Hopf bifurcation of an age-structured virus infection model

Abstract

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number R 0. In particular, when R 0 ≤ 1, the infection free equilibrium is globally asymptotically stable, and conversely if R 0 > 1, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.

Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth

“…Then the system (1.2) does not undergo Hopf bifurcation at the positive equilibrium E * (a). This conclusion is consistent with that of [20,24], the logistic growth of target cells is necessary for the observed oscillatory dynamics in the system (1.2). Furthermore, we have done some simulations (not shown).…”

“…As in Mohebbi et al [20], we can derive an equivalent discrete delay system as opposed to our age-structured model when t > τ. For the system (1.2), we let I = +∞ 0 i(t, a)da, and note that i(t, a) = e −d 2 τ i(t − τ , a − τ ), then the system (1.2) can be reformulated as the following DDEs when t > τ:…”

“…The age-structured models have been studied by many authors, such as [4,11,33]. Taking into account the effect of age-structured is thought to be very important for virus infection models [20,21,23,24,30]. Shu et al [24] provided some general results applicable to immune system dynamics and found that the logistic growth of uninfected cells on virus models with distributed delays could induce sustained oscillations.…”

“…Their results showed that the nonlinear target-cell growth rate was very important in causing oscillatory dynamics. Mohebbi et al [20] studied a new age-structured within-host virus model with logistic growth of target cells and viral absorption by infected cells. They presented that infection free steady state is globally asymptotically stable when the basic reproduction number was less than unity.…”

Bifurcation analysis of an age‐structured epidemic model with two staged‐progressions

“…1,[7][8][9][10][11] In addition to global stability, the destabilization of age-dependent positive steady state has attracted great attention recently, which may result in bifurcation behaviors. [12][13][14][15][16][17][18][19] To conduct analysis, the age-structured model can be rewritten as abstract Cauchy problem with a Lipschitz perturbation of a closed linear operator that is non-densely defined but satisfies the estimates of Hille-Yosida theorem. Due to the age-dependent parameters or infectious delay, Hopf bifurcation occurs in the population dynamics.…”