Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 16, pp. 1–26. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STABILITY ANALYSIS OF AN AGE-STRUCTURED VIRAL INFECTION MODEL WITH LATENCY CHUNYANG LI, XIU DONG, JINLIANG WANG Abstract. Age structure and cell-to-cell transmission are two major infection mechanisms in modeling spread of infectious diseases. We propose an age- structured viral infection model with latency, infection age-structure and cell- to-cell transmission. This paper aims to reveal the basic reproduction number and prove it to be a sharp threshold determining whether the infection dies out or not. Mathematical analysis is presented on relative compactness of the orbit, existence of a global attractor, and uniform persistence of system. We further investigate local and global stability of the infection-free and infection equilibrium. 1. Introduction In the past two decades, since the pioneering work of Perelson et al. [20], within- host virus dynamics has attracted considerable attention of researchers. Many mathematical models describing the dynamics inside the host of various infectious diseases such as HIV, HBV have been formulated and studied [15, 16, 18, 19, 22, 29, 35]. The investigation of such models can help better understanding the interaction mechanisms of target cells, infected cells, and free virus particles through time in an infected individual. The classical viral infection model in Perelson et al. [20] neglects certain features that may be important to consider for HIV, such as age structure in the infected cell component. By allowing for mortality rate and viral production rate of infected cells are functions of the infection age of the infected cells instead of constant, Nelson et al. [17], Huang et al. [6], Browne et al. [2], and Wang et al. [32] have studied age-structured model of HIV infection by considering infection age to be a continuous variable. These models generalizes the discrete and distributed delay viral infection model (modeling time delay between viral entry of a target cell and viral production from the newly infected cell. Further, such formulation of a hybrid system of ODEs and PDEs have already made the mathematical analysis very challenging in determining the threshold dynamics of equilibrium in viral infection model, which may allow us to have a good understanding on productively infected cells. 2020 Mathematics Subject Classification. 34K20, 92D25. Key words and phrases. Age-structured model; cell-to-cell transmission; uniform persistence; global stability; Lyapunov function; basic reproduction number. ©2022. This work is licensed under a CC BY 4.0 license. Submitted April 2, 2021. Published February 28, 2022. 1 2 C. LI, X. DONG, J. WANG EJDE-2022/16 Recently, HIV latency have been considered together in the viral infection model, which is the main reason that long-term low viral load persistence in patients on antiretroviral therapy. Usually, suppress the plasma viral load to below the detec- tion limit may last half life of months or years [22]. Recent studies reveals the decay dynamics of the latent reservoir, such as latency infected CD4+ T cells, see e.g. Muller et al. [14], Kim and Perelson [9] and Strain et al. [27]. Latently infected cells could be activated by specific antigen. Strain et al. [27] found that cells specific to frequently encountered antigens are activated soon while cells specific to rare antigens are not. It may be important for latently infected cells activation to be more general functions of cellular infection-age. Alshorman et al. [1] introduced latency infection age to model the heterogeneity of latently infected CD4+ T cells. Recently, based on the facts that HIV latency remains a major obstacle to viral elimination, Wang and Dong [31] considered the model dT (t) dt = h− dT (t)− βT (t)V (t),( ∂ ∂t + ∂ ∂a ) e(a, t) = −θ1(a)e(a, t),( ∂ ∂t + ∂ ∂b ) i(b, t) = −θ2(b)i(b, t), dV (t) dt = ∫ ∞ 0 p(b)i(b, t) db− cV (t), (1.1) with boundary and initial conditions e(0, t) = fβT (t)V (t), i(0, t) = (1− f)βT (t)V (t) + ∫ ∞ 0 ξ(a)e(a, t) da, T (0) = T0 ≥ 0, e(a, 0) = e0(a) ∈ L1 +(0,∞), i(b, 0) = i0(b) ∈ L1 +(0,∞), V (0) = V0 ≥ 0, where T (t), e(a, t), i(a, t), V (t) the concentration of uninfected CD4+ T cells at time t, latently infected T cells with latency age a at time t, productively infected cells with infection age b at time t, and virions in plasma at t, respectively. The parameter h, d, β and c are the production rate of uninfected CD4+ T cells, the per capita death rate of uninfected cells, infection rate of CD4+ T cells by infectious virus and the viral clearance rate, respectively. In model (1.1), a small fraction f ∈ (0, 1) is assumed to be latency infected cells and that the remaining 1 − f become productively infected cells (see also in [1]. θ1(a) is used to illustrate the decreasing effect of the pool size of latent infected cells when latently infected cells are activated. θ2(b) represents the death rate of productively infected cells. ξ(a) denotes the activation rate of latently infected T cells with latency age a.∫∞ 0 ξ(a)e(a, t) da denotes the total number of productively infected cells from the activation of latently infected cells. p(b) is the production rate of viral particles with infection age b. L1 +(0,∞) is the set of all integrable nonnegative functions on R+ := [0,+∞). In [31], the authors shows that (1.1) has a global attractor, and it is uniformly persistent if the basic reproduction number is greater than one. The threshold dynamics of infection-free and infection equilibrium subject to latently age and infection age are also addressed by Lyapunov functionals techniques. EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 3 However, more and more research pay attention to the fact that virus can also spread by direct cell-to-cell transmission [3, 4, 7, 10, 12, 23, 24, 31]. Viral parti- cles can be simultaneously transferred from infected target cells to uninfected ones through virological synapses during cell-to-cell transmission [7]. Some evidences reveal that cell-to-cell transmission may have a lower risk of being neutralized by neutralizing antibodies or cleared by cytotoxic T lymphocytes [12]. Dimitrov et al. [4] found that the infectivity of HIV-1 during cell-to-cell transmission is greater than the infectivity of cell-free viruses. Sigal et al. [24] claimed that cell-to-cell spread of HIV-1 does reduce the efficacy of antiretroviral therapy. Lai and Zou [10] formulated viral infection model incorporating both the virus-to-cell infection and cell-to-cell transmission in the form of discrete and distributed delay differen- tial equations and obtain the threshold dynamics in term of the basic reproduction number. Wang et al. [32] considered the variance in the infectivity with respect to the infection age of the infected cells in the cell-to-cell transmission. Thus, it is natural to consider a model incorporating Latency infection age, infection age and cell-to-cell transmission. This constitutes one motivation of the present paper. Our second motivation comes from a series of works on infection age within host models [17, 6, 2, 21, 32], which are devoted to understanding the joint effects of the age structure and the cell-to-cell transmission on threshold dynamics of these models. Our study is mainly motivated by [31, 32] where the authors proved that the threshold dynamics of the model. It is then interesting to see whether similar results hold for our present model. In this article, we propose and study the following age-structured HIV infection model with latency and and both cell-free and cell-to-cell transmission modes. dT (t) dt = h− dT (t)− β1T (t)V (t)− β2T ∫ ∞ 0 q(b)i(b, t) db,( ∂ ∂t + ∂ ∂a ) e(a, t) = −θ1(a)e(a, t),( ∂ ∂t + ∂ ∂b ) i(b, t) = −θ2(b)i(b, t), dV (t) dt = ∫ ∞ 0 p(b)i(b, t) db− cV (t), (1.2) with boundary conditions e(0, t) = f ( β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db ) , i(0, t) = (1− f) ( β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db ) + ∫ ∞ 0 ξ(a)e(a, t) da, (1.3) and initial conditions T (0) = T0 ≥ 0, V (0) = V0 ≥ 0, e(a, 0) = e0(a) ∈ L1 +(0,∞), i(b, 0) = i0(b) ∈ L1 +(0,∞), where β1 and β2 are the infection rate of CD4+ T cells by infectious virus and productively infected cells, respectively. The meaning of other parameters in (1.2) are the same as in (1.1). We make the following assumptions on the parameters and functions in (1.2). Assumption 1.1. (i) h, d, β1, β2, c > 0; 4 C. LI, X. DONG, J. WANG EJDE-2022/16 (ii) For 1 = 1, 2, q(·), θi(·), p(·), ξ(·) ∈ L∞+ (0,∞) satisfy the conditions: q̄ := ess supb∈[0,∞) q(b) <∞, θ̄i := ess supa∈[0,∞) θi(a) <∞, p̄ := ess supa∈[0,∞) p(a) <∞, ξ̄ := ess supa∈[0,∞) ξ(a) <∞, (iii) q(·), p(·), ξ(·) are Lipschitz continuous on R+ := [0,∞) with Lipschitz constants Mq, Mp, Mξ respectively; (iv) There exists µ0 ∈ (0, d] such that θ1(a), θ2(b) ≥ µ0 for all a ≥ 0; (v) There exists a maximum age b+ > 0 for the viral production such that p(b) > 0 for b ∈ (0, b+) and p(b) = 0 for b > b+. The remaining part of this article proceeds as follows. In the next Section, we give some preliminary results including solution semi-flow, Volterra formulation of solutions, boundedness of solutions, basic reproduction number and existence of equilibria. Section 3 is devoted to the relative compactness of solution semi-flow and the existence of global attractor. The uniform persistence of (1.2) is proved in Section 4. We obtain the local stability of the infection-free equilibrium and the infection equilibrium in Section 5. Then we establish their global attractivity in Section 6. 2. Preliminaries 2.1. Semi-flow solution. We define the state space of (1.2) as Y = R+ × L1 +(0,∞)× L1 +(0,∞)× R+ endowed with the norm ‖(x, ϕ, ψ, y)‖Y = |x|+ ‖ϕ‖L1 + ‖ψ‖L1 + |y| for (x, ϕ, ψ, y) ∈ Y. If any initial value X0 = (T0, e0(·), i0(·), V0) ∈ Y satisfies the coupling equations e(0, 0) = f ( β1T0V0 + β2T0 ∫ ∞ 0 q(b)i0(b) db ) , i(0, 0) = (1− f) ( βT0V0 + +β2T0 ∫ ∞ 0 q(b)i0(b) db ) + ∫ ∞ 0 ξ(a)e0(a)da, then (1.2) is well-posed under Assumption 1.1 according to Iannelli [8] and Magal [11]. In fact, it is easy to show that (T (t), e(·, t), i(·, t), V (t)) ∈ Y for each t ≥ 0. We still assume that the initial values satisfy the coupling equations in the remaining context. Thus we have a continuous solution semi-flow Φ : R+×Y → Y defined by Φ (t,X0) = Φt(X0) := (T (t), e(·, t), i(·, t), V (t)) , t ≥ 0, X0 ∈ Y. (2.1) For the ease of notation, we introduce Ω(a) = e− ∫ a 0 θ1(τ)dτ , Γ(b) = e− ∫ b 0 θ2(τ)dτ for a, b ≥ 0, (2.2) Q(t) = ∫ ∞ 0 q(b)i(b, t) db, M(t) = ∫ ∞ 0 ξ(a)e(a, t) da, N(t) = ∫ ∞ 0 p(b)i(b, t ) db. (2.3) EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 5 It follows from (ii) and (v) of Assumption 1.1 that for all a, b ≥ 0, 0 ≤ Ω(a) ≤ e−µ0a, 0 ≤ Γ(b) ≤ e−µ0b, Ω′(a) = −θ1(a)Ω(a), Γ′(b) = −θ2(b)Γ(b). (2.4) 2.2. Volterra formulation. Along the characteristic lines t − a = const. and t− b = const., the second and third equations of (1.2) can be calculated as e(a, t) =  f [ β1V (t− a) + β2Q(t− a) ] T (t− a)Ω(a) = e(0, t− a)Ω(a), if 0 ≤ a ≤ t, e0(a− t) Ω(a) Ω(a−t) , if 0 ≤ t ≤ a; (2.5) and i(b, t) = { i(0, t− b)Γ(b), if 0 ≤ b ≤ t, i0(b− t) Γ(b) Γ(b−t) , if 0 ≤ t ≤ b, (2.6) where i(0, t− b) = {(1− f) [β1T (t− b)V (t− b) + β2T (t− b)Q(t− b)] +M(t− b)}. 2.3. Boundedness of solutions. Proposition 2.1. Let us define Ξ := { X0 = (T0, e0, i0, V0) ∈ Y : T0 + ‖e0(a)‖L1 ≤ h µ0 , T0 + ‖e0(a)‖L1 + ‖i0(b)‖L1 ≤ h µ1 , V0 ≤ p̄h cµ0 + hp̄ξ̄ cµ2 0 , ‖Φt(X0)‖Y ≤ h µ̃0 } , where µ̃0 := µ0 1 + ξ̄ µ0 + p̄ c + p̄ξ̄ cµ0 , µ1 := µ0 1 + ξ̄ µ0 . Then Ξ is a positively invariant subset for Φ; that is, Φ(t,X0) ∈ Ξ for all t ≥ 0 and X0 ∈ Ξ. Moreover, Φ is point dissipative and Ξ attracts all points in Y. Proof. It follows from (2.5) and changing variables that ‖e(·, t)‖L1 = ∫ t 0 e(0, t− a)Ω(a) da+ ∫ ∞ t e0(a− t) Ω(a) Ω(a− t) da = ∫ t 0 e(0, σ)Ω(t− σ) dσ + ∫ ∞ 0 e0(τ) Ω(t+ τ) Ω(τ) dτ. Thus d‖e(·, t)‖L1 dt = e(0, t)Ω(0) + ∫ ∞ 0 e0(τ) Ω(τ) dΩ(t+ τ) dt dτ + ∫ t 0 e(0, σ) dΩ(t− σ) dt dσ. It follows from (2.4) and changing of variables that d‖e(·, t)‖L1 dt = e(0, t)Ω(0)− ∫ ∞ 0 e0(τ) Ω(τ) θ1(t+ τ)Ω(t+ τ)dτ − ∫ t 0 e(0, σ)θ1(t− σ)Ω(t− σ) dσ = e(0, t)Ω(0)− ∫ ∞ 0 θ1(a)e(a, t) da. (2.7) 6 C. LI, X. DONG, J. WANG EJDE-2022/16 This, combined with the first equation in (1.2) and (v) of Assumption 1.1, gives us d(T (t) + ‖e(·, t)‖L1) dt = h− dT (t)− β1T (t)V (t)− β2T ∫ ∞ 0 q(b)i(b, t) db + f [β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db]− ∫ ∞ 0 θ1(a)e(a, t) da ≤ h− µ0(T (t) + ‖e(·, t)‖L1) for t ≥ 0. An application of the variation of constants formula immediately yields T (t) + ‖e(·, t)‖L1 ≤ h µ0 − e−µ0t { h µ0 − (T0 + ‖e0‖L1) } , t ≥ 0, (2.8) which implies that if X0 ∈ Ξ then T (t) + ‖e(·, t)‖L1 ≤ h µ0 for t ≥ 0. Further, we can derive d‖i(·, t)‖L1 dt = (1− f)[β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db] + ∫ ∞ 0 ξ(a)e(a, t) da− ∫ ∞ 0 θ2(b)i(b, t) db. It follows that d(T (t) + ‖e(·, t)‖L1) + ‖i(·, t)‖L1) dt = h− dT + ∫ ∞ 0 ξ(a)e(a, t) da− ∫ ∞ 0 θ1(a)e(a, t) da− ∫ ∞ 0 θ2(b)i(b, t) db. From (2.7) and (v) of Assumption 1.1, d(T (t) + ‖e(·, t)‖L1) + ‖i(·, t)‖L1) dt ≤ h+ ξ̄‖e(·, t)‖L1 − µ0(T (t) + ‖e(·, t)‖L1 + i(·, t)‖L1) ≤ h+ ξ̄ h µ0 − µ0(T (t) + ‖e(·, t)‖L1 + i(·, t)‖L1). Using the variation of constants formula again gives ‖T (t) + ‖e(·, t)‖L1) + ‖i(·, t)‖L1 ≤ h+ ξ̄ hµ0 µ0 − e−µ0t {h+ ξ̄ hµ0 µ0 − (T (t) + ‖e(·, t)‖L1 + i(·, t)‖L1) } , t ≥ 0. (2.9) From (2.9), we have ‖i(·, t)‖L1 ≤ h µ0 + ξ̄h µ2 0 . Similarly, it follows from dV (t) dt ≤ p̄‖i(·, t)‖L1 − cV (t) ≤ p̄ ( h µ0 + ξ̄h µ2 0 ) − cV (t) that V (t) ≤ p̄( hµ0 + ξ̄h µ2 0 ) c − e−ct { p̄h cµ0 + hp̄ξ̄ cµ2 0 − V0 } ≤ p̄h cµ0 + hp̄ξ̄ cµ2 0 − e−µ0t { p̄h cµ0 + hp̄ξ̄ cµ2 0 − V0 } . (2.10) EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 7 Summing (2.8), (2.9) and (2.10), we conclude that if X0 ∈ Ξ, then for t ≥ 0, ‖Φt(X0)‖Y ≤ ( 1 + ξ̄ µ0 + p̄ c + p̄ξ̄ cµ0 ) h µ0 − e−µ0t {( 1 + ξ̄ µ0 + p̄ c + p̄ξ̄ cµ0 ) h µ0 − ‖X0‖Y } = h µ̃0 − e−µ0t { h µ̃0 − ‖X0‖Y } ≤ h µ̃0 . (2.11) Consequently, Ξ is positively invariant with respect to Φ. From (2.9) and (2.11) if follows that lim supt→∞[T (t) + ‖e(·, t)‖L1 ] ≤ h µ1 and lim supt→∞ ‖Φt(X0)‖Y ≤ h µ̃0 for any X0 ∈ Y, that is, Φ is point dissipative and Ξ attracts all points in Y. This completes the proof. � The following result is a direct consequence of Proposition 2.1, which will be used later. Proposition 2.2. Let A ≥ h/µ̃0 be given. If X0 ∈ Y satisfies ‖X0‖Y ≤ A, then the following statements hold for all t ≥ 0. (i) T (t), ‖e(·, t)‖L1 , ‖i(·, t)‖L1 , V (t) ≤ A; (ii) M(t) ≤ ξ̄A and N(t) ≤ p̄A; (iii) e(0, t) ≤ fβA2, i(0, t) ≤ (1− f)βA2 + ξ̄A, where β = β1 + β2q̄. 2.4. Existence of equilibria. System (1.2) admits an infection-free equilibrium P 0 = (T 0, e0(a), i0(b), V 0) := (hd , 0, 0, 0). An equilibrium (T ∗, e∗, i∗, V ∗) ∈ Y of (1.2) should satisfy h− dT ∗ − β1T ∗V ∗ − β2T ∗ ∫ ∞ 0 q(b)i∗(b) db = 0, d da e∗(a) = −θ1(a)e∗(a), d db i∗(b) = −θ2(b)i∗(b),∫ ∞ 0 p(b)i∗(b) db = cV ∗, e∗(0) = fβ1T ∗V ∗ + fβ2T ∗ ∫ ∞ 0 q(b)i∗(b) db, i∗(0) = (1− f)β1T ∗V ∗ + (1− f)β2T ∗ ∫ ∞ 0 q(b)i∗(b) db+ ∫ ∞ 0 ξ(a)e∗(a) da. (2.12) where T ∗, e∗(a), i∗(b), and V ∗ are not zero. We denote K = ∫ ∞ 0 ξ(a)Ω(a) da, J = ∫ ∞ 0 p(b)Γ(b) db, L = ∫ ∞ 0 q(b)Γ(b) db. We define the basic reproduction number of (1.2) as R0 = (1− f)β1T 0J c + (1− f)β2T 0L+ fβ1T 0KJ c + fβ2T 0KL. After a simple calculation, we see that if R0 > 1 then (1.2) has a unique infection equilibrium P ∗ = (T ∗, e∗(a), i∗(b), V ∗) with T ∗ = T 0 R0 , e∗(a) = fh ( 1− 1 R0 ) Ω(a), (2.13) 8 C. LI, X. DONG, J. WANG EJDE-2022/16 i∗(b) = (1− f + fK)h ( 1− 1 R0 ) Γ(b), V ∗ = 1 c ∫ ∞ 0 p(b)i∗(b) db. (2.14) Thus we arrive at the following result. Theorem 2.3. (i) System (1.2) always has an infection-free equilibrium P 0. (ii) If R0 > 1, then (1.2) admits a unique infection equilibrium P ∗ = (T ∗, e∗(a), i∗(a), V ∗) defined by (2.13). 3. Asymptotic smoothness of Φ(t,X0) In this section, we begin with showing the asymptotic smoothness of semiflows, then by [5, Theorem 3.4.6], the semiflow has a compact attractor. In what follows, we adopt the approach in [30, Theorem 4.2 of Chapter IV]. Definition 3.1 ([28]). A set A in Y is called a compact attractor of a set B ⊆ X if A is compact, invariant, and non-empty and Φt(B) → A as t → ∞. The last means that, for every open subset U of Y with A ⊆ U , there is some r > 0 such that Φt(B) ⊆ U for all t ≥ r (i.e. Φ([r,∞)×B) ⊆ U). The following proposition reveals that the functions M(t) and N(t) are Lipschitz continuous. The proof comes from using Proposition 2.1, Assumption 1.1 and [33, Proposition 4.1]. We omit it. Proposition 3.2. For any solution of (1.2), the functions M(t) , N(t) and Q(t) are Lipschitz continuous on R+ with Lipschitz coefficients LM , LN and LQ. Next we divide Φ : R+ × Y → Y into the following two operators Θ, Ψ : R+ × Y → Y: Θ(t,X0) := (0, ϕ̃e(·, t), ϕ̃i(·, t), 0), Ψ(t,X0) := (T (t), ẽ(·, t), ĩ(·, t), V (t)), where ϕ̃e(a, t) = { 0, if t > a ≥ 0, e(a, t), if a ≥ t ≥ 0; ϕ̃i(b, t) = { 0, if t > b ≥ 0, i(b, t), if b ≥ t ≥ 0; ẽ(a, t) = { e(a, t), if t > a ≥ 0, 0, if a ≥ t ≥ 0; ĩ(b, t) = { i(b, t), if t > b ≥ 0, 0, if b ≥ t ≥ 0. Then Φ(t,X0) = Θ(t,X0) + Ψ(t,X0) for t ≥ 0. Following the proof of [34, Propo- sition 3.13], we can arrive at the main result of this section. Theorem 3.3. For X0 ∈ Ξ, the orbit {Φ(t,X0) : t ≥ 0} has a compact closure in Y if the following two conditions hold: (i) There exists a function ∆ : R+ × R+ → R+ such that, for any r > 0, limt→∞∆ (t, r) = 0 and if X0 ∈ Ω with ‖X0‖Y ≤ r then ‖Θ (t,X0) ‖Y ≤ ∆ (t, r) for t ≥ 0; (ii) For t ≥ 0, Ψ (t, ·) maps any bounded sets of Ξ into sets with compact closure in Y. Proof. (i) Let ∆(t, r) = e−µ0tr, then limt→∞∆(t, r) = 0. By (2.5) and (2.6), ϕ̃e(a, t) = { 0, if t > a ≥ 0, e0(a− t) Ω(a) Ω(a−t) , if a ≥ t ≥ 0; EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 9 ϕ̃i(b, t) = { 0, if t > b ≥ 0, i0(b− t) Γ(b) Γ(b−t) , if b ≥ t ≥ 0. Then, for X0 ∈ Ξ satisfying ‖X0‖Y ≤ r and for t ≥ 0, we have ‖Θ (t,X0) ‖Y = |0|+ ‖ϕ̃e(·, t)‖L1 + ‖ϕ̃i(·, t)‖L1 + |0| = ∫ ∞ t |e0(a− t) Ω(a) Ω(a− t) | da+ ∫ ∞ t |i0(b− t) Γ(b) Γ(b− t) | db = ∫ ∞ 0 |e0(σ) Ω(σ + t) Ω(σ) | dσ + ∫ ∞ 0 |i0(σ) Γ(σ + t) Γ(σ) | dσ = ∫ ∞ 0 |e0(σ)e− ∫ σ+t σ θ1(τ)dτ | dσ + ∫ ∞ 0 |i0(σ)e− ∫ σ+t σ θ2(τ)dτ | dσ ≤ e−µ0t‖e0‖L1 + e−µ0t‖i0‖L1 ≤ e−µ0t‖X0‖Y . (ii) We next claim that Ψ (t, ·) maps any bounded sets of Ξ into sets with compact closure in Y. It follows from Proposition 2.1 that T (t) and V (t) remains in the compact set [0, h/µ̃0] ⊂ [0, A]. We only need to verify that ẽ (a, t) and ĩ (b, t) remain in a precompact subset of L1 + (0,∞), which is independent of X0 ∈ Ξ. We follow the method of [26, Theorem B.2]) to check the following conditions valid to ẽ (a, t) and ĩ (b, t), (i) The supremum of ‖ẽ (·, t) ‖L1 with respect to X0 ∈ Ξ is finite; (ii) limh→∞ ∫∞ h ẽ (a, t) da = 0 uniformly with respect to X0 ∈ Ξ; (iii) limh→0+ ∫∞ 0 |ẽ (a+ h, t)− ẽ (a, t) |da = 0 uniformly with respect to X0 ∈ Ξ; (iv) limh→0+ ∫ h 0 ẽ (a, t) da = 0 uniformly with respect to X0 ∈ Ξ. It follows from (2.5), (2.6), Proposition 2.2, and (2.4) that ẽ(a, t) ≤ fβA2e−µ0a, ĩ(b, t) ≤ [ (1− f)βA2 + ξ̄A ] e−µ0b. Thus, (i), (ii), and (iv) are satisfied. Next we verify condition (iii). For sufficiently small h ∈ (0, t), we have∫ ∞ 0 |ẽ(a+ h, t)− ẽ(a, t)| da = ∫ t−h 0 |e(a+ h, t)− e(a, t)|da+ ∫ t t−h |0− e(a, t)|da = ∫ t−h 0 ∣∣∣e(0, t− a− h)Ω(a+ h)− e(0, t− a)Ω(a) ∣∣∣ da+ ∫ t t−h ∣∣∣e(0, t− a)Ω(a) ∣∣∣da ≤ ∆1 + ∆2 + fβA2h, where ∆1 = ∫ t−h 0 e(0, t− a− h)|Ω(a+ h)− Ω(a)| da, ∆2 = ∫ t−h 0 |e(0, t− a− h)− e(0, t− a)|Ω(a) da. 10 C. LI, X. DONG, J. WANG EJDE-2022/16 We first estimate ∆1. Directly calculations give∫ t−h 0 |Ω(a+ h)− Ω(a)|da = ∫ t−h 0 ( Ω(a)− Ω(a+ h) ) da = ∫ t−h 0 Ω(a) da− ∫ t h Ω(a) da = ∫ t−h 0 Ω(a) da− ∫ t−h h Ω(a) da− ∫ t t−h Ω(a) da = ∫ h 0 Ω(a) da− ∫ t t−h Ω(a) da ≤ h, it follows from Proposition 2.2 that ∆1 ≤ fβA2h. Next we estimate ∆2. Rewriting ∆2 as ∆2 = ∫ t−h 0 |e(0, t− a− h)− e(0, t− a)|Ω(a)da = ∫ t−h 0 ∣∣∣(fβ1T (t− a− h)V (t− a− h) + fβ2T (t− a− h)Q(t− a− h) ) − ( fβ1T (t− a)V (t− a) + fβ2T (t− a)Q(t− a) )∣∣∣Ω(a) da ≤ ∫ t−h 0 ∣∣fβ1T (t− a− h)V (t− a− h)− fβ1T (t− a)V (t− a) ∣∣Ω(a) da + ∫ t−h 0 ∣∣fβ2T (t− a− h)Q(t− a− h)− fβ2T (t− a)Q(t− a) ∣∣Ω(a) da. Since T (t) and V (t) are both Lipschitz continuous on R+ with Lipschitz constants MT = h + dA + β1A 2 + β2q̄A 2 and MV = (p̄ + c)A, respectively. It follows from Proposition 3.2 and [13, Proposition 6] that T (t)V (t)is Lipschitz continuous with Lipschitz constants MTV = AMV +AMT and MTQ = AMQ+ q̄AMT . Denote that G = fβ1MTV + fβ2MTQ. Thus ∆2 ≤ Gh ∫ t−h 0 e−µ0ada ≤ Gh µ0 . Hence ∫ ∞ 0 |ẽ(a+ h, t)− ẽ(a, t)|da ≤ ( 2fβA2 + G µ0 ) h, and condition (iii) follows. As for ĩ (b, t), we have∫ ∞ 0 |̃i(b+ h, t)− ĩ(b, t)| db = ∫ t−h 0 |i(b+ h, t)− i(b, t)|db+ ∫ t t−h |0− i(b, t)|db = ∫ t−h 0 ∣∣i(0, t− b− h)Γ(b+ h)− i(0, t− b)Γ(b) ∣∣ db+ ∫ t t−h ∣∣i(0, t− b)Γ(b) ∣∣ db ≤ Υ1 + Υ2 + [ (1− f)βA2 + ξ̄A ] h, EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 11 where Υ1 = ∫ t−h 0 i(0, t− b− h)|Γ(b+ h)− Γ(b)| db, Υ2 = ∫ t−h 0 |i(0, t− b− h)− i(0, t− b)|Γ(b) db. Similarly, we have ∫ t−h 0 ∣∣∣Γ(b + h) − Γ(b) ∣∣∣db ≤ h. Hence from Proposition 2.2, we can conclude that Υ1 ≤ [(1− f)βA2 + ξ̄A]h. For Υ2, we have Υ2 ≤ (1− f) ∫ t−h 0 ∣∣∣β1 (T (t− b− h)V (t− b− h)− T (t− b)V (t− b)) + β2 (T (t− b− h)Q(t− b− h)− T (t− b)Q(t− b)) ∣∣∣Γ(b) db + ∫ t−h 0 |M(t− b− h)−M(t− b)|Γ(b) db. As before, MTV = AMV + AMT ,MTQ = AMQ + q̄AMT . Recall that M(t) is Lipschitz continuous on R+ with Lipschitz constants LM = (ξ̄fβA + ξ̄θ̄1 + Mξ)A. Set H = (1− f)(β1MTV + β2MTQ) + LM , By a zero-trick, then we have Υ2 ≤ Hh ∫ t−h 0 e−µ0bdb ≤ Hh µ0 . Finally, we have∫ ∞ 0 |̃i(b+ h, t)− ĩ(b, t)| db ≤ { 2[(1− f)βA2 + ξ̄A] + H µ0 } h, thus condition (iii) follows. This completes the proof. � According to Smith and Thieme [26], we arrive at the following theorem for the existence of global attractors of the semi-flow {Φ(t)}t≥0. Theorem 3.4. The semi-flow {Φ(t)}t≥0 has a global attractor A in Y, which at- tracts any bounded subset of Y. 4. Uniform persistence The aim of this section is to show that (1.2) is uniformly persistent when the basic reproduction number is greater than one. Let ê(t) := e(0, t) and î(t) := i(0, t). Then the first three equations of (1.2) can be rewritten as dT (t) dt = h− dT (t)− 1 f ê(t), e(a, t) = { ê(t− a)Ω(a), if t ≥ a ≥ 0, e0(a− t) Ω(a) Ω(a−t) , if a ≥ t ≥ 0; i(b, t) = { î(t− b)Γ(b), if t ≥ b ≥ 0, i0(b− t) Γ(b) Γ(b−t) , if b ≥ t ≥ 0, (4.1) 12 C. LI, X. DONG, J. WANG EJDE-2022/16 where ê(t) = f ( β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db ) , (4.2) î(t) = (1− f) ( β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db ) + ∫ t 0 ξ(a)Ω(a)ê(t− a) da+ ∫ ∞ t ξ(a) Ω(a) Ω(a− t) e0(a− t) da. (4.3) Lemma 4.1. If R0 > 1, then there exists a positive constant ε0 such that lim sup t→∞ ê(t) > ε0. (4.4) Proof. We first get an estimate on î(t) as follows. By (4.3), we have î(t) ≥ (1− f) ( β1T (t)V (t) + β2T ∫ ∞ 0 q(b)i(b, t) db ) + ∫ t 0 ξ(a)Ω(a)ê(t− a)da. (4.5) It follows from the fourth equation in (1.2) that V (t) ≥ ∫ t 0 e−c(t−τ) ∫ τ 0 p(b)i(b, τ) db dτ = ∫ t 0 e−c(t−τ) ∫ τ 0 p(b)Γ(b)̂i(τ − b) db dτ. This, combined with (4.5), gives us (1− f) ( β1T ∫ t 0 e−c(t−τ) ∫ τ 0 p(b)Γ(b)̂i(τ − b) db dτ + β2T ∫ t 0 q(b)̂i(t− b)Γ(b) db ) + fβ1 ∫ t 0 ξ(a)Ω(a)T (t− a) ∫ t−a 0 e−c(t−a−τ) ∫ τ 0 p(b)Γ(b)̂i(τ − b) dbdτ da (4.6) + fβ2 ∫ t 0 ξ(a)Ω(a)T (t− a) ∫ t 0 q(b)̂i(τ − b)Γ(b) dbda ≤ î(t). Since R0 > 1, there exists a sufficiently small ε1 > 0(ε1 = 1 f ε0) such that (1− f)β1 c h− ε1 d ∫ ∞ 0 p(b)Γ(b) db+ fβ1 c h− ε1 d ∫ ∞ 0 ξ(a)Ω(a) da ∫ ∞ 0 p(b)Γ(b) db + (1− f)β2 h− ε1 d ∫ ∞ 0 q(b)Γ(b) db (4.7) + fβ2 h− ε1 d ∫ ∞ 0 ξ(a)Ω(a) da ∫ ∞ 0 q(b)Γ(b) db > 1. We claim that (4.4) holds for this ε0. Otherwise, there exists a T > 0 such that ê(t) ≤ ε0 for all t ≥ T. Then it follows from (4.1) that dT (t) dt ≥ h− dT (t)− ε1 for t ≥ T . This implies that lim inft→∞ T (t) ≥ h−ε1 d . Thus there exists T̂ > T such that T (t) ≥ h−ε1 d for all EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 13 t ≥ T̂ and hence (4.6) becomes (1− f)β1 h− ε1 d ∫ t 0 e−c(t−τ) ∫ τ 0 p(b)Γ(b)̂i(τ − b) db dτ + (1− f)β2 h− ε1 d ∫ t 0 q(b)̂i(t− b)Γ(b) db + fβ1 h− ε1 d ∫ t 0 ξ(a)Ω(a) ∫ t−a 0 e−c(t−a−τ) ∫ τ 0 p(b)Γ(b)̂i(τ − b) dbdτ da + fβ2 h− ε1 d ∫ t 0 ξ(a)Ω(a) ∫ t 0 q(b)̂i(τ − b)Γ(b) db da ≤ î(t), (4.8) for all t ≥ T̂ . Without loss of generality, we can assume that (4.8) holds for all t ≥ 0 (just replace X0 by Φ(T̂ ,X0)). Then taking the Laplace transforms on both sides of (4.8), we obtain L[̂i] ≥ (1− f)β1 h− ε1 d 1 c+ λ ∫ ∞ 0 e−λbp(b)Γ(b) dbL[̂i] + (1− f)β2 h− ε1 d ∫ ∞ 0 e−λbq(b)Γ(b) dbL[̂i] + fβ1 h− ε1 d 1 c+ λ ∫ ∞ 0 e−λbp(b)Γ(b) db ∫ ∞ 0 e−λaξ(a)Ω(a) daL[̂i] + fβ2 h− ε1 d ∫ ∞ 0 e−λbq(b)Γ(b) db ∫ ∞ 0 e−λaξ(a)Ω(a) daL[̂i]. Here L[̂i] denotes the Laplace transform of î, which is strictly positive because of (4.2) and Assumption 1.1. Dividing both sides of the above inequality by L[̂i] and letting λ→ 0 give us 1 ≥ (1− f)β1 c h− ε1 d ∫ ∞ 0 p(b)Γ(b) db+ (1− f)β2 h− ε1 d ∫ ∞ 0 q(b)Γ(b) db + fβ1 c h− ε1 d ∫ ∞ 0 p(b)Γ(b) db ∫ ∞ 0 ξ(a)Ω(a) da + fβ2 h− ε1 d ∫ ∞ 0 q(b)Γ(b) db ∫ ∞ 0 ξ(a)Ω(a) da, which contradicts (4.7). This completes the proof. � We define a function ρ : Y → R+ on Y by ρ (x, ϕ, ψ, y) = fβ1xy + fβ2x ∫ ∞ 0 q(b)ψ(b) db, (x, ϕ, ψ, y) ∈ Y. We easily see that ρ(Φt(X0)) = ê(t) for t ≥ 0 and X0 ∈ Y. Then Lemma 4.1 tells us that if R0 > 1 then the semi-flow Φ is uniformly weakly ρ-persistent. Further, from Theorem 3.4 and the Lipschitz continuity of î and Smith and Thieme [26, Theorem 5.2], we conclude that the uniform weak ρ-persistence of the semi-flow Φ implies its uniform (strong) ρ-persistence. We have the following result. Theorem 4.2. If R0 > 1, then the semi-flow Φ is uniformly (strongly) ρ-persistent. 14 C. LI, X. DONG, J. WANG EJDE-2022/16 When R0 > 1, the uniform persistence of (1.2) immediately follows from The- orem 4.2. In fact, it follows from (4.1) that ‖e(·, t)‖L1 ≥ ∫ t 0 ê(t − a)Ω(a) da and hence from a variation of the Lebesgue-Fatou lemma [25, Section B.2] we obtain lim inf t→∞ ‖e(·, t)‖L1 ≥ ê∞ ∫ ∞ 0 Ω(a) da, where ê∞ = lim inft→∞ ê(t). Under Theorem 4.2, there exists a positive constant ε > 0 such that ê∞ > ε if R0 > 1 and hence the persistence of e(a, t) with respect to ‖ · ‖L1 follows. By a similar argument, we can prove that T (t) and V (t) are persistent with respect to | · | and i(a, t) is persistent with respect to ‖ · ‖L1 . To summarize, we obtain the following result. Theorem 4.3. If R0 > 1, then the semi-flow {Φ(t)}t≥0 is uniformly persistent in Y, that is, there exists a constant ε > 0 such that, for each X0 ∈ Y, lim inf t→∞ T (t) ≥ ε, lim inf t→∞ ‖e(·, t)‖L1 ≥ ε, lim inf t→∞ ‖i(·, t)‖L1 ≥ ε, lim inf t→∞ V (t) ≥ ε. 5. Local stability of infection-free and infection equilibrium We begin with the local stability of infection-free equilibrium P 0. Theorem 5.1. The infection-free equilibrium P 0 = (h/d, 0, 0) is locally asymptot- ically stable if R0 < 1 while it is unstable if R0 > 1. Proof. Linearizing (1.2) around the disease-free equilibrium P 0 under introducing the perturbation variables x1(t) = T (t)− h d , x2(a, t) = e(a, t), x3(b, t) = i(b, t), x4(t) = V (t), we obtain the system dx1(t) dt = −dx1(t)− β1 h d x4(t)− β2 h d ∫ ∞ 0 q(b)x3(b, t) db,( ∂ ∂t + ∂ ∂a ) x2(a, t) = −θ1(a)x2(a, t),( ∂ ∂t + ∂ ∂b ) x3(b, t) = −θ2(b)x3(b, t), dx4(t) dt = ∫ ∞ 0 p(b)x3(b, t) db− cx4(t), x2(0, t) = f ( β1 h d x4(t) + β2 h d ∫ ∞ 0 q(b)x3(b, t) db ) , x3(0, t) = (1− f) ( β1 h d x4(t) + β2 h d ∫ ∞ 0 q(b)x3(b, t) db ) + ∫ ∞ 0 ξ(a)x2(a, t)da, (5.1) We set x1(t) = x0 1e λt, x2(a, t) = x0 2(a)eλt, x3(b, t) = x0 3(b)eλt, x4(t) = x0 4e λt, (5.2) where x0 1, x 0 2(a), x0 3(b), x0 4 are to be determined. Plugging (5.2) into (1.1), we have λx0 1 = −dx0 1 − β1 h d x0 4 − β2 h d ∫ ∞ 0 q(b)x0 3(b) db, EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 15 λx0 2(a) + dx0 2(a) da = −θ1(a)x0 2(a), x0 2(0) = f ( β1 h d x0 4 + β2 h d ∫ ∞ 0 q(b)x0 3(b) db ) , (5.3) λx0 3(b) + dx0 3(b) db = −θ2(b)x0 3(b), x0 3(0) = (1− f) ( β1 h d x0 4 + β2 h d ∫ ∞ 0 q(b)x0 3(b) db ) + ∫ ∞ 0 ξ(a)x0 2(a)da, (5.4) λx0 4 = ∫ ∞ 0 p(b)x0 3(b) db− cx0 4, (5.5) By integrating the first equation in (5.3) and (5.4) from 0 to a, we obtain x0 2(a) = x0 2(0)e−λa− ∫ a 0 θ1(s)ds, (5.6) x0 3(b) = x0 3(0)e−λb− ∫ b 0 θ2(s)ds = [ (1− f) ( β1 h d x0 4 + β2 h d ∫ ∞ 0 q(b)x0 3(b) db ) + ∫ ∞ 0 ξ(a)x0 2(a)da ] e−λb− ∫ b 0 θ2(s)ds. (5.7) and from(5.3), (5.5), (5.6), and (5.7), we have x0 2(0) = 1− f λ+ c hβ1 d x0 2(0) ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)dsdb + f λ+ c hβ1 d x0 2(0) ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)ds db + (1− f)β2 h d x0 2(0) ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db + fβ2 h d x0 2(0) ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db. (5.8) It follows that H(λ) = 1, (5.9) where H(λ) = 1− f λ+ c hβ1 d ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)dsdb + f λ+ c hβ1 d ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)ds db + (1− f)β2 h d ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db + fβ2 h d ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db. Since H is a continuously differentiable with limλ→∞H(λ) = 0, limλ→−∞H(λ) = ∞, and H′(λ) < 0, it follows that (5.9) has a unique real root, say λ∗. Moreover, noting H(0) = R0, we have λ∗ < 0 if R0 < 1 and λ∗ > 0 if R0 > 1, which implies 16 C. LI, X. DONG, J. WANG EJDE-2022/16 that P 0 is unstable if R0 > 1. Now suppose that R0 < 1. Let λ = µ + νi be an arbitrary complex root of (5.9). Then 1 = |H(λ)| = |H(µ+ νi)| ≤ H(µ), which implies that 0 > λ∗ ≥ µ. In other words, all roots of (5.9) have negative real parts and hence P 0 is locally asymptotically stable if R0 < 1. This completes the proof. � Theorem 5.2. The unique infection equilibrium P ∗ of (1.2) is locally asymptoti- cally stable when R0 > 1. Proof. Linearizing the system (1.2) at P ∗ under introducing the perturbation vari- ables y1(t) = T (t)− T ∗, y4(t) = V (t)− V ∗, y2(a, t) = e(a, t)− e∗(a), y3(b, t) = i(b, t)− i∗(b), we obtain the system dy1(t) dt = −dR0y1(t)− β1T ∗y4(t)− β2T ∗ ∫ ∞ 0 q(b)y3(b, t) db,( ∂ ∂t + ∂ ∂a ) y2(a, t) = −θ1(a)y2(a, t),( ∂ ∂t + ∂ ∂b ) y3(b, t) = −θ2(b)y3(b, t), dy4(t) dt = ∫ ∞ 0 p(b)y3(b, t) db− cy4(t), y2(0, t) = fd(R0 − 1)y1(t) + fβ1T ∗y4(t) + fβ2T ∗ ∫ ∞ 0 q(b)y3(b, t) db, y3(0, t) = (1− f)d(R0 − 1)y1(t) + (1− f)βT ∗y4(t) + (1− f)β2T ∗ ∫ ∞ 0 q(b)y3(b, t) db+ ∫ ∞ 0 ξ(a)y2(a, t)da, (5.10) We set y1(t) = y0 1e λt, y2(a, t) = y0 2(a)eλt, y3(b, t) = y0 3(b)eλt, y4(t) = y0 4e λt, (5.11) where y0 1 , y 0 2(a), y0 3(b), y0 4 are to be determined. Substituting (5.11) into (5.10) yields λy0 1 = −dR0y 0 1 − β1T ∗y0 4 − β2T ∗ ∫ ∞ 0 q(b)y0 3(b) db, (5.12) λy0 2(a) + dy0 2(a) da = −θ1(a)y0 2(a), y0 2(0) = fd(R0 − 1)y0 1 + fβ1T ∗y0 4 + fβ2T ∗ ∫ ∞ 0 q(b)y0 3(b) db, (5.13) λy0 3(b) + dy0 3(b) db = −θ2(b)y0 3(b), y0 3(0) = (1− f)d(R0 − 1)y0 1 + (1− f)β1T ∗y0 4 + (1− f)β2T ∗ ∫ ∞ 0 q(b)y0 3(b) db+ y0 2(0) ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda, (5.14) λy0 4 = ∫ ∞ 0 p(b)y0 3(b) db− cy0 4 , (5.15) EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 17 Integrating the first equation of(5.13), (5.14) from 0 to a yields y0 2(a) = y0 2(0)e−λa− ∫ a 0 θ1(s)ds, (5.16) y0 3(b) = y0 3(0)e−λb− ∫ b 0 θ2(s)ds = [ (1− f)d(R0 − 1)y0 1 + (1− f)β1T ∗y0 4 + (1− f)β2T ∗ ∫ ∞ 0 q(b)y0 3(b) db ] × e−λb− ∫ b 0 θ2(s)ds + y0 2(0) ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda · e−λb− ∫ b 0 θ2(s)ds. and from (5.15), we have y0 4 = ∫∞ 0 p(b)y0 3(b) db λ+ c = 1− f λ+ c ( d(R0 − 1)y0 1 + β1T ∗y0 4 + β2T ∗ ∫ ∞ 0 q(b)y0 3(b) db ) × ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)dsdb + y0 2(0) λ+ c ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)ds da ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)ds db. (5.17) Substituting (5.12) and (5.13) in (5.17) yields the characteristic equation at P ∗, H(λ) = (λ+ d)H1(λ)− λ− dR0 = 0, (5.18) where H1(λ) = 1− f λ+ c β1T ∗ ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)dsdb + f λ+ c β1T ∗ ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 p(b)e−λb− ∫ b 0 θ2(s)ds db + (1− f)β2T ∗ ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db + fβ2T ∗ ∫ ∞ 0 ξ(a)e−λa− ∫ a 0 θ1(s)dsda ∫ ∞ 0 q(b)e−λb− ∫ b 0 θ2(s)ds db. It is sufficient to show that (5.18) has no roots with non-negative real parts. By way of contradiction, suppose that it has a root λ = µ + νi with µ ≥ 0. Then we have (µ+ νi+ d)H1(µ+ νi)− µ− iν − dR0 = 0. Separating the real part of the above equality gives Re H1(µ+ νi) = (µ+ dR0)(µ+ d) + ν2 (µ+ d)2 + ν2 > 1. (5.19) Noticing that H1(0) = T ∗ R0 T0 = 1 and H1 is a decreasing function, we have Re H1(µ+ νi) ≤ |H1(µ)| = H1(µ) ≤ H1(0) = 1, which contradicts with (5.19). This completes the proof. � 18 C. LI, X. DONG, J. WANG EJDE-2022/16 6. Global stability of the infection-free and infection equilibrium Set g(·) on (0,∞) defined by g(x) = x− 1− lnx for x ∈ (0,∞). Obviously, g(·) has a unique minimum at 1 with g(1) = 0. Theorem 6.1. The infection-free equilibrium P 0 of (1.2) is globally attractive if R0 < 1. Proof. By Theorem 5.1, it remains to show that P 0 is globally attractive in Y by the method of Lyapunov function. Consider candidate Lyapunov functional, LIFE(t) = L1(t) + L2(t) + L3(t) + L4(t), where L1(t) = T 0g (T (t) T 0 ) , L2(t) = ∫∞ 0 φ(a)e(a, t) da, L3(t) = ∫∞ 0 ψ(b)i(b, t) db, and L4(t) = β1T 0 c V (t). Here the nonnegative kernel functions φ(a) and ψ(b) will be determined later. The derivative of L1 along the solutions of (1.2) is calculated as follows, dL1(t) dt = ( 1− T 0 T )( h− dT − β1T (t)V (t)− β2T ∫ ∞ 0 q(b)i(b, t) db ) = ( 1− T 0 T )( dT 0 − dT − β1T (t)V (t)− β2T ∫ ∞ 0 q(b)i(b, t) db ) = −dT 0 (T 0 T + T T 0 − 2 ) − β1TV + β1T 0V − β2T ∫ ∞ 0 q(b)i(b, t) db+ β2T 0 ∫ ∞ 0 q(b)i(b, t) db. By using integration by parts, we have dL2(t) dt = ∫ ∞ 0 φ(a) ∂e(a, t) ∂t da = − ∫ ∞ 0 φ(a) [ θ1(a)e(a, t) + ∂e(a, t) ∂a ] da = −φ(a)e(a, t) ∣∣∣∞ 0 + ∫ ∞ 0 φ′(a)e(a, t) da− ∫ ∞ 0 φ(a)θ1(a)e(a, t) da = φ(0)e(0, t) + ∫ ∞ 0 ( φ′(a)− φ(a)θ1(a) ) e(a, t)da. Similarly, dL3(t) dt = ψ(0)i(0, t) + ∫ ∞ 0 ( ψ′(b)− ψ(b)θ2(b) ) i(b, t) db. It is easy to see that dL4(t) dt = β1T 0 c (∫ ∞ 0 p(b)i(b, t) db− cV ) = β1T 0 c ∫ ∞ 0 p(b)i(b, t) db− β1T 0V. Therefore, dL(t) dt = −dT 0 (T 0 T + T T 0 − 2 ) − e(0, t)− i(0, t) + ∫ ∞ 0 ξ(a)e(a, t)da+ β1T 0V + β2T 0 ∫ ∞ 0 q(b)i(b, t) db + d(L2(t) + L3(t) + L4(t)) dt EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 19 = −dT 0 (T 0 T + T T 0 − 2 ) + ( fφ(0) + (1− f)ψ(0)− 1 )( β1TV + β2T ∫ ∞ 0 q(b)i(b, t) db ) + ∫ ∞ 0 ( φ′(a)− φ(a)θ1(a) + ψ(0)ξ(a) ) e(a, t)da + ∫ ∞ 0 ( ψ′(b)− ψ(b)θ2(b) + β1T 0 c p(b) + β2T 0q(b) ) i(b, t) db. Now we choose ψ(b) = ∫ ∞ b (β1T 0 c p(u) + β2T 0q(u) ) e− ∫ u b θ2(ω)dωdu, φ(a) = ∫ ∞ a ψ(0)ξ(u)e− ∫ u a θ1(ω)dωdu. Then ψ(0) = β1T 0J c + β2T 0L, φ(0) = β1T 0KJ c + β2T 0KL, and ψ and φ satisfy ψ′(b)− ψ(b)θ2(b) + β1T 0 c p(b) + β2T 0q(b) = 0, φ′(a)− φ(a)θ1(a) + ψ(0)ξ(a) = 0. The derivative of LIFE along solutions of (1.2) is dLIFE(t) dt = −dT 0 (T 0 T + T T 0 − 2 ) + ( fφ(0) + (1− f)ψ(0)− 1 )( β1TV + β2T ∫ ∞ 0 q(b)i(b, t) db ) = −dT 0 (T 0 T + T T 0 − 2 ) + ( R0 − 1 )( β1TV + β2T ∫ ∞ 0 q(b)i(b, t) db ) = −dT 0 (T 0 T + T T 0 − 2 ) + ( R0 − 1 ) 1 f e(0, t). Notice that dLIFE(t) dt = 0 implies that T = T 0. It can be verified that the largest invariant set where dLIFE(t) dt = 0 is the singleton {P 0}. Therefore, by the invariance principle, P 0 is globally attractive when R0 ≤ 1. � The following result immediately follows from Theorem 5.1 and Theorem 6.1. Theorem 6.2. If R0 < 1, then the infection-free equilibrium P 0 of (1.2) is globally asymptotically stable. We next establish the global stability of the infection equilibrium. Lemma 6.3. Suppose that R0 > 1. Then every solution (T (t), e(a, t), i(b, t), V (t)) of (1.2) satisfies (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db )[ 1− e(0, t)i∗(0) e∗(0)i(0, t) ] + ∫ ∞ 0 ξ(a)e∗(a) [ 1− e(a, t)i∗(0) e∗(a)i(0, t) ] da = 0. (6.1) 20 C. LI, X. DONG, J. WANG EJDE-2022/16 Proof. We have (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db )e(0, t)i∗(0) e∗(0)i(0, t) + ∫ ∞ 0 ξ(a)e∗(a) e(a, t)i∗(0) e∗(a)i(0, t) da = ( (1− f)(βTV + β2T ∫ ∞ 0 q(b)i(b, t) db) + ∫ ∞ 0 ξ(a)e(a, t)da ) i∗(0) i(0, t) = i∗(0) and (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) + ∫ ∞ 0 ξ(a)e∗(a)da = i∗(0). This immediately gives (6.1). � Theorem 6.4. Assume that R0 > 1. Then the unique infection equilibrium P ∗ = (T ∗, e∗(a), i∗(a), V ∗) of (1.2) defined by (2.13) is globally asymptotically stable. Proof. By Theorem 5.2, it suffices to show that P ∗ is globally attractive. We show this by applying the Lyapunov technique again. Let G[x, y] = x− y − y ln x y , for x, y > 0. It is easy to see that G is non-negative on (0,∞)× (0,∞) with the minimum value 0 only when x = y. Furthermore, it is easy to verify that xGx[x, y] + yGy[x, y] = G[x, y]. Consider as a candidate the Lyapunov functional LEE(t) = H1(t) +H2(t) +H3(t) +H4(t), where H1(t) = G[T, T ∗], H2(t) = ∫ ∞ 0 φ1(a)G [ e(a, t), e∗(a) ] da, H3(t) = ∫ ∞ 0 ψ1(b)G [ i(b, t), i∗(b) ] db, H4(t) = β1T ∗ c G[V, V ∗], with ψ1(b) = ∫ ∞ b (β1T ∗ c p(u) + β2T ∗q(u) ) e− ∫ u b θ2(ω)dω du, φ1(a) = ∫ ∞ a ψ1(0)ξ(u)e− ∫ u a θ1(ω)dω du . (The reason of this choice is similar to that in the Proof of Theorem 6.1.) One can easily see that φ1(0) = β1T ∗KJ c + β2T ∗KL, ψ1(0) = β1T ∗J c + β2T ∗L, ψ′1(b)− ψ1(b)θ2(b) = −β1T ∗ c p(b)− β2T ∗q(b), φ′1(a)− φ1(a)θ1(a) = −ψ1(0)ξ(a). Next we calculate the derivative ofH along solutions of (1.2). Firstly, differentiating H1(t) along solutions of (1.2) yields dH1(t) dt = ( 1− T ∗ T )( h− dT − β1T (t)V (t)− β2T ∫ ∞ 0 q(b)i(b, t) db ) EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 21 = −dT ∗ (T ∗ T + T T ∗ − 2 ) + 1 f ( 1− T ∗ T ) (e∗(0)− e(0, t)). Secondly, using (2.5), we have H2(t) = ∫ t 0 φ1(a)G[e(0, t− a)Ω(a), e∗(a)]da + ∫ ∞ t φ1(a)G[e0(a− t)e− ∫ a a−t θ1(ω)dω, e∗(a)]da = ∫ t 0 φ1(t− r)G[e(0, r)Ω(t− r), e∗(t− r)]dr + ∫ ∞ 0 φ1(t+ r)G[e0(r)e− ∫ t+r r θ1(ω)dω, e∗(t+ r)]dr = B∞(t) + B∈(t). The derivative of B∞ and B∈ take the form dB∞(t) dt = φ1(0)G[e(0, t), e∗(0)] + ∫ t 0 φ′1(t− r)G [ e(0, r)e− ∫ t−r 0 θ1(ω)dω, e∗(t− r) ] dr − ∫ t 0 φ1(t− r)θ1(t− r) [ e(0, r)e− ∫ t−r 0 θ1(ω)dωGx [ e(0, r)e− ∫ t−r 0 θ1(ω)dω, e∗(t− r) ] + e∗(t− r)Gy [ e(0, r)e− ∫ t−r 0 θ1(ω)dω, e∗(t− r) ]] dr, and dB∈(t) dt = ∫ ∞ 0 φ′1(t+ r)G [ e0(r)e− ∫ t+r r θ1(ω)dω, e∗(t+ r) ] dr − ∫ ∞ 0 φ1(t+ r)θ1(t+ r) [ e0(r)e− ∫ t+r r θ1(ω)dωGx [ e0(r)e− ∫ t+r r θ1(ω)dω, e∗(t+ r) ] + e∗(t+ r)Gy [ e0(r)e− ∫ t+r r θ1(ω)dω, e∗(t+ r) ]] dr. We obtain the derivative of H2(t), dH2(t) dt = φ1(0)G[e(0, t), e∗(0)] + ∫ ∞ 0 [ φ′1(a)− φ1(a)θ1(a) ] G[e(a, t), e∗(a)] da = φ1(0)G[e(0, t), e∗(0)]− ∫ ∞ 0 ψ1(0)ξ(a)G[e(a, t), e∗(a)] da. A similar argument as in the derivative of H2, we calculate the derivative of H3, dH3(t) dt = ψ1(0)G[i(0, t), i∗(0)] + ∫ ∞ 0 [ ψ′(b)− ψ(b)θ2(b) ] G[i(b, t), i∗(b)] db = ψ1(0)G[i(0, t), i∗(0)]− ∫ ∞ 0 (β1T ∗ c p(b) + β2T ∗q(b) ) G[i(b, t), i∗(b)] db. We calculate the derivative of H4, dH4(t) dt = β1T ∗ c ∫ ∞ 0 p(b)i(b, t) db− β1T ∗V + β1T ∗V ∗ − β1T ∗V ∗ cV ∫ ∞ 0 p(b)i(b, t) db. 22 C. LI, X. DONG, J. WANG EJDE-2022/16 If follows from ψ1(0) = β1T ∗J c + β2T ∗L and φ1(0) = β1T ∗KJ c + β2T ∗KL that dLEE dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) + 1 f ( 1− T ∗ T ) (e∗(0)− e(0, t)) + φ1(0)G[e(0, t), e∗(0)]− ∫ ∞ 0 ψ1(0)ξ(a)G[e(a, t), e∗(a)]da + ψ1(0)G[i(0, t), i∗(0)]− ∫ ∞ 0 (β1T ∗ c p(b) + β2T ∗q(b) ) G[i(b, t), i∗(b)] db + ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db+ β1T ∗V ∗ − β1T ∗V − V ∗ V ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db. (6.2) Recall that (1− f) ( β1T ∗V ∗ − β1TV + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db− β2T ∫ ∞ 0 q(b)i(b, t) db ) + ∫ ∞ 0 ξ(a) ( e∗(a)− e(a, t) ) da = i∗(0)− i(0, t), and fφ1(0) + (1− f)ψ1(0) = (1− f)β1T ∗J c + (1− f)β2T ∗L+ fβ1T ∗KJ c + fβ2T ∗KL = ( (1− f)β1J c + (1− f)β2L+ fβ1KJ c + fβ2KL ) T 0 R0 = 1. Thus (6.2) becomes dLEE(t) dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) + 1 f ( 1− T ∗ T ) (e∗(0)− e(0, t)) + 1 f G[e(0, t), e∗(0)]− ∫ ∞ 0 β1T ∗ c p(b)G[i(b, t), i∗(b)] db + ψ1(0) [ (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) ln e(0, t)i∗(0) e∗(0)i(0, t) + ∫ ∞ 0 ξ(a)e∗(a) ln e(a, t)i∗(0) e∗(a)i(0, t) da ] + ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db+ β1T ∗V ∗ − β1T ∗V − V ∗ V ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db. EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 23 It follows that dLEE(t) dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) − 1 f e∗(0) (T ∗ T + ln e(0, t) e∗(0) ) + β2T ∗ ∫ ∞ 0 q(b)i(b, t) db − ∫ ∞ 0 (β1T ∗ c p(b) + β2T ∗q(b) ) G[i(b, t), i∗(b)] db + ψ1(0) [ (1− f) ( β1T ∗V ∗ + +β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) ln e(0, t)i∗(0) e∗(0)i(0, t) + ∫ ∞ 0 ξ(a)e∗(a) ln e(a, t)i∗(0) e∗(a)i(0, t) da ] + ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db+ β1T ∗V ∗ − V ∗ V ∫ ∞ 0 β1T ∗ c p(b)i(b, t) db. (6.3) Recall that e∗(0) = f ( β1T ∗V ∗ + β2T ∗ ∫∞ 0 q(b)i∗(b) db ) and ∫∞ 0 p(b)i∗(b) db = cV ∗ in (2.12). Collecting the terms of (6.3) yields dLEE(t) dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) + ψ1(0)(1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) ln e(0, t)i∗(0) e∗(0)i(0, t) + ψ1(0) ∫ ∞ 0 ξ(a)e∗(a) ln e(a, t)i∗(0) e∗(a)i(0, t) da − β2T ∗ ∫ ∞ 0 q(b) ( i(b, t)− i∗(b)− i∗(b) ln i(b, t) i∗(b) ) db + ∫ ∞ 0 β1T ∗ c p(b)i∗(b) ( 2 + ln i(b, t) i∗(b) − T ∗ T − ln e(0, t) e∗(0) − V ∗i(b, t) V i∗(b) ) db + β2T ∗ ∫ ∞ 0 q(b)i(b, t) db− β2T ∗ ∫ ∞ 0 q(b)i∗(b) (T ∗ T + ln e(0, t) e∗(0) ) db. Further, we have dLEE(t) dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) + β2T ∗ ∫ ∞ 0 q(b)i∗(b) ( 1− T ∗ T − ln e(0, t) e∗(0) + ln i(b, t) i∗(b) ) db + ψ1(0)(1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) × ( 1− e(0, t)i∗(0) e∗(0)i(0, t) + ln e(0, t)i∗(0) e∗(0)i(0, t) ) + ψ1(0) ∫ ∞ 0 ξ(a)e∗(a) ( 1− e(a, t)i∗(0) e∗(a)i(0, t) + ln e(a, t)i∗(0) e∗(a)i(0, t) ) da + ∫ ∞ 0 β1T ∗p(b) c i∗(b) ( 3− T ∗ T + ln T ∗ T − V ∗i(b, t) V i∗(b) 24 C. LI, X. DONG, J. WANG EJDE-2022/16 + ln V ∗i(b, t) V i∗(b) − TV e∗(0) T ∗V ∗e(0, t) + ln TV e∗(0) T ∗V ∗e(0, t) ) db − ψ1(0) { (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db )[ 1− e(0, t)i∗(0) e∗(0)i(0, t) ] + ∫ ∞ 0 ξ(a)e∗(a) [ 1− e(a, t)i∗(0) e∗(a)i(0, t) ] da } − ∫ ∞ 0 β1T ∗p(b) c i∗(b) ( 1− TV e∗(0) T ∗V ∗e(0, t) ) db. Recall that Lemma 6.3 holds. Putting (6.1) into the above inequality, we have dLEE(t) dt = −dT ∗ ( T T ∗ + T ∗ T − 2 ) − ψ1(0) [ ∫ ∞ 0 ξ(a)e∗(a)g (e(a, t)i∗(0) e∗(a)i(0, t) ) da + (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) g (e(0, t)i∗(0) e∗(0)i(0, t) ) db ] − ∫ ∞ 0 β1T ∗ c p(b)i∗(b) [ g (T ∗ T ) + g (V ∗i(b, t) V i∗(b) ) + g ( TV e∗(0) T ∗V ∗e(0, t) )] db − [ ∫ ∞ 0 β1T ∗p(b) c i∗(b) ( 1− TV e∗(0) T ∗V ∗e(0, t) ) db + β2T ∗ ∫ ∞ 0 q(b)i∗(b) ( 1− Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) ) db ] + β2T ∗ ∫ ∞ 0 q(b)i∗(b) ( 1− T ∗ T − ln e(0, t) e∗(0) + ln i(b, t) i∗(b) + 1− Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) ) db . Notice that ∫ ∞ 0 β1T ∗p(b) c i∗(b) ( 1− TV e∗(0) T ∗V ∗e(0, t) ) db + β2T ∗ ∫ ∞ 0 q(b)i∗(b) ( 1− Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) ) db = 0 and 1− T ∗ T − ln e(0, t) e∗(0) + ln i(b, t) i∗(b) + 1− Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) = 1− T ∗ T + ln T ∗ T + 1− Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) + ln Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) = −g (T ∗ T ) − g ( Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) ) . We have dLEE(t) dt EJDE-2022/16 AGE-STRUCTURED VIRAL INFECTION MODEL 25 = −dT ∗ ( T T ∗ + T ∗ T − 2 ) − ψ1(0) [ ∫ ∞ 0 ξ(a)e∗(a)g (e(a, t)i∗(0) e∗(a)i(0, t) ) da + (1− f) ( β1T ∗V ∗ + β2T ∗ ∫ ∞ 0 q(b)i∗(b) db ) g (e(0, t)i∗(0) e∗(0)i(0, t) ) db ] − ∫ ∞ 0 β1T ∗ c p(b)i∗(b) [ g (T ∗ T ) + g (V ∗i(b, t) V i∗(b) ) + g ( TV e∗(0) T ∗V ∗e(0, t) )] db − β2T ∗ ∫ ∞ 0 q(b)i∗(b) [ g (T ∗ T ) + g ( Ti(b, t)e∗(0) T ∗i∗(b)e(0, t) )] db ≤ 0 and dLEE(t) dt = 0 implies that T = T ∗ and i(b, t) i∗(b) = i(0, t) i∗(0) = V V ∗ = e(0, t) e∗(0) = e(a, t) e∗(a) , for all a, b ≥ 0. 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[33] Wang, J.; Zhang, R.; Kuniya T.; The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321–343. [34] Webb, G. F.; Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York and Basel, 1985. [35] Yan, Y.; Wang, W.; Global stability of a five-dimensional model with immune responses and delay, Discrete Cont. Dyn. Sys. B, 17 (2012), 401–416. Chunyang Li School of Mathematical Science, Heilongjiang University, Harbin 150080, China Email address: 2190977@s.hlju.edu.cn Xiu Dong School of Mathematics, Harbin Institute of Technology, Harbin 150001, China Email address: ngxiaoxiu@163.com Jinliang Wang (corresponding author) School of Mathematical Science, Heilongjiang University, Harbin 150080, China Email address: jinliangwang@hlju.edu.cn 1. Introduction 2. Preliminaries 2.1. Semi-flow solution 2.2. Volterra formulation 2.3. Boundedness of solutions 2.4. Existence of equilibria 3. Asymptotic smoothness of (t,X0) 4. Uniform persistence 5. Local stability of infection-free and infection equilibrium 6. Global stability of the infection-free and infection equilibrium Acknowledgments References