Global dynamics for a class of infection-age model with nonlinear incidence ∗

Yuji Li, Rui Xub,c,1, Jiazhe Lin Institute of Applied Mathematics, Army Engineering University, Shijiazhuang 050003, Hebei, China Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, China xur2020meu@163.com; rxu88@163.com


Introduction
Over the past few years, within-host virus models have been studied extensively to describe the dynamics inside the host of various infectious diseases such as HIV, HBV and so on.For it is not easy to obtain accurate information of patients, specific hypotheses testing based on clinical data is an arduous task.Therefore, many researchers have made great efforts by mathematical models in this area of research [4,8,10,13,[15][16][17]26], presenting assumptions that the death rate and virus production rate of infected cells are both constant in their works.However, biological observations show that the death rate of Y. Li et al. infected cells has been different during the period of infection, and there exists a maximal bud rate of viruses when virus particles gradually bud out of the host cell until the cell is dead.Thus, age structure is employed to make HBV infection model more realistic.In particular, the fact has been explained that the mortality rate and viral production rate of infected cells are functions of the infection age of the infected cells instead of constants.
It is worth noting that there exists direct cell-to-cell infection in vivo spread of the virus.Besides, the infection is more potent and efficient means of virus propagation than the virus-to-cell infection mechanism.Viral particles can be simultaneously transferred from infected target cells to uninfected ones through virological synapses during cell-to-cell infection.Thus, it is necessary to understand the viral dynamics in terms of applications.There exist less age-structured virus models to take both virus-to-cell and cell-to-cell infection into consideration.Recently, Wang et al. [20] established an HIV infection model containing the two modes of infection and allowing age-dependent death rate of infected cells and age-dependent production rate of virus.Meanwhile, considering antiretroviral therapy of HIV, Xu et al. [27] has proposed the following model: There is no certain observation suggesting that viruses infect cells with linear incidence rate.Motivated by this fact, several within-host virus dynamics models have been constructed to investigate the dynamics of models to take saturation incidence rate or other nonlinear incidence rate into consideration [2, 6, 7, 9, 19, 21-25, 28, 29].However, almost none of these investigations take both age structure and cell-to-cell infection into account.For biological consideration, we introduce a more general incidence rate to formulate the following model: https://www.mii.vu.lt/NA i(t, 0) = β 1 x(t)f (v with initial condition Here x(t), v I (t), v N I (t) denote the concentration of susceptible cells, infectious virions and noninfectious virions at time t, respectively.The density of infected target cells of infection age a (i.e., the time that has elapsed since an HBV virion has penetrated cell) at time t is denoted by i(t, a).λ is the recruitment rate of healthy susceptible cells, d is the per capita death rate of uninfected cells, δ(a) is the age-dependent remove rate of infected cells, µ is the clearance rate of virions, β 1 is the infection rate of free virus.η p denotes the efficacy of the PI inhibitor.We also assume that the efficacy of the PI inhibitor, which blocks the cell-to-cell infection is denoted by η Evidently, the last equation of system (1) is independent to the others because v N I (t) does not exist in the first four equations of system (1).Let k(a) = (1 − η Denote function space By the standard theory of age-structured model, it can be verified that system (3) with initial condition (2) has a unique nonnegative solution.Thus, system (3) generates a continuous semi-flow Φ : R + × Z → Z, which takes the form The organization of this paper is as follows.In Section 2, we present some preliminaries results of the system (3).Asymptotic smoothness of the semi-flow generated by system (3) is analyzed.Then we study the existence of equilibria and obtain the expression of the basic reproduction number R 0 .In Section 3, the global stability of equilibria is proved.More details concerning the global stability analysis of virus models, we refer readers to [5-7, 9, 14, 18-25, 28, 29].

Preliminaries
To study the global dynamics of the model, it is necessary to make assumptions about f (v I ) as follows.
Assumption 1.We assume that: https://www.mii.vu.lt/NASet I(t) = ∞ 0 i(t, a) da, which represents the total number of infected cells at time t.Biologically, there exists a finite maximum age, thus it is reasonable to assume that lim a→∞ i(t, a) = 0. Then from system (3) we have Thus, x(t) + ∞ 0 i(t, a) da λ/min{d, δ min }.According to the assumption of p(a) and the fourth equation of system (3), it is easy to get p )p + λ/(µ min{d, δ min }).Integrating the second equation of system (3) along the characteristic line t − a = const yields Obviously, i(a, t) remains nonnegative for nonnegative initial condition.Suppose that there exists t 0 such that x(t 0 ) = 0 and x(t) > 0 for t ∈ [0, t 0 ).Then x (t 0 ) = λ > 0, which implies that x(t) 0 for all t 0. Furthermore, from the forth equation of system (3) we have v I (t) + µv I (t) = ∞ 0 q(a)i(t, a) da, which gives d(v I (t)e µt )/dt = e µt ∞ 0 q(a)i(t, a) da, then we have v I (t) = e −µt v I (0)+ t 0 e −µ(t−s) ∞ 0 q(a)i(t, a) da ds.Thus, v I (t) 0 for all positive initial data, and f (v I (t)) 0 for all positive initial data, which implies that i(t, a) 0. Then the positive invariant set of system (3) can be given as p )p + λ µ min{d, δ min } .

Asymptotic smoothness
To analyze the global dynamics of system (3), it is necessary to prove the smoothness of the semi-flow generated by system (3).Firstly, we introduce some lemmas as follows.
Y. Li et al.
Lemma 2. If Φ(t) : Z → Z, t 0 is asymptotically smooth point dissipative and orbits of bounded sets are bounded, then there exists a global attractor A 0 .If Φ(t) is also oneto-one on A 0 , then Φ(t)/A 0 is a C r -group.In addition, if Z is a Banach space, then A 0 is connected.
By using the similar method in [1,12], we can state the following result, which shows that system (3) has a global compact attractor.Lemma 3. Assume that R 0 > 1, then there exists A 0 , a compact subset of Z 0 , which is a global attractor for the semi-flow of system (3) in Z 0 .Moreover, A 0 is invariant under the semi-flow, that is, •) for all t, s 0, and Ψ (0, •) being the identity map.In order to utilize Lemmas 1 and 2, we decompose the solution semi-flow into two parts Ψ = Ψ (t, χ 0 ) + Ψ (t, χ 0 ).This decomposition is done in such a way that lim t→∞ Ψ (t, χ 0 ) = 0 for every χ 0 ∈ Z 0 , and for a fixed t and any bounded set Ω in Z 0 , then the set { Ψ (t, χ 0 ): χ 0 ∈ Ω} is precompact.Here Ψ and Ψ are defined as follows: Notice that x(t) and v I (t) satisfy system (3) with i(t, a) = î(t, a) + ĩ(t, a).The function î(t, a) is the solution of the following system: and ĩ(t, a) is the solution of the following system: It is easy to obtain that î(t, a) and ĩ(t, a) are nonnegative.Let w(t) = ∞ 0 î(t, a) da.Thus, (5) implies that w (t) −δ min w(t).Therefore, we have lim t→∞ Ψ (t, χ 0 ) = 0 for every χ 0 ∈ Z 0 .In order to show that the set { Ψ (t, χ 0 ): x 0 ∈ Ω} is precompact for that fixed t and any bounded set Ω in Z 0 , we only need to verify the set { Ψ (t, χ 0 ): χ 0 ∈ Z 0 , fixed t} is precompact by utilizing Fréchet-Kolmogorov theorem.On the one hand, it holds that { Ψ (t, χ 0 ): χ 0 ∈ Z 0 , fixed t} ⊂ Z 0 , and { Ψ (t, χ 0 ): χ 0 ∈ Z 0 , fixed t} is bounded due to https://www.mii.vu.lt/NA that Z 0 is bounded.On the other hand, from (6) we have ĩ(t, a) = 0 for a > t.The third condition of Fréchet-Kolmogorov theorem is satisfied.Furthermore, in order to verify the second condition, we need to show that the L 1 -norm of ∂ ĩ(t, a)/∂a is bounded.Actually, from ( 6) we obtain that where Since, φ(t) is bounded for x 0 ∈ Z 0 , and x(t), v I (t) are bounded.Thus, we obtain from (7) that where ξ i (i = 1, 2, 3, 4) are constants that depend on the bounds of the parameters as well as the bounds of the solution.
Making use of Gronwall's inequality, we have Equation ( 7) implies that Together with ( 8) and ( 9), we have Thus, it is easy to show that the above integral can be made arbitrary small uniformly in the family of functions.This completes the proof of Lemma 3.

Existence and uniqueness
Define the basic reproduction number which means the average number of secondary infection produced by one infected cell during its period of infection.From the expression of R 0 it is easy to see that the virusto-cell infection always exists and the cell-to-cell infection can be prevented by increasing the dose of PI (protease inhibitor).System (3) always has a infection free steady state E 0 = (x 0 , i 0 (a), v 0 I ), where x 0 = λ/d, i 0 (a) = 0, v 0 I = 0.Moreover, there may exist a nonnegative steady state E * = (x * , i * (a), v * I ), where x * , i * (a), v * I are nonnegative and satisfy the following equations: From the first equation of ( 10) we get Solving the second equation of (10) yields From the third equation of (10) we have From the first and the fourth equations of (10) we get According to the Assumption 1, f (v) − vf (v) 0, g (v) remains negative for nonnegative initial condition Therefore, when R 0 > 1, there always exists a nonnegative v * I satisfying g(v * I ) = 0. Theorem 1. System (3) always has a steady state E 0 (x 0 , 0, 0); system (3) has a unique positive steady state

Stability analysis of steady states
In this section, we study the local and global stability of the infection-free steady state E 0 and the infection steady state of system (3).The local stability of the two steady states is studied by using the method of characteristic equations, while the global dynamics of system (3) is discussed by constructing Lyapunov functionals.

Stability of infection-free steady state
Theorem 2. If R 0 < 1, then the infection-free steady state E 0 of system (3) is locally asymptotically stable.Otherwise, it is unstable.
Linearizing system (3) at E 0 leads to the following system: To analyze the asymptotic behavior of E 0 , we set x 1 (t) = x 1 e ut , i 1 (t, a) = i 1 (a)e ut and v 1 (t) = v 1 e ut .Thus, we get the following equations: Solving (12) gives Substituting (13) into the last equation of ( 12), we can get Define a function G(u) that denotes the left hand of (14).Obviously, G(u) is a continuously differentiable function with lim u→∞ G(u) = 0.It is easy to see that G(0) = R 0 , and by direct computation, it shows that G (u) < 0, and therefore, G(u) is a decreasing function.Hence, any real solution of ( 14) is negative if R 0 < 1, and positive if R 0 > 1.
Thus, if R 0 > 1, the infection-free steady state E 0 is unstable.Next, we show that (3.3) has no complex solutions with nonnegative real part if R 0 < 1. Set Thus, we have https://www.mii.vu.lt/NAFor R 0 < 1, let u = ξ + ηi be an arbitrary complex root to (14) with ξ 0. Then It follows from ( 15) that ( 14) has a solution u = ξ + ηi only if ξ < 0. Thus, any solution of ( 14) must have a negative real part.Therefore, the infection-free steady state E 0 is locally asymptotically stable if R 0 < 1.
Theorem 3. If R 0 1, then the infection-free steady state E 0 of system (3) is globally asymptotically stable.
Proof.We consider the following Lyapunov functional V 1 = V 11 + V 12 + V 13 , where Here the nonnegative kernel function Φ(a) will be determined later.Using (10), differentiating V 1 along the solutions of system (3) yields Using (4), V 12 becomes Differentiating V 12 yields Also, using (10), differentiating V 13 along the solutions of system (3) yields According to Assumption 1, it is easy to get https://www.mii.vu.lt/NAAdding V 11 , V 12 , V 13 together gives
By differentiating the above equation, it can be verified that Notice that Φ(0) = R 0 .Hence, it follows that It can be verified that the largest invariant set where V 1 = 0 is the singleton E 0 .Thus, all solutions of system (3) converge to the infection-free steady state E 0 .Therefore, E 0 is globally asymptotically stable when R 0 1.Proof.To show the local stability, we linearize the system (3) around the infection steady state E * .In particular, we introduce the perturbation variables

Local stability of infection steady state
To analyze the asymptotic behavior of E * , we look for solutions of the form x 2 (t) = x 2 e ut , i 2 (t, a) = i 2 (a)e ut and v 2 (t) = v 2 e ut .Thus, we can consider the following eigenvalue problem: https://www.mii.vu.lt/NASolving ( 16), we have Substituting ( 17) into the last equation of ( 16) yields We rewrite the equation in the following form: It is not hard to see that for u with Re u 0, the right side of the characteristic equation ( 18) satisfies the following inequation: With respect to the left side of ( 18), for u with Re u 0, we have Hence, for u with Re u 0, the right side of ( 18) is strictly larger than one, while the left side of ( 18) is smaller than 1.Therefore, the contradiction implies that the characteristic equation ( 18) has no roots with non-negative real part.Thus, we have proved that the infection steady state E * is locally asymptotically stable.
To establish the global stability of the infection steady state E * , we define the following Lyapunov functional: where Before making use of the Lyapunov functional V 2 defined above to establish the global stability of infection steady state, it should be shown that the Lyapunov functional is well defined.To this end, we first show the uniform persistence of system (3).

Persistence
In this section, we investigate the uniform persistence of system (3) by using the persistence theory for infinite dimensional dynamical system.Define Since k(a), δ(a), q(a) ∈ L 1 + (0, ∞), we have ā1 , ā2 , ā3 > 0. Furthermore, let It is not difficult to verify the following result.
Furthermore, the following result is useful for the proof of uniform persistence.
Theorem 5.The disease-free steady state E 0 of system (3) is globally asymptotically stable for the semi-flow {Φ(t)} t 0 restricted to ∂Y.
Since x(t) λ/d as t tends to infinity, by comparison, we have i(t, a) ĩ(t, a), v I (t) ṽI (t), where ĩ(t, a) and ṽI (t) satisfy the following auxiliary system: Similar to (4), solving the first equation of (20) yields Nonlinear Anal.Model.Control, 24(1):47-72 where Substituting ( 21) into (22) yields where we have G(t) ≡ 0 for all t 0. From ( 23) we obtain that It is easy to show that (24) has a unique solution L(t) ≡ 0, in which ṽI (t) = 0. From (21) we have ĩ(t, a) = 0.For a t, it follows that which implies that ĩ(t, a) = 0 as t → ∞.Noting that i(t, a) ĩ(t, a), v I (t) ṽI (t), we have i(t, a) → 0 and v I (t) → 0 as t → ∞.It follows from the first equation of system (3) that x(t) → x 0 as t → ∞.Thus, E 0 is globally asymptotically stable in ∂Y.
Theorem 6.If R 0 > 1, then the semi-flow {Φ(t)} t 0 is uniformly persistent with respect to (Y, ∂Y), i.e., there exists an ε > 0, which is independent of initial values such that lim t→∞ Φ(t, z) Z ε for z ∈ Y. Furthermore, there is a compact subset A 0 ⊂ Y, which is a global attractor for {Φ(t, z)} t 0 in Y.
Proof.It follows from Theorem 5 that E 0 is globally asymptotically stable in ∂Y.Applying Theorem 4.2 in [3], we need only to show that W s (E 0 ) ∩ Y = ∅, where Otherwise, there exists a solution y ⊂ Y such that Φ(t, y) → E 0 as t → ∞.In this case, there exists a sequence {y n } ⊂ Y such that Φ(t, y n ) − E 0 Z < 1/n for t 0. Denote Φ(t, y n ) = (x n (t), i(t, •), v In (t)) and y n = (x n (0), i(0, •), v In (0)).Since R 0 > 1, we can choose n sufficiently large satisfying x 0 > 1/n and https://www.mii.vu.lt/NAwhere x 0 = λ/d.For such a n > 0, there exists a T > 0 such that for t > T , x 0 − 1/n < x n (t) < x 0 + 1/n.Consider the following auxiliary system: Looking for solutions of system (26) of the form î(t, a) = î(a)e ut and vI (t) = vI e ut , where the function î(a) and the constant vI will be determined later, we obtain the following linear eigenvalue problem: Solving the first equation of system (27) yields î(a) = î(0)e − a 0 (u+δ(s)) ds .
Substitution (28) into the last two equations of (27), we obtain the characteristic equation of system (3) at the steady state E 0 as follows: where Clearly, we have lim u→∞ f (u) = 0. From (25) and Assumption 1, there exists a n > 0 and a T > 0 such that Nonlinear Anal.Model.Control, 24(1):47-72 Hence, if R 0 > 1, (29) has at least one positive root.This implies that the solution ( î(t, •), vI (t)) of system ( 26) is unbounded.By comparison, the solution Φ(t, y n ) of system (3) is unbounded, which contradicts to the boundedness of system (3).Therefore, the semi-flow Φ(t) t 0 generated by system (3) is uniformly persistent.Furthermore, there is a compact subset A 0 ⊂ Y, which is a global attractor for Φ(t) t 0 in Y.This completes the proof.

Global stability of the infection steady state
Now we are ready to establish the global stability of the steady state E * .The following theorem summarizes the result.
Theorem 7. If R 0 > 1, then the infection steady state E * of system (3) is globally asymptotically stable.
Proof.Using (10), we take the derivative of each part of the Lyapunov functional V 2 defined in (19) along the solutions of system (3) separately Using (4), it follows that From (11) and the fact that xG x (x, y) + yG y (x, y) = G(x, y), differentiating V 22 yields Notice that Φ(0) = 1 and Then we have Similarly, differentiating V 23 along the solutions of system (3) yields Y. Li et al.
Then we can get Therefore, we have x * i(t, 0)i * (a) da . Obviously, It is easy to see that g(x) = x − 1 − ln x 0 for all x > 0 with equality holding if and only if x = 1.Then it can be verified that the largest invariant set of V 2 = 0 is the singleton E * .It then follows from [11] that the compact global attractor A 0 = E * , which implies E * is globally asymptotically stable.

Numerical simulations
In this section, to illustrate the valid of theoretical results of this paper, we present corresponding numerical simulations.The backward Euler and linearized finite difference method will be used to discretize the ODEs and PDE in system (3), and the integral will be numerically calculated using Simpson's rule.Furthermore, we focus on the ageinfection model with saturation incidence.Let f (v I ) = v I /(1 + αv I ).Following [1] Y. Li et al. and references therein [12,22], we fix the following coefficients: λ = 10, d = 0.09, β 1 = 0.0025, µ = 2.4.Furthermore, we set the maximum age for the viral production as â = 10 and δ(a) = 0.4(1 + sin((a − 5)π/10)), p(a) = 300(1 + sin((a − 5)π/10)), 0 a 10, so that each of the averages is equal to 0.4 and 300, respectively, which were used in [30].Then we observe the dynamical behavior of solutions as follows when α varies.
We obtain that basic reproduction number R 0 is approximately calculated as 0.8603 and less than one.From Theorem 2 we know that infection-free steady state is locally asymptotically stable.In fact, we can observe in Figs.2(a) and 3(a) that free virion v I (t) converges to 0.
In another case, through direct calculation, we get the basic reproduction number R 0 , which is near 86.0316 and greater than one.From Theorem 4 we obtain that the positive steady state is locally asymptotically stable.From Figs. see that the infected cells i(t, a) converges to the positive steady state whether α = 0 or α = 0.9, just reaching different peak level.

Theorem 4 .
If R 0 > 1, then the infection steady state E * of system (3) is locally asymptotically stable.Y. Li et al.

9 Figure 4 .
Figure 4.The dynamical behavior of infected cells i(t, a).