Gul Zaman

A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives

Smoking is one of the main causes of health problems and continues to be one of the worlds most significant health challenges. In this paper, the dynamics of a giving up smoking model containing fractional derivatives is studied numerically. The multistep generalized differential transform method (for short MSGDTM) is employed to compute accurate approximate solutions to a giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

Presentation of Malaria Epidemics Using Multiple Optimal Controls

An existing model is extended to assess the impact of some antimalaria control measures, by reformulating 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.

Global Stability of Vector-Host Disease with Variable Population Size

The paper presents the vector-host disease with a variability in population. We assume, the disease is fatal and for some cases the infected individuals become susceptible. We first show the local and global stability of the disease-free equilibrium, for the case when R 0 < 1. We also show that for R 0 < 1, the disease free-equilibrium of the model is both locally as well as globally stable. For R 0 > 1, there exists a unique positive endemic equilibrium. For R 0 > 1, the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for R 0 > 1. Numerical results are presented for the justifications of theoratical results.

The transmission dynamic and optimal control of acute and chronic hepatitis B

ABSTRACT 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 . We show that if 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 . 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.

Prevention of Influenza Pandemic by Multiple Control Strategies

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.

Optimal strategy of vaccination & treatment in an SIR epidemic model

In this work, we propose a susceptibleinfectedrecovered (SIR) epidemic model which describes the interaction between susceptible and infected individuals in a community and analyze the SIR epidemic model through the optimal control theory and mathematical analysis. In addition, we present some possible strategies to prevent the spread of some infection causing epidemic in the society. In order to do this, we introduce an optimal control problem with an objective functional, where two control functions, vaccination and treatment have been used as control measures for susceptible and infected individuals. We show the existence of an optimal control pair for the optimal control problem and derive the optimality condition. Finally we consider a smoking epidemic model to illustrate our theoretical results with some numerical simulations, which use real data collected in April and May 2004 from 300 male students at three vocational technical high schools in Korean metropolitan areas. Our analysis suggests that two control strategies are more effective than only one control strategy in controlling the increase of male student smokers in Korean metropolitan areas.

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

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.

Dynamical aspects of an age-structured SIR endemic model

The main idea of this work is to present and study the dynamical behavior of an age-dependent SIR endemic model. First, the age-dependent SIR endemic model is formulated from existing SIR epidemic models by adding age-dependent recruitment rate. The behavior of the model is analyzed by using the basic reproductive number R 0 . The stability theory is used for the analysis of disease-free and endemic equilibrium. Finally, finite difference method of characteristics is used to present numerical results for the suggested age-structured endemic model.

Lie group analysis of magnetohydrodynamic tangent hyperbolic fluid flow towards a stretching sheet with slip conditions

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.

Asymptotic behavior of HIV-1 epidemic model with infinite distributed intracellular delays

In this study, asymptotic analysis of an HIV-1 epidemic model with distributed intracellular delays is proposed. One delay term represents the latent period which is the time when the target cells are contacted by the virus particles and the time the contacted cells become actively infected and the second delay term represents the virus production period which is the time when the new virions are created within the cell and are released from the cell. The infection free equilibrium and the chronic-infection equilibrium have been shown to be locally asymptotically stable by using Rouths Hurwiths criterion and general theory of delay differential equations. Similarly, by using Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction ratio is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable. Finally, numerical results with conclusion are discussed.