GLOBAL STABILITY OF AN AGE-STRUCTURED CHOLERA MODEL

In this paper, an age-structured epidemic model is formulated to describe the transmission dynamics of cholera. The PDE model incorporates direct and indirect transmission pathways, infection-age-dependent infectivity and variable periods of infectiousness. Under some suitable assumptions, the PDE model can be reduced to the multi-stage models investigated in the literature. By using the method of Lyapunov function, we established the dynamical properties of the PDE model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number R0: if R0 < 1 the cholera dies out, and if R0 > 1 the disease will persist at the endemic equilibrium. Then the global results obtained for multi-stage models are extended to the general continuous age model.

1. Introduction.Several recent studies have focused on modeling the cholera dynamics ( [1,22,23,26,27,28]).In 2010, Tien et al [28] formulated a SIRW model to describe the cholera dynamics.In the SIRW model, all contaminated water or infectious individuals are assumed to be equally infectious during their periodic infectivity.Actually, laboratory studies suggest that the infectivity of Vibrio cholera existing outside the host decays in time [22], and the infectivity of infectious individuals is also different at the differential age of infection.The age of infection models are often formulated to describe the heterogeneity in infectious individuals.Recent studies on age of infection models show that the age of infection may play an important influence on transmission dynamics of infectious disease [13,14,21,25], and consequently we should incorporate age of infection into the modeling transmission dynamics of cholera.
Variability of infectiousness in time had been described in the literature by multistage models if age of infection is considered as multiple discrete infection stages.Based on the existing multi-stage cholera models formulated in [22] and [28] , in the rate that the cholera pathogen with age of infection θ loses the infectivity γ(a) the rate that an infected individual with age of infection a recovers or dies from the disease this paper we develop more general model to study the cholera dynamics, where the age of infection is considered as a continuous variable.The model presented here takes the following form: where i 0 (a), w 0 (θ) are given non-negative functions.All the parameters in the system (1) are positive.The functions β(a), α(θ) and ξ(a) belong to L ∞ + ((0, +∞), R) \ {0 L ∞ }, and δ(θ), γ(a) belong to L 1 loc ((0, +∞), R), δ(θ), γ(a) ≥ 0 in (0, +∞).We further assume that there exists δ w > 0 such that δ(θ) ≥ δ w for almost every θ ≥ 0. The definition of the different parameters in the system (1) are listed in Table 1.
Assume that the infected individuals are partitioned into n infection stages defined by the infection age intervals [a i−1 , a i ], where 0 = a 0 < a 1 < • • • < a n−1 < a n = ∞, and the contaminated water is categorized into m levels defined by the age intervals [θ i−1 , θ i ], where 0 = θ 0 < θ we also assume that the parameters β(a), γ(a), ξ(a), δ(θ), α(θ) are constant with Let Integrating the second and third equations of system (1) on the age interval [a i−1 , a i ) and We use c i to denote the transfer rate constants of the infected individuals between the age groups [a i−1 , a i ) and [a i , a i+1 ), and d i to denote the transfer rate constants of the pathogen between age groups , where c n = 0 and d m = 0.By using the boundary conditions in system (1), we can obtain the equations for S(t), I i (t), W i (t): Based on the above assumptions that parameters are constant within age groups, the PDE model (1) can be reduced to the ODE model investigated in paper [22] and [28].The global properties of the ODE model ( 4) have been investigated in paper [22], and the results show that the global dynamics of the ODE model is completely determined by the basic reproduction number.The main objective of the paper is to prove that the global results are also true for the continuous age case.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.In this paper, by using a class of global Lyapunov functions we prove that the dynamics of the age of infection cholera model are completely determined by the basic reproduction number R 0 : if R 0 < 1 the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, a unique endemic equilibrium is globally asymptotically stable.Thus the global results obtained for ODE system are extended to the general continuous age model.
The paper is organized as follows.In the next section we mainly present the basic reproduction number, investigate the existence of the equilibrium, and then state the main results of the paper.In the section 3 we mainly show that the disease free equilibrium is globally asymptotically stable if the basic reproduction number is less than one.In order to prove the results on the global stability of the endemic equilibrium of the system (1), in Section 4 we present some preliminary results about uniform persistence and about the existence of global attractors.In the section 5 the proof of the results on the global stability of endemic equilibrium is strictly proved when the basic reproduction number is greater than one.We summarizes our results and outline some future work in the final Section 6.

2.
Reproduction number and main results.Using standard methods we can verify the existence and uniqueness of solutions to the system (1) (see [11] and [30]).Moreover, we can show that all solutions with nonnegative initial conditions will remain nonnegative and bounded for all t > 0. In this section, we mainly present the reproduction number for system (1), investigate the existence of equilibria, and then state the main results of the paper.
In epidemiology, the reproduction number is one of the most useful threshold parameters.It is generally defined as the average number of secondary infections produced by a typical infected individual during the entire period of infection when introduced into a completely susceptible population [2,3,29].As will be shown in the next section that the qualitative and quantitative behaviors of the model ( 1) is completely determined by the reproduction number.
First let us derive the reproduction number from the biological meanings of the model parameters.To simplify expressions, we introduce the following notations: Notice that µ + γ(s) is the rate at which an infected individual of infection age s leaves the infectious class, it then follows that K I (a) represents the probability of remaining in the infected class for an infected case at the age of infection a. Then the average number of secondary cases directly produced by an infected individual can be expressed as and the average concentration of the cholera pathogen produced by an infected individual can be defined as Similarly, K W (θ) represents the probability of remaining the infectivity for the unit concentration of cholera pathogen with infection age θ since δ(s) is the rate at which the cholera pathogen with age of infection s loses the infectivity.Thus, the integral of the production of α(θ) and K W (θ) over all ages, gives the average number of secondary cases directly produced by the unit concentration of cholera pathogen.Therefore, the average number of the secondary cases indirectly produced by an infected individual can be expressed as In summary, if a typical infected individual is introduced into a purely susceptible population, then the average number of secondary cases directly or indirectly produced during the entire period of infection can be expressed as We can easily see that R 0 is the basic reproduction number for system (1).Now we are able to state the result on the existence of equilibria for system (1).
Theorem 2.1.The system (1) can have up to two equilibria.More precisely, we have (1) The disease free equilibrium E 0 ( Λ µ , 0, 0) always exists.(2) If R 0 > 1, there exists a unique endemic equilibrium E * (S * , i * (a), w * (θ)), where Proof.It is obvious that the disease free equilibrium E 0 ( Λ µ , 0, 0) always exists and is unique.Now we prove the second case.The endemic equilibrium E * = (S * , i * (a), w * (θ)) can be found by solving the following system: Integrating from 0 to a or θ the second and third equations in (5) yields Substituting ( 6) into the fourth and fifth equation in (5) gives It follows from (7) and the first equation in ( 5) that Thus, It can be easily see from the expressions of S * , i * (a) and w * (θ) that the endemic equilibrium E * exists if and only if R 0 > 1.This completes the proof of Theorem 2.1.
In order to state the main results of the paper, we set ã = inf{a : Since the functions β(a), α(θ) and ξ(a) belong to and define Now we are able to state the main results of the paper.
is the unique equilibrium of system (1), and it is globally stable.
3. The proof of Theorem 2.2.In this section, we mainly prove Theorem 2.2.First, let us consider the local stability of the DFE E 0 , and we have the following Theorem 3.1.
Proof.Introducing the perturbation variables and linearizing the system ( 1) about E 0 we obtain the following system Let where x 0 1 , x 0 2 (a), x 0 3 (θ) are to be determined.Substituting ( 10) into ( 9), we obtain Integrating the first equation of (11b) from 0 to a yields Substituting ( 12) into (11c) and solving (11c), we obtain Substituting ( 12) and ( 13) into the second equation of (11b) gives the characteristic equation Let H (λ) denote the right hand side of (14) .Then H (λ) is a continuously differential function with lim λ→+∞ can be checked that H (λ) < 0, which implies that H (λ) is a decreasing function.
Thus the equation ( 14) has a unique real root λ * .Noting that we have λ * < 0 if R 0 < 1, and λ * > 0 if R 0 > 1.Let λ = ξ + ηi be an arbitrary complex root to equation (14).Then which implies that λ * > ξ.Thus, all the roots of the equation ( 14) have negative real part if and only if R 0 < 1. Therefore we have shown that the DFE is local asymptotically stable if R 0 < 1 and unstable if R 0 > 1.This completes the proof of Theorem 3.1.
The result in Theorem 3.1 states only the local stability of the DFE E 0 .Now we are able to give the proof of Theorem 2.2.
Proof. of Theorem 2.2.We know from Theorem 3.1 that the DFE E 0 is locally asymptotically stable when R 0 < 1.It suffices to show that E 0 is a global attractor.
For convenience, let Integrating the second equation in system (1) along characteristic lines we get Similarity, integrating the third equation in system (1) along the characteristic lines yields From ( 16) and ( 17) we obtain the inequality Taking the lim sup when t → +∞ of both sides of inequality (18) and using the Fatou's Lemma yield From ( 19) that τ = 0 if R 0 < 1.This implies that Then from the first equation in (1) we have S(t) → Λ µ as t → +∞.This implies that the DFE E 0 is global attractor.The proof of Theorem 2.2 is completed.
4. Preliminary results and uniform persistence.In this section, we first reformulate the system (1) as a Volterra equation and as a non-densely defined semilinear Cauchy problem in order to apply integrated semigroup theory, and then by using the persistence theory for continuous dynamics system we present some results about uniform persistence and about the existence of global attractors.
The Volterra integral formulation of age-structured models has been used successfully in various contexts and provides explicit (or implicit) formulas for the solutions of age-structure models [19].The system (1) with the boundary and initial conditions (2) can be rewritten as the following Volterra type equations where λ(t) and χ(t) are the unique solution of the following system of Volterra equations: We now use the approach introduced by Thieme [24] to reformulate the system (1) with the boundary and initial conditions (2) as a semilinear Cauchy problem.
In order to remove the nonlinearity from the boundary conditions, we enlarge the state space and we consider endowed with the usual product norm, and set ), and We consider the linear operator A : Dom(A) ⊂ X → X defined by where W 1,1 is a Sobolev space, and we define F : X 0 → X by Then by defining , we can reformulate the system (1) with the boundary and initial conditions (2) as the following abstract Cauchy problem for t ≥ 0 and v(0) = v 0 ∈ X 0+ .By using the result in Thieme [24] and Magal [15], we derive that the existence and the uniqueness of the semiflow {U (t)} t≥0 on X 0+ .By identity (S(t), 0 R 2 , i(•, t), w(•, t)) with (S(t), i(•, t), w(•, t)), it can be shown that the semiflow coincides with the one generated by using the Volterra integral formulation.By setting and we deduce that N (t) and W (t) satisfy the following ordinary differential inequalities and where ξ max = ess sup θ∈(0,∞) ξ(θ).By using the comparison theorem, it follows that are satisfied for some t = t 0 then they are satisfied for all t ≥ t 0 .Thus, the system (1) leaves the set positively invariant.Consequently, it then follows that the set is positively invariant absorbing set under the semiflow {U (t)} t≥0 on X 0+ , i.e., U (t)B ⊆ B and for each x ∈ (S 0 , 0, 0, i, w) ∈ X 0+ , This means that the semiflow {U (t)} t≥0 is bound dissipative on X 0+ (see Hale [7]).Furthermore, it follows from [16,30] that the semiflow {U (t)} t≥0 is asymptotically smooth.As a consequence of the results on the existence of global attractors in Hale [7], we have the following theorem.
Theorem 4.1.The system (22) generates a unique continuous semiflow {U (t)} t≥0 on X 0+ that is asymptotically smooth and bounded dissipative.Furthermore, the semiflow {U (t)} t≥0 has a global attractor A in X 0+ which attracts the bound sets of X 0+ .
From Theorem 4.1, it then follows that the semiflow {U (t)} t≥0 is asymptotically smooth, point dissipative and that the forward trajectory of a bound set is bounded.Furthermore, the DFE E 0 is globally asymptotically stable in ∂ M0 .Thus, Theorem 4.2 of Hale and Waltman [8] implies the semiflow {U (t)} t≥0 is uniformly persistent with respect to (∂ M0 , M0 ).This completes the proof of the first result.
Next, let us prove the second result.Since where Then for sufficiently large t we have Let (S(t), 0, 0, i(•, t), w(•, t)) be the solution to the system ( 22) with initial condition (S 0 , 0, 0, i 0 , w 0 ) ∈ A 0 .Then for sufficiently small > 0, we have S(t) > δ for all t ≥ 0. Furthermore, since the solution (S(t), 0, 0, i(•, t), w(•, t)) is in the attractor A 0 , we have for all t ≥ 0, where ( î(a, t), ŵ(θ, t)) satisfies the following auxiliary system Similarly to the discussion of system (30), we obtain where T (t) is the semigroup generated by the linear system (33) and λ * is the unique real eigenvalue of the system (33), and ψ(a) > 0, φ(θ) > 0 for all a, θ > 0 since (i 0 , w 0 ) ∈ M0 .Thus there exists T > 0 such that i(a, t) > 0, w(θ, t) > 0 for all a, θ > 0 and t > T .It then follows that Recall that the mapping (t, (S 0 , 0, 0, i 0 , w 0 )) → U (t)(S 0 , 0, 0, i 0 , w 0 ) is continuous.Also, the mapping is continuous in the L 1 norm.Using the compactness of A 0 and A 0 is invariant under the semiflow {u(t)} t≥0 , it follows that for each (S 0 , 0, 0, i 0 , w 0 ) ∈ A 0 we have Using the continuity of the map (34) again and the compactness of A 0 , the second result follows.This complete the proof of Theorem 4.3.
Proof.Introducing the perturbation variables y 1 (t) = S(t) − S * , y 2 (a, t) = i(a, t) − i * (a), y 3 (θ, t) = w(θ, t) − w * (θ), and linearizing the system (1) about E * we obtain the following system  Proof of Theorem 2.3.We only need to prove the second results because the first result can be easily seen from Theorem 4.2.If R 0 > 1, it then follows from Theorem 2.1 that the system (1) has a unique positive equilibrium E * (S * , i * (a), w * (θ)), and S * , i * (a), w * (θ) satisfy the following equations: We can easily check that and the function G(x) has the global minimum at x = 1 and G(1) = 0.
Let u(t) = (S(t), i(•, t), w(•, t)) ba a complete solution to system (1) that lies in the attractor A 0 .From Theorem 4.3, we know there exist δ 1 , δ 2 > 0 such that for all t ∈ R and a, θ ≥ 0. Now let us define the following Lyapunov function From ( 46) we can easily see that the function V is bounded when restricted to A 0 .Since the function G(x) is nonnegative for all x > 0, and has the global minimum at x = 1, it then follows that the function V (t) is nonnegative and the point E * is the global minimum point.We can also easily see that the function V (t) is continuously differentiable.Differentiating V (t) along solutions to (1) in A 0 , using (43),(44) and collecting terms, we obtain where i a (a, t) = ∂i(a, t) ∂a , i * a (a) = di * (a) da , by using (43) it then follows that Thus, by using integration by parts we have Here, we have used the second equation in (43) and the equation that Substituting the equation (49) into the second term in equation (48) gives Similarly, by using the third equation in (43), the second equation in (44) and the equation that the third term in equation ( 48) can be expressed as Substituting ( 50) and ( 51) into (48) yields that (52) Putting the equations (45) into (52), using ( 2) and (44), and rearranging the terms, we have Straight forward computation yields that Observe also that i * (0) By using the first equation in (44), we have We have shown that the function V is non-increasing along any complete solution u(t) in the attractor A 0 .Consider a point P (S P , i P , w p ) in the alpha limit commonly used to prove the global stability of nonlinear dynamical systems.In this paper, by constructed a class of global Lyapunov functions, and proved that the dynamics of the age of infection cholera model are completely determined by the basic reproduction number R 0 .Lyapunov functions of this type has been widely used for analyzing the ODE models in the literature (e.g., [5], [6], [9]) and was recently rediscovered ( [10], [19]) to study the global stability of endemic equilibrium for the epidemic models with age of infection.We should point out that the techniques used for the PDE models are quite different from the techniques used for the ODE models, thus the process that we prove the global stability of the age-structured model is not the trivial extension used for analyzing the multi-stage ODE models in [22,27] despite the conclusions are the same.Finally, in this paper the modeling of cholera transmission dynamics is relatively simple compared with the modeling of many other infectious diseases.Thus it is necessary to improve the mathematical modeling for describing the transmission dynamics of waterborne diseases, and analyze the qualitative behavior of the improved models so that we can have a better understanding of the waterborne disease dynamics.In addition, as we all know that the top priority of global public safety is to prevent and contain the spread of infectious diseases, therefore we are also interested in exploring the effectiveness and implication of various preventive and control strategies on the transmission dynamics of waterborne diseases.We leave these for future investigations.

Table 1 .
Definitions of frequently used symbols