Anwar Zeb

On fuzzy fully regular ordered

We have introduced and characterized a fully regular (V-regular) class of an ordered AG-groupoid in terms of fuzzy (left, right, two-sided, interior, bi-, generalized biand quasi) ideals. We have shown that all these fuzzy ideals need not to be coincide in an ordered AG-groupoid S even if S has a left identity but on the other hand, it has been shown that all these fuzzy ideals coincide in a V-regular ordered AG-groupoid with a left identity.

\mathcal{AG}

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.

-groupoids

This paper deals with the optimal control problem in the giving up smoking model of fractional order. For the eradication of smoking in a community, we introduce three control variables in the form of education campaign, anti-smoking gum, and anti-nicotive drugs/medicine in the proposed fractional order model. We discuss the necessary conditions for the optimality of a general fractional optimal control problem whose fractional derivative is described in the Caputo sense. In order to do this, we minimize the number of potential and occasional smokers and maximize the number of ex-smokers. We use Pontryagin’s maximum principle to characterize the optimal levels of the three controls. The resulting optimality system is solved numerically by MATLAB.

Approximating a Giving Up Smoking Dynamic on Adolescent Nicotine Dependence in Fractional Order.

We show the erroneous assumptions and reasoning by introducing the migration effect of individuals in the transmission model of Hepatitis B virus. First, some false results related to the eigenvalues and reproductive number in the recent literature in mathematical biology will be presented. Then, it will be proved that the product of the matrices in the next generation method to obtain the reproductive number R 0 is not correct and the local and global stability results based on the reproductive number R 0 are considered false.

Optimal Campaign Strategies in Fractional-Order Smoking Dynamics

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.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.

Comment on “Transmission Model of Hepatitis B Virus with Migration Effect”

In this paper, we discuss the asymptotic behavior of the optimal condition to control Ebola virus by considering the SEIRS model. First, we show stability and dynamical behavior of the model. Then, we find the conditions on parameters used in model which would minimize number of infected individuals. Finally, we give the graphical representations for different data.

Dynamical analysis of cigarette smoking model with a saturated incidence rate

In a recent paper (Zeb et al. in Appl Math Model 37(7):5326–5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge–Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

Optimal Conditions for the Control of Ebola Viral Disease

Mathematical modeling has become a valuable tool for the analysis of dynamics of infectious disease and for the support of control strategies development in recent years. This work highlights the conceptual ideas and mathematical tools needed for infectious diseases modeling.Themain convergence of this was on the dynamics of infectious diseases, the analysis of transmission patterns in various populations, and methods to assess the effectiveness of control strategies such as HIV, childhood infections, influenza, and vector borne infections. It was concerned with qualitative behaviors of infectious diseasemodel.The qualitative behavior ofmodel includes positivity, uniqueness, local stability, global stability, bifurcation analysis, control of diseases, and existence of solutions.This study provided a platform for the discussion of the major research challenges and achievements on qualitative behaviors of infectious diseases and their control. Due to the availability of a lot of applications of this study, many authors contributed. G. R. Phaijoo and D. B. Gurung demonstrated that dengue is spreading in new areas due to people movement. They considered a multipatch model to assess the impact of temperature and human movement on the transmission dynamics of dengue disease. Dynamics of vector and host populations are investigated with different humanmovement rates and different temperature levels. N. Pipatsart et al. discussed adaptive random network models to describe human behavioral change during epidemics and performed stochastic simulations of SIR epidemic models on adaptive random networks. C. Burgess et al. derived risk-based immunization by deployment to polio-endemic regions, which is sufficient to prevent transmission among both deployed and nondeployed US military populations. T. Tilahun et al. analyzed a compartmental nonlinear deterministic mathematical model for the typhoid fever outbreak and optimal control strategies in a community with varying population. L. Worden et al. extended an approximation technique of the long-term behavior of a supercritical stochastic epidemic model, using the WKB approximation and a Hamiltonian phase space, to the subcritical case. S. Anis et al. discussed some of the genetic properties on the basis of algebra. Finally, A. Miao et al. proposed a stochastic SIR model with vertical transmission and vaccination.They showed that large noise can lead to the extinction of infectious diseases, which is conducive to epidemic diseases control.

Comparing Two Numerical Methods for Approximating a New Giving Up Smoking Model Involving Fractional Order Derivatives

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

Mathematical Modeling and Control of Infectious Diseases

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model by taking into account both standard and bilinear incidence rates of fractional order. First, the nonnegative solution of the SEIR model of fractional order is presented. Then, the multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. Finally, the obtained results are compared with those obtained by the fourth-order Runge-Kutta method and non-standard finite difference (NSFD) method in the integer case.