Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

Abstract

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.

Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission

“…If a free virus particle encounters a susceptible cell, it has a chance to infect the cell, enabling the virus to spread through its target cell population [6]. Denoting by T(t) the concentration of CD4 + T cells at time t. The following target cells dynamics has been proposed by many literatures (see, for example [9,14,16,22] and the references cited therein), dT dt…”

“…Constant N is assumed to be the average number of virus particles produced by each infected cell (termed as the burst size). In order to be more realistic, incorporating time delays to represent that the recruitment of virusproducing cells at time t given by the number of cells that were newly infected at time t − τ and are still alive at time t (see, [8,9]). In [9], Li and Shu proposed the following DDEs to understand joint effects of target cells and intracellular delay on viral dynamics in vivo: (1.2) where the delay τ presents the lag from the time of initial infection to the production of new virions by infected CD4 + cells.…”

“…In order to be more realistic, incorporating time delays to represent that the recruitment of virusproducing cells at time t given by the number of cells that were newly infected at time t − τ and are still alive at time t (see, [8,9]). In [9], Li and Shu proposed the following DDEs to understand joint effects of target cells and intracellular delay on viral dynamics in vivo: (1.2) where the delay τ presents the lag from the time of initial infection to the production of new virions by infected CD4 + cells. Constant s is the death rate of infected but not yet virusproducing cells, and e −sτ describes the probability of infected target cells surviving the period of intracellular delay from t − τ to t. It is often of interest to study whether Hopf bifurcations can occur for the case where the target-cell dynamics having a mitosis component given by a logistic term.…”

Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

“…and α(0) < 0 by assumption (a) , therefore if τ c > 0 exists such that α(τ c ) = 0 then by the continuity (Michael Y. Li and Hogying Shu [10]) of α we have:…”

“…This form has the advantage that it simplifies the analytical work and also it is the form present in many population dynamical models involving delays [6], [7], [9], [10]. The DDE (2) may or may not have equilibrium points (or steady states) and these will depend on the values of µ.…”