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Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if Rv is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter β2 that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when ψ=1 for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (R0≤1) and endemic cases (R0>1). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number R0≈1.2591. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.

Mathematical modeling of infectious diseases with non-integer order getting attentions from scientists and researchers day by day. It is obvious that classical models in epidemiology can only be described through a fixed order while models in fractional order derivative are of not fixed order. Having non fixed order the fractional derivative becomes more powerful in modeling real life problems. In the recent era, different novel concepts regarding fractional operators such as the exponential decay and the Mittag–Leffler kernel have been introduced which overcome the limitations of the previous fractional order derivatives. These new operators have been found effective in modeling problems arising in science and engineering. A more recent operator in fractional calculus was introduced that is known as fractal-fractional operator. In this study, we consider this novel approach and apply it to an epidemic model of dengue fever and explore their dynamics. We show some important analysis for the dengue epidemic model in the presence of this new operator. The uniqueness and existence results will be shown. We show the simulation results for the considered model with a novel numerical approach which is not yet considered by anyone for such epidemic model. We obtain results for fractal model when fractional order is one and will have fractional solution when fractal order is one and have when both are present. We show that the fractal-fractional approach is much suitable for an epidemic model rather than fractional operator.

Natural symmetries exist in several processes of chemistry, physics, and biology. Symmetries possess interesting dynamical characteristics that cannot be seen in non-symmetric systems. The present paper investigates the competition between two banking systems, rural and commercial, in Indonesia, in parameterized fractional order Caputo derivative. A novel numerical method is used to discretize the competition system using the real data of rural and commercial banks in Indonesia for the period 2004–2014. The new scheme is more suitable and reliable for data fitting results and has good accuracy. The integer model is formulated in Caputo derivative and their stability results are presented. With the available parameters, the data for the model is analyzed using various scenarios. We shall compare the result with the previous method used in the literature and show that the present method is better than the previous method in the literature. It is shown that fractional order α and the parameter ρ involved in the numerical scheme provide excellent fitting.

The regular frequent pattern mining (RFPM) approaches are aimed to discover the itemsets with significant frequency and regular occurrence behavior in a dataset. However, these approaches mainly suffer from the following two issues: (1) setting the frequency threshold parameter for the discovery of regular frequent patterns technique is not an easy task because of its dependency on the characteristics of a dataset, and (2) RFPM approaches are designed to mine patterns from the static datasets and are not able to mine dynamic datasets. This paper aims to solve these two issues by proposing a novel top-K identical frequent regular patterns mining (TKIFRPM) approach to function on online datasets. The TKIFRPM maintains a novel synopsis data structure with item support index tables (ISI-tables) to keep summarized information about online committed transactions and dataset updates. The mining operation can discover top-K regular frequent patterns from online data stored in the ISI-tables. The TKIFRPM explores the search space in recursive depth-first order and applies a novel progressive node’s sub-tree pruning strategy to rapidly eliminate a complete infrequent sub-tree from the search space. The TKIFRPM is compared with the MTKPP approach, and it found that it outperforms its counterpart in terms of runtime and memory usage to produce designated topmost-K frequent regular pattern mining on the datasets following incremental updates.

The human immunodeficiency virus (HIV) is a major global public health issue and causes millions of deaths around the globe. The most severe phase of HIV infection is known as AIDS. In recent years, a number of mathematical models based on classical integer-order derivative have been developed to analyze the insight dynamics of HIV/AIDS. This paper presents the transmission dynamics of HIV/AIDS using fractional order (FO) and a fractal-fractional order compartmental model with the power-law kernel. In the first phase, the proposed model is formulated using the Caputo-type fractional derivative. The basic properties such as the solution positivity and existence as well as uniqueness of the fractional model are presented. The equilibria and the basic reproductive number [Formula: see text] are evaluated. Further, using fractional stability concepts the stability of the model (both local and global) around the equilibrium is presented in the disease-free case. In addition, the fractional model is solved numerically, and the graphical results with many values of [Formula: see text] are shown. In the second phase, the concept of a fractal-fractional (FF) operator is applied to obtain a more generalized model that addresses the dynamics of HIV/AIDS. The uniqueness and existence of the solutions of the FF-based model are shown via the Picard–Lindelof approach while the modified Adams–Bashforth method is utilized to present the numerical solution. Detailed numerical simulations are presented for various values fractional as well as the fractal orders, [Formula: see text] and [Formula: see text] respectively. The graphical results reveal that the FF-based model provides biologically more feasible results than the models in fractional and classical integer-order cases.

Imprisonment is a restriction of human liberty; Punishment of imprisonment is accepted form in Islamic and modern western world. Evidences of imprisonment are found in Quran and the Sunnah as during the life time of the Holy Prophet (PBUH), people used to be detained in the mosque. First jail was established during the time of Caliph Ali, in Islamic History. Islam allows to impose imprisonment on accused and convicted offenders and imprisonment even can be for a short time or in some cases it might be for life time. Islam considerers imprisonment as a discretionary punishment; the judge has an option to exercise punishment of imprisonment against any person if he thinks better for the reformation of any criminal. Islamic teachings urge to provide prisoners with other rights as food, clothing, education and medical treatment in case of any health-related issue along with healthy environment for residing. Initially, in the west there was no concept of imprisonment, first time jail was established in 1552 in the UK. In the beginning prisoners were deprived of basic rights as proper residence, food, clothing and sanitation. In addition to it, they were maltreated by jail authorities. In 18th century, the reforms were introduced in the western world where new laws were made in favor of prisoners and prisoners were divided into different categories on which different facilities were provided to them. Imprisonment is considered as a primary punishment; it is being exercised against all sort of criminals whether it is in accordance with their nature of crimes or not.

We formulate and analyze a new financial chaotic model in fractional stochastic differential equation in Atangana-Baleanu operator. The model is constructed initially in integer case and then the application of fractional and stochastic approach are used for the model generalization. We study with care some of the mathematical results for the model in integer case by identifying the suitable values of parameters and initial values for the state variables for which the model behaves chaotic. Further, using the concept of stochastic differential equation, the model is further extended to the stochastic fractional differential equations. The equilibriums possibly the model posses are shown and analyzed. Lyapanove exponent are investigated for the model. We give a new method that is introduced recently in literature to find the model solution numerically for the case of fractional stochastic differential equation. We obtained many interesting graphical results for the model by considering different scenarios.

Just as Allah is the Sustainer of all the worlds, the Holy Quran is the remembrance for the worlds, so His Beloved Hazrat Muhammad (PBUH) is the mercy for the worlds, The Holy Qur'an is the path of guidance and the biography of the Prophet is sustenance and viaticum of that journey. As Islam is incomplete without the Holy Quran, so the practice of humankind, according to the spirit of Qur'an, is impossible without the seerah of the Prophet (PBUH). Therefore, the relationship between the biography of the Prophet and the Quran is integral. That is why, just as Allah almighty has protected the Quran in terms of its words. He has also taken care of its meaning in the form of the Seerah of His prophet Muhammad, till the Day of Resurrection. So that life for humanity in every field, at the individual and collective levels, the Seerah of prophet, as the best way of life has always been available. Quran is the most important and irrefutable source of the biography of the Prophet. How Allah almighty himself does introduces and present His Prophet? It is very important question. In last few years, great attention has been paid to the biography of the Prophet (PBUH) at the official level in Islamic Republic of Pakistan, which was due on the Islamic scholars of the nation. Therefore, in order to get the right benefit from the Seerah of the Prophet (peace and blessings of Allah be upon him), we must first turn our attention to the most important source of it; the Quran. To be aware of the teachings of the Seerah; which are given in the principle form by the Holy Qur'an, we need to notice the concept presented by the Quran itself. In this paper, an attempt has to be made to shed light of the Seerah of the prophet (PBUH) from the guidelines provided in the Holy Quran according to the need of the time.

In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction number ℛ 0 < 1 . Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.

The present effort elaborates the fractional analyses for Darcy-Forchheimer hybrid nanoliquid flow over a porous spinning disk. Temperature and concentration slip conditions are utilized at the surface of the spinning disk. A specific type of nanoparticles known as Silver-Ag and Magnesium-oxide MgO is added to the base fluid, to synthesis the hybrid nanoliquid. By using Karman’s approach, the system of partial differential equations is depleted into a dimensionless system of differential equations. The obtained equations are further diminished to the first-order differential equation via selecting variables. To develop the fractional solution, the proposed model has been set up by Matlab fractional code Fde12. For accuracy and validity of the resulting framework, the outputs are compared with the fast-approaching numerical Matlab scheme boundary value solver (bvp4c). The impact of several flow constraints versus velocity, mass and thermal energy profiles have been portrayed and discussed. Magnesium oxide MgO compound is consists of Mg2+ and O2− ions, together bonded by a strong ionic bond, which can be synthesized by pyrolysis of magnesium hydroxide Mg (OH) 2 and MgCO3 (magnesium carbonate) at a very high temperature (700–1500 °C). It is more convenient for refractory and electrical applications. Similarly, the antibacterial upshots of silver Ag nano-size particles could be used to manage bacterial growth in several applications, such as dental work, burns and wound treatment, surgery applications and biomedical apparatus.

A dynamical model of SARS-CoV-2 in fractional derivative using the cases of coronavirus of the fourth wave is presented. We construct basically the model in an integer case, and later it is extended to a fractional-order system by applying the Atangana–Baleanu operator definition. We give some background definitions and results for the fractional-order model. We present for the disease-free case that the model is locally asymptotically stable when [Formula: see text]. The global dynamics of the fractional model are given when [Formula: see text] for the disease-free case. The model is further extended to fractional stochastic piecewise equations in the Atangana–Baleanu case. The reported cases from the fourth wave in Pakistan starting from July 1 up to November 16, 2021 are considered for the estimation of the parameters. We fitted our model to the suggested data and obtained the numerical value of the basic reproduction number [Formula: see text] for fractional order. We give the data fitting to both the fractional and piecewise stochastic differential equations, and show them both as having a good fitting to the data. We use further the numerical values of the model parameters and present its numerical results graphically using the effective numerical approaches. Some sensitive parameters that are reasonable for disease eliminations are used to obtain the graphical results.

Asymptomatic carriers play an important role in modelling infectious diseases. Asymptomatic infected people have no symptoms but can infect other people and spread the disease among other people. The laboratory confirmed that asymptomatic hepatitis B may infect individuals and may generate other infected cases. Due to this significant role of asymptomatic carriers, we are considering a new mathematical model for hepatitis B virus with asymptomatic carriers to study its dynamic analysis. First, we briefly discussed the formulation of the model, and then used the Caputo derivative to generalize the model. Using the definition of fractional stability analysis, we study the results of the model, and show that the model is locally asymptotically stable for disease-free cases when R 0 1 . We also give the result so that the model is globally asymptotically stable when R 0 1 . The endemic equilibrium of the model is obtained when R 0 > 1, which demonstrates the existence of a unique endemic equilibrium. Additionally, we use a new numerical scheme that is introduced for fractional differential equations to get numerical results. We use a number of fractional order values and present the graphical results of the model.

Stage classification is a significant important task for scene understanding, 3D TV, autonomous vehicle, and object localization. Images can be categorized into a limited number of 3D scene geometries, called stages, and each one of them is having a unique depth pattern to provide a specific context for stage objects. Moreover, convolutional neural networks (CNN) have shown high performance of scene classification due to their powerful perspective of feature learning and reasoning. However, we found that edge-preserving Laplacian filter (LF) based on Laplacian pyramids, which enhances the edge details of image scene owing to this, it can improve the performance of stage classification. We introduce a novel method of stage classification based on two-stream CNN model in which one stream is encoded by LF, and another stream is normal RGB images and their output is fused at the decision level. This proposed method is evaluated on two different stage datasets: first ‘stage-1209’ contains 1209 images, and second, ‘12-scene’ image dataset contains 12,000 images. Results exhibited that LF encoded images have a positive influence on stage classification accuracy. Following this, while using product rule the proposed method obtains the most significant improvement in the stage classification for both datasets. It improves particularly 7.96% stage accuracy on 12-scene image dataset, compared to the state-of-the-art method.

A complex SIR epidemic dynamical model using nonlinear incidence rate and nonlinear recovery rate is established to consider the impact of available hospital beds and interventions reduction on the spread of infectious disease. Rigorous mathematical results have been established for the model from the point of view of stability and bifurcation. The model has two equilibrium points when the basic reproduction number R 0 > 1 ; a disease-free equilibrium E 0 and a disease endemic equilibrium E 1 . We use LaSalle’s invariance principle and Lyapunov’s direct method to prove that E 0 is globally asymptotically stable if the basic reproduction number R 0 1 , and E 1 is globally asymptotically stable if R 0 > 1 , under some conditions on the model parameters. The existence and nonexistence of limit cycles are investigated under certain conditions on model parameters. The model exhibits Hopf bifurcation near the disease endemic equilibrium. We further show the occurring of backward bifurcation for the model when there is limited number of hospital beds. Finally, some numerical results are represented to validate the analytical results.

The image scene geometry recognition is an important element for reconstructing the 3D scene geometry of a single image. It is useful for computer vision applications, such as 3D TV, video categorization, and robot navigation system. A 3D scene geometry with a unique depth represents a rough structure of 2D images. An approach to efficient implementation and achieving high recognition accuracy of 3D scene geometry remains a significant challenges in the computer vision domain. Existing approaches attempt to use the pre-trained deep convolutional neural networks (CNN) models as feature extractor and also explore the benefits of multi-layer features representation for small or medium-size datasets. However, these studies pay little attention to building a discriminative feature representation by exploring the benefits of low-level features fusion with multi-layer feature from a single CNN model. To address this problem, we propose a novel model of image scene geometry recognition in which the low-level handcrafted features are integrated with deep CNN multi-stage features (HF-MSF) by using the feature-fusion and score-level fusion strategies. The low-level features contain rich discriminative information of 3D scene geometry, including shape, color, and depth estimation. In feature-fusion, the multi-layer features at different stages and handcrafted features are fused at an early phase, and in score-level fusion, the handcrafted features are integrated with multi-layer feature of a single CNN model at different stages and each stage is connected with a classifier and then score-level fusion of these classifiers is performed automatically to recognize the scene geometry type. For validation and comparison purposes, two well-known deep learning architectures, namely GoogLeNet and ResNet are employed as a backbone of proposed model. Experimental results exhibited that by taking the advantages of both types of fusion, the proposed HF-MSF model has an improved recognition accuracy of 12.21% and 4.96% when compared to G-MS2F model for 12-Scene and 15-Scene image datasets, respectively. Similarly, it improves the accuracy by 3.85% when compared with the FTOTLM model for the 15-Scene dataset.

Buruli is a neglected tropical disease that can now be found in many countries including developed countries like Australia. It is a skin disorder that usually occurs in the arms and legs. The disease has been identified in a number of mammals, particularly possum. Adequate eradication and control programs are needed to minimize infection and its spread to less developed countries before it becomes an epidemic. In this work, a SIR type possum epidemic model is proposed. The properties of the model are thoroughly studied and obtained its stability results. We determine the stability of the model at its fixed points and show that the model is locally and globally asymptomatically stable. The stability of the disease-free case is shown for $ \mathcal{R}_0 < 1 $ and the endemic case is examined for $ \mathcal{R}_0 > 1$. We further extend the model using the control variables and obtain an optimal control system and using the optimal control theory to characterize the necessary condition for controlling the spread of Buruli ulcer (BU). The model results are plotted to determine the best strategies for disease elimination. Numerical simulation has shown that the useful strategy consists in implementing all suggested controls.

3D layout extraction of a single image is an important aspect of artificial intelligence (AI) applications, e.g., 3D TV, object localization, and scene understanding. The pixel-level 3D layout of a single image can be extracted by using the prior information of image scene geometry. State-of-the-art Lou’s method introduces predefined templates corresponding to 3D scene geometry which use as prior information to extract the pixel-level 3D layout from a single image. 3D scene geometries called as Stages, are rough models of indoor and outdoor scenes. We introduce a new method of 3D layout extraction by using template-based segmentation as prior information and further template-based-segmentation is used as weighting map in random walk method to enhance the edges between two regions. Next, for each sub-region, multiple seeds are initialized by random walk method and their output segments are combined to generate 3D layout of a single image. Proposed algorithm is evaluated on the indoor and outdoor images dataset. The experimental results show that the proposed method has improved the accuracy by 5.31% versus the state-of-the-art methods. Additionally, this algorithm is extended for road scene extraction where instead of using template-based segmentation, the magnitude value of Gaussian derivative uses as weighting map. It is evaluated on KITTI road scene dataset and quantitatively compared to previous efforts on a publically available, provides superior results.

In ad hoc networks, the communication is usually made through multiple hops by establishing an environment of cooperation and coordination among self-operated nodes. Such nodes typically operate with a set of finite and scarce energy, processing, bandwidth, and storage resources. Due to the cooperative environment in such networks, nodes may consume additional resources by giving relaying services to other nodes. This aspect in such networks coined the situation of noncooperative behavior by some or all the nodes. Moreover, nodes sometimes do not cooperate with others due to their social likeness or their mobility. Noncooperative or selfish nodes can last for a longer time by preserving their resources for their own operations. However, such nodes can degrade the network's overall performance in terms of lower data gathering and information exchange rates, unbalanced work distribution, and higher end-to-end delays. This work surveys the main roots for motivating nodes to adapt selfish behavior and the solutions for handling such nodes. Different schemes are introduced to handle selfish nodes in wireless ad hoc networks. Various types of routing techniques have been introduced to target different types of ad hoc networks having support for keeping misbehaving or selfish nodes. The major solutions for such scenarios can be trust-, punishment-, and stimulation-based mechanisms. Some key protocols are simulated and analyzed for getting their performance metrics to compare their effectiveness.