Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.


Introduction
Cholera is an ancient intestinal disease for humans. It has a renowned place in epidemiology with John Snow's famous investigations of London cholera in 1850's which established the link between contaminated water and cholera outbreak. Cholera is caused by bacterium vibrio cholerae. The disease transmission consists of two routes: indirect environment-to-human (through ingesting the contaminated water) and direct person-to-person transmission routes. Even though cholera has been an object of intense study for over a hundred years, it remains to be a major public health concern in developing world; the disease has resulted in a number of outbreaks including the recent devastating outbreaks in Zimbabwe and Haiti, and renders more than 1.4 million cases of infection and 28,000 deaths worldwide every year [35].
It is well known that the transmission and spread of infectious diseases are complicated by spatial variation that involves distinctions in ecological and geographical environments, population sizes, socio-economic and demographic structures, human activity levels, contact and mixing patterns, and many other factors. In particular, for cholera, spatial movements of humans and water can play an important role in shaping complex disease dynamics and patterns [6,18]. There have been many studies published in recent years on cholera modeling and analysis (see, e.g., [1,3,4,11,16,17,20,25,26,29,30,31,32,37]). However, only a few mathematical models among this large body of cholera models have considered human and water movement so far. Specifically, Bertuzzo et al. incorporated both water and human movement and formulated a simple PDE model [1,19] and a patch model [2], in which only considered indirect transmission route. Chao et al. [5] proposed a stochastic model to study vaccination strategies and accessed its impact on spatial cholera outbreak in Haiti by using the model and data, for which both direct and indirect transmission were included. Tien, van den Driessche and their collaborators used network ODE models incorporating both water and human movement between geographic regions, and their results establish the connection in disease threshold between network and regions [7,27]. Wang et al. [31] developed a generalized PDE model to study the spatial spread of cholera dynamics along a theoretical river, employing general incidence functions for direct and indirect transmission and intrinsic bacterial growth and incorporating both human/pathogen diffusion and bacterial convection.
In the present paper, we shall pay our attention to a reaction-diffusion-convection cholera model, which employs a most general formulation incorporating all different factors. This PDE model was first proposed in [31] and received investigations [31,37]. Let us now describe this model explicitly in the following section.

Statement of Main Results
We study the following SIRS-B epidemic PDE model for cholera dynamics with x ∈ [0, 1], t > 0: (cf. [31]) subjected to the following initial and Neumann and Robin boundary conditions respectively: S(x, 0) = φ 1 (x), I(x, 0) = φ 2 (x), R(x, 0) = φ 3 (x), B(x, 0) = φ 4 (x), where each φ i (i = 1, 2, 3, 4) is assumed to be nonnegative and continuous in space Here S = S(x, t), I = I(x, t), and R = R(x, t) measure the number of susceptible, infectious, and recovered human hosts at location x and time t, respectively. B = B(x, t) denotes the concentration of the bacteria (vibrios) in the water environment. The definition of model parameters is provided in Table 1.
We assume all of these parameters to be positive. Hereafter let us write ∂ t , ∂ x , ∂ 2 xx for ∂ ∂t , ∂ ∂x , ∂ 2 ∂x 2 , respectively. To state our results clearly, let us denote the solution We also denote the Lebesgue spaces L p with their norms by · L p , p ∈ [1, ∞]. Finally, we denote the space of R 4 -valued functions continuous in x ∈ [0, 1] with the usual sup norm We define analogously Understanding the global dynamical behavior of cholera modeling problems is crucial in order to suggest effective measures to control the growth of the disease. To the best of our knowledge, the existing literature has only studied local dynamics of solutions of this general PDE model. The focus of the present work is global disease threshold dynamics, which will be established in terms of the basic reproduction number R 0 [12,24,33]. To that end, we conduct a rigorous investigation on the disease using the model, and analyze both model parameters and the system dynamics for a better understanding of disease mechanisms. Particularly, we perform a careful analysis on the global threshold dynamics of the disease.
In review of previous results, firstly the authors in [32] defined R ODE 0 for the SIRS-B ODE model, which can be extended to the SIRS-B PDE model as follows: denoting where is same except that the operators Θ 1 , Θ 2 in (7) would have no diffusive operators ∂ 2 xx . Moreover, the authors in [32] proved that when R ODE 0 ≤ 1, the model has the disease-free-equilibrium (DFE) (S, I, R, B) = (m * , 0, 0, 0) which is globally asymptotically stable (see Theorem 2.1 of [32]). On the other hand, when R ODE 0 > 1, it was proven that this ODE model has two equilibriums, namely the DFE which is unstable and endemic equilibrium which is globally asymptotically stable (see Theorem 2.1 [32]). For the SIRS-B PDE model with diffusion, the authors in [37] used spectral analysis tools from [24] to show that when R P DE 0 < 1, the DFE is locally asymptotically stable while if R P DE 0 > 1, then there exists η > 0 such that any positive solution of (1a) linearized at the DFE satisfies lim sup We emphasize here that both these stability and persistence results were local; specifically the results were obtained via analysis on the (S, I, R, B) that solves the system (1a) linearized at the DFE (m * , 0, 0, 0), not necessary the actual system (1a). The major difficulty was that because by definition R P DE 0 gives information only on the linearized system (see the definition (7), (13), (14)), it seemed difficult to utilize the hypothesis that R P DE 0 > 1 or R P DE 0 < 1 to deduce any information on the actual system (1a) (see e.g. Theorem 4.3 (ii) of [34]).
In this paper, we overcome this major obstacle and extend these stability results to global; moreover, we obtain the uniform persistence result. We also extend Lemma 1 of [13], which have proven to be useful in various other papers (e.g. Lemma 3.2, [28]) to the case with convection, which we believe will be useful in many future work. For simplicity, let us hereafter denote R 0 R P DE 0 , and by u(x, t, φ) the solution at (x, t) ∈ [0, 1] × [0, ∞) that initiated from φ: Then the system (1a) subjected to (2), (3) admits a unique global nonnegative solution u(x, t, φ) such that u(x, 0, φ) = φ(x). Moreover, if R 0 < 1, then the DFE (m * , 0, 0, 0) is globally attractive.
(1) We remark that typically the persistence results in the case R 0 > 1 requires a hypothesis that the solution is positive (see e.g. Theorem 4.3 (ii) of [34] and also Theorem 2.3 (2) of [37]). In the statement of Theorem 2.2, we only require that φ 2 (·) ≡ 0 or φ 4 (·) ≡ 0. Due to the Proposition 6.1, we are able to relax these conditions. Moreover, we note that sup in (8) is replaced by inf in (9). (2) The proof was inspired by the work of [13,28,33].
(3) We remark that it remains unknown what happens when R 0 = 1; for this matter, not global but even in the local case, it remains an open problem (see Theorem 2.3 [37]).
(4) In the system (1a), we chose a particular case of h represent the direct, indirect transmission rates, intrinsic growth rate of bacteria respectively (see [32,31]). We remark for the purpose of our subsequent proof that defining this way, f 1 , f 2 , h are all Lipschitz. It is clear from the proof that some generalization is possible.
The rest of the article is organized as follows. The next section presents preliminary results of this study. Section 4 verifies a key proposition as an extension of Lemma 1 of [13], which has proved to be useful in various context. Our main results are established in Sections 5-6. By employing the theory of monotone dynamical systems [38], we prove that (1) the disease free equilibrium (DFE) is globally asymptotically stable if the basic reproduction number R 0 is less than unity; (2) there exists at least one positive steady state and the disease is uniformly persistent in both the human and bacterial populations if R 0 > 1. Additionally, we identify a precise condition on model parameters for which the system admits a unique nonnegative solution, and study the global attractivity of this solution. In the end, a brief discussion is given in Section 7, followed by Appendix.

When there exists a constant
Following [21,37], we let A 0 i , i = 1, 2, 3 denote the differentiation operator , respectively. We can then define A i , (i = 1, 2, 3, 4) to be the closure of A 0 i so that A i on X i generates an analytic semigroup of bounded linear operator that is, for i = 1, 2, 3, It follows that each T i is compact (see e.g. pg. 121 [21]). Moreover, by Corollary 7.2.3, pg. 124 [21], because X + i = C([0, 1], R + ), each T i (t) is strongly positive (see Definition 3.2).
We now let and We recall some relevant definitions

compact and invariant (f (A) = A), and A attracts some open neighborhood of itself. A global attractor for f is an attractor that attracts every point in
Let U ⊂ P be nonempty, closed, and order convex. Then a continuous map Remark 3.1. This lemma remains valid even if the Laplacian is replaced by a general second order differentiation operator; in fact, all results from Chapter 7, [21] remain valid for a general second order differentiation operator (see pg. 121, [21]). In relevance we also refer readers to Theorem 1.1, [15], Corollary 8.1.3 [36] for similar general well-posedness results.

admits a unique nonnegative mild solution on the interval of existence
Remark 3.2. In the statement of Theorems 2.1, 2.2 of [37], we required the initial regularity to be in 1])} and obtained higher regularity beyond C([0, 1], R 4 ); here we point out that to show the global existence of the solution u(t) ∈ X + ∀ t ≥ 0, it suffices that the initial data is in X + . For completeness, in the Appendix we describe the estimate more carefully than that of Proposition 1 in [37] that is needed to verify this claim. [38]) Let E be an ordered Banach space with positive cone P such that int(P ) = ∅, U ⊂ P be nonempty, closed and order convex set. Suppose f : U → U is strongly monotone, strictly subhomogeneous and admits a nonempty compact invariant set K ⊂ int(P ). Then f has a fixed point e ≫ 0 such that every nonempty compact invariant set of f in int(P ) consists of e.
any metric space, is completely continuous for t > t 1 and point dissipative, then there exists a global attractor A. If Y is a Banach space, then A is connected and if t 1 = 0, then there is an equilibrium point of T (t).
and q be a generalized distance function for semiflow Ψ. Assume that (1) Ψ has a global attractor A, of pairwise disjoint, compact and isolated invariant sets in ∂Y 0 with the following properties Then there exists δ > 0 such that for any compact chain transitive set L that satisfies L ⊂ K i ∀ i = 1, . . . , n, min y∈L q(y) > δ holds. Moreover, a continuous mapping f : Y → Y, Y any metric space, is α-condensing (α-contraction of order 0 ≤ k < 1) if f takes bounded sets to bounded sets and It is well-known that a compact map is an α-contraction of order 0, and an α-contraction or order k is α-condensing. Moreover, by Lemma 2.3.5, [10], any α-condensing maps are asymptotically smooth.
where M 0 is nonempty and convex, and Ψ t a continuous-time semiflow on M such
Proof. Firstly, by Lemma 3.2, the unique nonnegative solution u(t, φ) exists on [0, ∞). As already used in the proof of Theorem 2.1, we know that (17) admits a unique positive steady state m * = b d . This implies that, as S, I, R ≥ 0, there exists δ−g ; here we used the hypothesis that g < δ. Hence, the solution semiflow Φ t is point dissipative (see Definition 3.1).
Finally, we know as shown in the proof of Proposition 6.2, that Φ t is compact so that it is asymptotically smooth by Lemma 3.6. Moreover, as we already showed that Φ t (W 0 ) ⊂ W 0 , by Proposition 6.4, we see that Φ t is ρ-uniformly persistent. We also know due to Proposition 6.2 that Φ t : X + → X + has a global attractor A. Thus, by Lemma 3.7, Remark 3.3, Φ t : W 0 → W 0 has a global attractor A 0 .
This implies that because we already showed that Φ t (W 0 ) ⊂ W 0 ∀ t ≥ 0, Φ t is compact so that it is α-condensing by Lemma 3.6, due to Lemma 3.8, we see that Φ t has an equilibrium a 0 ∈ A 0 . By Proposition 6.1, it is clear that a 0 is a positive steady state. This completes the proof of Theorem 2.2.

Conclusion
In this article, we have studied a general reaction-diffusion-convection cholera model, which formulates bacterial and human diffusion, bacterial convection, intrinsic pathogen growth and direct/indirect transmission routes. This general formation of the PDE model allows us to give a thorough investigations on the interactions between the spatial movement of human and bacteria, intrinsic pathogen dynamics and multiple transmission pathways and their contribution of the spatial pattern of cholera epidemics.
The main purpose of this work is to investigate the global dynamics of this PDE model (1a). To achieve this goal, we have established the threshold results of global dynamics of (1a) using the basic reproduction number R 0 . Our analysis shows that if R 0 > 1, the disease will persist uniformly; whereas if R 0 < 1, the disease will die out and the DFE is globally attractive when the diffusion rate of susceptible, infectious and recovered human hosts are identical. These results shed light into the complex interactions of cholera epidemics in terms of model parameters, and their impact on extinction and persistence of the disease. In turn, these findings may suggest efficient implications for the prevention and control of the disease.
Besides, we would like to mention that there are a number of interesting directions at this point, that haven't been considered in the present work. One direction is to study seasonal and climatic changes. It is well known that these factors can cause fluctuation of disease contact rates, human activity level, pathogen growth and death rates, etc., which in turn have strong impact on disease dynamics. The other direction is to model spatial heterogeneity. For instance, taking the diffusion and convection coefficients and other model parameters to be space dependent in 2 dimensional spatial domain (instead of constant values in 1 dimensional region) will better reflect the details of spatial variation. These would make for interesting topics in future investigations. 8. Appendix 8.1. Proof of Lemma 3.2. In this section, we prove Lemma 3.2 for completeness. The local existence of unique nonnegative mild solution on [0, σ), σ = σ(φ), as well as the blow up criterion that if σ = σ(φ) < ∞, then the sup norm of the solution becomes unbounded as t approaches σ from below is shown in the Theorem 2.1 [37]. To show that σ = ∞, we assume that σ < ∞, fix such σ and show the uniform bound which contradicts the blow up criterion. Specifically we show that by performing energy estimates more carefully, keeping track of the dependence on each constant, we may extend Proposition 1 of [37] to the case p = ∞. For brevity, we write L p to imply L p ([0, 1]) below for p ∈ [1, ∞].
Taking p → ∞ on the right hand side first and then the left hand side shows that sup t∈[0,σ) due to Minkowski's inequalities and (2). Next, a similar procedure shows that, as described in complete in detail in the proof of Proposition 1 of [37], we obtain Thus, Gronwall's inequality type argument shows that via Hölder's inequality, Now taking p → ∞ on the left hand side and then on the right hand side gives ∀ t ∈ [0, σ) where we used (40). Taking sup over t ∈ [0, σ) on the left hand side completes the proof.
By continuity in space of the local solution in [0, σ), the proof of Lemma 3.2 is complete.