One of the control measures available that are believed to be the most reliable methods of curbing the spread of coronavirus at the moment if they were to be successfully applied is lockdown. In this paper a mathematical model of fractional order is constructed to study the significance of the lockdown in mitigating the virus spread. The model consists of a system of five nonlinear fractional-order differential equations in the Caputo sense. In addition, existence and uniqueness of solutions for the fractional-order coronavirus model under lockdown are examined via the well-known Schauder and Banach fixed theorems technique, and stability analysis in the context of Ulam–Hyers and generalized Ulam–Hyers criteria is discussed. The well-known and effective numerical scheme called fractional Euler method has been employed to analyze the approximate solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many studies recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.

The basic idea underneath the generalized intuitionistic fuzzy soft set is very constructive in decision making, since it considers how to exploit an extra intuitionistic fuzzy input from the director to make up for any distortion in the information provided by the evaluation experts, which is redefined and clarified by F. Feng. In this paper, we introduced a method to solve decision making problems using an adjustable weighted soft discernibility matrix in a generalized intuitionistic fuzzy soft set. We define the threshold functions like mid-threshold, top-bottom-threshold, bottom-bottom-threshold, top-top-threshold, med-threshold function and their level soft sets of the generalized intuitionistic fuzzy soft set. After, we proposed two algorithms based on threshold functions, a weighted soft discernibility matrix and a generalized intuitionistic fuzzy soft set and also to show the supremacy of the given methods we illustrate a descriptive example using a weighted soft discernibility matrix in the generalized intuitionistic fuzzy soft set. Results indicate that the proposed method is more effective and generalized over all existing methods of the fuzzy soft set. Mathematics 2019, 7, 742 2 of 21 such as economics, medical science, social science, environmental science and engineering. In recent years, model vagueness has become interested in many authors. Many classical theories such as fuzzy set theory [1], probability theory, vague set theory [2], rough set theory [3], intuitionistic fuzzy set [4] and interval mathematics [5] are well known and effectively model uncertainty. These approaches show their inherent difficulties as pointed out by Molodtsov [6], because of intensive quantity and type of uncertainties. In Reference [6], Molodtsov defines the soft set which is a new logical instrument for dealing uncertainties.Soft set theory attracts many authors because it has a vast range of applications in many areas like the smoothness of functions, decision making, probability theory, data analysis, measurement theory, forecasting and operations research [6][7][8][9][10]. Nowadays, many authors work to hybridize the different models with soft set and achieved results in many applicable theories. Maji defines the fuzzy soft set and intuitionistic fuzzy soft set [11,12]. Then the further extensions of soft sets like the generalized fuzzy soft set [13], the interval-valued fuzzy soft set [14], the soft rough set [15], the vague soft set [16], the trapezoidal fuzzy soft set [17], the neutrosophic soft set [18], the intuitionistic neutrosophic soft set [19], the multi-fuzzy soft set [20] and the hesitant fuzzy soft set [21] are introduced. Agarwal defines the generalized intuitionistic fuzzy soft set (GIFSS) [22] which has some problems pointed out by Feng [23] and redefined GIFSS.In Reference [24], Coung defines the picture fuzzy set which is an extension of the fuzzy soft set and intuitionistic fuzzy set. In Reference [25], Sing defines the correlation coefficients of PFS and their applications in clustering anal...

In this paper, a generalized picture fuzzy soft set is proposed, which is an extension of the picture fuzzy soft sets. We investigate the basic properties of picture fuzzy soft sets and define an F-subset, M-subset, extended union, extended intersection, restricted union, restricted intersection and also prove the De Morgan’s laws for picture fuzzy soft information. We investigate upper and lower substitution for both picture fuzzy sets and generalized picture fuzzy soft sets. Meanwhile, the related proofs are given in detail. Finally, we propose an algorithm to deal with generalized picture fuzzy soft information. To show the supremacy and effectiveness of the proposed technique, we illustrate a descriptive example using generalized picture fuzzy soft information. Results indicate that the proposed technique is more generalized and effective over all the existing structures of fuzzy soft sets.

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The object of this study was to demonstrate the ability of machine learning (ML) methods for the segmentation and classification of diabetic retinopathy (DR). Two-dimensional (2D) retinal fundus (RF) images were used. The datasets of DR—that is, the mild, moderate, non-proliferative, proliferative, and normal human eye ones—were acquired from 500 patients at Bahawal Victoria Hospital (BVH), Bahawalpur, Pakistan. Five hundred RF datasets (sized 256 × 256) for each DR stage and a total of 2500 (500 × 5) datasets of the five DR stages were acquired. This research introduces the novel clustering-based automated region growing framework. For texture analysis, four types of features—histogram (H), wavelet (W), co-occurrence matrix (COM) and run-length matrix (RLM)—were extracted, and various ML classifiers were employed, achieving 77.67%, 80%, 89.87%, and 96.33% classification accuracies, respectively. To improve classification accuracy, a fused hybrid-feature dataset was generated by applying the data fusion approach. From each image, 245 pieces of hybrid feature data (H, W, COM, and RLM) were observed, while 13 optimized features were selected after applying four different feature selection techniques, namely Fisher, correlation-based feature selection, mutual information, and probability of error plus average correlation. Five ML classifiers named sequential minimal optimization (SMO), logistic (Lg), multi-layer perceptron (MLP), logistic model tree (LMT), and simple logistic (SLg) were deployed on selected optimized features (using 10-fold cross-validation), and they showed considerably high classification accuracies of 98.53%, 99%, 99.66%, 99.73%, and 99.73%, respectively.

A new condition on positive membership, neutral membership, and negative membership functions give us the successful extension of picture fuzzy set and Pythagorean fuzzy set and called spherical fuzzy sets ( SFS ) . This extends the domain of positive membership, neutral membership, and negative membership functions. Keeping in mind the importance of similarity measure and application in data mining, medical diagnosis, decision making, and pattern recognition, several studies on similarity measures have been proposed in the literature. Some of those, however, cannot satisfy the axioms of similarity and provide counter-intuitive cases. In this paper, we proposed the set-theoretic similarity and distance measures. We provide some counterexamples for already proposed similarity measures in the literature and shows that how our proposed method is important and applicable to the pattern recognition problems. In the end, we provide an application of a proposed similarity measure for selecting mega projects in under developed countries.

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