{'fraction_non_alphanumeric': 0.05649346383761414, 'fraction_numerical': 0.01524009722130986, 'mean_word_length': 4.364341085271318, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
THRESHOLD DYNAMICS OF A DELAYED NONLOCAL REACTION-DIFFUSION CHOLERA MODEL

. Taking account of spatial heterogeneity, latency in infected individuals, and time for shed bacteria to the aquatic environment, we build a delayed nonlocal reaction-diﬀusion cholera model. A feature of this model is that the incidences are of general nonlinear forms. By using the theories of monotone dynamical systems and uniform persistence, we obtain a threshold dynamics determined by the basic reproduction number R 0 . Roughly speak- ing, the cholera will die out if R 0 < 1 while it persists if R 0 > 1. Moreover, we derive the explicit formulae of R 0 for two concrete situations.


1.
Introduction. Cholera is an infectious disease that causes severe watery diarrhea, which can lead to dehydration and even death if untreated. It is caused by the bacterium Vibrio cholerae. According to the World Health Organization (WHO), cholera persists as a global threat and continues to challenge countries where access to safe drinking water and adequate sanitation is not assured. In fact, the WHO estimates 1.3 to 4.0 million cholera cases per year with 21,000 to 143,000 deaths worldwide [28]. Mathematical modeling has been playing a central role in understanding the transmission dynamics of cholera and the obtained theoretical results can help public agencies to draw appropriate control measures.
Like other waterborne diseases, cholera has multiple transmission pathways: it can be transmitted directly to humans by person-to-person contact or indirectly to humans via contaminated water. In [23], Tien and Earn proposed a model described by ordinary differential equations, which extends the classical SIR framework by adding a compartment (W) to track the pathogen concentration in water. The obtained SIRW model incorporates both direct and indirect transmissions. For this model, they not only derived fundamental quantities such as the basic reproduction number and the final outbreak size but also examined the dependence of them on transmission parameters for different pathways. In order to study differential infectivity in direct transmission, they also generalized the SIRW model to a staged progress model. However, differential infectivity is also crucial for indirect transmission. For example, laboratory studies suggest hyperinfectivity of freshly shed pathogen [9]. Though one can add compartments for multiple stages of infectious individuals and multiple infection stages of pathogen to consider these features, it is more realistic to use continuous age structures. In [3], Brauer et al. formulated the following partial differential equation (PDE) cholera model that incorporates simultaneously the age-of-infection structure of individuals and the age structure of pathogen with infectivities given by kernel functions, and initial conditions S(0) = S 0 > 0, i(0, ·) = i 0 (·) ∈ L 1 + (0, ∞), p(0, ·) = p 0 (·) ∈ L 1 + (0, ∞). Here S(t) denotes the number of susceptible individuals at time t ≥ 0, i(t, a) denotes the number of infected individuals of infection age a ≥ 0 at time t ≥ 0, and p(t, b) denotes the quantity of pathogen of age b ≥ 0 at time t ≥ 0 in the contaminated water. We refer to [3] for the meanings of the parameters. A threshold dynamics determined by the basic reproduction number is proved by constructing suitable Lyapunov functionals. Then Lin et al. [11,12] established same results for cases with saturation incidences and with both saturation incidences and imperfect vaccination, respectively.
Besides temporal heterogeneity, another heterogeneity affecting cholera dynamics is spatial heterogeneity. Though lots have been done in recent years on cholera modeling and analysis (see, for example, the related references in [25,27]), spatial heterogeneity has been rarely considered [2,4,5,16,25,27,32,33]. Among them, patch models and PDE models are used in [2,5,16] and [4,25,27,32,33], respectively. In particular, Wang and Wang [25] proposed the following PDE cholera model, One feature of model (1.2) is that intrinsic bacterial dynamics is incorporated and another is that general transmission rates f 1 (I) and f 2 (B) are employed. They investigated the effect of diffusive spatial spread on the disease spread and found that diffusive spatial spread does not produce a Turing instability to some extent.
It should be mentioned that, in applications, the domain is not necessarily to be one-dimensional. This motivates us to consider a PDE cholera model in a general bounded spatial domain. Another motivation comes from the fact that the parameters of PDE models for infectious diseases generally are space-dependent (see [1,10,24,30,31] as examples of some recent publications). However, in the above mentioned PDE cholera models, only Wang [27] considered the case where the transmission rates are dependent on both time and positions. As discussed above, age structure and spatial heterogeneity are important characteristics in modeling cholera. To the best of our knowledge, so far there is no work on cholera to incorporate both. In this paper, we modify (1.1) to incorporate the latent period and diffusion to capture the movement of human hosts and bacteria in a heterogeneous environment. We first derive the model in Section 2. Due to the assumption that parameters are space-dependent, we build a delayed reaction-diffusion cholera model with nonlocal effects, which are caused by the mobility of the latent individuals and bacteria. As far as we know, this is the first time a delayed nonlocal reaction-diffusion cholera model has been built. Then we analyze the resulting model in Section 3. The main result is a threshold dynamics, which is determined by the principal eigenvalue of a linear scalar non-local reactiondiffusion equation. Roughly speaking, if the principal eigenvalue is less than 0 then the disease-free steady state is globally attractive while if the principal eigenvalue is larger than 0 then the model is uniformly persistent. It is shown in Section 4 that the principal eigenvalue has the same sign as that of R 0 − 1. Here R 0 is the basic reproduction number, which is defined as the spectral radius of the next infection operator. We further find the explicit expressions of R 0 for two special cases with the bilinear incidence and Beddington-DeAngelis incidence. The paper ends with a brief conclusion.
2. Model formulation. To build the model, we assume that the population lives in a bounded but still connected region Ω. The population is divided into three disjoint compartments: susceptible, infected, and recovered. We assume that recovered individuals get permanent immunity and hence there is no need to consider the evolution of them. Let S(t, x) denote the number of susceptible individuals in location x at time t ≥ 0 and i(t, a, x) be the number of infected individuals of infection age a ≥ 0 in location x at time t ≥ 0. Moreover, p(t, b, x) denotes the concentration of Vibrio cholerae of biological age b ≥ 0 in location x at time t ≥ 0. Suppose that τ 1 is the average incubation period for infected individuals and τ 2 is the average time for shed bacteria to the aquatic environment, that is, an individual infected by the disease becomes infectious τ 1 time units after being infected while cholerae shed from infectious individuals can contact with susceptible individuals only τ 2 time units after being shed and hence they cannot infect susceptible individuals for b < τ 2 . Then are respectively the exposed individuals and infectious individuals in location x at time t, and are respectively the bacteria in location x at time t but not in the aquatic environment and in the aquatic environment.
We assume that there is a recruitment λ(x) of the population and all recruitment are susceptible. The susceptible diffuses at the rate D S (x) and has a natural death rate µ(x). A susceptible can be infected directly with person-person contacts and indirectly with bacteria with the incidences f 1 (x, S(t, x), i I (t, x)) and f 2 (x, S(t, x), p I (t, x)) at location x, respectively. Then S(t, x) is governed by the following reaction-diffusion equation, where ∇ is the gradient with respect to the spatial variable x. Employing the standard argument on age-structured population and spatial diffusion (see, for example, [15,6]), we get and Here D(a, x) and H(b, x) are the diffusion rates of the infected individuals and bacteria, respectively; µ i (x) and µ p (x) are their natural death rates, respectively; θ(a, x) and δ(b, x) represent respectively the recovery rate of infected individuals and the clearance rate of bacteria; and ξ(x) represents the shedding rate of an infectious individual at x. From the biological point of view, we consider a closed environment in the sense that the fluxes for the population and bacteria on the boundary of Ω, ∂Ω, are all zero, that is, we impose the following Neumann boundary conditions for S(t, x), i(t, a, x), p(t, b, x), For the ease of presentation, in the sequel, we assume for a ∈ (τ 1 , ∞], x ∈ Ω; Under these assumptions, we obtain a system of reaction-diffusion equations with nonlocal effects as follows. Integrating both sides of (2.2) from 0 to τ 1 and from τ 1 to ∞ yields and respectively. Similarly, and Biologically, we assume that i(t, ∞, x) = 0 and p(t, ∞, x) = 0. It then remains to determine i(t, τ 1 , x) and p(t, τ 2 , x), which are achieved below by applying the method of characteristics. For any r ≥ 0, consider solutions of (2.2) along the characteristic line t = a + r by letting v(r, a, x) = i(a + r, a, x). Then, for a ∈ [0, Regarding r as a parameter and solving (2.4) with (2.5) produce v(r, a, x) = Ω G 2 (a, x, y)[f 1 (y, S(r, y), i I (r, y)) + f 2 (y, S(r, y), p I (r, y))]dy, (2.6) where G 2 is the Green function of the operator ∇·[D E (·)∇]−(u i (·)+θ E (·)) associated with zero flux boundary condition. Evaluating (2.6) at a = τ 1 (hence r = t − τ 1 ) gives us Similarly we can get where G 3 is the Green function of the operator ∇·[H E (·)∇]−(u p (·)+δ E (·)) associated with zero flux boundary condition. , and also taking into consideration of (2.1), we obtain the following delayed nonlocal reaction-diffusion cholera model, Note that i E (t, x) and p E (t, x) are decoupled from the equations of S(t, x), i I (t, x), and p I (t, x). It suffices to consider the following delayed nonlocal reaction-diffusion system, For convenience, we denote Then in the sequel, we focus on the following system and the initial conditions To study the dynamics of (2.9), we impose the following assumptions on the parameter functions and incidences. (ii) f i is monotone non-decreasing and continuously differentiable with respect to the last two variables; For a continuous function φ on Ω, we denote

3.
A threshold dynamics. The purpose of this section is to establish the main result of this paper, a threshold dynamics. We start with the well-posedness of (2.9).
Let X = C(Ω, R 3 ) be the Banach space of all continuous functions from Ω to Then both (X, X + ) and (C τ , C + τ ) are strongly ordered Banach spaces. Given a continuous function u : , and ∇ · [D 3 (·)∇] − δ(·) respectively, subject to the zero flux boundary conditions. It is well-known that for any ϕ ∈ Y, t ≥ 0, respectively, subject to the zero flux boundary conditions. Moreover, for each t > 0,

.3]). Let
Then system (2.9) can be rewritten as the following integral equation, . Furthermore, it is easy to verify that for any φ ∈ C + τ and k ≥ 0, The above inequality implies that φ(0, x) + kF (φ)(x) ∈ C + τ when k is sufficiently small. Then it follows that Therefore, the following statement follows immediately from [14,Corollary 4]  Lemma 3.1. For every initial value function φ ∈ C + τ , system (2.9) admits a unique mild solution u(t, ·, φ) on its maximal interval of existence [0, and u(t, ·, φ) is a classical solution of (2.9) for all t > τ .
The following result plays an important role in the coming discussion. With the help of Lemma 3.2, we can obtain a solution semiflow of (2.9).
Theorem 3.1. For any φ ∈ C + τ , system (2.9) admits a unique solution u(t, ·, φ) on R + with u 0 = φ, and the solution semiflow Φ(t) = u t (·) : C + τ → C + τ , t ≥ 0, generated by (2.9) takes the following form which has a global compact attractor in C + τ . Proof. By way of contradiction, we assume that t φ < ∞. Then on [0, t φ ), we have Let ω(t, x) be the solution of On the other hand, by Lemma 3.2, system (3.3) admits a unique positive steady state ω * (x), which is globally asymptotically stable in Y + . Thus we can conclude that there is 1, 2, 3). Then by the results above, there exists a constant C > 0 such that or equivalently, where |Ω| is the volume of Ω. Similarly, with the assistance of (3.5), we can get . It follows that u 2 (t) + |Ω|u 1 (t − τ 1 ) and hence u 2 (t) is bounded on [0, t φ ). This proves the claim. This claim implies that there exists a positive number Q such that Again, it follows from Lemma 3.2 and the comparison principle that u 3 (t, ·, φ) is bounded on [0, t φ ).
The following results will be used in proving the persistence of (2.9).
where B is the positive constant specified in the proof of Theorem 3.1.
Then V(S(t)ψ) is the distribution of newly infectious individuals and shed bacteria at time t. It follows that represents the distribution of the total infectious individuals and shed bacteria during the infection period and hence L is the next infection operator. The spectral radius of L is defined as the basic reproduction number R 0 of (2.9), that is, Proof. The result follows from the general results in [21] and the same arguments in [26,Lemma 2.2].
Recall that V(x) is a matrix-valued function defined by and {S(t)} t≥0 : Y → Y is the solution semigroup associated with the following linear system: Let us define the following matrix-valued function We then see that  for any φ ∈ C + τ with φ i (0, ·) = 0 for i = 2, 3. Note that the basic reproduction number R 0 is defined in terms of the spectral radius of the operator L, which is inconvenient in application. As a result, we compute it in the case where all parameters except the diffusion rates are position independent, that is, we assume that λ(x) ≡ µ, µ(x) ≡ µ, θ(x) = θ, δ(x) ≡ δ, ξ(x) ≡ ξ, and µ i (x) + θ E (x) ≡ θ 1 , µ i (x) + δ E (x) ≡ δ 1 .

5.
Conclusion. In this paper, for the first time, we incorporated the spatial heterogeneity and temporal heterogeneity (for age structures in populations of individuals and bacteria) in the modeling of cholera spread. This results in a delayed reaction-diffusion model with nonlocal effects. Note that the incidences are in general nonlinear form. By employing the theories of monotone dynamical systems and persistence, we established a threshold dynamics completely determined by the principal eigenvalue of a linear nonlocal reaction-diffusion equation. In fact, the threshold dynamics is also characterized by the basic reproduction number, which is the spectral radius of the next infection operator. For the convenience of application, we also calculated the explicit formulae of the basic reproduction number in two particular cases of the incidences, bilinear and Beddington-DeAngelis ones.