This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried o...

We study chemostat models in which multiple species compete for two or more limiting nutrients. First we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equil...

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R(0) is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)≤1, and the disease is uniformly persistent and there...

In this paper, we propose a within-host HIV-1 epidemic model with cell-to-virus and cell-to-cell transmission. By mathematical analysis, we obtain the basic reproduction numberR0, which determines the viral persistence and the basic reproduction numberRcc 0 with respect to cell-to-cell transmission which is not strong enough, i.e., it is less than 1. If the basic reproduction number is less tha...

In this paper, we study a viral infection model with an immunity time delay accounting for the time between the immune system touching antigenic stimulation and generating CTLs. By calculation, we derive two thresholds to determine the global dynamics of the model, i.e., the reproduction number for viral infection R0 and for CTL immune response R1. By analyzing the characteristic equation, the ...

We consider a model of evolution in games in which a revising agent observes the actions of a random number of randomly sampled opponents and then chooses a best response to the distribution of actions in the sample. We provide a condition on the distribution of sample sizes under which an iterated p-dominant equilibrium is almost globally asymptotically stable under these dynamics. We show und...

This note deals with the constant control problem for homogeneous cooperative and irreducible systems. These systems serve as models for positive systems. A necessary and sufficient condition for global asymptotic stability of the zero solution of this class of systems is known. Adding a constant control allows to shift the equilibrium point from zero to a point in the first orthant. We prove t...

Abstract. In this paper, the global dynamics of a class of HIV-1 infection models with different infection rates and latently infected cells are investigated. We first modify the basic virus infection model and propose two models with bilinear infection rate and saturation infection rate, respectively, which take HIV-1 latency into consideration, and then study a model with a general nonlinear ...

We consider the global dynamics of a virus propagation model in which infective cells pass through several successive stages, assuming that the incidence of infection and the removal rate of the virus are nonlinear functions given in an abstract, unspecified form. Suitable Lyapunov functionals are constructed to establish the existence of a threshold parameter, the basic reproduction number R0 ...

We introduce a new notion of stability for nonlinear systems called Rational Stability (RS). Rational Stability is defined for autonomous systems that are globally asymptotically stable as a property of the induced norm of a scaling operator that maps one solution of the system to another. We give an Index called Rational Stability Index (RSI) which is a positive number that quantifies the amou...