Analysis and comparative study of a deterministic mathematical model of SARS-COV-2 with fractal-fractional operators: a case study.

In this paper, we investigate a fractal-fractional-order mathematical model with the influence of hospitalized patients and the impact of vaccination with fractal-fractional operators. The respective derivatives are considered in the Caputo, Caputo Fabrizio, and Atangana-Baleanu senses of fractional order α and fractal dimension τ . For the proposed problem, some results regarding basic reproduction number and stability are given. Using the next-generation matrix approach, we have investigated the global and local stability of several types of equilibrium points. We provide a detailed analysis of the existence and uniqueness of the solution. Moreover, we fit the model with the real data of Pakistan from June 01, 2020, till March 24, 2021. Then, we use the fractal-fractional derivative to find a numerical solution for the model. MATLAB software is used for numerical illustration. Graphical presentations corresponding to different parameteric values are given as well.

Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan.

In this paper, a five-compartment model is used to explore the dynamics of the COVID-19 pandemic, taking the vaccination campaign into account. The present model consists of five components that lead to a system of five ordinary differential equations. In this paper, we examined the disease from the perspective of a fractal fractional derivative in the Caputo sense with a power law type kernal. The model is also fitted with real data for Pakistan between June 1, 2020, and March 8, 2021. The fundamental mathematical characteristics of the model have been investigated thoroughly. We have calculated the equilibrium points and the reproduction number for the model and obtained the feasible region for the system. The existence and stability criteria of the model have been validated using the Banach fixed point theory and the Picard successive approximation technique. Furthermore, we have conducted stability analysis for both the disease-free and endemic equilibrium states. On the basis of sensitivity analysis and the dynamics of the threshold parameter, we have estimated the effectiveness of vaccination and identified potential control strategies for the disease using the proposed model outbreaks. The stability of the concerned solution in Ulam-Hyers and Ulam-Hyers-Rassias sense is also investigated. For the proposed problem, some results regarding basic reproduction numbers and stability analysis for various parameters are represented graphically. Matlab software is used for numerical illustrations. Graphical representations are given for different fractional orders and for various parametric values.

Biliary cystadenoma: Case report with MRI findings and surgical confirmation.

Biliary cystadenoma is a very uncommon benign cystic neoplasm involving the liver and the biliary tract. Most common presentations include right upper quadrant pain, nausea, vomiting, obstructive jaundice, and enlarging liver size. It can mimic many more commonly occurring diseases such as hepatic cyst, hepatic abscess, hydatid disease of the liver, and hepatic tuberculosis. Hence it becomes very challenging for physicians to correctly diagnose it due to its rarity and similarity with other conditions. Furthermore, very few pieces of literature guide physicians in correctly identifying the disease. Based on his physical examination and detailed investigation, we present a case report of a 72-year-old female diagnosed with biliary cystadenoma. We hope that this case report will significantly add to the existing literature on this subject.

Integral effects of brassinosteroids and timber waste biochar enhances the drought tolerance capacity of wheat plant.

Drought stress is among the major threats that affect negatively crop productivity in arid and semi-arid regions. Probably, application of some additives such as biochar and/or brassinosteroids could mitigate this stress; however, the mechanism beyond the interaction of these two applications is not well inspected. Accordingly, a greenhouse experiment was conducted on wheat (a strategic crop) grown under deficit irrigation levels (factor A) i.e., 35% of the water holding capacity (WHC) versus 75% of WHC for 35 days while considering the following additives, i.e., (1) biochar [BC, factor B, 0, 2%] and (2) the foliar application of 24-epibrassinolide [BR, factor C, 0 (control treatment, C), 1 (BR1) or 3 (BR2) µmol)]. All treatments were replicated trice and the obtained results were statistically analyzed via the analyses of variance. Also, heat-map conceits between measured variables were calculated using the Python software. Key results indicate that drought stress led to significant reductions in all studied vegetative growth parameters (root and shoot biomasses) and photosynthetic pigments (chlorophyll a, b and total contents) while raised the levels of oxidative stress indicators. However, with the application of BC and/or BR, significance increases occurred in the growth attributes of wheat plants, its photosynthetic pigments, especially the combined additions. They also upraised the levels of enzymatic and non-enzymatic antioxidants while decreased stress indicators. Furthermore, they increased calcium (Ca), phosphorus (P) and potassium (K) content within plants. It can therefore be deduced that the integral application of BR and BC is essential to mitigate drought stress in plants.

Vitamin D Supplementation to Treat Statin-Associated Muscle Symptoms: A Review.

For the past 15 years, cardiovascular disease (CVD) has been the leading cause of death for both men and women in the United States and worldwide. With an aging population, there has been increasing use of statin therapy to reduce the risk of CVD. However, statin-associated muscle symptoms (SAMS) remain an obstacle to this treatment, leading to discontinuation and nonadherence to statin therapy. Signs and symptoms of SAMS include muscle pain, tenderness, and increased serum creatine kinase. Despite the idiopathic pathophysiology of SAMS, some studies have shown an association between vitamin D deficiency and SAMS; the use of vitamin D supplements can lead to relief of these symptoms. The purpose of this review was to critique evidence for the association between low serum vitamin D and SAMS and the use of vitamin D supplementation for treatment.