A NOTE ON DYNAMICS OF AN AGE-OF-INFECTION CHOLERA MODEL

. A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model , Math. Biosci. Eng., 10, 2013, 1335{1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show (cid:12)rst the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be in(cid:12)nite if i ( a;t ) =i (cid:3) ( a ) = 0 on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.

1. Introduction. In this note, we will revisit an age-of-infection cholera model which is presented and studied in [1] by F. Brauer, Z. Shuai and P. van den Driessche. The model takes the following form with boundary conditions Here, individuals in the general population enter the susceptible population at a rate, A, die at a natural death rate µ. It is assumed that β 1 and β 2 are the direct transmission coefficient and indirect transmission coefficient, respectively. δ(b) represents the removal rate of the pathogen of age b, and θ(a) = µ +α(a)+ γ(a) in which the last two terms represent the disease induced death rate and the recovery rate for infected individuals of infection age a, respectively. ξ(a) represents the shedding rate of an infected individual of infection age a.
Model (1) is formulated by incorporating simultaneously the age-of-infection structure of individuals and the age structure of pathogen with infectivity given by kernel functions. The nonnegative kernel functions k(a) and q(b) measure the infectivity of infected individuals of infection age a and pathogen of age b, respectively.
From (2) and (3), integration of the second and third equation in (1) along the characteristic line t − a = const. and t − b = const. yields and System (1) always has a disease-free equilibrium P 0 = (S 0 , i 0 (a), p 0 (b)), where S 0 = A/µ, i 0 (a) = p 0 (b) = 0. Now let us investigate the positive equilibrium of system (1). For any positive equilibrium P * = (S * , i * (a), p * (b)) of system (1), it should satisfy the following equations Denote and After simple calculation, one can obtain that is defined as the basic reproduction number for system (1). Thus, (1) has a unique endemic equilibrium P * = (S * , i * (a), p * (b)) if and only if ℜ 0 > 1. The basic reproduction number ℜ 0 is proved to be a sharp threshold parameter, completely determining the global dynamics of (1). The main theorem obtained in [1] is (9).
the endemic equilibrium P * is global asymptotically stable with respect to solutions with initial conditions S 0 > 0 and i 0 (a), p 0 (b) > 0 bounded away from zero.
In (1), age-of-infection is considered as a continuous variable. Continuous agestructured in the infectious class allows the infectivity to truly be a function of the duration spent in class [5]. Because the continuous age model is described by first order PDEs, it is difficult to analyze the dynamics of the PDE models, particularly the global stability. The global stability for equilibria of (1) (Theorem 1.1) is proved by constructing suitable Lyapunov functions, which was adopted originally in [2,3] to get the global dynamical properties of some age-structured epidemiological models. Two Lyapunov functions are constructed to show the global stability of the disease-free and endemic equilibria. In [1], the authors then studied the final size problem for a simplified version of (1), then extended the result to a staged progression model. Recently McCluskey's work [5] has drawn much attention and elegantly established the global stability problems to two-dimensional continuous age-structured epidemic models.
The results presented in [1] gave a global attracting analysis of equilibria of model (1), but leaving out the necessary arguments, including relative compactness of orbit generated by system (1) and uniform persistence, which are two major challenges in applying the main results in [8] to particular models. This provides us with one motivation to conduct our work. The object of this note is to show that, under some assumptions, system (1) can be reformulated as a Volterra integral equation in order to apply functional analysis theory, and then we present some results about uniform persistence and about the existence of global attractors. The methods of theoretical analysis follow the techniques laid out in the new book [8].
As an application of the methods in [8], it is expected that calculations here help to demonstrate the usefulness of the techniques given in [8], and can be applicable to more age-structured epidemic models.
The paper is organized as follows. In section 2 we describe preliminary results and some notations providing the context where this paper is to be read. The relative compactness of orbit and uniform persistence for ℜ 0 > 1 is shown in section 3 and 4-the key results of this paper. Section 5 contributes to the stability analysis of system (1).

Preliminary results.
We make the following assumption on parameters, which is thought to be biologically relevant.
Let us define a functional space for system (1): + is the space of functions on (0, ∞) that are non-negative and Lebesgue integrable, equipped with the norm The initial condition (3) for the system can be rewritten as Next, we first define the continuous semi-flow associated with this system. It follows from (4), (5) and Assumption 2.1, we easily see that system (1) has a unique nonnegative solution for any initial condition Y 0 ∈ Y. Thus, we can obtain a continuous semi-flow Φ : R + × Y → Y defined by system (1) such that Thus Without loss of generality, for a ≥ 0, we denote It follows from (ii) and (v) of Assumption 2.1 that It follows that Ω ′ (a) = −θ(a)Ω(a) and Thus (4) and (5) can be rewritten as and It is useful to note that and the boundary conditions given in (2) can be rewritten as i(t, 0) = β 1 S(t)P (t) + β 2 S(t)Q(t) and p(t, 0) = M (t). Letμ and define the state space for system (1) by (17) The following proposition holds true: Proposition 2.2. Let Φ and Ω be defined by (10) and (17), respectively. Ω is positively invariant for Φ, that is,

Moreover, Φ is point dissipative and Ω attracts all points in Y.
Proof. First we have It follows from equation (14) that We make the substitution a = t − σ in the first integral,and a = t + τ in the second integral, and differentiating by t, yields d dt Notes that Ω(0) = 1 and Ω ′ (a) = −θ(a)Ω(a), we have d dt Similarly, we can get d dt Adding the first equation of (1) and (19), we have from (v) of Assumption 2.1 that d dt Hence, it follows from the variation of constants formula that This implies that for any solutions of (1) satisfying Y 0 ∈ Ω, Then, it follows from (20), (22) Hence, it follows from the variation of constants formula that Adding (21) and (23), we have From (22) and (24) it follows that for any solutions of (1) satisfying Y 0 ∈ Ω, Φ t (Y 0 ) ∈ Ω for all t ≥ 0. This implies the positive invariance of set Ω for semi-flow Φ. Moreover, it follows from (21) and (23) that lim sup Therefore, Φ is point dissipative and Ω attracts all points in Y. This completes the proof.
Recall that (v) of Assumption 2.1, we have the following proposition, which is direct consequences of Proposition 2.2.
, then the following statements hold true for all t ≥ 0: As presented in (iii) of Assumption 2.1, it is assumed that the coefficient functions k(·), q(·) and ξ(·) be Lipschitz continuous. This allows the initial conditions for i and p to be taken in L 1 + (0, ∞). Then, the functions P (t), Q(t) and M (t), related to the boundary conditions i(t, 0) and p(t, 0), can be shown to be Lipschitz continuous.

Proposition 2.4. The functions P (t), Q(t) and M (t) are Lipschitz continuous on
Let t ≥ 0 and h > 0. We can check that By applying k(a) ≤k, i(t + h − a, 0) ≤βC 2 and Ω(a) ≤ 1 for the first integral, and making the substitution σ = a − h for the second integral to (25), we get It follows from (16) that Thus, Combining the above relations together, we obtain it follows that P (t) is Lipschitz continuous with coefficient M P = (kβC +kθ + M k )C. The proof of Lipschitz continuous of Q(t) and M t is similar to that of P (t).
The proof is completed.

JINLIANG WANG, RAN ZHANG AND TOSHIKAZU KUNIYA
The following Proposition will be used in next section, which come from [5].
3. Relative compactness of the orbit. In [1], the proof of the global stability of each equilibrium utilized Lyapunov functional technique combined with the invariance principle. Since we are now concerned with the infinite dimensional Banach space Y including L 1 (0, ∞), according to [6, Theorem 4.2 of Chapter IV], we have to show first the relative compactness of the orbit {Φ(t, Y 0 ) : t ≥ 0} in Y in order to make use of the invariance principle. To this end, we first decompose Φ : R + × Y → Y into the following two operators Θ, Ψ : R + × Y → Y: wherẽ Then we have Φ (t, Y 0 ) = Θ (t, Y 0 ) + Ψ (t, Y 0 ) , ∀t ≥ 0. Note thatĩ(t, a) andp(t, b) can be written as Following the line of [7, Proposition 3.13], we are now in the position to state and prove the following main Theorem of this section. (i) There exists a function ∆ : To show that the conditions (i) and (ii) in Theorem 3.1 hold, we first prove the following lemma.
We next prove the following Lemma, which is based on Theorem B.2 from [8].

Proof. From Proposition 2.2, it is easily seen that S(t) remains in the compact set
Thus, we only have to show thatĩ (t, a) andp (t, b) remain in a precompact subset of L 1 + (0, ∞), which is independent of Y 0 ∈ Ω. To this end, it suffices to verify the following conditions (see e.g., [8,Theorem B.2]). We just verify that the following conditions valid forĩ (t, a), and the conditions can be similarly verified forp (t, b).

t−a−h)P (t−a−h)+β 2 S(t−a−h)Q(t−a−h))|Ω(a+h)−Ω(a)|da
Recall that 0 ≤ Ω(a) = e − ∫ a 0 θ(τ )dτ ≤ e −µ0a , and Ω(a) is non-increasing function with respect to a, we have Hence, combing above with (35) yields For Ξ 1 , combining Proposition 2.3 with the expression for dS(t) dt , we find that | dS(t) dt | is bounded by M S = A + µC + β 1k C 2 + β 1q C 2 , and therefore S(·) is Lipschitz on [0, ∞) with coefficient M S . By Proposition 2.5, there exists two Lipschitz coefficients M P , M Q for P, Q respectively. Thus, S(·)P (·) and S(·)Q(·) is Lipschitz which converges to 0 as h → 0 + . Let C 0 ⊂ Y be a bounded closed set and C > A/µ 0 be a bound for C 0 . We note that M depends on C, which depends on the set C 0 , but not on Y 0 . Therefore, this inequality holds for any Y 0 ∈ C 0 . Thus,ĩ remains in a pre-compact subset C i of L 1 + . Similarly,p remains in a pre-compact subset   (4) and (5) can be rewritten as and where Ω(a) and Γ(b) are defined by (12). Substituting (36) and (37) into the boundary condition (2), we obtain the following system of integral equations ofî(t) andp(t): In addition, note that the first equation of (1) can be rewritten as so we have the following result. Proof. From (38)-(39) and the positivity of coefficients, we obtain inequalitieŝ Combining (42) and (43), we obtain the following integral inequality ofî(t).
In what follows, we prove that for the solutionî(t) satisfying (44), there exists a positive constant ϵ > 0 such that (41) holds. Now it follows from (7)- (9) and (12) that if ℜ 0 > 1, then there exists a sufficiently small ϵ > 0 such that For such ϵ, we show that (41) holds. Suppose for the contrary, if there exists a sufficiently large constant T > 0 such that Then, it follows from (40) that Performing the variation of constants formula, we have Then, (44) becomeŝ for all t ≥ T . Now, without loss of generality, we can perform the time-shift of system (1) with respect to T . That is, replacing the initial condition of system (1) by Y 1 := Φ(T, Y 0 ), we can consider the long time behavior of the system. Then, (46) holds for t ≥ 0 and by taking the Laplace transform of both sides, we have where L[î] denotes the Laplace transform ofî, which is strictly positive because of (38) and Assumption 2.1. Dividing both sides by L[î] and letting λ → 0, we obtain inequality which contradicts to (45).
We prove the following lemma. Proof. Suppose that S(r * ) = 0 for a number r * ∈ R and show a contradiction. In this case, it follows from the first equation of (47) and (48) that dS(r * )/dr = A > 0. This implies that S(r * − η) < 0 for sufficiently small η > 0 and it contradicts to the fact that the total Φ-trajectory ϕ remains in Y. Consequently, S(r) is strictly positive on R.
By changing the variables, we can rewrite (48) as follows.
The total Φ-trajectory ϕ enjoys the following nice properties: Proof. From the second statement of Lemma 4.2, by performing appropriate shifts, we see thatî(r) = 0 for all r ≥ r * ifî(r) = 0 for all r ≤ r * , where r * ∈ R is arbitrary. This implies that eitherî(r) is identically zero on R or there exists a decreasing sequence {r j } ∞ j=1 such that r j → −∞ as j → ∞ andî(r j ) > 0. In the latter case, lettingî j (r) :=î(r + r j ), r ∈ R, we have from (48) that where S := inf r∈R S(r) > 0 and Then, sinceĵ j (0) =î(r j ) > 0 andĵ j (r) is continuous at 0, it follows from Corollary B.6 of Smith and Thieme [8] that there exists a number r * > 0, which depends only on β 1 Sk(a)Ω(a), such thatî j (r) > 0 for all r > r * . From the definition ofî j , this implies thatî(r) > 0 for all r > r * + r j . Since r j → −∞ as j → ∞, we obtain that i(r) > 0 for all r ∈ R by letting j → ∞. Consequently,î(r) is strictly positive on R.
Now, let us define a function ρ : Then, it follows from the previous argument that Then, Lemma 4.1 implies the uniform weak ρ-persistence of semi-flow Φ for ℜ 0 > 1. Moreover, from Theorem 3.4 and Lemmas 4.2-4.3 and the Lipschitz continuity of i (which immediately follows from Proposition 2.4), we can apply Theorem 5.2 of Smith and Thieme [8] to conclude that the uniform weak ρ-persistence of semi-flow Φ implies the uniform (strong) ρ-persistence. In conclusion, we obtain the following theorem.
The uniform persistence of system (1) for ℜ 0 > 1 immediately follows from Theorem 4.4. In fact, it follows from (36) that for i 0 ∈ L 1 + (0, ∞), to ∥·∥ L 1 follows. By a similar argument, we can prove that S(t) and p(t, a) are also persistent with respect to | · | and ∥·∥ L 1 . Consequently, we have the following theorem.
To prove that the Lyapunov functional used by Brauer et al. [1] is well-defined, it suffices to show that are finite for all t ≥ 0. To this end, we make the following assumption on the initial conditions. It is obviously true from Theorem 4.5 that S * ln S(t)/S * is finite for all t ≥ 0. Since it follows from (6) and (14) that for t − a > 0, Note that from Theorem 4.5, the first term is finite for all t ≥ 0. For a − t ≥ 0, Hence, under Assumption 4.6, we see from (49)-(50) that i * (a) ln i(t, a)/i * (a) is finite and converges to zero as a → +∞, which implies that the integration ∫ ∞ 0 i * (a) ln i(t, a)/i * (a)da is finite. In a similar way, we can prove that ∫ ∞ 0 p * (b) ln p(t, b)/p * (b)db is finite. Consequently, we see that the Lyapunov functional used by Brauer et al. [1] is well-defined. 5. Stability analysis of system (1). In this section, we present the stability results of system (1), including the local stability and global stability.
Proof. Linearizing the system (1) at disease-free equilibrium P 0 under introducing the perturbation variables which implies that λ * > ξ. Thus, all the roots of the equation (58) have negative real part if and only if ℜ 0 < 1. Therefore we have shown that the disease-free equilibrium P 0 is local asymptotically stable if ℜ 0 < 1 and unstable if ℜ 0 > 1. This completes the proof of Theorem 5.1.