Dynamics of a diffusive age-structured HBV model with saturating incidence.

In this paper, we propose and investigate an age-structured hepatitis B virus (HBV) model with saturating incidence and spatial diffusion where the viral contamination process is described by the age-since-infection. We first analyze the well-posedness of the initial-boundary values problem of the model in the bounded domain Ω ⊂ Rn and obtain an explicit formula for the basic reproductive number R0 of the model. Then we investigate the global behavior of the model in terms of R0: if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable, whereas if R0 > 1, then the infected steady state is globally asymptotically stable. In addition, when R0> 1, by constructing a suitable Lyapunov-like functional decreasing along the travelling waves to show their convergence towards two steady states as t tends to ∞, we prove the existence of traveling wave solutions. Numerical simulations are provided to illustrate the theoretical results.


1.
Introduction. Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV). It is a major global health problem as it can cause chronic infection and puts people at high risk of death from cirrhosis and liver cancer [2,51]. Chronic HBV infection is often the result of exposure early in life, leading to viral persistence in the absence of strong humoral and/or cellular immune responses [12]. The prevalence of chronic HBV infection in areas of high endemicity is at least 8% with 10-15% prevalence in Africa/Far East. As of 2010, China has 120 million infected people, followed by India and Indonesia with 40 million and 12 million, respectively [26]. According to World Health Organization (WHO), an estimated 600,000 people die every year related to the hepatitis B infection [17]. In an effort to model HBV infection dynamics, Nowak et al. [36] introduced a basic viral infection model within-host. After then, dynamical properties of HBV models have been studied by many authors, see, for example, [3,42,37].
Notice that for HBV infection, susceptible host cells and infected host cells are both hepatocytes and cannot move under normal conditions, but viruses can move freely in liver. Based on the basic model of [36,3,42], Wang and Wang [48] proposed the following system to simulate HBV infection with spatial dependence: where u(x, t), w(x, t) and v(x, t) represent the densities of uninfected cells, infected cells and free virus at location x and time t, respectively; susceptible cells are produced at rate λ, die at rate au, and become infected at rate βuv; infected cells are produced at rate βuv and die at rate bw; free viruses are produced from infected cells at rate kw and are removed at rate dv. The parameters λ, a, β, b, k, d are all positive constants. ∆v = n i=1 , n = 1, 2 or 3 is the Laplacian operator, D is the diffusion coefficient. For system (1), Wang and Wang [48] established the existence of traveling waves.
Considering the fact that there exists a intracellular time delay between infection of a cell and production of new virus particles, Wang et al. [49] modified system (1) as a diffusive HBV model with delay (HBV production lags by a delay τ behind the infection of a hepatocyte), and studied its stability when the space is assumed to be homogeneous and inhomogeneous. For the same model, the traveling wave solutions results are established in Gan et al. [13]. The readers can also refer to [18,52,27,28] and the references cited therein for related studies.
Note that the rate of infection in most virus infection models is assumed to be bilinear in the free virus v and the uninfected cells u. However, experiments reported in [11] have shown that the infection rate of microparasitic infections is an increasing function of the parasite dose, and is usually sigmoidal in shape. Thus, it is reasonable to assume that the infection rate of HBV is given by the saturation response [45,53]. Xu and Ma [53] considered a diffusive HBV infection model with time delay and saturating incidence, They obtained the global stability results of the infected and the uninfected steady states. More recently, Zhang and Xu [54] considered a diffusive HBV model with delayed Beddington-DeAngelis response, they showed that there exist traveling wave solutions connecting the infected steady state and the uninfected steady state when the basic reproduction number is larger than unit. McCluskey and Yang [34] proposed a diffusive virus dynamics model with general incidence function and time delay, and they obtained the global stability results of the model. Recently, the study of age-structured models has attracted many authors' attention. Since in 1927 Kermack and McKendrick [23,24,25] introduced the age-sinceinfection in some epidemic models, then the age variable is more and more widely used to describe either the age of individuals or the age since infection (see for example, [46,30,32,6,29]). In virus dynamics models, recent observations suggest that the death rate of infected cells should vary over their life span [1,14] and the virion production rate is initially low and increases with the age of the infected cell [41] (where age is defined as the time that has passed since the infection of the cell). Nelson et al. [35] developed an age-structured model of HIV infection, in which the production rate of viral particles and the death rate of productively infected cells are allowed to vary and depend on two general functions of age, respectively. The authors studied the local stability of the model. Its global stability results were proved by Huang et al. in [22]. Instead of using the bilinear infection rate kT V in [35], recently, Wang et al. [50] proposed a class of age-infection HIV models with nonlinear infection rate type F (T )G(V ), they also obtained the global stability results of the model. One can refer to [43,39,55] and the references therein for more related studies on age-structured viral infection models. We can see from these studies that the age since infection plays an important role in the study of viral infection models.
Motivated by above works, by incorporating an age-since-infection and saturating incidence into model (1), in this paper, we consider the following model: for t > 0, x ∈ Ω, with homogeneous Neumann boundary conditions and the initial and boundary conditions: In model (3)-(5), θ is the infection age, the time that has elapsed since an HBV virion has penetrated cell; w(x, θ, t) is the density of infected cells of infection age θ at location x and at time t. βuv/(1 + αv) is the saturating incidence of HBV infection, where α, β > 0. Ω is a bounded domain in R n with smooth boundary ∂Ω, ∂/∂ν denotes the outward normal derivative on ∂Ω. The boundary conditions in (4) imply that the virus populations do not move across the boundary ∂Ω. The positive constants s and µ are the recruitment and death rate of uninfected cells, respectively. The functions p(θ) and δ(θ) denote respectively the infection agespecific viral production rate and the age-specific death rate of productively infected cells.
One of the most challenging problems in the analysis of models for viral infection is determining sharp conditions for the global stability of steady states. Yet such results are necessary for derivation of parameter thresholds for clearing infections. On the other hand, for age-structured reaction-diffusion models, traveling wave solutions are important since in many situations they determine the long-term behavior of other solutions and account for propagation of patterns and domain invasion of species in population biology. Recently, there has been some progress on the study of traveling wave solution for age-structured reaction-diffusion equations, see, for example [7,8,9,44,10]. But to the best of our knowledge, there are few results on the global stability for age-structured reaction-diffusion equations in the literatures.
In the present paper, we are mainly interested in the local stability as well as the global stability of the two steady states and the existence of traveling wave solutions connecting the two steady states of system (3)- (5). The organization of this paper is as follows. In the next section, we present some basic results of model (3)-(5). In Section 3, we first discuss the local stability of the uninfected and infected steady states by analyzing the corresponding characteristic equations. Then by constructing Lyapunov like functionals, we further investigate the global stability of the two steady states. In Section 4, by constructing a pair of upper and lower solutions and following the frameworks established in [13] and [10], we prove the existence of traveling wave solutions for system (3) when R 0 > 1, and the non-existence of traveling wave solutions when R 0 < 1. Some numerical simulations are performed in Section 5 to illustrate the main results. Finally, a brief discussion is presented in the last section.
2. The well-posedness of system (3). Throughout this work, we make the following assumption.
Since δ(θ) denotes the infection age-specific death rate of productively infected cells, it is reasonable to assume that δ min ≥ µ. Then we study the well-posedness for the initial-boundary value problem of system (3)- (5).
Defined a threshold value as R 0 is called the basic reproduction number of system (3)- (5). Denote that Π(θ) = p(θ)e − θ 0 δ(τ )dτ , then the total number of viral particles produced by an infected cell in its life span equals which is finite since that +∞ 0 p(θ)dθ is finite. By a direct computation, we get the following conclusion.
Let C := BU C(Ω, R) be the set of all bounded and uniformly continuous functions fromΩ to R, C + := BU C(Ω, R + ). C 2 (Ω) denotes the set of all functions in C whose derivatives up to order 2 and C 2 + (Ω) denotes the set of all functions in C + whose derivatives up to order 2.

Lemma 2.2. For any given initial values
Since the first two equations in system (3)-(5) have no diffusion terms, its solution map Ψ(t) is not compact. In order to deal with this case, we use the Kuratowski measure of noncompactness (see [5]), K, which is defined by K(B) := inf{r : B has a finite cover of diameter < r}, for any bounded set B. We set K(B) = ∞ whenever B is unbounded. It is easy to see that B is precompact (i.e.,B is compact) if and only if K(B) = 0. Then we have the following results. Proof. By Lemma 2.3, it follows that Ψ(t) is K-contracting on X + . By Lemma 2.2, it follows that Ψ(t) is point dissipative on X + , and forward orbits of bounded subsets of X + for Ψ(t) are bounded. By Theorem 2.6 in [31], Ψ(t) has a global attractor that attracts each bounded set in X + .
3. Stability of steady states. In this section, we first discuss the local stability of the steady states E 0 and E * by analyzing the corresponding characteristic equations. Then by use of the method of constructing Lyapunov-like functionals, we investigate the global stability of the two steady states.

DYNAMICS OF A DIFFUSIVE AGE-STRUCTURED HBV MODEL 941
3.1. Local stability analysis of steady states. Using integration, w(x, θ, t) satisfies the following Volterra formulation: Then system (3)−(5) can be modified as the following equal system for t > 0, x ∈ Ω, with homogeneous Neumann boundary conditions (4). Substituting (9) into the third equation of system (10), we have It follows from Assumption 2.1, p(θ) belongs to L ∞ + ((0, +∞), R) \ {0 L ∞ }, that F w (x, t) equals to zero when t is large enough. Thus system (11) can be written as for t > 0, x ∈ Ω. System (12) always has an uninfected steady state E 0 1 = (s/µ, 0); If R 0 > 1, then system (12) has a unique infected steady state E * 1 = (u * , v * ). Let E(u 0 , v 0 ) represent any feasible steady state of system (12). Then the linearization of system (12) Let 0 = µ 1 < µ 2 < · · · be the eigenvalues of the operator −∆ on Ω with the homogenous Neumann boundary conditions, and E(µ i ) be the eigenspace corre- Substituting u(x, t) = e λt φ(x) and v(x, t) = e λt ψ(x) into system (13), we have the associated eigenvalue problem where Π(λ) represents the Laplace transform of the function Π(θ). It follows from the proof of Lemma 2.2 in [21], we have that the problem (14) has a principal eigenvalue, denoted by λ * and λ * satisfies the following equation For E 0 1 , the principal eigenvalue λ * satisfies the following equation Since λ 1 = −µ is a root of Eq. (16), we need only consider the roots of the following In fact, if λ is a root of Eq. (17) with ℜλ ≥ 0, then we have It follows from above inequalities and Eq. (17) we have that So, if R 0 < 1, all the roots of Eq. (17), and therefore Eq. (16) have negative real parts. Accordingly, the principal eigenvalue λ * has negative real part, the uninfected steady state E 0 1 (s/µ, 0) of system (11) is locally asymptotically stable if R 0 < 1.
For E * 1 , the principal eigenvalue λ * satisfies the following equation Whether the value of µ equals that of (d + µ i D) for some i can determine the types of the roots to Eq. (18). Now we are in a position to consider two cases.
(i) If µ = d + µ i D for all i ∈ N + then λ 1 = −µ is not the root of Eq. (18). So that Eq. (18) can be written as where Z = (λ+d+µiD) (λ+µ) . If λ is a root of Eq. (19) with ℜλ ≥ 0, then we have ℜZ ≥ 0, Combing the above inequalities and Eq. (19) we have that So we need only to consider the following If λ is a root of Eq. (20) with ℜλ ≥ 0, then it leads to the contradiction: To sum up, all the roots of Eq. (18) have negative real parts if R 0 > 1. Accordingly, the principal eigenvalue λ * has negative real part, the infected steady state (11) is locally asymptotically stable. Summarizing the above discussions, we can arrive at the following results.
Theorem 3.1. If R 0 < 1, the uninfected steady state E 0 of system (3) is locally asymptotically stable, otherwise it is unstable; if R 0 > 1, the infected steady state E * of system (3) is locally asymptotically stable.

3.2.
Global stability of the steady states. Motivated by the method used in [16], in this subsection, by constructing two Lyapunov functionals and using LaSalle's invariant principle, we investigate the global stability of the uninfected steady state E 0 and the infected steady state E * .
The following theorem is about the global stability of the uninfected steady state E 0 .
Proof. Since the local stability of system (3) has been proved in Theorem 3.1, then we need only to prove the omega limit set contain only the uninfected steady state E 0 . We set function H(z) = z−1−ln z, z ∈ R + and consider the following Lyapunov functional where Obviously, the function f (θ) is bounded and satisfies By use of u 0 = s/µ, we have .
which is bounded due to Lemma 2.2 and the boundedness of function f (θ).
Recalling that Ω D∆vdx = 0, we have If R 0 ≤ 1, it is easy to see that dL(t)/dt ≤ 0 with the equality holding if and only if at E 0 . We conclude that the largest invariant set [15]), E 0 is globally asymptotically stable when R 0 ≤ 1. This completes the proof of Theorem 3.2.
In the following, we prove the global stability of the infected steady state E * by considering a Lyapunov functional for R 0 > 1. For this, it is important to establish some type of positivity of solutions so that the Lyapunov functional will be known to be finite. The following definition characterizes initial conditions that will be used in the future discussion.
Proof. By the first equation of system (3), we have that for each x ∈ Ω, t > 0. Thus, u(x, ·) is increasing if it is less than αs αµ+β . Furthermore, by solving this differential inequality for each x, we find that Therefore, for any positive We note that v is non-negative. From Definition 3.3, we know that v(0) > 0. Next we show that v(t) is positive for t > 0. By use of Ω D∆vdx = 0, we have that .
By Lemma 5.1 in [29], v(t) is unbounded. Since Ω is connected, the diffusion term causes the support of v(·, t) to spread instantaneously to all of the int(Ω), the interior of Ω. Thus, v(x, t) > 0 for all x ∈ int(Ω), t > 0. We now show that v is positive on ∂Ω. Suppose v(x 0 , t) = 0 for some x 0 ∈ ∂Ω, t > 0. By Proposition 13.3 in [19] (a version of the Maximum Principle used for the boundary), it follows that ∂v ∂ν | x=x0 < 0, contradicting the boundary condition (4). Thus, v(x, t) > 0 for all x ∈ ∂Ω, t > 0, and therefore for all x ∈ Ω, t > 0.

4.
Existence of traveling wave solutions. Based on the stability results obtained above, in this section, we will study the existence and non-existence of traveling wave solutions describing the spatial invasion of the hepatitis B virus within a liver, where the infection is initially absent. In Subsection 4.1, we first make some basic preparations and state the main results obtained in this section, the proof of which are deferred to Subsection 4.2.
For 1 < R 0 < 1 + α * , we define the minimal speed by where α * is the unique solution of the equation Now, we are in a position to state the main results in this section. (ii) If R 0 < 1, then system (27) has no travelling wave solution.
The existence of travelling wave solution of system (27) can help us to understand the viral contamination process. The condition 1 < R 0 < 1 + α * in Theorem 4.2 (i) cannot be modified as 1 < R 0 , since the viral release strategy adopted in system (3) (therefore in system (27)) is "budding", which means the death of infected cells and release of virions are independent processes (see [36,48,49,38]). To complete the discussion, we need consider another type of viral release strategy, "bursting", which means there is a coupling of the release of free virions and burst of infected cells (see [53,54]). Mathematically, p(θ) = N δ(θ) in system (3), then the number of virions will not increase until the lysis of the infected cells and release of virions. Proof. Based on Theorem 4.2, we need only to prove that R 0 > 1 implies that R 0 < 1 + α * . From p(θ) = N δ(θ) and (25) we have that ρ 2 (θ) = N ρ 1 (θ), where N = βsN µ 2 . It follows from R 0 = K ρ3 , (28) and (31) that This completes the proof of Corollary 4.1 .

Existence result.
In this subsection we prove assertion (i) of Theorem 4.2, the proof of which consists of several steps. We will need the following definition of upper and lower solutions to system (30).
Then the solutions of system (30) are all bounded. It follows that the convergence towards the infected steady state E * can be solved by building a suitable Lyapunov like functional (see for instance, [10]). Therefore, the first thing to do in this subsection is to construct a pair of upper and lower solutions to system (30).
Proof. For φ(t), we consider the following two cases.
For ψ(t), we consider the following two cases.
Next, for ϕ(·, t). Similarly, we also consider the following three cases.
In order to complete the proof of Theorem 4.2 (i), it remains to prove the convergence of the solutions to the infected steady state E * as t → +∞. This result can be obtained by using a suitable Lyapunov functional. Let From (28) we have g(0) = K. By using (29), we define a set and for each (φ, ϕ, ψ) ∈ C consider a functional V (φ, ϕ, ψ)(t) : R → R defined by where Then we have the following result.
Then there exists some constant m > 0 (only depending on M ) such that Proof of Lemma 4.6. From Lemmas 4.4-4.5, we obtain that (42) holds. Since v is bounded, we obtain that v is also bounded in W 2,∞ (R). Therefore we have From the definition of function H we have 0 ≤ W (u, w, v)(t) for all t ∈ R. Now we can claim that To this end, we need only to prove that To check this, let t 0 > 0 be given. Since w(0, t) > 0 for all t ∈ R, there exists ǫ > 0 such that We consider λ T ∈ R the eigenvalue and j an associated eigenvector of the following problem: Moreover, we assume that 0 < j(t) ≤ ǫ on (−t 0 , t 0 ) and set i(θ, t) = e λT θ j(t), θ ≥ 0, t ∈ [−t 0 , t 0 ].
Let us now show that the map t → V (u, w, v)(t) is decreasing.
XICHAO DUAN, SANLING YUAN AND KAIFA WANG Noticing that by simple calculations, we have that It then follows from 0 ≤ u(t) ≤ 1, 0 ≤ u * ≤ 1 we have that and therefore the map t → V (u, w, v)(t) is decreasing. This completes the proof of Lemma 4.6. Now, we shall use Lemma 4.6 to prove the final step in the proof of Theorem 4.2 (i).
Proof of Theorem 4.2 (i). As in Lemma 4.6, we again point out that the condition (42) holds. Now, we denote {t n } n≥0 as an increasing sequence of positive real numbers, and t n → +∞ as n → +∞. Then we obtain the following sequences of functions: u n (t) = u(t + t n ), w n (θ, t) = w(t + t n , θ), v n (t) = u(t + t n ).
Due to exponential and elliptic estimates, one may assume that the sequences {u n }, {v n } and {w n } converge towards some functions u, v and w for the topology of C 1 loc (R 2 , (0, ∞)) × C 1,1 loc ([0, ∞) × R, (0, ∞)). Then, by use of (42), we have that functions u, v and w satisfy Moreover, since the map t → V (u, w, v)(t) is decreasing we obtain that for each n ≥ 0 From Lemma 4.6, it is bounded and then there exists ζ ∈ R such that lim t→+∞ V (u n , w n , v n )(t) = ζ, ∀ t ∈ R.
By using (42), (46) and Lebesgue convergence theorem we have and therefore V ( u, w, v)(t) = ζ for t ∈ R. Finally, we can find that ( u, w, v) is a solution of system (30). It then follows from the last part of Lemma 4.6 that That is to say, This completes the proof of Theorem 4.2 (i).

4.2.2.
Non-existence result. Assume that R 0 < 1, and there is a non-negative solution (u, w, v) of system (30) such that 0 ≤ u(t) ≤ 1, Since 0 ≤ u(t) ≤ 1, we have where v(t) is bounded for t ∈ R. Using the comparison principle we obtain that for any θ ≥ 0 sup Thus, It then follows from R 0 < 1 and the maximum principle that v ≡ 0. Accordingly, and therefore w ≡ 0.
5. Numerical simulations. In this section, by taking the explicit function forms of p(θ) and δ(θ) used in [35,20], we perform some numerical simulations to illustrate the theoretical results obtained in Sections 3 and 4. The infection age-specific viral production rate takes the form where b controls how rapidly the saturation level, P max , is reached. The term d 1 represents the delay in viral production; that is, it takes time d 1 days after initial infection for the first viral particles to be produced. The death rate of infected cells takes the form where δ 0 is the background death rate, δ 0 + δ m is the maximal death rate, c controls the time to saturation, and d 2 is the delay between infection and the onset of cellmediated killings. From Assumption 2.1, the functions p(θ) and δ(θ) belong to L ∞ + ((0, +∞), R) \ {0 L ∞ }, then there exist constantsθ 1 ,θ 2 > 0 satisfyinḡ For sake of convenience, we assume thatθ 1 =θ 2 =θ in the sequel. We take the parameter values in (47) When the age factor is presented, it is difficult to simulate system (3) directly. Notice that the liver can be separated into several segments, including the left lobe of the liver, spigelian lobe, quadrate lobe of the liver and the right lobe of the liver. For the sake of mathematical simplification, we suppose that the space is homogeneous in each segment and the length of each segment is equivalent. Thus, we can use a lattice differential equation to replace system (3). Now, we divide the space into four lattices and let U i (t) denote the variable in the ith lattice at time t, i = 1, 2, 3, 4. Since the Neumann boundary conditions (5) describe a situation where viral particles do not leave the domain, they are "reflected" at the boundary. Similarly to [49,4], we define the discrete Laplacian as Therefore, system (3) can be expressed as lattice differential equations (see Eq. (4.4) in [49]), and the dimension of Ω is one in this situation. For simplification, we take the initial conditions as follows: u 1 (0) = 3 × 10 6 , w 1 (θ, 0) = 0, v 1 (0) = 500, As in [53] we take in system (3) s = 10 7 , β = 5 × 10 10 , µ = 0.1, α = 0.002, d = 5, and take a day as the unit time.  Example 5.1. In system (3), we set P max = 50,θ = 20 and D = 0.5. Then system (3) with parameter values (51)-(52) has an unique uninfected steady state E 0 = (s/µ, 0, 0). It follows from the direct computation that R 0 = 0.5785 < 1. By Theorem 3.1, we see that E 0 is globally asymptotically stable. Numerical simulation illustrates our result (see Fig. 1).
Example 5.2. In system (3), we set P max = 350,θ = 20 and D = 0.5. Then system (3) with parameter values (51)-(52) has a infected steady state E * = (u * , w * , v * ) (defined in Lemma 2.1). It follows from the direct computation that R 0 = 4.0493 > 1. By Theorem 3.2, we see that E * is globally asymptotically stable. It follows from Theorem 4.2 that system (3) always has a traveling wave solution with speed c > c * connecting E 0 and E * . Numerical simulation illustrates our result (see Fig.  2).
Next, we will simulate system (3) to study the effect of age factor under different diffusion coefficients. Here, we take the baseline parameter values as in (52)   Corollary 4.1 tells us that the condition 1 < R 0 < 1 + α * in Theorem 4.2 can be simplified as R 0 > 1 provided p(θ) = N δ(θ). This is true when p(θ) and δ(θ) take respectively the forms of (47)- (48). To show this, we need only to check it by varying the parameter values in the function p(θ). With the help of Matlab, we can respectively calculate a series of values for α * and R 0 as the values of P max , c and d 2 vary, and then plot them in Fig. 4, from which we can see that the inequality R 0 < 1 + α * naturally holds provided R 0 > 1. 6. Conclusion and discussion. In this paper, a more general diffusion HBV model is studied in which only the free virus particles have diffusivity. The model incorporates the saturating incidence βuv/(1 + αv) (α > 0), the infection-agedependent virion production rate and the death rate of productively infected cells. Mathematically, the results obtained in the present paper can be extended to the limit case when α = 0. For this model, we derive the basic reproduction number R 0 . Under the Neumann boundary condition, we discuss the local and global stability of the uninfected steady state and the infected steady state of system (3), respectively. More precisely, the uninfected steady state is globally asymptotically stable when R 0 ≤ 1, and the infected steady state is globally asymptotically stable when R 0 > 1. This implies that the basic reproduction number R 0 plays a sharp threshold role in the HBV infection. On the other hand, by constructing a pair of upper and lower solutions and applying Lemma 2.9 developed by Ducrot et al. [10], we prove the  existence of traveling wave solutions for system (27). Specifically, there exists a c * := √ Dα * , where α * is defined in (31), such that system (27) has traveling wave solutions with positive wave speed c > c * which connecting the two steady states when 1 < R 0 < 1 + α * , which reduces to R 0 > 1 in the case when p(θ) = N δ(θ).
Theorems 3.2-3.5 show that R 0 may be used to design the control strategies of the virus infection and to estimate the infection level. In the case when R 0 > 1, we are concerned with the age distribution and spatial velocity of infection (i.e., spreading speed), and their interactions as viruses move freely in a liver. Theorem 4.2 and Corollary 4.1 confirm the existence of the spreading speed of HBV in system (27). By comparison, it is shown that "bursting" may be more easily result in traveling wave solutions than "budding". The closely relationship between wave speed and age factor can be clearly seen from the expression of c * and (31), i.e., the age-sinceinfection may effect the spatial velocity of infection.
A natural question is whether c * is the minimal wave speed c min , that is, there does not exist traveling wave solution connecting the two steady states for 0 < c < c min . Although Ducrot and Magal [10] have obtained some general results on the asymptotic speeds of spread for a class of age-structured epidemic system with diffusion which two speeds indeed coincide, their methods are not valid for system   (27). It is of interest to study the relation of the minimal wave speed and the asymptotic speeds of spread for system (27). We leave it for further investigation.