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

Abstract In this paper, we investigate an age-structured HIV infection model with logistic growth for target cell. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analysing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al., we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our results.


Introduction
The HIV (Human Immunodeficiency Virus) that causes AIDS (Acquired Immune Deficiency Syndrome) has attracted the attention of numerous researchers. According to the [28] report, there are more than 36.7 million people infected worldwide and 1.1 million deaths in 2015. The virus main attacks CD4 + T cell (target cell), which is very important to the human immune system. The HIV pandemic is one of the greatest challenges in the field of global public health.
Recently, it has been realized that mathematical models are very necessary to understand HIV infection. Ho et al. [7] and Perelson et al. [22] considered two different ordinary differential equations (ODEs) to describe the interaction of susceptible cell, infected cell and free virus. Since then, a lot of mathematical models of HIV infection or other infectious disease have been extensively investigated by incorporating various factors [2,3,5,[8][9][10]13,27,29,30,32,34,35,37]. These models include ODEs, DDEs (delay differential equations) and PDEs (partial differential equations).
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. When the basic reproduction number was greater than unity, the system underwent a Hopf bifurcation through which the infection equilibrium lost the stability and periodic solutions appeared. They also found that the logistic growth of uninfected cells is the key factor which can induce sustained oscillations.
By using centre manifold theory and Hopf bifurcation theorem of non-densely defined Cauchy problems in [17] and [14], Liu et al. [15] showed that age-structured model of consumer-resource mutualism undergoes a Hopf bifurcation at the positive equilibrium under some conditions. Wang and Liu [31] considered an age-structured compartmental pest-pathogen model. Their results showed that Hopf bifurcation occurred at a positive steady state as bifurcating parameter passed some values.
Motivated by above, we study an age-structured HIV infection model with logistic target-cell growth and employ Hopf bifurcation theorem in Liu et al. to study that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values in this paper. Our model consists of three variables: susceptible CD4 + T cell whose concentration at time t is denoted by T(t); infected CD4 + T cell whose concentration at time t with age a is denoted by i(t, a); V(t) represents the concentration of free virus at t. We make the following assumptions in this paper: (i) The infected CD4 + T cell is assumed to be age-structured, whereas the susceptible CD4 + T cell and the free virus are not age-structured. (ii) The constant recruitment rate of the susceptible CD4 + T cell is and the natural death rate is d 1 . (iii) Once stimulated by antigen, susceptible CD4 + T cell undergoes mitosis immediately, and the mitosis can be described by where K is the maximum level of CD4 + T cells in the body, r is the intrinsic growth rate. (iv) Free viral particle can infect susceptible CD4 + T cell, and the infection rate is β. The natural death rate of infected cell is d 2 .
(v) α(a) is new free virus particle production rate of an infected cell with infected age a and defined by let K 0 := +∞ 0 α(a)e −d 2 a da, i.e. α * = d 2 K 0 e d 2 τ , where τ represents the time from initial infection to release of new free virus particle, d 3 is the clearance rate of free virus. K 0 represents the total number of virus particles produced by an infected cell during its life span. Based on the above discussion and assumptions, we obtain the following HIV infection model with logistic target-cell growth and virus-to-cell infection.
The rest of this paper is organized as follows. In the next section, we summarize the main results on Hopf bifurcation theorem obtained in [14]. In Section 3, the stability of the steady state and the existence of Hopf bifurcation are investigated. In Section 4, we perform numerical simulations to verify our analytical results. Finally, in Section 5, a brief conclusion is given.

Preliminaries
We recall the Hopf bifurcation theorem in [14] for the following non-densely defined abstract Cauchy problem: where μ ∈ R is the bifurcation parameter, A : D(A) ⊂ X → X is a linear operator on a Banach space X with D(A) not dense in X and A is not necessary to be a Hille-Yosida operator, F : R × D(A) → X is a C k map with k ≥ 4. Set and A 0 is the part of A in X 0 , which is defined by We denote by {T A (t)} t≥0 the C 0 semigroup of bounded linear operators on X (respectively {S A (t)} t≥0 the integrated semigroup) generated by A. The essential growth bound ω 0,ess (L) ∈ (−∞, +∞) of L by where L(X) is the space of bounded linear operators from X into X, T L (t) ess is the essential norm of T L (t) defined by T L (t) ess = κ(T L (t)B X (0, 1)), where B X (0, 1) = {x ∈ X : x X ≤ 1}, and each bounded set B ⊂ X,κ(B) = inf{ε > 0: B can be covered by a finite number of balls of radius≤ ε} is the Kuratovsky measure of non-compactness.
We make the following assumptions on the linear operator A and the nonlinear map F. Assumption 2.1: Assume that A : D(A) ⊂ X → X is a linear operator on a Banach space (X, · ) such that there exist two constants ω A ∈ R and M A ≥ 1, such that (ω A , +∞) ⊂ ρ(A) and the following properties are satisfied Assumption 2.1 implies that A 0 is the infinitesimal generator of a C 0 semigroup {T A 0 (t)} t≥0 of bounded linear operator on X 0 and By Proposition 2.6 in [17], we know if Assumption 2.1 is satisfied, then A generates a unique integrated semigroup {S A (t)} t≥0 . If we assume in addition that A is a Hille-Yosida operator, then we have Next, we consider the non-homogeneous Cauchy problem where f ∈ L 1 ((0, τ ), X).    By Corollary 2.11 in [17], we know if Assumptions 2.1 and 2.2 hold, then for each τ > 0, A is a Hille-Yosida operator, then Assumption 2.8 in [18] holds. By Theorem 2.9 in [18], we know Assumption 2.2 holds.

Theorem 2.1: Let assumptions 2.1-2.3 be satisfied. Then there exist
, which is an integrated solution of (2.1} with the parameter value μ(ε) and the initial value x ε . So for each t ≥ 0, u ε satisfies Moreover, we have the following properties:

function and we have the Taylor expansion
where ω(0) is the imaginary part of λ(0) defined in Assumption 2.3.

Stability of equilibria and existence of Hopf bifurcation
In this section, we will study stability of equilibria and existence of Hopf bifurcation for (1.2)

The transformation of the cauchy problem
In system (1.2), by setting we can rewrite system (1.2) as the following age-structured model: Define the linear operator L : D(L) → X by we notice L is non-densely defined since Now we can rewrite PDEs system 3.1 as the following non-densely defined abstract Cauchy problem , The global existence and uniqueness of solutions of system 3.2 follow from the results of [16] and [18].

Existence of equilibria
If w(a) = 0x(a) ∈ X 0 is an equilibrium of 3.2, we have Hence we obtain It is easy to see system 3.3 has always the equilibriumx 1 Furthermore, we define where R 0 is called the basic reproduction number of model 1.2. Biologically, R 0 represents the total number of newly infected cells resulted from a single infected cell. When R 0 > 1 holds, system 3.3 has a unique positive equilibrium

Linearized equation
Let y(t) := w(t) − w(a), then system (3.2) is equivalent to the following system: , 5) and the equilibrium w(a) of system (3.3) is transformed into the zero equilibrium of system (3.5).
The linearized system of system (3.5) at the equilibrium 0 is as follows: where A = L + DF(w). Then system (3.5) can be written as By applying the results of Liu, Magal and Ruan [14], we obtain the following result.

Lemma 3.2:
For λ ∈ , λ ∈ ρ(L) and It is readily checked that so L is a Hille-Yosida operator and Define the part of L in D(L) by L 0 , and we know Then, we can claim that L 0 is the infinitesimal generator of a C 0 semigroup {T L 0 (t)} t≥0 on D(L) and for each t ≥ 0 the linear operator T L 0 (t) is defined by Now we estimate the essential growth bound of the C 0 semigroup generated by A 0 which is the part of A in D(A). We observe that for any 0 ϕ ∈ D(L), Then DF(w) : D(L) ⊂ X → X is a compact bounded linear operator. From (3.6), we obtain Then we have By applying the perturbation results in [6], we obtain Thus, by the above discussion and Theorem 3.5.5 in [1], we obtain the following proposition.

Proposition 3.1:
The linear operator A is a Hille-Yoside operator, and the essential growth rate of the strongly continuous semigroup generated by A 0 is strictly negative, that is, In order to apply Theorem 2.1, we remain to precise the spectral properties of A 0 . Setting C := DF(w), and let λ ∈ . Since(λI − L) is invertible, it follows that λI − Consider the equation From the above discussion and by using the proof of Lemma 3.5 in [31], we obtain the following lemma. From the above discussion, we know that the linear operator A satisfies Assumptions 2.1, 2.2 and 2.3.

Stability of the boundary equilibrium
Now, we consider the stability of the boundary equilibrium E 1 (T 1 , 0, 0), we obtain Thus we obtain the characteristic equation We know that the coefficients a 3 not depend on τ by our assumption. It is easy to see that which have a negative eigenvalue, and the other eigenvalues are given by When τ = 0, Equation (3.8) becomes λ 2 + a 1 λ + a 2 + a 3 = 0 with a 1 > 0. If R 0 < 1, it is readily obtain that a 2 + a 3 > 0. Hence, equilibrium E 1 is locally asymptotically stable when τ = 0. When τ = 0, let λ = iω(ω > 0) is the solution of Equation (3.8), then separating in real and imaginary parts, we obtain −ω 2 + a 2 = −a 3 cos(ωτ ),

Stability of the positive equilibrium and Hopf bifurcation
When R 0 > 1, the characteristic equation of system (1.2) about the positive equilibrium E * (a) can be rewritten as Since K 0 is a constant, we know that the coefficients a, b, c, d and e have nothing to do with τ . It is easy to see that Thus we have i.e.

Numerical simulations
In this section, we present some numerical simulations to verify the main results by using MATLAB programs. On the one hand, we simulate the system (1.

Conclusion and discussion
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: (5.1) We can study the Hopf bifurcation of the system (5.1) using similar methods of [12,19,25,26,36].
In this paper, we use a different method, that is, by using the centre manifold theory and Hopf bifurcation theorem of non-densely defined Cauchy problems in [17] and [14], to study the Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth and virus-to-cell infection. By choosing τ as the bifurcation parameter and analysing the corresponding characteristic equation, we can conclude that the local asymptotically stability of infection-free equilibrium E 1 is completely determined by the basic reproduction number R 0 . If R 0 < 1, then the infection-free equilibrium E 1 is local asymptotically stable for τ ≥ 0. If R 0 > 1, the sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Finally, numerical simulations are given to verify the theoretical analysis. From these waveforms and the phase trajectories above, it is shown that these results are in accord with the theoretical analysis.
Our analytical results indicate that introduction of parameter τ can affect the dynamic behaviour of the system (1.2) and that there is a threshold τ 0 , when τ = τ 0 , the model exhibits Hopf bifurcation. Through these results, it can be concluded that delay τ plays an important role in the HIV spread and induces Hopf bifurcation with Logistic growth in our model.
For studying that the logistic growth of target cells is necessary for oscillation with or without intracellular delay. We let r = 0, that means, there is no logistic growth rate in system (1.2), we know that the system (1.2) has only a boundary equilibrium E 1 = ( /d 1 , 0, 0), when basic reproduction number R 0 = β K 0 /d 1 d 3 < 1, if R 0 > 1, besides E 1 , the system (1. By simple calculation, we deduce that a 2 − 2b > 0, (a 2 − 2b)(b 2 − e 2 − 2ac) > (c 2 − d 2 ) > 0, by the Routh-Hurwitz criterion, we know that Equation (5.6) has no positive roots, which is a contradiction with ω > 0. Thus we have the following conclusion. If R 0 > 1, then E * (a) is locally asymptotically stable for all τ ≥ 0.
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). We did not find other critical parameters for Hopf bifurcation. Above results suggest that delay and the logistic growth of uninfected cells are both the factor which can induce sustained oscillations.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work is supported by the NNSF of China (11861044 and 11661050), the NSF of Gansu Province (148RJZA024) and the Development Program of HongLiu first-class disciplines in Lanzhou University of Technology.