Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate Eric Avila-Vales ⇑ , Noé Chan-Chí, Gerardo García-Almeida Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje Cat. 13615, Col. Chuburná de Hidalgo Inn, Mérida, Yucatán, Mexico article info Article history: Received 17 June 2014 Accepted 25 August 2014 abstract In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptoti- cally stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The study of epidemic and viral dynamics via mathe- matical modelling has been an interesting topic to investi- gate in the last decades. Researchers have constructed mathematical models which could play a significant role in better understanding diseases and drug therapy strate- gies to fight against them. During the process of viral infection, as soon a virus invades host cells, Cytotoxic T Lymphocytes (CTL’s) play an important role in responding to the aggression. Lymphocytes are programmed to kill the infected cells through the lysine of the infected ones. To model the immune response during a viral infection, taking into account the CTL response, researchers consider the following set of differential equations _ x ¼ s  dx  bxy; _ y ¼ bxy  ay  pyz; _ z ¼ f ðy; zÞ bz; where variable x; y and z represent the populations of unin- fected cells, infected cells, and number of CTL’s by ml of peripheral blood, respectively. The parameter s represents a constant source of susceptible cells, b is the infection rate constant, we assume that a susceptible cell become infected at rate proportional to the number of infected cells. Constants d and a represents the death rates of sus- ceptible and infected respectively. Infected cells are killed at a rate p by the CTL immune response. The function f ðy; zÞ describes the rate of immune response due to virus activation. In this paper we consider f ðy; zÞ¼ cy  myz, the term myz represents an immune impairment according to [1], the CTL cells proliferate at a rate c and decay at rate m. Linear and bilinear immune response have been consid- ered in [2–5]. In [4,6,7] time delays have been incorporated for immune response, since antigenic stimulation generating CTLs may need a period of time, that is, the activation rate of CTL response at time t may depend on the population of antigen at a previous time. On the other hand, it has been realised recently [8,9,13] that there are also delays in the process of cell infection and virus production, and thus, delays should be incorporated into the infection equation and/or the virus production equation of a model. In this paper, we consider the following model, http://dx.doi.org/10.1016/j.chaos.2014.08.009 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +52 (999) 942 31 40x1108. E-mail addresses: avila@uady.mx (E. Avila-Vales), noe.chan@uady.mx (N. Chan-Chí), galmeida@uady.mx (G. García-Almeida). Chaos, Solitons & Fractals 69 (2014) 1–9 Contents lists available at ScienceDirect Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos