In LaSalle's book "The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on ℝ is a global attractor, and proposes to study the natural extensions of these conditions in ℝⁿ. Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor...

A model of viral infection of monocytes population by the dengue virus is formulated as a system of four ordinary differential equations. The model takes into account the immune response and nonlinear incidence rate of susceptible and free virus particles. Global stability of the uninfected steady state is investigated. Such a steady state always exists. If it is the only steady state, then it is globally asymptotically stable. If any infected steady state exists, then the uninfected...

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique....

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB is diabetes (DM), which is predicted to increase dramatically over the next two decades, particularly in low and middle income countries, where TB is widespread. This study aimed to assess the strength of the association between TB and DM. We present a deterministic model for TB in a community in order to determine the impact of DM in the spread of the disease....

In this article a method is presented to find systematically the domain of attraction (DOA) of hybrid non-linear systems. It has already been shown that there exists a sequence of special kind of Lyapunov functions $V_n$ in a rational functional form approximating a maximal Lyapunov function $V_M$ that can be used to find an estimation for the DOA. Based on this idea, an improved method has been developed and implemented in a Mathematica-package to find such Lyapunov functions $V_n$ for a class of hybrid (piecewise...

An impulsive differential equation with time varying delay is proposed in this paper. By using some analysis techniques with combination of coincidence degree theory, sufficient conditions for the permanence, the existence and global attractivity of positive periodic solution are established. The results of this paper improve and generalize some previously known results.

A Volterra model with mutual interference concerning integrated pest management is proposed and analyzed. By using Floquet theorem and small amplitude perturbation method and comparison theorem, we show the existence of a globally asymptotically stable pest-eradication periodic solution. Further, we prove that when the stability of pest-eradication periodic solution is lost, the system is permanent and there exists a locally stable positive periodic solution which arises from the pest-eradication...

A Volterra model with mutual interference concerning integrated pest management is proposed and analyzed. By using Floquet theorem and small amplitude perturbation method and comparison theorem, we show the existence of a globally asymptotically stable pest-eradication periodic solution. Further, we prove that when the stability of pest-eradication periodic solution is lost, the system is permanent and there exists a locally stable positive periodic solution which arises from the pest-eradication...

This paper presents mathematical models for tuberculosis and its dynamics under the implementation of the direct observation therapy strategy (DOTS) in Nigeria. The models establish conditions for the eradication of tuberculosis in Nigeria based on the fraction of detected infectious individuals placed under DOTS for treatment. Both numerical and qualitative analysis of the models were carried out and the effect of the fraction of detected cases of active TB on the various epidemiological classes...

We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio...

Sufficient conditions are established for the global stability of solutions of certain third-order nonlinear differential equations. Our result improves on Tunc’s [10].

This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system $$x^{\text{'}}=-e\left(t\right)x+f\left(t\right){\phi }_{p^{*}}\left(y\right),y^{\text{'}}=-g\left(t\right){\phi }_p\left(x\right)-h\left(t\right)y,$$ where $p>1$, $p^{*}>1$ ($1/p+1/p^{*}=1$), and ${\phi }_q\left(z\right)={\left|z\right|}^{q-2}z$ for $q=p$ or $q=p^{*}$. The coefficients are not assumed to be positive. This system includes the linear differential system ${𝐱}^{\text{'}}=A\left(t\right)𝐱$ with $A\left(t\right)$ being a $2\times 2$ matrix as a special case. Our results are new even in the linear case ($p=p^{*}=2$). Our results also answer the question whether the zero solution of the linear system is asymptotically stable even when Coppel’s condition does not hold...