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In this research article, we investigated a coronavirus (COVID-19) epidemic model with random perturbations, which was mainly constituted of five major classes: the susceptible population, the exposed class, the infected population, the quarantine class, and the population that has recovered. We studied the problem under consideration in order to derive at least one, and only one, nonlocal solution within the positive feasible region. The Lyapunov function was used to develop the necessary result of existence for ergodic stationary distribution and the conditions for the disease's extinction. According to our findings, the influence of Brownian motion and noise effects on epidemic transmission were powerful. The infection may diminish or eradicate if the noise is excessive. To illustrate our proposed scheme, we numerically simulated all classes' findings.

Abstract In this paper, we propose a mathematical model to describe the influence of the SARS-CoV-2 virus with correlated sources of randomness and with vaccination. The total human population is divided into three groups susceptible, infected, and recovered. Each population group of the model is assumed to be subject to various types of randomness. We develop the correlated stochastic model by considering correlated Brownian motions for the population groups. As the environmental reservoir plays a weighty role in the transmission of the SARS-CoV-2 virus, our model encompasses a fourth stochastic differential equation representing the reservoir. Moreover, the vaccination of susceptible is also considered. Once the correlated stochastic model, the existence and uniqueness of a positive solution are discussed to show the problem’s feasibility. The SARS-CoV-2 extinction, as well as persistency, are also examined, and sufficient conditions resulted from our investigation. The theoretical results are supported through numerical/graphical findings.

Breast cancer is a frequently encountered and potentially lethal illness that can affect not only women but also men. It is the most common disease affecting women globally, and is the main cause of morbidity and death. Early and accurate detection of this risky disease is crucial. A timely and precise identification of breast cancer can decrease death rate and can also protect people from additional damage. The traditional methods used for breast cancer detection are very expensive in term of time and cost. The goal of this study is to develop a system which can detect the breast cancer accurately and at an early stage. The primary objective of this research study is to make use of histopathological images to identify breast cancer correctly and faster. In the proposed research work, we have developed a model with the name Breast Cancer Detection Network (BCDecNet), which comprises eleven learnable layers, i.e., eight convolution layers and three fully connected (FC) layers. The architecture has a total of twenty-nine layers, including one input layer, seven leaky ReLu (LR) layers, four ReLu layers, five maximum-pooling layers, six batch-normalization (BN) layers, one cross-channel normalization layer and three dropout layers, a softmax layer, and a classification layer. The proposed work uses image-based data taken from the Kaggle online repository. The suggested model achieved 97.33% accuracy, 96% precision, 96.5% recall and a 96.25% F1 score. Furthermore, the result of the proposed model was compared with other hybrid approaches used for diagnosis of breast cancer at early stages. Our model achieved a more satisfactory result than all other approaches used for breast cancer detection. Additionally, the proposed BCDecNet model can be generally applied to other medical-image datasets for diagnosis of various diseases.

The efficiency, temperature distribution, and temperature at the tip of straight rectangular, growing and decaying moving exponential fins are investigated in this article. The influence of internal heat generation, surface and surrounding temperatures, convection-conduction, Peclet number and radiation-conduction is studied numerically on the efficiency, temperature profile, and temperature at the tip of the fin. Differential transform method is used to investigate the problem. It is revealed that thermal and thermo-geometric characteristics have a significant impact on the performance, temperature distribution, and temperature of the fin's tip.The results show that the temperature distribution of decaying exponential and rectangular fins is approximately 15 and 7% higher than growing exponential and rectangular fins respectively. It is estimated that the temperature distribution of the fin increases by approximately 6% when the porosity parameter is increased from 0.1 to 0.5. It is also observed that the decay exponential fin has better efficiency compared to growing exponential fin which offers significant advantages in mechanical engineering.

In this article, we study moving longitudinal and exponential fins with variable thermal conductivity, thermal emissivity, and heat transfer coefficient. The problem is solved numerically using shooting technique. To compare the performance of trapezoidal fins with exponential fins, we introduce a novel idea of volume adjusted fin heat transfer rate, fin effectiveness, and fin efficiency. The result shows that heat transfer from the exponential fin is higher than from the trapezoidal fin. It is observed that the capability of an exponential fin is higher than the trapezoidal fin. With increasing thermal conductivity and radiation-conduction number as well as convection-conduction number, the fin’s efficiency increases. The results illustrate the effect of heat transfer coefficient, thermal emissivity, radiation-conduction and convection-conduction parameters, dimensionless radiation, and convection sink temperatures on the fin’s performance.

Abstract In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form.

Abstract The performance and thermal properties of convective–radiative rectangular and moving exponential porous fins with variable thermal conductivity together with internal heat generation are investigated. The second law of thermodynamics is used to investigate entropy generation in the proposed fins. The model is numerically solved using shooting technique. It is observed that the entropy generation depends on porosity parameter, temperature ratio, temperature distribution, thermal conductivity and fins structure. It is noted that entropy generation for a decay exponential fin is higher than that of a rectangular fin which is greater than that of a growing exponential fin. Moreover, entropy generation decreases as thermal conductivity increases. The results also reveal that entropy generation is maximum at the fin’s base and the average entropy production depends on porosity parameters and temperature ratio. It is further reveal that the temperature ratio has a smaller amount of influence on entropy as compared to porosity parameter. It is concluded that when the temperature ratio is increases from 1.1 to 1.9, the entropy generation number is also increase by $$30\%$$ 30 % approximately. However, increasing porosity from 1 to 80 gives 14-fold increase in average entropy generation.

In the present manuscript, an age-structured heroin epidemic model is formulated with the assumption that susceptibility and recovery are age-dependent. Keeping in view some control measures for heroin addiction, a control problem for further analysis is presented. The main results are the existence of control variables, sensitivities, adjoint system and the setting of an optimal control problem. We used the techniques of weak derivatives and a general principle of Pontryagin's type for obtaining the optimal control problem. To compare our results, we demonstrated sample simulations which show the effect of control on the entire population.

Abstract Main concern of current research is to develop a novel mathematical model for stagnation-point flow of magnetohydrodynamic (MHD) Prandtl–Eyring fluid over a stretchable cylinder. The thermal radiation and convective boundary condition are also incorporated. The modeled partial differential equations (PDEs) with associative boundary conditions are deduced into coupled non-linear ordinary differential equations (ODEs) by utilizing proper similarity transformations. The deduced dimensionless set of ODEs are solved numerically via shooting method. Behavior of controlling parameters on the fluid velocity, temperature fields as well as skin friction and Nusselt number are highlighted through graphs. Outcome declared that dimensionless fluid temperature boosts up for both the radiation parameter and Biot number. It is also revealed that the magnitude of both heat transfer rate and skin friction enhance for higher estimation of curvature parameter. Furthermore, comparative analysis between present and previous reports are provided for some specific cases to verify the obtained results.

Abstract Infection of hepatitis B virus (HBV) is a global health problem. We provide the study about hepatitis B virus dynamics that can be controlled by education campaign (awareness), vaccination, and treatment. Initially we bring constant controls in considerations for treatment, vaccination, and education campaign (awareness). In the case of constant controls, we study the stability and existence of the disease-free and endemic equilibria model’s solutions. Afterwards, we take time as a control and formulate the suitable optimal control problem, acquire optimal control strategy in order to reduce the number of humans that are infected and the costs associated. At the end, results of numerical simulations show that the optimal combination of education campaign (awareness), treatment, and vaccination is the most efficient way to control the infection of hepatitis B virus (HBV) infection.

The current work is devoted to bring out a detail analysis including qualitative and semi-analytical study of Hepatitis B model under the Caputo- Fabrizio fractional derivative (CFFD). For the required results, fixed point theory is used to establish the conditions for existence and uniqueness of solution to the considered model. On the other hand, for semi analytical solutions, we use decomposition method of Adomian coupled with integral transform of Laplace. Moreover, the concerned solutions are presented via graphs to analyze the dynamics of different compartments of the model.

Novel coronavirus disease is a burning issue all over the world. Spreading of the novel coronavirus having the characteristic of rapid transmission whenever the air molecules or the freely existed virus includes in the surrounding and therefore the spread of virus follows a stochastic process instead of deterministic. We assume a stochastic model to investigate the transmission dynamics of the novel coronavirus. To do this, we formulate the model according to the charectersitics of the corona virus disease and then prove the existence as well as the uniqueness of the global positive solution to show the well posed-ness and feasibility of the problem. Following the theory of dynamical systems as well as by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions of the extinction and the existence of stationary distribution. Finally, we carry out the large scale numerical simulations to demonstrate the verification of our analytical results.

The pandemic of SARS-CoV-2 virus remains a pressing issue with unpredictable characteristics which spread worldwide through human interactions. The current study is focusing on the investigation and analysis of a fractional-order epidemic model that discusses the temporal dynamics of the SARS-CoV-2 virus in a community. It is well known that symptomatic and asymptomatic individuals have a major effect on the dynamics of the SARS-CoV-2 virus therefore, we divide the total population into susceptible, asymptomatic, symptomatic, and recovered groups of the population. Further, we assume that the vaccine confers permanent immunity because multiple vaccinations have commenced across the globe. The new fractional-order model for the transmission dynamics of SARS-CoV-2 virus is formulated via the Caputo-Fabrizio fractional-order approach with the maintenance of dimension during the process of fractionalization. The theory of fixed point will be used to show that the proposed model possesses a unique solution whereas the well-posedness (bounded-ness and positivity) of the fractional-order model solutions are discussed. The steady states of the model are analyzed and the sensitivity analysis of the basic reproductive number is explored. Moreover to parameterize the model a real data of SARS-CoV-2 virus reported in the Sultanate of Oman from January 1st, 2021 to May 23rd, 2021 are used. We then perform the large scale numerical findings to show the validity of the analytical work.

In this article, a non-integer nonlinear mathematical model for toxoplasmosis disease in human and cat population is proposed and studied. The basic concepts of the model's dynamic are given. The study of qualitative dynamics is done by the basic threshold parameter $ R_{0} $ . Local and global stabilities are done and the system's disease free equilibrium point is an attractor when $ R_{0}1 $ . The sensitivity analysis of $ R_{0} $ shows which parameter has positive/negative impact on the model. Numerical simulation of the model for the parameters occurred in threshold parameter is also discussed. The techniques of Adams Bashforth Moulton will be considered to justify all the derived theoretical results which will help in understanding to study the effect of various parameters to both the transient and steady-state dynamics of the disease infection.

This paper represents a non-integer-order Vallis systems in which we applied the Grünwald-Letnikov tactics with Binomial coefficients in order to realize the numerical simulations to a set of equations. Recently researchers reported in the literature that it is the generalization of integer order dynamical model. Several cases involving non-integer and integer analysis with different values of non-integer order have been applied to Vallis systems to see the behavior of simulations. To visualize the effect of non-integer order approach, the time histories and phase portraits have been plotted. The consequences expose that the non-integer-order Vallis model can reveal a genuine equitable comportment to Vallis systems and might bid greater perceptions towards the understanding of such complex dynamic systems.

In this article, a stochastic SACR hepatitis B epidemic model is taken to be under consideration. We develop a stochastic epidemic model by considering the effect of environmental fluctuation on the hepatitis B dynamics and distribute the transmission rate by white noise. Using the stochastic Lyapunov function theory, we have shown the existence and uniqueness of the global positive solution. The extinction and persistence for our proposed model are derived with sufficient conditions. The numerical simulations are carried out using first-order Itô-Taylor stochastic scheme in the last section for the verification of our theoretical results.

We propose a theoretical study to investigate the spread of the SARS-CoV-2 virus, reported in Wuhan, China. We develop a mathematical model based on the characteristic of the disease and then use fractional calculus to fractionalize it. We use the Caputo-Fabrizio operator for this purpose. We prove that the considered model has positive and bounded solutions. We calculate the threshold quantity of the proposed model and discuss its sensitivity analysis to find the role of every epidemic parameter and the relative impact on disease transmission. The threshold quantity (reproductive number) is used to discuss the steady states of the proposed model and to find that the proposed epidemic model is stable asymptotically under some constraints. Both the global and local properties of the proposed model will be performed with the help of the mean value theorem, Barbalat’s lemma, and linearization. To support our analytical findings, we draw some numerical simulations to verify with graphical representations.

Middle East respiratory syndrome coronavirus (MERS-CoV) remains an emerging disease threat with regular human cases on the Arabian Peninsula driven by recurring camels to human transmission events. In this paper, we present a new deterministic model for the transmission dynamics of (MERS-CoV). In order to do this, we develop a model formulation and analyze the stability of the proposed model. The stability conditions are obtained in term of R0, we find those conditions for which the model become stable. We discuss basic reproductive number R0 along with sensitivity analysis to show the impact of every epidemic parameter. We show that the proposed model exhibits the phenomena of backward bifurcation. Finally, we show the numerical simulation of our proposed model for supporting our analytical work. The aim of this work is to show via mathematical model the transmission of MERS-CoV between humans and camels, which are suspected to be the primary source of infection.

In this paper, we formulate a stochastic model for hepatitis B virus transmission with the effect of fluctuation environment. We divide the total population into four different compartments, namely, the susceptible individuals in which the disease transmission rate is distributed by white noise, the acutely infected individuals in which the same perturbation occur, the chronically infected individuals and the recovered individuals. We use the stochastic Lyapunov function theory to construct a suitable stochastic Lyapunov function for the existence of positive solution. We also then establish the sufficient conditions for the hepatitis B extinction and the hepatitis B persistence. At the end numerical simulation is carried out by using the stochastic Runge–Kutta method to support our analytical findings.

In this article, the Adomian decomposition method (ADM) is applied to find the solution of HIV infection model of latently infected CD4+T cells. This method investigates the solution of ordinary differential equation which is calculated in the form of the components of an infinite series. These components can be easily calculated. The efficiency and the reliability of proposed method is demonstrated in different time intervals by numerical example. The derived results indicate that the approximate solution by using the ADM can be obtained in a more efficient way. All computations have been carried out by computer code written in Mathematica.

Middle East respiratory syndrome coronavirus has been persistent in the Middle East region since 2012. In this paper, we propose a deterministic mathematical model to investigate the effect of media coverage on the transmission and control of Middle Eastern respiratory syndrome coronavirus disease. In order to do this we develop model formulation. Basic reproduction number R0 will be calculated from the model to assess the transmissibility of the (MERS-CoV). We discuss the existence of backward bifurcation for some range of parameters. We also show stability of the model to figure out the stability condition and impact of media coverage. We show a special case of the model for which the endemic equilibrium is globally asymptotically stable. Finally all the theoretical results will be verified with the help of numerical simulation for easy understanding.

In this article we propose a stochastic model to discuss the dynamics of novel corona virus disease. We formulate the model to study the long run behavior in varying population environment. For this purposes we divided the total human population into three epidemiological compartments: the susceptible, covid-19 infected, recovered and recovered along with one class of reservoir. The existence and uniqueness of the newly formulated model will be studied to show the well-possedness of the model. Moreover, we investigate the extinction analysis as well as the persistence analysis to find the disease extinction and disease persistence conditions. At the end we perform simulation to justify the investigation of analytical work with the help of graphical representations.

In this work, we propose a mathematical model to analyze the outbreak of the Coronavirus disease (COVID-19). The proposed model portrays the multiple transmission pathways in the infection dynamics and stresses the role of the environmental reservoir in the transmission of the disease. The basic reproduction number R0is calculated from the model to assess the transmissibility of the COVID-19. We discuss sensitivity analysis to clarify the importance of epidemic parameters. The stability theory is used to discuss the local as well as the global properties of the proposed model. The problem is formulated as an optimal control one to minimize the number of infected people and keep the intervention cost as low as possible. Medical mask, isolation, treatment, detergent spray will be involved in the model as time dependent control variables. Finally, we present and discuss results by using numerical simulations.

Nowadays in the world, the biggest problem is how to control the transmittable disease. Mathematical biology is one of the useful areas to study and investigate the control of transmittable diseases. In the current study, we developed a mathematical model which reflects the dynamics of hepatitis B virus by using the tools of stochastic modeling and particularly considering the environmental noise. For a better understanding of the disease, we divide the whole population into four disjoint classes. The main focus of the study is to show the effect of stochasticity on the dynamics of the disease as well as to obtain the required threshold parameter. The stochastic Lyapunov function is formulated, whose analysis shows that there exists a positive solution to the model. The study was further enriched by investigating the sufficient conditions of hepatitis B extinction and stationary distribution. In order to support theoretical results, at the end of the study, we present sample simulations of the model by using the stochastic Runge-Kutta method.

We investigate and analyze the dynamics of hepatitis B with various infection phases and multiple routes of transmission. We formulate the model and then fractionalize it using the concept of fractional calculus. For the purpose of fractionalizing, we use the Caputo–Fabrizio operator. Once we develop the model under consideration, existence and uniqueness analysis will be discussed. We use fixed point theory for the existence and uniqueness analysis. We also prove that the model under consideration possesses a bounded and positive solution. We then find the basic reproductive number to perform the steady-state analysis and to show that the fractional-order epidemiological model is locally and globally asymptotically stable under certain conditions. For the local and global analysis, we use linearization, mean value theorem, and fractional Barbalat’s lemma, respectively. Finally, we perform some numerical findings to support the analytical work with the help of graphical representations.

Jordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, $x^{3}y=y^{3}x\ \forall \ x,y\in Q$ and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.

We propose an epidemic model for the transmission of hepatitis B virus along with the classification of different infection phases and hospitalized class. We formulate the model and discuss its basic mathematical properties, e.g. existence, positivity, and biological feasibility. Exploiting the next generation matrix approach, we find the basic reproductive number of the model. We perform sensitivity analysis to illustrate the effect of various parameters on the transmission of the disease. We investigate stability of the equilibria of the model in terms of the basic reproduction number. Conditions for the stability of the proposed model are obtained using various approaches. Finally, we perform the numerical simulations to discuss sensitivity analysis and to support our analytical work.

This study proposes a mathematical model of Anthroponotic visceral leishmaniasis epidemic with saturated infection rate and recommends different control strategies to manage the spread of this disease in the community. To do this, first, a model formulation is presented to support these strategies, with quantifications of transmission and intervention parameters. To understand the nature of the initial transmission of the disease, the reproduction number $ R_0 $ is obtained by using the next-generation method. On the basis of sensitivity analysis of the reproduction number $ R_0 $ , four different control strategies are proposed for managing disease transmission. For quantification of the prevalence period of the disease, a numerical simulation for each strategy is performed and a detailed summary is presented. Disease-free state is obtained with the help of control strategies. The threshold condition for globally asymptotic stability of the disease-free state is found, and it is ascertained that the state is globally stable. On the basis of sensitivity analysis of the reproduction number, it is shown that the disease can be eradicated by using the proposed strategies.

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number $ R_0 $ . We show that if $ R_0\leq 1, $ then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if $ R_0>1 $ . To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.

This paper presents some new exact solutions corresponding to unsteady fractional Jeffrey fluid produced by a flat plate between two side walls perpendicular to the plate. The fractional calculus approach in the governing equations is used. The exact solutions are established by means of the Fourier sine transform and discrete Laplace transform. The series solution of velocity field and the associated shear stress in terms of Fox H-functions, satisfying all imposed initial and boundary conditions, have been obtained.

This article studies the dynamical behavior of the analytical solutions of the system of fraction order model of HIV-1 infection. For this purpose, first, the proposed integer order model is converted into fractional order model. Then, Laplace-Adomian decomposition method (L-ADM) is applied to solve this fractional order HIV model. Moreover, the convergence of this method is also discussed. It can be observed from the numerical solution that (L-ADM) is very simple and accurate to solve fraction order HIV model.

New exact solutions for unsteady magnetohydrodynamic (MHD) flows of a generalized second-grade fluid due to uniform accelerating plate are derived. The generalized second-grade fluid saturates the porous space. Fractional derivative is used in the governing equation. The analytical expressions for velocity and shear stress fields are obtained by using Laplace transform technique for the fractional calculus. The obtained solutions are expressed in series form in terms of Fox H-functions. Similar solutions for ordinary second-grade fluid passing through a porous space are also recovered. Moreover, several figures are sketched for the pertinent parameters to analyze the characteristics of velocity field and shear stress.

Right nuclear square loops are loops satisfying y zz))==(xy)(zz). We construct an infinite family of non-associative non-commutative right nuclear square loops whose smallest member is of order 12.

We study the heat transfer of a Magneto Hydrodynamic (MHD) boundary layer flow of a Newtonian fluid over a pervious shrinking sheet under the influence of thermal slip. The flow allows electric current to pass through. The governing PDEs are transformed into self-similar ODEs via Lie group analysis. We study the variations in the dimensionless quantities like velocity and temperature of the flow in terms of the different parameters involved in the problem. We discuss the thickness of the boundary layers under the influence of various parameters involved in the flow. Numerical simulations are carried out to explain and support the results obtained.

In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.

Hepatitis B infection caused by the hepatitis B virus. It is one of the serious viral infection and a global health problem. In the transmission of hepatitis B infection different phases, i.e., acute and chronic carrier stages play an important role. The chronic carries individuals do not exhibit any symptoms and are able to transmit the infection. Here we assessed the transmissibility associated with different infection stages of hepatitis B and generated an epidemic model with nonlinear incidence rate. In order to do this, first we formulate the model by splitting the infectious class into two subclasses, namely acutely infected and chronic carries with both horizontal and vertical transmission. The basic properties of the proposed model are presented. The basic reproductive number is obtained by using the next generation matrix approach. Biological sense of the threshold condition is investigated and discussed in detail. We also find the conditions to investigate all possible equilibria of the model in terms of the basic reproduction number. Finally, we perform numerical simulations to support our analytical work.

In this paper, we studied MHD two dimensional flow of an incompressible tangent hyperbolic fluid flow and heat transfer towards a stretching sheet with velocity and thermal slip. Lie group analysis is used to develop new similarity transformation, using these similarity transformation the governing nonlinear partial differential equation are reduced into a system of coupled nonlinear ordinary differential equation. The obtained system is solved numerically by applying shooting method. Effects of pertinent parameters on the velocity and temperature profiles, skin friction, local Nusselt number are graphically presented and discussed. Comparison between the present and previous results are shown in special cases.

There have been anecdotal reports of tuberculous cardiac involvement, mainly in cases of military tuberculosis or immune deficient individuals. The spectrum of clinical presentations of tuberculous cardiac involvements includes incidental detection of single and multiple well-circumscribed tuberculomas, symptomatic obstructive lesions, AV conduction abnormalities, and even sudden death. We present a case of cardiac tuberculoma in an immune-competent person who presented with worsening dyspnea. The unique morphology of this mass posed an imaging challenge that required 4-dimensional (4D) echocardiography and cardiac magnetic resonance (CMR) detail to differentiate the mass from an anterior mitral leaflet (AML) aneurysm. Histological examination after surgical resection confirmed its tuberculous etiology.

In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider the generalized Euler method (GEM) and multi-step generalized differential transform method (MSGDTM). We also show the unique positive solution for this model and present numerical results graphically.

[This corrects the article DOI: 10.1371/journal.pone.0103617.].

This paper is focused on the transmission dynamics and optimal control of Anthroponotic Cutaneous Leishmania. The threshold condition R0 for initial transmission of infection is obtained by next generation method. Biological sense of the threshold condition is investigated and discussed in detail. The sensitivity analysis of the reproduction number is presented and the most sensitive parameters are high lighted. On the basis of sensitivity analysis, some control strategies are introduced in the model. These strategies positively reduce the effect of the parameters with high sensitivity indices, on the initial transmission. Finally, an optimal control strategy is presented by taking into account the cost associated with control strategies. It is also shown that an optimal control exists for the proposed control problem. The goal of optimal control problem is to minimize, the cost associated with control strategies and the chances of infectious humans, exposed humans and vector population to become infected. Numerical simulations are carried out with the help of Runge-Kutta fourth order procedure.

Left alternative loops are loops satisfying (xy)=(xx)y. We construct an infinite family of non-associative non-commutative left alternative loops whose smallest member is of order

We present the prevention of avian influenza pandemic by adjusting multiple control functions in the human-to-human transmittable avian influenza model. First we show the existence of the optimal control problem; then by using both analytical and numerical techniques, we investigate the cost-effective control effects for the prevention of transmission of disease. To do this, we use three control functions, the effort to reduce the number of contacts with human infected with mutant avian influenza, the antiviral treatment of infected individuals, and the effort to reduce the number of infected birds. We completely characterized the optimal control and compute numerical solution of the optimality system by using an iterative method.

We present the prevention of influenza pandemic by using multiple control functions. First, we adjust the control functions in the pandemic model, then we show the existence of the optimal control problem, and, by using both analytical and numerical techniques, we investigate cost-effective control effects for the prevention of transmission of disease. To do this, we use four control functions, the first one for increasing the effect of vaccination, the second one for the strategies to isolate infected individuals, and the last two for the antiviral treatment to control clinically infectious and hospitalization cases, respectively. We completely characterized the optimal control and compute the numerical solution of the optimality system by using an iterative method.

Although Lrp5 is known to be an important contributor to the mechanisms regulating bone mass, its precise role remains unclear. The aim of this study was to establish whether mutations in Lrp5 are associated with differences in the growth and/or apoptosis of osteoblast-like cells and their proliferative response to mechanical strain in vitro. Primary osteoblast-like cells were derived from cortical bone of adult mice lacking functional Lrp5 (Lrp5(-/-)), those heterozygous for the human G171V High Bone Mass (HBM) mutation (LRP5(G171V)) and their WT littermates (WT(Lrp5), WT(HBM)). Osteoblast proliferation over time was significantly higher in cultures of cells from LRP5(G171V) mice compared to their WT(HBM) littermates, and lower in Lrp5(-/-) cells. Cells from female LRP5(G171V) mice grew more rapidly than those from males, whereas cells from female Lrp5(-/-) mice grew more slowly than those from males. Apoptosis induced by serum withdrawal was significantly higher in cultures from Lrp5(-/-) mice than in those from WT(HBM) or LRP5(G171V) mice. Exposure to a single short period of dynamic mechanical strain was associated with a significant increase in cell number but this response was unaffected by genotype which also did not change the 'threshold' at which cells responded to strain. In conclusion, the data presented here suggest that Lrp5 loss and gain of function mutations result in cell-autonomous alterations in osteoblast proliferation and apoptosis but do not alter the proliferative response of osteoblasts to mechanical strain in vitro.

An existing model is extended to assess the impact of some antimalaria control measures, by re-formulating the model as an optimal control problem. This paper investigates the fundamental role of three type of controls, personal protection, treatment, and mosquito reduction strategies in controlling the malaria. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. The optimality system is solved numerically by a competitive Gauss-Seidel-like implicit difference method. Finally, numerical simulations of the optimal control problem, using a set of reasonable parameter values, are carried out to investigate the effectiveness of the proposed control measures.