OF AN AGE-STRUCTURED VIRUS INFECTION MODEL

. 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.


1.
Introduction. Within-host virus dynamics has been studied by many authors using differential equations to model the coupled changes in target cell, infected cell and virus populations; e.g. Perelson and Nelson [23], De Leenheer and Smith [8], Nowak and May [21], Culshaw and Ruan [7]. In particular, age-structured virus models have been of recent interest in the literature. Nelson et. al. [20], proposed an age structured model for HIV-1 infection, which generalizes the standard delay differential equation models by allowing for infected cell death rate and viral production to vary with age since infection of an infected cell. Rong et. al. [25] considered two age-structured models to study HIV-1 infection dynamics which extend the existing age-structured models by Nelson et. al. [20] and Kirschner and Webb [17] by incorporating combination therapies. The age related models are normally in the form of partial differential equations (PDEs). Rong et. al. [25] proved the local stability of the positive equilibrium when the basic reproduction number is greater than unity, and global stability results for a general age-structured model of intra-individual HIV dynamics are proved by Huang et. al. [15]. Recently the global dynamical properties of PDEs have also attracted interest in the literature, e.g. Browne and Pilyugin [6], McCluskey [19], Yang et. al. [31].
In virus models, the stability of the positive equilibrium is of interest, since researchers investigate whether certain factors may lead to sustained oscillations in the viral concentration. De Leenheer and Smith found that the infection equilibrium can be destabilized via a Hopf bifurcation when the underlying growth rate has logistic form, rather than the simpler linear form often assumed in virus models [8]. Previous studies have shown that target cell growth may be density dependent and follow a logistic-like growth equation [2]. In recent work on virus models with distributed delays, sustained oscillations were found to be induced by logistic growth of uninfected cells, showing the importance of the nonlinear target cell growth rate in causing oscillatory dynamics [14].
In modeling virus dynamics, the interaction between the virus and uninfected cells is usually described by a linear mass action function [21]. In many studies the (mass-action) loss of virus due to entrance into uninfected and infected cells are often neglected because the uninfected and infected cells are considered to change on a much slower time scale than the free virus [23]. This simplification may be valid in HIV-1 infection, but not in other infections such as Polio infection [3]. Even when modeling HIV, explicit inclusion of viral absorption into target cells can affect the dynamics [29]. Additionally, the proportion of infected cells co-infected by multiple instances of viral absorption may be rather large, and is important to quantify for estimating the viral recombination rate [9,16]. Recombination within co-infected cells can play a large role in viral diversity and evolution within a host. Some recent works have extended the standard HIV model to include viral absorption by infected cells [9,10,24]. Based on these points, we think that it is reasonable to consider virus dynamics with consideration of depletion of viruses due to entrance into the uninfected and infected cells.
In this paper, we extend previous models to study the behavior of viral infection with explicit cell infection-age structure, logistic target cell growth rate, and viral absorption by infected cells. We consider the following system: T * (t, 0) = kV (t)T (t), T * (0, a) = T * (a) ∈ L 1 Here T (t) and V (t) denote the concentrations of healthy cells and free virus particles at time t, respectively. T * (t, a) denotes the density of infected cell concentration with respect to age a since infection at time t and L 1 + (0, ∞) is the non-negative cone of L 1 (0, ∞) and R + is non-negative real numbers. In this model it is assumed that T cells are only created by proliferation of existing T cells as in vitro. We present the proliferation by a logistic function in which s = e−g where e is the proliferation rate and g is natural death rate of uninfected cells. Moreover e T max = s T 0 , where T max is the T cell population density at which proliferation shuts off and T 0 is infection free steady state of healthy cells. This function is used to model proliferation, both, in vitro and in vivo in some literature such as [28,2]. The parameters k and d are stand for the infection rate and natural death rate of free viruses respectively. The function δ(a) represents the age dependent death rate of infected cells and p(a) is the virion production rate of an infected cell of age a. Both, δ(a) and p(a) are assumed to be in the non-negative cone of L ∞ (0, ∞), moreover 0 < b ≤ δ(a) and p(a) ≤ κ a.e. for some constants b and κ. Free virus can re-infect the previously infected cells, described by the mass action term γV (t) ∞ 0 T * (t, a)da inserted into the right hand side of V (t) equation, where γ is re-infection rate and ∞ 0 T * (t, a)da is the total population of infected cells.
The paper is organized as follows. In section 2, the reproduction number as a threshold is obtained and the local stability of disease-free equilibrium and infection equilibrium are studied. Section 3 is devoted to persistent analysis of the system (1)-(4). In section 4, by constructing the suitable Lyapunov function, the global stability of uninfected equilibrium for R 0 ≤ 1 is stablished. In section 5, special cases of system (1)- (4) showing Hopf bifurcation are studied, along with quantification of viral re-infection and recombination rates. Finally section 6 is devoted to a brief discussion.
2. Local stability. By integrating along characteristics and applying boundary condition, the PDE in (3) will be in the following form: where φ(a) = e − a 0 δ(s)ds , is the probability that an infected cell survives till age a. By (5), the system (1)-(4) is equivalent to the following system: where 1 {} is the indicator function.
Preliminary results. In the following, we give two theorems about existence, uniqueness, boundedness and nonnegativity of solutions of system (6)- (8). Due to similarity of system (6)-(8) and system (3) in [6], our proofs are almost similar to the related proofs in [6] in some cases.
Theorem 2.1. Let x 0 = (T (0), V (0), T * (0, a)) ∈ R 2 + × L 1 + (0, ∞). Then there exists > 0 and a neighborhood B 0 ⊂ R 2 + × L 1 + (0, ∞), with x 0 ∈ B 0 such that there exists a unique continuous function, ψ : is the solution to system (6)- (8). Proof. Solutions to the system (6)-(8) must satisfy the following equation: Let Y be the set of continuous functions , a neighborhood containing x 0 , are to be determined. Let D ⊂ Y , contain functions whose range is contained in ball B = B((T (0), V (0), T * (0, a)), r), for some radius r. Note that D is complete metric space. At the following, we define operator Γ on D and show that: ii) Γ (η) is a contraction map on complete metric space D, whose fixed point is the unique solution of the system (6)-(8) with initial point x 0 .
Now let x = (x 1 , x 2 , l (a)) ∈ B 0 and g(t, x) = (g 1 (t, x), g 2 (t, x), g 3 (t, x)) ∈ D, and define operator Γ as follows: To prove the first part of (i), it is clear that Γ 1 (η)(t, x) ∈ R and from the assumption where, 0 < b ≤ δ(a) on [0, ∞) for some constant b. To prove the second part of (i), take B 0 = B((T (0), V (0), T * (0, a)), r/2). Then By Dominated Convergence Theorem (DCT), J1 → 0 as t → 0. Hence J1 ≤ r 16 for all t ∈ [0, ] provided that sufficiently small. We note that, set of continuous functions with compact support is dense in L 1 . Now let ζ be continuous function with compact support in [0, ∞] such that T * (0, .) − ζ < r 32 . Then Hence for constants M 1 , M 2 and sufficiently small. Therefore Γ (η) : D → D. To prove (ii), Let g, h ∈ D. Then where M is a constant. Therefore Γ is a contraction mapping on D for sufficiently small. By the Contraction Mapping Theorem there exists a unique fixed point of Γ in D that we denote this function by ψ(t, x), which solves the initial value problem and is continuous on [0, ] × B 0 .
Theorem 2.2. Solution to the system (6)-(8) remain nonnegative for almost every a ≥ 0 and bounded in forward time.
To prove the boundedness of solutions, we note that our assumption on the parameters imply that 0 ≤ p(a) ≤ κ a.e., and 0 < b ≤ δ(a) a.e., ∀a ∈ [0, ∞). Also, from the nonnegativity of solutions of system (6) for all t ≥ 0 in the interval of its existence. Now, consider the following: Now, boundedness follow from nonnegativity of solutions.

Local analysis. System (6)-(8) has an infection free equilibrium
) satisfying the following equations, where T * (0) = kV T . Substituting (11) into (10), we get solving (9) in T , and substituting into (12), we have where (14) in V , we have the following: Now we define the basic reproduction numbers which governs the stability of infection free equilibrium and the existence of infection equilibrium, The above quantity, R 0 , can be interpreted as the average number of infected cells produced by a single infected cell in a T cell population at infection-free level T 0 . The following cases are established: i) If R 0 < 1, then the positive root for the equation (14) is from (13), the related T must satisfy the next relation as follows: in this case also, there is not infection equilibrium. ii) If R 0 > 1, we have two positive roots for (14) but only smaller root leads to non-negative T with, and In this case also, the system (1)-(5) has a unique infection equilibrium E = (T 2 , V 2 , T * 2 (a)), that we simply denote by E = (T , V , T * (a)), where T * (a) is given by (11). iii) If R 0 = 1, which implies that N > 1, then one root of the above equation is V 3 = 0, whose related equilibrium is E 0 = (T 0 , 0, 0) and the other root is and the related T 4 must satisfy the condition (13). Then we have: hence the necessary condition for the admisibility of T 4 is N ≤ 1 which is a contradiction. Therefore (T , V , T * (a)) = (T 0 , 0, 0) and the only equilibrium point in this condition is E 0 = (T 0 , 0, 0).
Proof. First, we study the local stability of E 0 . Linearizing system(1)-(5) about E 0 , we obtain: where a). Consider the exponential solutions T 1 (t) = T 1 e λt , V 1 (t) = V 1 e λt and T * 1 (t, a) = T * 1 (a)e λt of the system (18). Then we have: With substituting (22) into (21), we have Then, by substituting (23) in (20), and letting λ = u + iv as a complex number, the next follows as below: Now let λ = u + iv, then the real-part of the above equation is then if u > 0, the righthand side of the above equation is negative that is a contradiction. Now define H (u) as follows: This implies that equation (24) has at least one positive root. Hence, E 0 is unstable.
Proof. Suppose that γ = 0. The proof is similar and simpler for the case γ = 0. If γ = 0, from (17), Now we investigate the local stability of E when R 0 > 1. We show that unique infection equilibrium E of system (6)-(8) can be stable or unstable depending on parameter values.
In order to investigate the local stability of E = (T , V , T * (a)), let Then the exponential solution of linearized system about E in the form of T 1 (t) = T 1 e λt , V 1 (t) = V 1 e λt and T * 1 (t, a) = T * 1 (a)e λt , satisfies the following equations: Doing some simplification, we have the following: Substituting (26) and (28) into (27), the next follows as below: So the above equation simplifies to the following: We note that E = (T , V , T * (a)) = (T 2 , V 2 , T 2 * (a)) whose amounts are given by (16), (17) and (11). From (10) we have: Clearly from (30) and (29), λ = 0 cannot be a solution of (29). Therefore, in the case that the equation (29) has solution where the real part of λ changes sign by changing parameter values, then Hopf bifurcation occurs. To compute Hopf critical point, let complex number λ = ic be the solution of (29). Therefore from (29), we have: The real part and imaginary part of the above equation is as follows respectively: Imaginary part: c γ Using (33) in (32), we have the following rough equation where existence of real solution of c is equivalent to withĒ be the Hopf critical point.
In the numerical section, Section 5, for the case 1 < R 0 , we have chosen the parameter values in such ways that (29) has solutions with negative or non-negative real part for different values of parameters. Therefore, E will be stable or unstable, respectively.
3. Uniform persistence. Persistent theory provides a tool for determining whether a species will ultimately persists or extinct in a dynamical system.
We first give some preliminary definitions from [13] and [11] as follows: The semigroup S(t) is said to be uniformly persistent (with respect to X 0 and ∂X 0 ) if there exists η > 0, such that for any x ∈ X 0 , lim inf t→∞ d( The semigroup S(x) is said to be asymptotically smooth , if for any bounded subset B ⊂ X, for which S(t)B ⊂ B for t ≥ 0, there exists a compact set K such that d(S(t)B, K) → 0 as t → ∞.
The semigroup S(t) is said to be point dissipative in X if there is a bounded nonempty set B in X such that, for any x ∈ X, there is a t 0 = t 0 (x, B) such that S(t)x ∈ B for t ≥ t 0 .
A set A in X is said to be global attractor if it is compact, invariant and for any bounded set in X, d( where A ∂ is global attractor in ∂X, and ω(x) = {y ∈ X : ∃t n ↑ ∞, such that S(t n )x → y} is omega limit set of x. Also, a set A in X is said to be a global attractor if it is compact, invariant and for any bounded set B in X, d(S(t)B, A) → 0 as t → 0. A set A in X is said to be invariant if S(t)A = A for t ≥ 0 and if S(t)A ⊂ A, it is said to be forward invariant. The stable (or attracting) set of a compact invariant set A is denoted by W s (A) = {x|x ∈ X, ω(x) = ∅, ω(x) ⊂ A}. The unstable (or repelling) set of a compact invariant set A, W u is defined by W u (A) = {x|x ∈ X, there exists a backward orbit γ − (x) such that α γ (x) = ∅, α γ (x) ⊂ A}, where α γ (x) = {y ∈ X : ∃t n ↓ −∞, such that S(t n )x → y} is alpha limit set of x. It should be noted that, there is no backward uniqueness, hence the definitions will possibly consider multiple backward orbits from a point.
To consider uniform persistence of S(t) and existence of global attractor in X 0 , we will use the following theorems, from Hale and Waltman [13].  [13]). Suppose metric space X is a closure of an open set X 0 ; that is X = X 0 ∪ ∂X 0 , and S(t) satisfies condition: and we have the following:  To verify the asymptotically smoothness of semigroup generated by system (6)-(8), we use the following lemma: ). For each t ≥ 0, suppose R(t) = U (t) + C(t) : X → X has the property that C(t) is completely continuous function and there is a continuous function k : R + × R + → R + such that k(t, r) → 0 as t → 0 and U (t)x ≤ k(t, r) if x ≤ r. Then R(t), t ≥ 0, is asymptotically smooth.
We note that, a continuous mapping S : X → X is said to be compact (completely continuous) if S maps any bounded set to a precompact set in M (i.e., M is compact). A semigroup C(t) is completely continuous if for each t ≥ 0 and each bounded set B ⊂ X, we have, {C(s)B, 0 ≤ s ≤ t} is bounded and C(t)B is precompact. Since one of the components of S(t) is L 1 + (0, ∞), which is infinite dimentional space, boundedness does not imply precompactness. Therefore, we use the following lemma: Lemma 3.4 ([1]). Let K ⊂ L p (0, ∞) be closed and bounded where p ≥ 1. Then K is compact if and only if the following holds: Proposition 3.5. The semigroup S(t), generated by system (6)-(8), is asymptotically smooth.
Proof. Let S(t) = U (t) + C(t) where,  )) ∈ X. Moreover π 1 S(t) is projection of S(t) on R 2 + . Let B ⊂ X be a bounded set. Then from the theorem (2.2), π 1 S(t)B is bounded in R 2 + , therefore is precompact. The projection of S(t) on L 1 + (0, ∞) is given by π 2 S(t) = U 2 (t) + C 2 (t). Let x ≤ r, then we have: where k(t, r) = re −bt . Clearly, U (t)x ≤ k(t, r) → 0 as t → 0. To show that C 2 (t)B satisfies compactness condition, we apply lemma (3.4). Let B ⊂ X be closed and bounded, and x ≤ r for all x ∈ B. Now we check the first condition of lemma (3.4). We have: where M 1 is a constant. Existance of such a constant M 1 , follows from Theorem (2.2). By integral formation of solutions and Theorem (2.2), there exist constants M 2 and M 3 such that: Hence, by (36), (37) and (38), we have: This converges uniformly to 0 as h → 0. Therefore condition (i) is proved for C 2 (t)B. Notice that from the definition of (C 2 (t)x)(a), ∞ h |(C 2 (t)x)(a)|da = 0 as h ≥ t, and therefore condition (ii) is also satisfied. Hence by lemma (3.4), C 2 (t)B is compact, and therefore C(t)B ⊂ π 1 S(t)B × C 2 (t)B is precompact. Now we can apply lemma (3.3), and conclude that S(t) is asymptotically smooth.
We will need the following result about linear Volterra integro-differential equations.
Theorem 3.8. Assume R 0 > 1, the semiflow {S(t)} t≥0 generated by system (1)-(4) is uniformly persistent with respect to the sets ∂X 0 , X 0 ; that is, there exists > 0, such that for each x ∈ X 0 , where X 0 and ∂X 0 are as in Lemma (3.6). Furthermore, there exists a compact subset A 0 ⊂ X 0 which is global attractor for {S(t)} t≥0 in X 0 .
Proof. To prove the uniform persistence and the existence of compact attractor, we will apply Theorem (3.1), and Theorem (3.2) or Theorem 3.7 in [18]. We will use the same argument as in the proof of theorem 3.6 in [6].
To complete the prove that x 0 is acyclic, we need only to prove W s ({x 0 }) ∩ X 0 = ∅.
Suppose by the way of contradiction, that there exists Since {C(t n )} is precompact, there exists a convergent subsequent{t n k }, such that a)) and x n = (T n (0), V n (0), T * n (0, a)). Then we have: Then, from equations (7) and (8), and comparison principle, we have: y n (0) = V (0).
Since V N ≥ y N , hence, S(t)x N is unbounded, which is a contradiction. Therefore, W s ({x 0 }) ∩ X 0 = ∅. Then by Theorem (3.1), S(t) is uniformly persistent, and by Theorem (3.2), there exists a compact set A 0 ⊂ X 0 which is global attractor for 4. Global analysis. In this section, we construct suitable Lyapunov function to investigate the global stability of the infection free equilibrium for system (1)-(5). Then, we have: Note that the last inequalities follow from the fact that R 0 ≤ 1 and The latter implies lim sup t→∞ T (t) ≤ lim t→∞ y(t) = T 0 , whereẏ = sy − sy 2 T 0 is a Bernoulli equation with lim t→∞ y(t) = T 0 . Therefore dW (t) dt = 0, implies that V (t) = 0 or R 0 = 1 and T (t) = y(t). It is easy to show that the largest invariant set where dW (t) dt = 0 is the singleton {E 0 }. Thus, when R 0 ≤ 1, by Liapunov-LaSalle asymptotic stability theorem, {E 0 } is globally asymptotically stable for system (1)-(5).

5.
Numerical results and Hopf bifurcation. In this section we study numerical examples, and apply the results obtained in the previous sections to two special cases. The parameter values for some infectious models in literatures are listed in Table 1, in which s = e − g where e is the proliferation rate and g is natural death rate of uninfected cells, and e T max = s T 0 , where T max is the T cell population density at which proliferation shuts off as noted in introduction. In order to check our computation in this paper, we choose arbitrary values for certain parameters in some cases.
Example 1. Suppose that δ(a) = δ, p(a) = p and I(t) := T * (t) = ∞ 0 T * (t, a)da, then the system (1)-(4) becomes similar to the system (1) in [24] with k 2 = 0 as follows: (40) Example 2 (DDE). Now we consider infected cell death rates δ(a) and viral production rates p(a) of the following piecewise form: Note that by defining J(t) = τ 0 T * (t, a) da and I(t) = ∞ τ T * (t, a) da, then (1)-(4) turn to the following delay differential equations As in [4,25], we think of τ as the intracellular delay between cell infection and viral production. Experimental results have shown that for HIV infection this delay τ is between 1 and 2 days, we utilize τ = 2 days in our simulations of this model. The defined variables J(t) and I(t) represent infected cells in their eclipse phase and productive phase, respectively.  1 and 4, the parameter values are chosen such that R 0 < 1. In this cases, E 0 is globally asymptotically stable. In other figures R 0 > 1 and E 0 is unstable. In Figures 2 and 5 , E is stable and in Figures 3 and 6, E is unstable and solutions approach to a periodic solution. We note that in both examples the Hopf bifurcation occurs when we increase R 0 by altering a couple of the parameter values (see captions for details). While the condition for Hopf bifurcation, Eq. (36), is too complicated to formally analyze, we can conjecture that the logistic growth of target cells is necessary for the observed oscillatory dynamics. Indeed, the model analyzed in [24] is similar to system (38), except without logistic growth rate, and did not display periodic solutions. Additionally, Shu et al. found that a similar delay model to (41)-(44) (without cell re-infection) only displayed oscillations when logistic growth was used for uninfected cells [14] In further simulations (not shown), we varied re-infection rate γ and did not find it to be a critical parameter for Hopf bifurcation. Thus, evidence suggests that the logistic growth of uninfected cells is the factor which can induce sustained oscillations. In addition, we quantify the probability of an infected cell being re-infected before infection-age σ ≤ τ , where σ is the cell infection-age when reverse transcription is complete. The interest in this quantity lies in the fact that viral recombination within a co-infected cell can only occur before completion of the reverse transcription, and the recombination rate is of significance for within-host HIV evolution [16]. In general, the probability of an infected cell being re-infected before infection-age σ ≤ τ at the positive equilibrium, similar to calculations in [30], is given by For the piecewise function in this example, π(σ) = γV µ+γV 1 − e −(µ+γV )σ . In general, the number of infected cells which have been re-infected before age σ at any time t, denoted by R(t, σ), is given by the following expression: The above formula is similar to equations derived when the age-structured model includes explicit immune response in [5]. At the positive equilibrium, it can be shown  that the above equations are related as probability of reinfection is equal to the proportion of re-infected cells (re-infected at age a ≤ σ): π(σ) =R(σ)/ ∞ 0T * (a)da. For each of the cells re-infected before reverse transcription (before age σ), there is a probability that recombination can occur. Experimental estimates of the proportion of re-infected cells have recently been measured to be 10-30% of infected cells in for HIV [16,10]. For the parameters in Figures 5 and 6, we calculate the probability of re-infection of an infected cell during the eclipse phase (calculated at the positive equilibrium) to be π(τ ) = 0.23 and π(τ ) = 0.2882, respectively. A fraction of re-infected cells will undergo recombination during reverse transcription.
6. Discussion. The global analysis of age-structured within-host virus model was studied by many authors such as [31] and [6]. To study the global stability of infection equilibrium, Browne and Pilyugin [6] assumed the "sector" condition, which does not hold for our model that incorporates proliferation of healthy cells. Also, they assumed that the proportion of viruses which entered into the cells are small in comparison to the free viruses or can be absorbed into the viral death rate, a common assumption for HIV models. However, this model does not work for all infections, such as poliovirus, where the amount of virus entered into the cells is notable to consider. In addition, for HIV infection, the re-infection of already infected cells is important to consider for quantifying viral recombination which may contribute significantly to in-host evolution [10]. In this paper, we introduced a model which accounts for viral entry into infected cells, which amplifies the complexity of analysis of the model.
In this paper, we find reproduction number and analyzed the model for R 0 ≤ 1 and 1 < R 0 . For R 0 ≤ 1, we proved that the infection-free equilibrium is globally asymptotically stable. Conversely, we showed that in the case 1 < R 0 , the infection free equilibrium, there exists a unique infection equilibrium, and the system is uniformly persistent. By choosing different parameters values the unique infection equilibrium can destabilize through a Hopf bifurcation. In this unstable case, we demonstrate sustained oscillations in numerical examples, along with quantifying the amount of viral re-infection.
There are several limitations to our study which should be noted. Although the model predicts clearance of the virus when R 0 ≤ 1, current treatment for HIV cannot eradicate the virus due to latently infected cells which are not targeted by antiviral therapy. Recent studies have modeled HIV persistence and the latent reservoir [26], which provides motivation for future work on extending our model to include latency. Our model also neglects virus mutations, which occur very frequently for HIV. This is particularly relevant for viral resistance to host immune response or drug treatment, and viral recombination in co-infected cells may effect the evolution of resistance. Finally, we note that the Hopf bifurcation condition (36) was too complex for rigorous analysis at this point. Future work will further analyze the observed oscillatory behavior and determine how cell re-infection affects viral dynamics.