DYNAMICS OF A PARTIALLY DEGENERATE REACTION-DIFFUSION CHOLERA MODEL WITH HORIZONTAL TRANSMISSION AND PHAGE-BACTERIA INTERACTION

. We propose a cholera model with coupled reaction-diﬀusion equations and ordinary diﬀerential equations for discussing the eﬀects of spatial heterogeneity, horizontal transmission, environmental viruses and phages on the spread of vibrio cholerae . We establish the well-posedness of this model which includes the existence of unique global positive solution, asymptotic smoothness of semiﬂow, and existence of a global attractor. The basic reproduction number R 0 is obtained to describe the persistence and extinction of the disease. That is, the disease-free steady state is globally asymptotically stable for R 0 ≤ 1, while it is unstable for R 0 > 1. And, the disease is persistence and the model has the phage-free and phage-present endemic steady states in this case. Further, the global asymptotic stability of phage-free and phage-present endemic steady states are discussed for spatially homogeneous model. Finally, some numerical examples are displayed in order to illustrate the main theoretical results and our opening questions.


Introduction
Cholera is an acute intestinal infectious disease which is caused by the bacteria vibrio cholerae; it has been around for hundreds of years. Currently, cholera is transmitted in two main ways: one is environmental transmission (environment to human), that is, a person becomes infected by ingesting water or food contaminated with vibrio cholerae; the other is horizontal transmission (human to human), such as close contact with people infected with cholera, or contact with the excrement of cholera patients, etc., may be followed by infection. When people get infected with cholera, it causes symptoms like vomiting, muscle cramps, severe copious watery diarrhea, and so on, if not treated promptly, the infection may lead to death after 1 or 2 days. Although modern technology, medicine and public health conditions have improved dramatically compared to the past, but cholera is still not been eliminated and remains endemic in Asia, India, Africa, and Latin America. This is a major threat to the public health of low-income groups in developing countries in particular. In recent years, several areas in large-scale outbreak of cholera: nearly 100,000 cases are reported in Zimbabwe from 2008 to 2009 [6,19,31]; 545,000 cases is reported in Haiti from 2010 to 2012 [2,37,47]; 1,115,378 cases and 2310 deaths are reported in Yemen from 2017 to 2018 [5].
Recently, many research works have studied the transmission mechanism of cholera and put forward effective prevention and control measures. In particular, from the aspect of mathematical dynamics modeling, Codeço [9] incorporated cholera bacteria in aquatic reservoirs into SIR epidemic model, proposed a SIR-B model to simulate the transmission of cholera, and proved the stability of diseasefree and endemic equilibria. Considering the movement of humans, Capone et al. [4] employed a reaction-diffusion model based on the model in [9], and studied the impact of population movements on the existence and stability of disease-free and endemic equilibrium; here they neglected the human-to-human transmission of vibrio cholerae. Wang et al. [49] conducted a diffusive cholera model combining horizontal and environmental transmission and investigated the effect of diffusive spatial spread on the transmission of this disease spread; They revealed that incorporating spatial diffusion does not produce a Turing instability in some extent. Chen et al. [7] purposed a reaction-diffusive cholera model with nonlinear incidence rate, and obtained the global stability of the disease-free and endemic equilibria. In addition, Zhou et al. [56] proposed a reaction-diffusive model with waterborne pathogen and general incidence rate, and investigated the extinction and persistence of disease which are described by the basic reproduction number. Note that the above mentioned models are discussed in a homogeneous space. However, differences in spatial location, water availability and sanitation have a important impact on the transmission of diseases, so it is necessary to consider reaction-diffusion models with spatial heterogeneity [48,51,52,54].
Since the frequent outbreak of cholera brought the local public health a heavy burden, how to prevent and intervene cholera is particularly important. Common intervention methods of cholera include rehydration therapy, antibiotics, vaccination and water treatment [3,13,28,29,30,38,46]. In terms of reducing vibrio cholerae in the environment, Misra et al. [30] introduced a delay SIRS-B compartment model to simulate the transmission of water-born disease, and discussed the effects of disinfectants on the control of diseases. Note that, phages, the natural organisms presents in the aquatic reservoirs, are viruses that live on vibrio cholerae, which injects its genetic material into bacterial cells and replicates within the host cell. When phages reproduce to a certain number, it can cause the bacterial cells lysis and releases additional phage into the environment, and interaction of phages and bacteria can be described by the predator-prey relationship. Now, phages had been proved in modulating cholera epidemics to play a crucial role, so the introduction of phages in the environment is helpful for combating cholera [11,12]. Besides, Malik et al. [24] pointed out that using the basic laboratory equipment preparation phages can be quickly and easily. Alternatively, one can use the freeze-dried, spray drying, emulsification and micro capsule preparation phages, these methods can keep the stability of the years. These also imply that introducing phages may be an effective strategy for vibrio cholerae control.
To discuss the role of phages in cholera prevention and control, Jensen et al. [18] further extended the model in [9] by including a phage compartment P , and revealed that incorporating phages can effectively decrease the concentration of bacteria, and then it can reduce infection in humans. And then, Misra et al. [26] investigated a reaction-diffusion cholera model with phages control, investigated the global dynamics of this model through constructing an appropriate Lyapunov function. Other studies on the dynamical models of cholera with phages can be found in [20,27] and the references therein, name just a few.
Based on the above discussion and the interrelationship between vibrio cholerae, phages and hosts, we propose a novel dynamical model with mutually coupled reaction-diffusion equations and ordinary differential equations. Here, we consider only the spatial dispersal behavior of the host because of the relatively large movement of the host and the relatively small movement of vibrio cholerae and phages in the environment. This results in the solution semiflow of the proposed model is not compact since these equation of vibrio cholerae and phages have no diffusion terms. The rest of this article is organized as follows. In Section 2, we present this model and prove the well-posedness. The basic reproduction number and the global asymptotic stability of the disease-free steady state are obtained in Sections 3 and 4, respectively. In Sections 5 and 6, we focus on the existence and stability of phage-free and phage-present endemic steady states for spatially heterogeneous or homogeneous cases, respectively. Some numerical simulations are performed to support theoretic results and conjectures in Section 7. A brief conclusions are found in Section 8.

Model formulation and well-posedness
Let Ω be a bounded domain with smooth boundary ∂Ω in R n , and ∂ ∂ν is the normal derivative along the outward ν to ∂Ω. Based on the idea of compartment modeling, the host population at domain Ω is divided into three classes: susceptible, infected and recovered, and their quantities or density are denoted by S(x, t), I(x, t) and R(x, t) for location x and time t, respectively. Further, V (x, t) and P (x, t) correspond to the concentration of vibrio cholerae and phages. From the interrelationships between vibrio cholerae, phages and hosts, the cholera model with horizontal and environmental transmission reads with the Neumann boundary condition and the equation of recovered class Here, the horizontal transmission described by the mass action α(x)SI and environmental transmission described by the saturation incidence β(x)SV k(x)+V . The meaning of other model parameters is as follows: D S > 0, D I > 0 and D R > 0 are diffusion coefficients of susceptible, infected and recovered hosts, respectively; Λ(x) is the recruitment rate of susceptible hosts; d(x), µ 0 (x) are the natural death rates of susceptible, infected and recovered hosts, bacteria, respectively; δ(x) is the loss rate of phages; γ(x) is the removal rate of infected hosts expect for natural death; α(x) is the horizontal transmission rate from infected hosts to susceptible hosts; β(x) is the environmental transmission rate from bacteria to susceptible hosts; k(x) is the half-saturation coefficient which depicts the concentration of bacteria that leads 50% chance of contracting vibrio cholerae; η(x) is the shedding rate of bacteria from infected hosts; µ(x) is the self-growth rate of bacteria; ξ(x) is the adsorption rate of phages; θ(x) is burst size which represents the concentration of phages produced per vibrio cholerae; m(x) is the half-saturation coefficient.
For (S, I, V, P ) ∈ X + , let Because the V and P in equation (2.1) without the diffusion terms, the solution semiflow Φ(t) is not compact. To solve this problem, one shall prove that Φ(t) is asymptotically smooth by introducing the Kuratowski measure κ of noncompactness, which is determined as κ(B) := inf{r : B has a finite cover of diameter less than r}, for any bounded B ⊂ X + . It is not hard to see that B is precompact if and only if κ(B) = 0.
For the sake of convenience, we let x := (S, I), y := (V, P ) and define g(x, t, x, y) := (g 1 (x, t, I, V, P ), g 2 (x, t, I, V, P )), Then the Jacobian of g(x, t, x, y) with respect to y is calculated as follows .
The following lemma is to claim Φ(t) satisfies κ-contraction condition.
Lemma 2.4. If there exists a r > 0 such that
Proof. Similar to the proof of [17,Lemma 4.1], we can prove that Φ(t) is asymptotically compact on B in the sense that for any sequences φ n ∈ B and t n → ∞, there exists subsequences φ n k and t n k → ∞ such that Φ(t n k )φ n k converges in X as k → ∞. Next, we consider the omega limit set of B for solution semiflow Φ(t) on X + , which is defined as where dist(Φ(t)B, ω(B)) is described as the distance from Φ(t)B to ω(B). Thus, Φ(t) is κ-contracting. The proof is complete. In particular, on the well-posedness of (2.2), the following corollary is obvious from Theorems 2.3 and 2.6. Corollary 2.7. For model (2.2), the following results hold.

Basic reproduction number
For model (2.1) there exists a unique disease-free steady state E 0 = (Ū (x), 0, 0, 0). Now, we calculate the basic reproduction number R 0 by the results in [53,Section 3], which describes the average number of the secondary infections while introducing a single infection in a completely susceptible host population.
Linearizing (2.1) at the steady state E 0 , one can obtain obtain a linear system, for t > 0, (3.1) Note that S(x, t) and P (x, t) are decoupled from the equations of I(x, t) and V (x, t), thus we only discuss the subsystem of (3.1), for t > 0, , we can obtain the eigenvalue problem and denote Π(t) by the solution semigroup of (3.2). Since system (3.2) is cooperative, Π(t) is a positive C 0 -semigroup generated by B. Let Π(t) is the semigroup generated by B, then B and B are closed and resolvent positive operators from the Theorem 3.12 in [45]. To derive the basic reproduction number of (2.1), suppose that the bacteria is invaded at time t = 0, and the initial distribution of infected hosts described by φ(x) = (φ 2 (x), φ 3 (x)) T . Therefore, as times evolves, the distribution of new infected hosts is F (x) Π(t)φ(x) at time t. Hence, the total distribution of new infected hosts is Then L is a continuous and positive operator of (2.1), which maps the initial infected hosts distribution φ(x) to the total new infected hosts distribution during the average illness period. Based on [53], the spectral radius of L as the basic reproduction number of (2.1), that is, (3.4) Furthermore, by [53, Theorem 3.1(i)], the following lemma is valid.
For the sake of convenience in discussing, we verify that R 0 has the relationship with the important indicators λ 0 and τ 0 .
From conclusion (i) in Lemma 3.2, R 0 has the variational formula

and the basic reproduction number is
.
Note that because ofthe second equation in (3.2) without diffusion term, Π(t) is not compact. The following results shows the existence of the principal eigenvalue for (3.3).
Lemma 3.4. If R 0 ≥ 1, then the principal eigenvalue of (3.3) is s(B) which associates with a strongly positive eigenfunction.
Proof. Let Note that the eigenvalue problem has a principal eigenvalueω 0 associated with the positive eigenvector ϕ 0 . We denote by λ * the larger root of the algebraic equation Hence, By [53, Theorem 2.3(i)], the proof is complete.
Remark 3.6. Lemma 3.5 is the special case of Lemma 3.4, and s(B) is the principal eigenvalue of (3.3) without any limitation. But if µ and µ 0 are both related on x, whether the Lemma 3.5 still holds or not is unknown for R 0 < 1.
Lastly, we need to prove that A = {E 0 }. When V → 0, from the forth equation in (2.1), it also known that P → 0 whether P 0 = 0 or not. This together with above arguments, implies that {E 0 } is globally attractive in ∂X 1 . In addition, {E 0 } is the only compact invariant subset in ∂X 1 . Further, for any φ ∈ A, ω(φ) ⊂ ∂X 1 , we obtain ω(φ) = {E 0 }. Because A is compact invariant in X + by Theorem 2.6, this together with the stability of {E 0 } and Lemma 4.2 imply that A = {E 0 }. Local stability and global attractivity of E 0 indicate that E 0 is globally asymptotically stability. This proof is complete.
For the spatially homogeneous case, we have the following result.
Corollary 5.1. Let W := C(Ω, R 3 ) be the Banach space and its positive cone is denoted by W + .
The following lemma is about the uniform weak repulsion of the disease-free steady stateĒ 0 , which is necessary to verify the persistence of (5.1).
Finally, by [ Remark 5.5. Although we obtain that the existence of endemic steady state without phages E 1 = (S 1 (x), I 1 (x), V 1 (x), 0) for (2.1), it is unknown about its uniqueness and local/global stability. Fortunately, if the heterogeneous space degenerates to the homogeneous space, i.e., model (2.1) degenerates to (2.2), then we can obtain the uniqueness and stability of E 1 .
For model (2.2), if R 0 > 1, then there is a phage-free endemic steady state and I 1 is the positive root of f (I) = AI 2 + BI + C. Here, Clearly, we have f (0) = C > 0 for R 0 > 1. From A < 0, equation f (I) = 0 exists two real roots, one is positive and the other is negative. Furthermore, if R 0 ≤ 1, then B < 0 and f (0) = C ≤ 0, which implies that df (I) dI < 0 for all I > 0. This implies that f (I) has no positive roots if R 0 ≤ 1. Thus, (2.2) has a unique phagefree endemic steady state E 1 = ( S 1 , I 1 , V 1 , 0) for R 0 > 1. Further, we have the following result on the global stability of the phage-free endemic steady state E 1 .
Remark 5.7. Because of the limitations of the study methodology, we did not obtain global stability of the phage-free endemic steady state E 1 = (S 1 (·), I 1 (·), V 1 (·), 0) of (2.1); however, we pose an interesting open question.

Phage-present endemic steady state
The existence and stability of the phage-present endemic steady state of model (2.1) are difficult to obtain because of the spatial heterogeneity and the saturation incidence (the transmission rate of vibrio cholerae from environment to host), so we only discuss the existence and stability of the phage-present endemic steady state for space homogeneous form, i.e., model (2.2).
Suppose that for (2.2) there exists the phage-present endemic steady state E * = ( S * , I * , V * , P * ), then Simple calculations yield and I * is the positive root of h(I) =ĀI 2 +BI +C, wherē SinceĀ < 0 andC > 0, equation h(I) = 0 has two real roots, one is positive and one is negative. We define the phage invasion reproduction number as When R p 0 > 1, it ensure that P * > 0. Then (2.2) has a unique phage-present endemic steady state E * = ( S * , I * , V * , P * ). To establish the global asymptotic stability of E 1 of (2.2), we need to do the following preparatory work.
Proof. From V 1 > V * , we can choose a small ε 3 > 0 such that Suppose, by contradiction, there is ψ 0 ∈ X 0 such that lim sup t→∞ Φ(t)ψ 0 − E 1 X + < ε 3 . This inequality in the sense that there exists t 3 > 0 such that V 1 − ε 3 < V (x, t; ψ 0 ). Thus, from the forth equation of (2.2), we obtain By the above discussion on X 0 , we have P (x, t) > 0 for x ∈Ω and t > 0. Therefore, there is a positive constant b such that P (x, t 3 ; ψ 0 ) ≥ bP 0 (x). Utilizing the standard comparison principle, we obtain From (6.2) we have lim t→∞ P (x, t) = ∞. This is a contradiction with the boundedness of P (x, t) by Corollary 2.7. The proof is complete.
Finally, we discuss the global stability of E * = ( S * , I * , V * , P * ). The following assumption is necessary.
Proof. We choose the Lyapunov function where constants the c 1 > 0 and c 2 > 0 will be determined below, and g(x) = x − 1 − ln x. By calculating the derivative of L 4 (t), we have With the help of (6.1), we obtain Noticing the fact that 1 − x ≤ − ln x for all x > 0, we have As the choice of c 1 and c 2 in Theorem 5.6, let c 1 = β S * V * /(η I * (k + V * )) and Therefore = 0} is the singleton { E * }. Consequently, from the invariable principle [15], we conclude that E * is globally asymptotically stable if R 0 > 1 and R p 0 > 1. The proof is complete. On the existence and stability of the phage-present endemic steady state of (2.1), we also propose an interesting opening question. Conjecture 6.5. If R 0 > 1 and min{V 1 (x)} > max{ δ(x)m(x) θ(x)ξ(x) } for all x ∈Ω, then (2.1) has a unique steady state E * = (S * (x), I * (x), V * (x), P * (x)) which is globally asymptotically stable.

Numerical simulations
In this section, we present some numerical examples to illustrate the main results and verify two opening questions, as well as to investigate the effects of the strength of spatial heterogeneity on basic reproduction number R 0 . 7.1. Spatially homogeneous case. In this subsection, we illustrate the dynamics of (2.2), that is, Theorems 4.5, 5.6, and 6.4. To simply the discussion, we choose Ω = [0, 10] ⊂ R. According to the biological significance of our model and the relevant references, some main model parameters are fixed as Table 1.
Example 7.1. For the stability of the disease-free steady state E 0 , we choose α = 3.4286 × 10 −5 , β = 0.018, k = 1 × 10 7 , η = 1.2, m = 2.2 × 10 6 . It follows Diffusion coefficient of infected hosts 0.008 that R 0 ≈ 0.9957 < 1 by direct calculations. This implies that the bacteria will be eliminated and the disease is extinct in host population. From Figure 1(a), we note that the distribution of susceptible host tend to the stable value Λ d . And the plot in Figure 1(b) shows that regardless of the initial values of the infected host, environmental viruses and phages, the disease eventually converges to 0 as t → ∞. This suggests that as long as the basic reproduction number is less than 1, the disease eventually disappears from the population regardless of the initial value status at the time of the outbreak.
Example 7.2. For the stability of the phage-free endemic steady state E 1 , we choose α = 2 × 10 −4 , β = 0.05, k = 1.1 × 10 7 , η = 6, m = 2.3 × 10 6 . It follows that R 0 ≈ 5.8211 > 1, and V 1 ≈ 8729.4789 ≤ δm θξ = 13570 by direct calculations. All conditions of Theorem 5.6 hold; therefore, the phage-free endemic steady state E 1 of (2.2) is globally asymptotically stable. This is shown in Figures 2 (a) and (b), due to the low rate of phages transformation, phages eventually become extinct regardless of their initial state. While the disease forms endemically in the host population, the susceptible, infected and environmental viruses tend to their respective steady states.

Conclusion and discussion
We developped a cholera models with coupled reaction-diffusion equations and ordinary differential equations to discuss the effects of spatial heterogeneity, environmental viruses and phages on disease transmission. Here, we consider not only the horizontal transmission of vibrio cholerae between hosts, but also the transmission of vibrio cholerae between the environment and hosts, and the interaction between vibrio cholerae and phages in the environment. Since the diffusion of vibrio cholerae and phages in the environment are not considered, this makes the solution semiflow of our model lacking compactness, while creating some difficulties in analyzing the dynamics.
By using the comparison principle, the Kuratowski measure of noncompactnes and other methodological techniques, we verify the existence of nonnegative solution, the point dissipation and the asymptotic smoothness of the solution semiflow. Further, we obtain the basic reproduction number R 0 , which is identified as the spectral radius of next generation operator. In addition, the variational formula of R 0 for spatially heterogeneous case and the expression of R 0 for spatially homogeneous case are calculate. Of course, our basic reproduction number also perfectly portrays the persistence and extinction of the disease. Specifically, if R 0 < 1, the disease-free steady state E 0 is globally asymptotically stable, which indicates that the bacteria is eliminated. We also confirm the global stability of E 0 in a critical case that R 0 = 1, which is the novelty of this paper. Further, the global dynamics of our model for R 0 > 1 is also analyzed in detail. This includes, the existence and global stability of phage-free endemic steady state for heterogeneous or homogeneous space, which implies that cholera becomes endemic by persisting in the host and the environment, while environmental phages tend to become extinct due to their low reproduction rate. Further, we discuss the uniform persistence of phages, bacteria, susceptible and infected hosts and the global stability of phage-present endemic steady state at R 0 > 1 and some other technical conditions. The numerical simulations explain the main conclusions, especially our two conjectures about the global asymptotic stability of the phage-free and phage-present endemic steady states. In addition, numerical simulations also discuss the sensitivity of the main parameters of this model with respect to the basic reproduction number and the influence of diffusion coefficients on the distribution of infectious diseases. In the era of increasingly global economy, the spread of infected hosts allows pathogens to reach all corners of the global village, resulting in increasing epidemic risk levels in some low-epidemic areas and higher epidemic levels in highepidemic areas because of the influx of susceptible hosts. Therefore, reducing the necessary movement of people and increasing local control measures during periods of high outbreaks is one of the effective means to eliminate outbreaks throughout the region.
Notice that we have only demonstrated the global stability of phage-free and phage-present endemic steady states in a homogeneous environment, while the situation in a heterogeneous environment become two interesting open questions. In addition, the activity of vibrio cholerae in the environment is closely related to the time of its shedding [32,35,40,50,52], so it becomes significant to consider the effect of vibrio cholerae activity on disease transmission. These are all topics that deserve further consideration in the future.