Boundedness Properties Of Solutions Research Articles

New Results on a Partial Differential Equation with General Piecewise Constant Argument

There have been very few analyses on partial differential equations with piecewise constant arguments and as far as we know, there is no study conducted on heat equation with piecewise constant argument of generalized type. Motivated by this fact, this study aims to solve and analyse heat equation with piecewise constant argument of generalized type. We obtain formal solution of heat equation with piecewise constant argument of generalized type by separation of variables. We apply the Laplace transform method using unit step function and method of steps on each consecutive intervals. We investigate stability, oscillation, boundedness properties of solutions.

Stability and optimal control of a disease model with vertical transmission and saturated incidence

This paper studies a nonlinear SI-model featuring vertical transmission and saturated incidence terms. The model is analyzed by utilizing some dynamical system tools to assess the dynamic behavior of solutions to a system of first order nonlinear ordinary differential equations arising from the spread of a vertically transmitted disease in a population. The mathematical well-posedness of the governing model is proven through positivity and boundedness properties of solutions. Lyapunov functions with LaSalle’s invariance principle are used to establish the global asymptotic dynamics of the system about the disease-free and endemic steady states. Additionally, sensitivity analysis is conducted to show how the epidemiological threshold (basic reproduction number) of the model is affected by variations in parameters. The study is extended to explore the significance of a time-dependent optimal control for blocking vertical transmission route using Pontryagin’s maximum principle of the optimal control theory. The obtained qualitative results are supplemented by numerical simulations. It is shown that incorporation of the optimal control helps in preventing vertical transmission of infection, and in turn diminishes the spread of the disease in the population.

A stochastic delay model of HIV pathogenesis with reactivation of latent reservoirs

Abstract In this paper, we propose a stochastic HIV model with latency to describe the factors that are responsible for extinction and persistence of the infection. This model incorporates latent period of reservoirs that inhibits the eradication of the virus. It is shown that this model leads naturally to a stochastic system of differential delay equations. We derived the global existence, positivity and boundedness properties of solutions of the proposed stochastic system. Analyzing the model, sufficient conditions for the eradication and the persistence of the disease are also obtained. Furthermore, numerical simulations are performed to study the impact of several strategies of drug therapy on the persistence and the extinction of HIV virus. Our results constitute an important step towards control strategies to reduce HIV infection.

Perturbing liapunov functions and ultimate boundedness in terms of two measures

A family of Liapunov functions with each member of the family satisfying weaker requirements is employed to study ultimate boundedness properties of solutions of differential equations in terms of two measures. The generality of the approach is indicated through various remarks