Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

Abstract

We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD4 + T cells by a logistic function and the infected CD4 + T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some parameter values such that this HIV infection model has a non-trivial periodic solution which bifurcates from the positive equilibrium. The numerical simulations are also carried out. .cn (Z. Liu).where T , I and V denote the population of uninfected CD4 + T cells, productively infected CD4 + T cells and virus, respectively. They treated the logistic proliferation term rT 1 − T +I K in which r is the maximum proliferation rate and K is the uninfected CD4 + T cells population density at which proliferation shuts off. Since the proportion of productively infected CD4 + T cells I is very small and thus it is reasonable to ignore this correction. They eventually represented the proliferation of uninfected CD4 + T cells by a logistic function rT 1 − T K . However, the authors only consider the classical virus-to-cell infection and neglect the direct cell-to-cell transmission.[12] incorporated the two modes (virus-to-cell infection and cell-to-cell transmission) of transmission into a classic model and considered the following model system+∞ 0 q(a)i(t, a)da, ∂i(t,a) ∂t + ∂i(t,a) ∂a = −σ(a)i(t, a), dV (t) dt = ∞ 0 p(a)i(t, a)da − cV (t),

Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge

“…Based on the above mathematical framework, different kinds of predator-prey models with age structure were considered in [16,36,39,40,41,42,44,48] and it is proved that these models undergo Hopf bifurcation at the equilibrium and non-trivial periodic orbits bifurcate from the equilibrium. Similar conclusions were also obtained in the virus model, epidemic model, consumer-resource symbiosis model, alcoholism model and other models with age structure [2,5,7,8,20,22,38,46,47]. We also refer to [17,21,23,43] for the studies on the Bogdanov-Takens bifurcation and Zero-Hopf bifurcation problems of relevant models.…”

On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes

“…Hence, many scholars have made strenuous efforts, so as to find out the impact of age on evolution. For more details, please see [29,19,12,1,28,23,35,36,30,31,10,34,37,33,32] and the references therein for related works. Unfortunately, most of these equations barely get the existence, boundedness and positivity of the solution for the age-dependent version predation equations.…”

Existence of periodic wave trains for an age-structured model with diffusion

“…In many situation, population dynamical models become considerably intricate when age structure will be considered in individual interactions. In recent years, considering the significance of age structure in populations has become more and more prevalent, the literature on all aspects of the interacting populations with age structure has proliferated [4,5,19,11,27,28,30,31,32,15].…”