Theoretical study of diffusive model of HIV-1 infection and its analytical solution.

T his article studied a mathematical model for the diffusive human immunodeficiency virus-type 1 (HIV-1) infection combining with stem cell therapy. The HIV-1 infection is a chronic disease and the viral replication continues if the patient stopes use the antiretroviral therapy (cART). Therefore, it is important to seek the cure of HIV-1 infection and some medical trials showed the cure by stem cell therapy and there are others failure to achieve the cure of HIV-1 with same treatments. The novelty of this paper is constructing a mathematical model with adding diffusion terms to study the effect of spread of virus and other cells in the body. Theoretical analysis such as boundedness, positivity, stability (local/global) of the HIV-1 model is presented. The model is solved analytically by the tanh expansion method. The results show that the tanh expansion method is a very useful technique, that can give a good prediction of the effect of stem cell therapy on infected cells on the spread of the virus. The results further demonstrated that the best way to control the disease is by limiting the spread of the virus; more so than the spread of other components.

Theoretical and semi-analytical simulation for a two-predator-one-prey model during the mating period.

The article introduces a new application which is a system of equations of two predators and one prey with the term of interaction between male and female of predators and prey. Such term appears when male and female of predators feed on the same prey during their mating period. The mathematical model has been studied theoretically and semi-analytically. The positivity, boundedness, local and global stability are proved for the system. The logarithm of multistage differential transform method (MsDTM) is used to study this new application. The MsDTM is used because it globally converges to the solution, it is a highly accurate, fast and simple approach. The stability analysis as well as semi-analytical solutions of the system are obtained to understand the dynamic of the model. Moreover, the effects of several parameters in the system are presented. As a results, we obtain the periodic solution when when the growth rate of prey is larger than the growth rate of both type of predators.

On fractional numerical simulation of HIV infection for CD8+ T-cells and its treatment.

The AIDS is a chronic disease and the researchers still exert their high efforts to reach the cure of HIV infection. The most common treatment is the antiretroviral therapy (cART) and the virus can be more effected if the patients stop using cART. The other problem is that the CD8+ T cells might be exhausted by persistent immune activation by cART. This paper introduces fractional-order into a mathematical model of HIV infection combining with stem cell therapy and control the infection by the immune system cells (CD8+ T cells). The paper introduced the numerical solutions for the mathematical model. The results show that the stem cell therapy with the activation of immune system cells might causes the cure for a HIV patient. This results are consistent with medical studies. Also, we proposed the effect of the fractional order (α) on the figures of the components.

Exponential Time Differencing Method for Studying Prey-Predator Dynamic during Mating Period.

This paper addresses a first numerical simulation to the nonlinear dynamic system of equations that describes the prey-predator model at the predator mating period. Some male species accompany the females during the mating period. In this case, both male and female feed on the same prey. The presented work shows the numerical solution for this specific case of the prey-predator mathematical model via an exponential time differencing method. In addition, the paper provides the biological implication of the solution.

Simulation study on nonlinear structures in nonlinear dispersive media.

In this work, the dynamic mechanism scenario of nonlinear electrostatic structures (unmodulated and modulated waves) that can propagate in multi-ion plasmas with the mixture of sulfur hexafluoride and argon gas is reported. For this purpose, the fluid equations of the multi-ion plasma species are reduced to the evolution (nonplanar Gardner) equation using the reductive perturbation technique. Until now, it has been known that the solution of nonplanar Gardner equation is not possible and for stimulating our data, it will solve numerically. At that point, the present study is divided into two parts: the first one is analyzing planar and nonplanar Gardner equations using the Adomian decomposition method (ADM) for investigating the unmodulated structures such as solitary waves. Moreover, a comparison between the analytical and numerical simulation solutions for the planar Gardner equation is examined, showing how powerful the ADM is in finding solutions in the short domain as well as its fast convergence, i.e., the approximate solution is consistent with the analytical solution for the planar Gardner equation after a few iterations. Second, the modulated envelope structures such as freak waves (FWs) are investigated in the framework of the Gardner equation by transforming this equation to the nonlinear Schrödinger equation (NLSE). Again, the ADM is used to solve the NLSE for studying FWs numerically. Furthermore, the effect of physical parameters of the plasma environment (e.g., Ar+-SF5 +-F--SF5 - plasma) on the characteristics of the nonlinear pulse profile is elaborated. These results help in a better understanding of the fundamental mechanisms of fluid physics governing the plasma processes.

Analytical Solutions of a Modified Predator-Prey Model through a New Ecological Interaction.

The predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animal population within a certain phenomenon. Due to the sexual interaction of the predator in the mating period, the males and females feed together on one or more preys. This scenario describes the ecological interaction between two predators and one prey. In this study, the nonlinear diffusive predator-prey model is presented where this type of interaction is accounted for. The influence of this interaction on the population of predators and preys is predicted through analytical solutions of the dynamical system. The solutions are obtained by using two reliable and simple methods and are presented in terms of hyperbolic functions. In addition, the biological relevance of the solutions is discussed.