Spatial dynamics of a generalized cholera model with nonlocal time delay in a heterogeneous environment

In this work, we mechanistically formulate a generalized cholera model with nonlocal time delay to study the impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment. Mathematical challenges lie in the fact that (i) the generalized cholera model considers the intrinsic growth of short-lived hyperinfectious (HI vibrios) state of V. cholerae and lower-infectious (LI vibrios) state of V. cholerae simultaneously; and (ii) this article originally derives the detailed classiﬁcations of spatial dynamics for the cholera model with some generally functional response functions, non-uniformness of diffusion rates and nonlocal time delay. We introduce three basic reproduction numbers: one is for HI state of V. cholerae , the other is for LI state of V. cholerae , and another is for the cholera disease in the host population. Based on these basic reproduction numbers, we further establish the global threshold dynamics. Under some conditions, the basic reproduction number of infection is strictly decreasing with respect to the diffusion coefﬁcients of infectious hosts. © 2024 The


Introduction
Cholera is an ancient disease characterized by severe intestinal infection caused by the bacterium Vibrio cholerae, and its source of infection is mainly contaminated water.Despite extensive theoretical and clinical researches, it remains a serious public health burden in developing countries [1,48].According to the World Health Organization [49], there are 1.3 to 4.0 million cases of cholera each year and 21,000 to 143,000 deaths from cholera worldwide.Researches have shown that vibrio cholerae, shed freshly from the infected human host through the gastrointestinal tract, can survive for several hours and possess a high level of infectivity, known as the short-lived HI vibrios state of V. cholerae [3,18].Nelson et al. [32] demonstrated that HI vibrios can be up to 700 times more infectious than vibrio cholerae, which grows in the environment for several months (known as the LI vibrios state of V. cholerae).This hyperinfectivity is key to understanding the explosive nature of human-to-human transmission in outbreaks [18].Furthermore, since the decay from the HI state occurs within hours, Hartley et al. [18] suggested that rapid local transmission through the direct route is responsible for rapid spread and the "explosive" nature of cholera epidemics while the much slower environmental route accounts for much slower dynamics.
A lot of mathematical models have been proposed to provide insights into cholera prevention [7,8,12,14,18,59,63]. Capasso and Paveri-Fontana [9] firstly proposed an ordinary differential equation (ODE) model of cholera, which concerned the evolution of the infected individuals and bacteria population.Later, Codeço [11] extended the Capasso and Paveri-Fontana's cholera model [9] with an additional equation for the susceptible individuals and illustrated the influence of aquatic hosts on the dynamics of cholera transmission.Joh et al. [21] developed a family of iSIR (indirectly transmitted SIR) models by considering the minimum infection dose (MID) into the incidence term and showed that an outbreak can result from noninfinitesimal introductions of either infected individuals or additional pathogens in the reservoir.Tien and Earn [45] proposed a waterborne disease model incorporating environment-human and human-human infection pathways simultaneously.Jensen et al. [20] modified the model in [11], with the new model including phage compartment P and dividing the infectors into bacteria and phage-infected individuals and phage-infected individuals.Subsequently, Kong et al. [24] improved the model in [21] by incorporating bacteriophage and showed that oscillating trajectories could exist at the microbial and population scales.Shuai et al. [40] developed a cholera model that combines hyperinfectivity and temporary immunity using distributed delays.Capasso and Maddalena [10] proposed a partial differential equation (PDE) model to study the spatial spread of bacterial diseases by assuming that the bacteria diffuse randomly in the habitat.Bertuzzo et al. [4] introduced the spatial motion of concentrated pathogens in cholera epidemics and calculated the transmission velocity of cholera waves under different topologies.Zhou et al. [62] established a reaction-diffusion waterborne pathogen model with general incidence rate in a homogeneous environment.Wang et al. [52] extended the model in [62] by using space-dependent coefficients in a heterogeneous environment.

Model formulation
The nonlocal infection demonstrates that individuals in the latency period can be mobile, resulting in infectious individuals being present at any location in the habitat [15,26,39].One of the immediate ways to address this nonlocal infection is to introduce the concept of age of infection.Suppose that host population is divided into four categories: susceptible (S := S(x, t)), exposed (E := E(x, t)), infectious (I := I (x, t)) and recovered (R := R(x, t)) class, and all humans live in a bounded spatial habitat with a smooth boundary ∂ .The concentrations of HI and LI state of V. cholerae are denoted by B 1 := B 1 (x, t) and B 2 = B 2 (x, t), respectively.Applying the standard SIR model, we have the following model Ξ(x, t, 0) = f 1 (S(x, t), I (x, t)) + f 2 (S(x, t), B 1 (x, t)) + f 3 (S(x, t), B 2 (x, t)), (1.1) for x ∈ , t > 0 and a I > 0, where D i (x, a I ) > 0 stands for the diffusion rate at location x with infection age a I .d(x, a I ) > 0 includes natural and disease-induced death rates and recovery rate of infectious individuals at location x and age a I .From the biological point of view, we assume that Ξ(x, t, ∞) = 0, and the average infection period is fixed by τ , D i (x, a I ) and d(x, a I ) are given as follows: From model ( and z(x, g, g) = Ξ(x, g, 0) = f 1 (S(x, g), I (x, g)) + f 2 (S(x, g), B 1 (x, g)) + f 3 (S(x, g), B 2 (x, g)), g ≥ 0, z(x, 0, g) = Ξ(x, 0, −g), g < 0.
Taking g as a parameter and solving the above equation: T 1 (t − g)Ξ (•, g, 0), for x ∈ , 0 ≤ t − g ≤ τ and t ≥ g ≥ 0, respectively, with Neumann boundary condition on .Further, we get Denote γ (x, y, t) as the kernel function of Thus, we have Note that the equations of E and R are decoupled from the model system.Denote From the discussions above, we get the following generalized cholera model with nonlocal time delay: for x ∈ , t > 0, associated with Neumann boundary condition and nonnegative initial condition where It is worth mentioning that our mathematical model (1.2) is very generalized, which includes some nice existing models such as those described in [38,52,61,62], as particular cases.Considering the properties of T 1 (t), we generalize model (1.2) in a more general sense of kernel function.We still choose γ (x, y, τ ) to represent a general nonnegative kernel function, which satisfies the following basic assumption: (H0) For τ 0, γ (x, y, τ )dy is continuous in x ∈ ¯ , γ (x, y, τ )dx is continuous in y ∈ ¯ , and γ (x, y, τ )ζ(y)dy > 0 for any By Cauchy inequality, we get (1.4).
Throughout this paper, we make the following basic assumptions: (H1) n(x, z 1 ) ∈ C 0,1 ( × R + ) and ∂ z 1 n(x, z 1 ) < 0 for x ∈ and z 1 ≥ 0. There is a unique The assumption (H2) is given due to technique requirements.Note that the assumption is also reasonable from a biological point of view, since the bilinear incidence rate β i z 1 z i (i = 2, 3, 4), the Holling type II functional response β i z 1 z i /(1 + a i z i ) (i = 2, 3, 4), and the Beddington-DeAngelis functional response β i z 1 z i /(1 +a i z 1 +b i z i ) (i = 2, 3, 4) are included as special cases.

Motivation and goal
Bacterical hyperinfectivity is a nonnegligible factor in cholera dynamics [18,35].Recently, Wang and Wang [54] proposed a reaction-advection-diffusion model by considering bacterical hyperinfectivity, and studied the impact of bacterical hyperinfectivity on the spread of cholerea.
Wang and Wu [53] further considered a diffusive cholera model with bacterial hyperinfectivity, and investigated the threshold-type results and asymptotic profiles when the dispersal rate of susceptible humans approaches zero or infinity, where the diffusion of HI and LI state of V. cholerae are ignored.In epidemiology, the basic reproduction number is a critical threshold for disease outbreak [2,47,57,65].Thus, one natural question is how do the basic reproduction number for HI state of V. cholerae, LI state of V. cholerae and infectious host work together to determine the threshold dynamics of cholera.
In fact, the spread of infectious diseases is significantly influenced by spatial heterogeneity, such as spatial location, water availability, and sanitation.Thus, it is essential to consider spatial heterogeneity for disease prevention and control [4,23,34,46].Mukandavire et al. [31] estimated the basic reproduction numbers for 10 provinces in Zimbabwe and showed that the transmission mode of cholera varies widely across the country.Tuite et al. [46] established different basic reproduction numbers for 10 administrative departments of Haiti.Wang et al. [50] developed a host-pathogen model in which the susceptible and infected hosts have the same diffusion coefficient but the pathogen does not spread.Wang et al. [55] proposed a reaction-convection-diffusion model with time-periodic coefficients to study the dynamics of cholera transmission.
Inspired by the above discussions, the main purpose of this paper is to address these questions: How does the bacterial hyperinfectivity affect the global dynamics of cholera epidemics in heterogeneous environments?How to quantify the infection risk of cholera in a spatially heterogeneous environment?Specifically, the existence and stability of infection-free steady state and endemic steady state of model (1.2) are investigated by exploring the effects of the basic reproduction numbers for the HI state of V. cholerae (R e1 ), LI state of V. cholerae (R e2 ) and cholera disease in the host population (R 0 ) on the global dynamics of model (1.2).Many interesting and important phenomena have been found in this work, which are briefly summarized as follows: (F1) When max {R e1 , R e2 } < 1 and R 0 ≤ 1, the infection-free steady state is globally asymptotically stable (see Theorem 5.1).In biology, when there are few HI vibrios recently shed from individuals (R e1 < 1) and few LI vibrios in the environment (R e2 < 1), under the assumption that the expected number of secondary cases produced by an infective individual is less or equal than one (R 0 ≤ 1), then the disease will die out.(F2) When Case 1: R e1 ≥ 1 or R e2 ≥ 1, or Case 2: max {R e1 , R e2 } < 1 and R 0 > 1 is satisfied, the disease will persist and there exists at least one endemic steady state (see Theorem [5.2]).Moreover, under some conditions, the infection steady state is globally attractive in some special case (see Theorem [5.9]).In biology, when Case 1: there are a large number of HI vibrios recently shed from individuals (R e1 ≥ 1) or a large number of LI vibrios in the environment (R e2 ≥ 1), or Case 2: there are few HI vibrios recently shed from individuals (R e1 < 1) and few LI vibrios in the environment (R e2 < 1) under the assumption that the expected number of secondary cases produced by an infective individual is larger than one (R 0 > 1), then the disease will persist and become endemic.(F3) When Case 1: R e1 ≥ 1 and R e2 ≥ 1, or Case 2: max {R e1 , R e2 } < 1 and R 0 > 1 is satisfied, the unique positive homogeneous steady state is globally asymptotically stable (see Theorem [6.2]).In biology, when Case 1: there are a large number of HI vibrios recently shed from individuals (R e1 ≥ 1) and a large number of LI vibrios in the environment (R e2 ≥ 1), or Case 2: there are few HI vibrios recently shed from individuals (R e1 < 1) and few LI vibrios in the environment (R e2 < 1) under the assumption that the expected number of secondary cases produced by an infective individual is larger than one (R 0 > 1), then the disease will persist in a homogeneous environment.
(F4) Under some conditions, R 0 is a decreasing function in D 2 (see Proposition [4.5]).In biology, if the habitat is homogeneous, the diffusion of infectious hosts will reduce cholera infection.
The remainder of the work is organized as follows.Section 2 focus on the well-posedness of the model.In Section 3, we investigated the dynamics of cholera model without shedding resource and obtain the expressions of R e1 and R e2 .In Section 4, we define the basic reproduction number for infection, denoted by R 0 .In Section 5, we investigate the global dynamics of the cholera model with nonlocal time delay.In Section 6, the unique positive steady state of a homogeneous model is considered.In Section 7, conclusions and discussions are given.
By (H3), there exist positive constants b 0 , b 3 , b 4 such that for m ≥ 0. Especially, H2) and continuity of [ (τ )1](x) = γ (x, y, τ )dy in ¯ , there exist positive constants p 2 , p 3 and p 4 satisfying where p i0 (i = 2, 3, 4) > 0 and p i1 (i = 3, 4) > 0 are constants.We substitute the second and third inequalities into the first one to get where M 1 and M 2 are both positive constants.By Gronwall's inequality, we have Integrating the equations of z 1 and z 2 , we get where According to the comparison principle, we obtain lim By (2.2), we integrate the equation of z 3 and obtain Similarly, by (2.3), we integrate the equation of z 4 and get In order to estimate Z 2,2 (t), Z 3,2 (t) and Z 4,2 (t) for t > t 4 .We first multiple the equation for z 2 by z 2 and integrate on .By (1.4) and (2.4), we have where Similarly, multiplying the equation for z 3 (resp.z 4 ) by z 3 (resp.z 4 ) and integrating on .By (2.2), (2.3) and Cauchy inequality, we have where ( For any v ∈ W 1,2 ( ) and small ε > 0, adding the above three inequalities, we have where ) are positive constants.Then, by comparison principle, we have Finally, let in order to estimate J 2h , multiplying the equation for z 2 by 2hz 2h−1 2 and integrating on .Define by (2.4), Young inequality and h ≥ 1, one gets then multiplying the equation for z 3 by 2hz 2h−1 3 and integrating on , by (2.2), Young inequality and h ≥ 1, we get Similarly, multiplying the equation for z 4 by 2hz 2h−1 4 and integrating on , by (2.3), Young inequality and h ≥ 1, we obtain by (2.5), adding the above three inequalities, one gets 2 , where M 3 is independent of h and φ.Now, by induction, we prove that Therefore, we get x ∈ , and there exists a constant k 0 > 0 independent of φ satisfying Furthermore, if there are some x 0 ∈ and t 0 ≥ 0 such that z 2 (x 0 , t 0 ) > 0 or z 3 (x 0 , t 0 ) > 0 or z 4 (x 0 , t 0 ) > 0, then z i (x, t) > 0 for i = 2, 3, 4, t > t 0 + τ and x ∈ .

Dynamics of environment model without shedding source
Without shedding source, the dynamics of HI state of V. cholerae is determined by the dynamics of LI state of V. cholerae is determined by Note that h 1 (x, 0) = 0, linearizing model (3.1) at the trivial steady state 0, we have where h 1 (x) = ∂h 1 (x,0) ∂B 1 .Similarly, since h 2 (x, 0) = 0, we linearize model (3.2) at the trivial steady state 0, we have where with Neumann boundary condition.We define the spectral radius of the next generation operator − h 1 A −1 z 3 as the basic reproduction number for HI state of V. cholerae, that is, similarly, we also define the basic reproduction number for LI state of V. cholerae as follows Recall that T z 3 and T z 4 are the solution semigroups associated with A z 3 and A z 4 , respectively.Assume that ϕ(x) and ψ(x) are the initial densities of HI state of V. cholerae and LI state of V. cholerae, respectively.[T z 3 (t)ϕ](x) and [T z 4 (t)ψ](x) represent the densities of survived HI and LI state of V. cholerae at time t.Then the densities of newly generated HI and LI state of V. cholerae are h 1 (x)[T z 3 (t)ϕ](x) and h 2 (x)[T z 4 (t)ψ](x), respectively.Therefore, the total densities of next generation HI and LI state of V. cholerae during the life cycle of initial bacteria are calculated as respectively, which indicates that both − h 1 A −1 z 3 and − h 2 A −1 z 4 are the next generation operators.It follows from [13,Remark 1.6] and [57, Theorem 3.2] that 1/R e1 is the principal eigenvalue of the following equation: and 1/R e2 is the principal eigenvalue of the following equation: Additionally, R e1 and R e2 have the following variational representations, respectively By Krein-Rutman theorem, the spectral bounds of h 1 + A z 3 and h 2 + A z 4 are the same as their principal eigenvalues, respectively, and have the following variational representations: Note that A z i (i = 3, 4) are resolvent-positive with s(A z i ) < 0 (i = 3, 4
Suppose that max {R e1 , R e2 } < 1, we define the spectral radius of FV −1 as the basic reproduction number for model (1.2), that is, Since max {R e1 , R e2 } < 1, the operator −V is resolvent-positive with s(−V ) < 0. By [44, Theorem 3.12], F − V is resolvent-positive because it generates a positive semigroup.Thus, by [44, Theorem 3.5], R 0 − 1 has the same sign as s(F − V ).It follows from Lemma [4.1] that s(F − V ) has the same sign as λ * .We get the following result.
Next, we want to find another expression of R 0 such that the direct and indirect transmission mechanism are clearly separated in the expression.
⎞ ⎠ be a resolvent-positive operator with s(B) < 0. Then we have where and Proof.Let ψ = Fφ and φ = −B −1 ϕ.Then thus, we have Consequently, and where By iteration, it follows that Thus, we get By Gelfand's formula and the squeeze theorem, one gets r(−FB −1 ) = r(L 1 ), where By Lemma [4.4], we obtain where and Now, we analyze the dependence of R 0 on D 2 .Assume that D 2 is a constant and (x, y, τ ) is a constant multiplication of delta function satisfies Krein-Rutman theorem that R 0 is a principal eigenvalue of A d + A i with a positive eigenfunction φ(x).We have Since max {R e1 , R e2 } < 1, by strong maximum principle, ξ , ϕ and ψ are positive functions.We get ) ) Taking D 2 as a variable and taking the derivatives of D 2 on both sides of the above equations yield ) Multiplying (4.5) and (4.8) by ξ and ξ , respectively, and then integrating the difference, one gets Similarly, multiplying (4.6) and (4.9) by ϕ and ϕ, respectively, we have Proof.Recall that R 0 − 1 and s(F − V ) have the same sign as λ * , where λ * is the principal eigenvalue of e −λ * τ F − V with a positive eigenfunction (ξ, ϕ, ψ).We have Note that if R 0 ≤ 1, then λ * ≤ 0. Given any solution z = (z 1 , z 2 , z 3 , z 4 ), following [6], for t ≥ 0, we define If z 2 or z 3 or z 4 is not identically zero, it follows from the strong maximum principle that there exists t 0 ≥ 0 such that z i (x, t) > 0 (i = 2, 3, 4) for t ≥ t 0 − τ .Furthermore, if z 1 (x, t) ≤ u * (x) for t ≥ −τ , by (H2) and (H3), we have It follows from strong maximum principle that z 2 (x, t) < l(t 1 , z)e λ * t ξ(x), z 3 (x, t) < l(t 1 ; z)e λ * t ϕ(x), z 4 (x, t) < l(t 1 , z)e λ * t ψ(x), for t > t 1 ≥ t 0 .Thus, l(t, z) is strictly decreasing in t.We claim that z i (i = 2, 3, 4) → 0 as t → ∞.If λ * < 0, the claim is clear.If λ * = 0, denote l = lim t→∞ l(t, z), the claim is true when l = 0.When l > 0, there is a subsequence t n such that z(x, t + t n ) → z(x, t) when n → ∞ and z i (x, t) (i = 2, 3, 4) is not identically zero.Furthermore, z 1 (x, t) ≤ u * (x) for t ≥ −τ .Similarly, we get that l(t; z) is strictly decreasing for all sufficiently large t.However, l(t; z) = lim n→∞ l(t + t n ; z) = l, which leads to a contradiction.Thus, we have proved the claim.
Proof.We first apply a contradiction argument to indicate that the set M for the first time at x = x 0 and t = t 0 , we obtain z 1 (x 0 , t 0 ) = M and ∂z 1 (x 0 , t 0 )/∂t ≥ 0. By model (5.1), we have 0 ≤ ∂z 1 (x 0 , t 0 )/∂t < n(x 0 , M) < n(x 0 , z 1 (x 0 )) = 0, which leads to a contradiction.Adding the first two equations of model ( 5.1), we can similarly prove that the set x ∈ ¯ and t ≥ 0. It follows from (H3) and comparison theorem that for x ∈ ¯ and t ≥ 0. Thus, the positive invariance of A M is obtained.
Let Q be any bounded subset in X + , we can find a large M > z 1 such that Q ⊂ A M .By the positive invariance of A M , we can immediately obtain the boundedness of γ + (Q).For each ψ ∈ Q, there exists a ṫ ≥ 0 such that t ψ ∈ A M for t ≥ ṫ.We next have to show that the choice of ṫ is independent of ψ .If M ≤ M, the result is clear by choosing ṫ = 0. Thus, we only assume that M > M. Since z 1 < M, we may choose ε > 0 sufficiently small such that Consider ∂e 1 ∂t = n(x, e 1 (x, t)) with e 1 (x, 0) = M .It follows from the comparison principle that z 1 (x, t) ≤ e 1 (x, t) for t ≥ 0 and x ∈ ¯ .Due to n(x, e 1 ) ≤ n(x, M 1 ) ≤ max x∈ ¯ n(x, M 1 ) < 0 whenever e 1 ≥ M 1 , we choose . Therefore, z 3 (x, t) ≤ e 3 (t) for t ≥ ṫ1 + ṫ2 and x ∈ ¯ .Furthermore, since e 3 (t) ≤ − με whenever e 3 (t) ≥ M 3 , we choose to get z 4 (x, t) ≤ e 4 (t) ≤ M 4 for t ≥ ṫ1 + ṫ2 + ṫ3 + ṫ4 and x ∈ ¯ .Let t = ṫ1 + ṫ2 + ṫ3 + ṫ4 .Therefore, we get t Q ∈ A M for t ≥ ṫ.
Apparently, model (5.1) always admits a unique infection-free steady state ( z 1 (x), 0, 0, 0).The linear operator for the linearized system of model (5.1) is decomposed as A = F − V , where for V to be well-defined, we impose the following assumption: Suppose that R e1 < 1 and (H4) holds, since −V is resolvent-positive with s(−V ) < 0, F is positive and A is also resolvent-positive, by [44,Theorem 3.5], we obtain that R 0 − 1 has the same sign as s(A), where R 0 is defined as the spectral radius of FV −1 , that is, R 0 = r(FV −1 ).
By Lemma [5.3] amd Lemma [5.4] that the semiflow t of model (5.1) is point dissipative and the orbit of any bounded set is also bounded.In order to use [17, Theorem 2.1], we need to show that t is asymptotically smooth.Due to the equations of z 1 , z 2 and z 4 in model (5.1) have no diffusion terms, the solution semiflow t loses its compactness.Thus, we introduce the Kuratowski measure R, for any bounded set Q, denoted by be the vector field corresponding to the equations of z 1 , z 2 and z 4 in model (5.1).The Jacobian of G with (z 1 , z 2 , z 4 ) is .
Lemma 5.5.t is asymptotically smooth and R-contracting if there is a r > 0 satisfying Remark 5.6.A sufficient condition for (5.2) is Let P t : E → E be the solution semiflow of the linearized system of model (5.1),where E := C( ¯ , R 3 ) is a functional space for the linearized system of model (5.1), we have P t ψ = (z 2 (•, t, ψ), z 3 (•, t, ψ), z 4 (•, t, ψ)) for t ≥ 0 and ψ ∈ E. Apparently, P t is a positive C 0semigroup on E and its infinitesimal generator A = F − V is closed and resolvent positive.Note that by (H5), we have d 4 (x) > h 2 (x), which means that (H5) contains (H4).Lemma 5.7.If R e1 < 1 < R 0 and (H5) holds, then s(A) is the principal eigenvalue of the eigenvalue problem and there is a strongly positive eigenfunction associated with s(A).
Proof.It follows from R 0 > 1 that s(A) > 0. Define L(t) and N (t) on E as and follows from (H5) that a > 0.Then, the operator L(t) can be estimated as Let Ṫz 3 (t) = e (D 3 + h 1 (•)−σ (•))t be the compact and strongly positive C 0 semigroup generated by this together with the boundness of Furthermore, we get ρ e (P t ) ≤ e −at < 1 ≤ e s(A)t = ρ(P t ) for t > 0, where ρ e (P t ) and ρ(P t ) are the essential spectral radius and spectral radius of P t , respectively.P t is a strongly positive and bounded operator on E. By the generalized Krein-Rutman theorem [33], we get that s(A) is the principal eigenvalue of model ( 5.3) with a strictly positive eigenfunction.
Proof.Set J (s) = s − 1 − ln s ≥ J (1) = 0, for any s > 0. Let where is strictly positive in , and We calculate the time derivative of Y(t) along the solution of model (5.1): Thus, one gets where .

Global stability of the cholera model in a homogeneous case
We assume that f i (m, n) = mf i (n) (i = 1, 2, 3) and consider the following homogeneous model where [ (τ )1](x) = (τ ) for x ∈ ¯ .Note that both heat kernel and delta kernel satisfy model (6.1).Thus, we get u * (x) = z 1 , where z 1 is the unique positive solution of n( z 1 ) = 0.The formulas (3.3) and (3.4) can be simplified as where , by Krein-Rutman theorem, A d + A i is a compact and positive operator with a positive eigenfunction 1 corresponding to a positive principal eigenvalue , Let z 1 ∈ (0, z 1 ) be an independent variable, and regard z i (i = 2, 3, 4) as functions of z 1 defined as follows H1) and (H3), the second equation admits a unique solution for z i ≥ z i (i = 3, 4), where A homogeneous positive steady state exists if and only if V has a root in (0, z 1 ).Obviously, This implies that V (z) admits at least one root in z * 1 ∈ (0, z 1 ).For the critical condition R e1 = R e2 = 1, we still get z 2 = z 3 = z 4 = 0, when z 1 = z 1 .Consequently, V ( z 1 ) = 0 and As z 1 approaches z 1 from the left, z 3 and z 4 approach to 0 from the right, and σ − h 1 (z 3 ) and d 4 − h 2 (z 4 ) approach zero from the right, thus, V (z 1 ) → ∞.Especially, V (z 1 ) < 0 for z 1 close to z 1 .It follows that V (z) has at least one root z * 1 ∈ (0, z 1 ).Let z * 2 = (τ )n(z * 1 )/d 2 , z * 3 and z * 4 be the unique positive solutions of respectively.Hence, the existence of positive homogeneous steady state is proved.
In order to prove the uniqueness, we consider the following model 2) The set of positive homogeneous steady states of model (6.1) is the same as the set of positive equilibria of model (6.2).Set J (s) = s − 1 − ln s ≥ J (1) = 0, for any s > 0. Let where We calculate the time derivative of W(t) along the solution of model (6.2): Therefore, .
Note that n is decreasing, we get By (H3), we have Moreover, since f i > 0 and f i ≤ 0 (i = 1, 2, 3), we have These together with the nonnegativity of J , we obtain dW dt ≤ 0, and the largest invariant set of dW dt (z) = 0 is a singleton {z * }.It follows from LaSalle invariance principle that z * is globally attractive, which indicates that it is the unique positive homogeneous steady state of model (6.1).The proof is complete.Theorem 6.2.If Case 1: R e1 ≥ 1 and R e2 ≥ 1, or Case 2: max {R e1 , R e2 } < 1 and R 0 > 1 is satisfied, then the positive homogeneous steady state of model (6.1) is globally asymptotically stable.
Proof.We divide the proof into two claims.
Claim 1: The positive homogeneous steady state z * is globally attractive.

Concluding remarks
In this paper, we proposed a generalized cholera model with nonlocal time delay to study the impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment.Main features of our model system are summarized as follows: (i) we simultaneously considered the intrinsic growth of HI and LI state of V. cholerae; (ii) some generally functional response functions, non-uniformness of diffusion rates and nonlocal time delay of the model system are incorporated.
We derived three basic reproduction numbers for HI state of V. cholerae (R e1 ), LI state of V. cholerae (R e2 ) and cholera disease in the host population (R 0 ).The impact of the diffusion of infectious hosts on cholera dynamics was discussed under some conditions.It is shown that R 0 is a decreasing function in D 2 when D 2 , β z 3 /μ and β z 4 /σ μ are constant functions.
Furthermore, the detailed classifications of spatial dynamics for the proposed model were investigated: (i) when max {R e1 , R e2 } < 1 and R 0 ≤ 1, the infection-free steady state is globally asymptotically stable; (ii) when Case 1: R e1 ≥ 1 or R e2 ≥ 1, or Case 2: max {R e1 , R e2 } < 1 and R 0 > 1 is satisfied, the disease will persist and there exists at least one endemic steady state; (iii) when Case 1: R e1 ≥ 1 and R e2 ≥ 1, or Case 2: max {R e1 , R e2 } < 1 and R 0 > 1 is satisfied, the unique positive homogeneous steady state is globally asymptotically stable.In general, in a heterogeneous environment, the analysis of global stability of the infection steady state is highly challenging.When the diffusion coefficient of HI state of V. cholerae (D 3 ) is a positive constant independent of x, and the nonlocal time delay and the spatial diffusion of susceptible hosts (D 1 ), infectious hosts (D 2 ) and LI state of V. cholerae (D 4 ) are not considered, we obtained the global attractivity of the infectious steady state.Namely, assume that (H5) and (H6) hold, if R e1 < 1 < R 0 , then the infection steady state of model (5.1) is globally attractive.
The innovation of mathematical results lies in (i) the global dynamics is determined by three basic reproduction numbers; (ii) the global attractiveness of a simplified model in a heterogeneous environment is proved by constructing appropriate Lyapunov functional; (iii) our analytical approach works for general functional response functions and thus can be applied to specific waterborne disease models, such as those described in [38,52,61,62].
One of the further steps to take with this model is to study the global stability of the infection steady state in a heterogeneous environment when D 1 > 0, D 2 > 0, and D 4 > 0. Meanwhile, the model system can be extended to incorporate other important epidemiological features, such as seasonality [27,55], human behavior [56], bacteriophage [5,24] and immunological threshold (a minimum dose of bacteria is required to yield an infection) [21,25,27].

0 Ξ
D i (x, a I ) = D E (x), for x ∈ and a I ≤ τ, D I (x), for x ∈ and a I > τ, d(x, a I ) = d E (x), for x ∈ and a I ≤ τ, d I (x), for x ∈ and a I > τ, where D E , D I , d E and d I are continuous and positive on ¯ .Denote E(x, t) = τ (x, t, a I )da I , I (x,t) = ∞ τ Ξ(x, t, a I )da I .

Lemma 4 . 1 .
Let F and V be given as in(4.2).The spectral bound s(e −λτ F − V ) is a continuous and decreasing function of λ.Let λ * ∈ R be the unique solution of λ * = s(e −λ * τ F − V ), we get λ * = ω(U).Moreover, λ * is the principal eigenvalue of e −λ * τ F − V with positive eigenfunction and it has the same sign as s(F − V ).

( 4 . 4 )
with Neumann boundary condition.Multiplying the second equation of model (4.3) and the first equation of model (4.4) by ϕ 3 and ϕ 0 , respectively, one gets