fast solutions. We also compare the method with a numerical method and find that it gives very accurate prices not only for the option value, but also for the critical stock prices.