Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics

Abstract

This paper investigates a class of viral infection models with a nonlinear infection rate and two discrete delays, one of which represents an intracellular latent period for the contacted target cell with virus to begin producing virions, the other of which represents the time needed in cytotoxic T cells (CTLs) response before immune becomes effective after a novel pathogen invades. Since immune system is a complex network of cells and signals that have evolved to respond to the presence of pathogens, we further assume two situations for immune activation delay. When both delays are ignored, the global stability for the ordinary differential equations model are established. While both delays are included, the positivity and boundedness of all solutions of the delay differential equations model are proved. Utilizing Lyapunov functionals and LaSalle invariance principle, the global dynamical properties are also studied. In particular, stability switch is shown to occur as immune delay increasing by bifurcation theory. Our results exhibit that the intracellular delay does not affect the stability of equilibria. However, the immune activation delay is able to destabilize the interior equilibrium and brings periodic solutions. Numerical simulations are performed to verify the theoretical results and display the different impacts of two type delays in two cases. Those analysis give us some useful suggestions on new drugs to fight against

Global dynamics for a class of age-infection HIV models with nonlinear infection rate

“…There have been numerous works on pathogen in-host dynamics describing HIV infection. The pioneer works of Nowak and May [27], Perelson and Nelson [28] have been further developed by incorporating some biological features as a realistic attempt, such as, immune response and delays (see e.g., Hellriegel [8], Hetzel and Anderson [9], Bairagi and Adak [1] and Huang et al [12]). However almost all works had not taken into account an important picture of the mortality rate and viral production rate of infected cells dependent on the infection age of cells.…”

“…It is well known that nonlinear incidence rates are frequently used to describe the viral infection process based on experiments data and reasonable assumptions [23] and it is important to account for a number of nonlinear features of the biological phenomena involved, which is influenced by the availability of susceptible cells and by the force of infection of viral cells. For example, Holling type II functional response [12], saturation infection rate [20,38], Beddington-DeAngelis functional response [10], Crowley-Martin functional response [39,40] and general nonlinear incidence c(x)f (v) [6], where c(x) denotes the contact rate function at concentration of the target cells x and f (v) denotes the force of infection by virus at concentration v. Motivated by the works mentioned above (see, [6,10,13,[20][21][22]24,38]), in this paper, we develop the model (1.2) with nonlinear incidence rate and investigate the global stability of its equilibrium.…”

Stability and Hopf bifurcation analysis of immune response delayed HIV type 1 infection model with two target cells

“…A limitation to the previous studies of HIV‐1 infection models is that the majority ignore the effects of the cause of differentiation in the CTLs and the effect of time delays. One such extension includes the addition of drug therapy (one can refer to and references therein), usually included as a constant term. On the basis of these extensions, our proposed model may help to interpret events occurring during the disease process.…”

“…In modeling of many biological processes, time delays are usually introduced for the purpose of accurate representations of the phenomena. Time delay plays a significant role in modeling the interaction between healthy cells and HIV‐1 (see and references therein). In the investigation of HIV dynamics, it has been assumed that there may be a time interval in the activation of immune response after the invasion of the foreign agents which may lead mathematical models to delay differential equations.…”

Effects of randomness on viral infection model with application

“…Modeling the viral infections in vivo has caught the attention of scientists in recent years with the discoveries of many new members in the family of viral diseases like SAARS (Severe Acute Respiratory Syndrome) despite the presence of senior citizens (AIDS -Acquired Immune Deficiency Syndrome, conjunctivitis etc.). The different approaches adopted by mathematical biologists like Huang et al in Huang, Yokoi, Takeuchi, Kajiwara, andSasaki (2011), Herz et.al in Herz, Bonhoeffer, Anderson, May, and tried their hand with Delay Differential Equations and Fractional Differential Equations in Zhang, Huang, Liu, and Fan (2015) with regard to intracellular delay in viral dynamics and HIV infection Models with Non-linear incidence rates respectively while M. Pitchaimani et al in Pitchaimani and Monica (2015) focussed on HIV-1 infection model with three time delays. Also, Pitchaimani and Rajaji contributed works in Stochastic Nowak -May Model analysis as in Pitchaimani and Rajaji (2016).…”