key: cord-0561031-5m9xe5kq
authors: Yin, Hong-Ming
title: On a Reaction-Diffusion System Modeling Infectious Diseases Without Life-time Immunity
date: 2020-11-17
journal: nan
DOI: nan
sha: 9af592ceb3c8d1e2fcf2a66c548e072a6923b911
doc_id: 561031
cord_uid: 5m9xe5kq

In this paper we study a mathematical model for an infectious disease such as Cholera without life-time immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction-diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behavior of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving {em apriori} estimates. The analysis developed in this paper can be employed to study other epidemic models in biological and ecological systems.

In an ecological environment various infectious diseases in human or animals occur frequently (see [31] ). The current global Covid-19 pandemic is one such an example. Other recent examples include epidemics caused by the HIV, Cholera and Zika viruses. Scientists often use the well-known Susceptibility-Infection-Recovery (SIR) model ( [16] ) to describe how a virus spreads and evolves in future. The SIR model and its various extensions have been studied by many scientists ( see [1, 2, 8, 12, 17, 19, 23] , for examples). A good review for the model written by Hethcote in 2000 can be seen in [11] . The most of these studies focus on understanding the complicated dynamics of interaction between different hosts and viruses. However, this SIR model and its extensions cannot include the mobility of human hosts around different geographical regions. In order to take the movement of human hosts into consideration, one must develop a new mathematical model which can reflect these factors (see [9, 15, 30] ). Toward this goal, considerable progress has been made for different types of infectious diseases. In particular, many researchers have studied the mathematical model for the epidemic caused by the Cholera virus (see [3, 8, 12, 13, 21, 23, 25, 26, 27] , etc.).

In this paper we consider the mathematical model for the cholera epidemic with diffusion processes. A novel feature for the model is that there is no life-time immunity. This leads to a coupled reaction-diffusion system with different diffusion coefficients. We begin by describing the model system recently studied in [21, 25, 27, 28] .

Let Ω be a bounded domain in R n with C 2 -boundary ∂Ω. Let Q T = Ω×(0, T ] for any T > 0. When T = ∞, we denote Q ∞ by Q. Let S(x, t), I(x, t) and R(x, t) represent, respectively, susceptible, infected and recovered human hosts. Let B(x, t) be the concentration of bacteria. Then by the population growth and the conservation laws we see that S, I, R and B satisfy the following reaction-diffusion system:

(1.1)

subject to the following initial and boundary conditions:

(∇ ν S, ∇ ν I, ∇ ν R, ∇ ν B) = 0, (x, t) ∈ ∂Ω × (0, T ], (1.5) (S(x, 0), I(x, 0), R(x, 0), B(x, 0)) = (S 0 (x), I 0 (x), R 0 (x), B 0 (x)), x ∈ Ω, (1.6) where ν represents the outward unit normal on ∂Ω, h(B) is a differential function with h(0) = 0, 0 ≤ h(B) ≤ 1.

A typical example used in [27] for h(B) is

For reference, we list various parameters in the model as in [25, 26] . We would like to point out that the analysis for the reaction-diffusion system (1.1)-(1.4) is often difficult since there is no comparison principle ( [14, 20] ). Some basic questions such as global existence and uniqueness for the system are extremely challenging since Eq.(2.1) and Eq.(2.2) contain some quadratic terms in the system ( [5, 6, 7] ). Many of the quantitative properties are still open. For example, from the physical point of view, the concentration, S(x, t), I(x, t), R(x, t) and B(x, t) must be nonnegative and bounded in Q T . However, it appears that there has been no rigorous proof in the previous research when the space dimension is greater than 1. Answering these open questions is one of the motivations for this paper. Moreover, with different diffusion coefficients in a reaction-diffusion system, the dynamics of a solution may be very different from that of an ODE system. The most striking example is the Turing phenomenon in which the solution of an ODE system is stable while the solution of the corresponding reaction-diffusion system is unstable when one diffusion coefficient is much larger than the other ( [10] ).

When the space dimension is equal to 1, the authors of [13, 19, 25, 27, 28] studied the system (1.1)-(1.6). The global well-posedness is established for the model. Moreover, some global dynamical analysis for the solution is carried out in these papers. However, when the spatial dimension n is greater than 1, the analysis becomes much more complicated. The purpose of this paper is to study the reaction-diffusion system (1.1)-(1.6) when the space dimension is greater than 1. Moreover, the diffusion coefficients in the model for susceptible, infected and recovered human hosts are different. Furthermore, the diffusion coefficients in our model allow to depend on space and time, which are certainly more realistic than in the previous model. This fact is observed from the recent Covid-19 pandemic in which the mobility for infected patients is close to zero due to required global quarantine and travel restriction. It is also clear that for flu-like viruses the rate of infection is much higher in the winter than in the summer.

In this paper we establish a global existence result for any space dimension n. Particularly, we prove that the solution is actually classical if the dimension is less than or equal to 3 with no restriction on all parameters. Hence, global well-posedness for the model problem (1.1)-(1.6) is established when the space dimension is less than or equal to 3. Moreover, under a condition on some parameters, we are able to prove that the solution is uniformly bounded and converges to the steady-state solution (global attractor, see [22] ) for any space dimension as time evolves. These results improve the previous research obtained by others where the well-posedness is proved when the space dimension is equal to 1 (see [19, 26, 27, 28] ). The main idea for establishing global existence is to derive various apriori estimates (see [30] ). Our analysis in this paper uses a lot of very delicate results for elliptic and parabolic equations ( [4, 14, 18, 29] ). Particularly, we use a subtle form of Galiardo-Nirenberg's inequality to derive a uniform bound in Q for the solution when the space dimension is less than or equal to 2 without any restriction on parameters. The global asymptotic analysis is based on accurate energy estimates for the solution of the system. The method developed in this paper can also be used to deal with different epidemic models caused by some viruses such as avian influenza for birds ( [24] ).

The paper is organized as follows. In Section 2 we first recall some basic function spaces and then state the main results. In Section 3, we prove the first part of the main results on global solvability of the system (1.1)-(1.6) (Theorem 2.1 and Theorem 2.2). The long-time behavior of the solution in the second part is proved in Section 4 (Theorem 2.3 and Theorem 2.4). In Section 5, we give some concluding remarks.

For reader's convenience, we recall some basic Sobolev spaces which are standard in dealing with elliptic and parabolic partial differential equations.

Let α ∈ (0, 1). We denote by C α, α 2 (Q T ) the Hölder space in which every function is Hölder continuous with respect to (x, t) with exponent (α, α 2 ) inQ T . Let p > 1 and V be a Banach space with norm || · || v , we denote

When V = L p (Ω), we simply use

Moreover, the L p (Q T )-norm is denoted by || · || p for simplicity.

Sobolev spaces W k p (Ω) and W k,l p (Q T ) are defined the same as in the classical books such as [10] and [18] .

Let

We first state the basic assumptions for the diffusion coefficients and known data. All other parameters in Eq.(1.1)-(1.4) are assumed to be positive automatically throughout this paper. Since there is no essential difference for the analysis in this paper, we set b k = 0 in Eq. H(2.1). Assume that d i (x, t) ∈ L ∞ (Ω × (0, ∞)) for all i. There exist two positive constants d 0 and D 0 such that

For brevity, we set

We use Z(x, t) = (u 1 , u 2 , u 3 , u 4 ) to be a vector function defined in Q. The right-hand sides of the equations (1.1), (1.2), (1.3) and (1.4) are denoted by f 1 (Z), f 2 (Z), f 3 (Z), f 4 (Z), respectively. With the new notation, the system (1.1)-(1.6) can be written as the following reaction-diffusion system:

subject to the initial and boundary conditions:

Let q > 1. We define a product space X equipped with the standard product norm:

The conjugate dual space of X is denoted by X * . Definition 2.1 We say Z(x, t) to be a weak solution to the problem (2.1)-

for any test function φ k , φ kt ∈ X * with φ k (x, T ) = 0 on Ω for all k = 1, 2, 3, 4. 

. A much better result can be proved when the space dimension is less than or equal to 2. With more regularity for the coefficients and other known functions, we can obtain the classical solution. 

With a certain condition for some parameters, we can prove that the solution exists globally and is uniformly bounded in Q for any dimension n.

Moreover, the weak solution is smooth, provide all coefficients d k and b(x, t), Z 0 (x) are smooth.

With the result of Theorem 2.3, we can establish the global asymptotic behavior of the solution.

is the steady-state solution for the following elliptic equation

That is (S ∞ (x), 0, 0, 0) is a global attractor. 

From the physical model of view, the concentration must be nonnegative. However, this fact has not been proved rigorously in the literature. Here we first prove this fact by using the strong maximum principle. The proof is rather technical.

Proof. First of all, we assume that all initial functions are positive overΩ. Namely, we assume that there exists a small number a 0 > 0 such that

By the continuity, we know that there is a number T 0 > 0 such that

If T * = ∞, then the conclusion holds. If T * < ∞, then at least one of quantities attains 0 at some point (x * , T * ) for some x * ∈Ω. Suppose

from the boundary condition we can use Hofp's lemma to conclude that x * must not be located on the boundary ∂Ω. On the other hand, the strong maximum principle implies that x * can not be located at an interior point of Ω. It follows that min

which is a contradiction. From Eq. (2.2), we note that

We can use the strong maximum principle to conclude that

which is a contradiction with the assumption. This implies that T * = ∞. Now when initial functions do not have a positive lower bound, for any ε > 0 we simply use (S 0 (x) + ε, I 0 (x) + ε, R 0 (x) + ε, B 0 (x) + ε) to replace the original initial vector function Z(x, 0). Then we know that the corresponding solution (u 1ε (x, t), u 2ε (x, t), u 3ε (x, t) and u 4ε (x, t) are positive in Q T for any T > 0. By taking the limit as ε → 0, we obtain the nonnegetivity of u k (x, t) in Q for k = 1, 2, 3, 4.

As in the standard analysis in deriving an apriori estimate for solution of a partial differential equation we may assume that the solution is smooth in Q T , A special attention is paid to be what a constant C depends precisely on known data in the derivation. We will denote by C, C 1 , C 2 · · · , etc.,the generic constants in the derivation. Those constants may be different from one line to the next as long as their dependence is the same. 

where C 1 and C 2 depends only on know data. In addition C 2 also depends on the upper bound of T . Proof: We take integration over Ω for Eq.(2.1) to Eq.(2.3) and then add up to see

where C 1 depends only on known data. Now we integrate of Ω for Eq.(2.1) again to find

It follows that

Next we take integration over Ω for Eq.(2.4) to obtain

Thus, we obtain the desired L 1 (Ω)-estimate for u 4 . Q.E.D. To derive more apriori estimates, we need to use a very delicate result from the theory of parabolic equations with measure data. The reader can find the proof from [18] for n ≤ 2 and [4] for n ≥ 3. Lemma 3.3. Let a(x, t) be a measurable function with 0 < a 0 ≤ a(x, t) ≤ a 1 < ∞ in Q and f (x, t) ∈ L ∞ (0, ∞; L 1 (Ω)). Let u(x, t) be a weak solution of the following parabolic equation

Then, (a) If n = 1, then u(x, t) is uniformly bounded in Q. Moreover, sup 0<t<∞ ||u|| ∞ ≤ C 3 ;

(b) If n = 2, then u(x, t) ∈ L p (Q) for any p ∈ (1, ∞). Moreover,

(c) If n ≥ 3, then for any p ∈ (1, n(n+2) n 2 −2 ) and q ∈ (1, n+2 n+1 ) the solution u(x, t) has the following regularity in Q T for any T > 0:

Moreover, there exists a constant C 5 such that

where C 3 , C 4 and C 5 depend on ||f || L ∞ (0,∞;L 1 (Ω)) , ||u 0 || W 1,q (Ω) , the upper bound of T and known data. In addition, C 5 also depends on p.

For n = 1 and n = 2, the conclusions of (a) and (b) are well-known from [18] . For n ≥ 3, the conclusion (c) is proved in [4] .

Q.E.D. 

where r and C 6 depends only on known data. Proof. We assume that n ≥ 3 since we already obtained a much stronger estimate for n ≤ 2 by Lemma 3.3. We derive the estimate by using the standard energy method. Indeed, let r > 0 to be chosen. We multiply Eq.(2.1) by u r 1 and integrate over Ω to obtain:

where at the final step we have used Young's inequality with a small parameter ε > 0. Now we choose r = q − 1 > 0 and ε = d 2 to obtain

which is uniformly bounded by Lemma 2.2 and 2.3 as long as r ∈ (0, n(n+2) n 2 −2 − 1). Thus, the proof of Lemma 3.4 is completed.

Q.E.D. With the estimates from previous lemmas, we are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. There are several ways to prove the existence of a weak solution for the system (2.1)-(2.6). For any ε > 0, we denote by χ(u) the Heaviside function. Set

and f 4 (Z) are the same as before. Now we consider the following approximated reaction-diffusion system:

subject to the same initial and boundary conditions (2.5)-(2.6). We claim that the above approximate system has a unique weak solution in V 4 2 L ∞ (Q T ) for every small ε > 0 and ||u 3 || L ∞ (Q T ) ≤ 1 ε . Indeed, for every sufficiently small ε > 0, if

subject to the same initial and boundary conditions. The maximum principle implies that

where C depends only on known data. From Eq.(3.2), by using the maximum principle we see that u 2 will be uniformly bounded in Q T with the bound which has the same dependency as u 1 . Thus, by Eq. (3.3) , we apply the maximum principle again to see that u 3 would be bounded by a constant C which depends only on known data. This is a contradiction, which implies that u 3 ≤ 1 ε in Q T for a sufficiently small ε.

(see [10] or [18] ). Moreover,

Furthermore, it is clear that all apriori estimates from Lemma 3.1 to 3.4 hold and these bounds are independent of ε. Note that Z εt ∈ L p ′ (0, T ; W 1,−p ′ (Ω)) with p ′ = 1+r r . By using the weak compactness of L 1+r (Q T ), we can extract a subsequence of ε if necessary, as ε → 0, that for k = 1, 2, 3, 4,

Moreover, by using Egorov's theorem, we see, as ε → 0,

On the other hand, since u 3 ∈ L ∞ (0, T ; W 1,p (Ω)), we see

is sufficiently small, provided that ε is chosen to be sufficiently small. It follows that for any test function φ(x, t) ∈ L 2 (0, T ;

Consequently, for all k we have, as ε → 0,

a.e. in Q T and strongly in L 1 (Q T ).

For any test function φ k ∈ X * with φ k ∈ L ∞ (Q T ) and φ kt ∈ X * , φ k (x, T ) = 0, we have for all k = 1, 2, 3, 4,

After taking limit as ε → 0, we obtain a weak solution Z(x, t) ∈ X. Moreover, by Lemma 3.4, we see u 3 ∈ W 2,1 p (Q T ). Q.E.D.

Proof of Corollary 2.1. Leq n ≤ 2. To see the uniform boundedness of Z(x, t) in Q, we note that u 3 (x, t) ∈ L ∞ (0, ∞; L 1+r (Ω)) for any r ∈ (0, 2n n+1 ). We apply a result of Lemma 2.6 from [6] that sup

subject to a homogeneous Neumann boundary condition and an initial condition. Hence, since u 3 is uniformly bounded in Q we see by using the maximum principle that

where C depends only on known data. To prove the uniform boundedness for u 2 , we use the same energy method as for u 1 to obtain

where we have used the uniform boundedness of u 1 and C depends only on known data. We recall a Gagliardo-Nirenberg's inequality:

where p, q, s and θ satisfy

For n = 2, we choose q = 2, s = 1, θ = 1 2 and p = 2,

If we choose ε = rd 0 (1+r) 2 , we immediately obtain

where

where C depends only on known data. As u 1 and u 2 are uniformly bounded, we see

where C depends only on known data. Consequently, for n = 2 we use Lemma 2.6 in [6] to have

where C depends only on known data. Once u 2 is uniformly bounded in Q, from Eq.(2.4) we can apply the maximum principle to obtain

where C depends only on known data.

Moreover, when f k (Z) is bounded for k = 1, 2, 3, 4,, then the Hölder continuity of the weak solution Z(x, t) directly comes from the standard DiGorgi-Nash's estimate for parabolic equations ( [18] ). The uniqueness of weak solution for n ≤ 2 is straight forward by the energy method since u k for all k is bounded in Q. Hence, the proof of Corollary 2.1 is completed.

Q.E.D. To prove Theorem 2.2, we need additional regularity conditions for d 3 (x, t) and other known data. In the rest of this section we assume H(2.4) holds and n = 3. t) ), i = 1, 2, · · · n. 

where C 6 depends only on known data.

Proof. First of all, we note that for any p ∈ (1, n+2 n+1 )

where C depends on known data and upper bound of T . By W 2,1 p (Q T )-estimate, we obtain

We first take derivative for Eq.(2.3) with respect to x k to see that U 3k satisfies

Since U 2k ∈ L q (Q T ) and u 3 ∈ W 2,1 p (Q T ) by Lemma 3.4, we see

where C depends only on known data. Hence, we immediately obtain the interior W 2,1 p (Q T )-estimate for U 3k ( [29] ). Namely, for any Ω 0 ⊂ Ω and Q 0T = Ω 0 × (0, T ], there exists a constant C such that

where C depends only on the known constants and the distance Ω 0 and ∂Ω.

To derive the global W 2,1 p -estimate for U 3k , we first assume that x 0 ∈ ∂Ω and ∂Ω is flat near x 0 . Let B r (x 0 ) be a small ball centered at x 0 with radius r and 

It follows that, for k = 1, 2, · · · , n − 1,

That is, for k = 1, 2, · · · , n − 1,

Moreover, for k = n we see

Now we can apply the global W 2,1 p -estimate for U 3k for k = 1, 2, · · · , n to obtain

where C depends only on known data, r and the upper bound of T . When ∂Ω is not flat near x 0 , since ∂Ω ∈ C 2+α we use the standard transformation to convert Γ 0 to be a flat boundary near x 0 in a new coordinate and then use the same argument as above to obtain the desired W 2,1 q -estimate. We shall skip the step. The reader can find the detailed calculation for a general elliptic equation in [10] . Finally, since ∂Ω is compact, after using a finite number of covering we can obtain the W 2,1 q -estimate near ∂Ω. Thus, the proof for Lemma 3.5 is completed.

Q.E.D. By using the standard embedding theorem for Sobolev spaces (see [10] ), we immediately obtain the following consequence. Corollary 3.6. Let n ≤ 3. There exists a constant C 7 such that

where q is the number in Lemma 3.5, α = n(q−1) n+2 ∈ (0, 1) and C 7 depends only on known data.

Lemma 3.7. Let n ≤ 3. There exists a constant C 8 such that

where C 8 depends only on known data. Proof: Note that u 1 (x, t), u 2 (x, t) ≥ 0. It follows that

Since u 3 is bounded in Q T by Corollary 3.6, we apply the standard the comparison principle ( [18] ) to obtain

where C 8 depends only on known data and the upper bound of T .

Lemma 3.8. Let n ≤ 3. There exists a constant C 9 such that

where C 9 depends only on known data. Proof: Since u 1 is uniformly bounded in Q T and 0 ≤ h(B) ≤ 1, we apply the standard the maximum principle to conclude that u 2 is uniformly bounded:

which is uniformly bounded by Lemma 3.7. When u 2 is uniformly bounded, we can use the maximum principle method to obtain an L ∞ (Q T )-bound for u 4 .

Q.E.D. With the L ∞ -bounds for u k , k = 1, 2, 3, 4 in hand, we can use DiGiorgi-Nash's estimate to obtain the following estimate in Hölder space ( [18] ). 

where C 10 depends only on known data. Q.E.D.

Proof of Theorem 2.2: With the apriori estimate in Lemma 3.9, we can prove the global existence by using either a bootstrap method ( [30] ) or a fixed-point method. We use Leray-Schauder's fixed point theorem. Choose a Banach space Y = L ∞ (Q T ). For any (λ, J) ∈ [0, 1] × Y, we consider the following reaction-diffusion system:

subject to the following initial and boundary conditions:

For every J(x, t) ∈ Y and λ ∈ (0, 1], under the assumption H(2.1)-(2.4) the standard theory for parabolic equation ( [18] ) implies that the system (3.5)-(3.11) has a unique solution

Note that M 0 [J] = 0 and a fixed-point of M 1 along with u * 1 , u * 3 , u * 4 forms the solution of the original system (2.1)-(2.6). It is a routine to show that M λ is a continuous mapping from Y into Y . Moreover, since the embedding operator from C α, α 2 (Q T ) into Y is compact, the estimate in Lemma 3.9 implied that M λ is a compact mapping from Y into Y . Furthermore, it is clear that all estimates from Lemma 3.6-3.9 hold for all fixed-points of M λ [u 2 ] = u 2 and any λ ∈ [0, 1]. By applying Leray-Schauder's fixed-point theorem, we see that M λ has a fixed-point. Particularly, when λ = 1 the fixed point along with u 1 , u 3 and u 4 forms a solution of the problem (2.1)-(2.6). The uniqueness is obvious since the solution is uniformly bounded in Q T .

Q.E.D.

In this section we study the global asymptotic behavior of the solution and prove Theorem 2.3 and Theorem 2.4. In order to see a clear physical meaning for all species, we use the original variables S, I, R, B instead of u 1 , u 2 , u 3 , u 4 in this section.

Proof of Theorem 2.3: Keep in mind that we shall always derive an estimate for S, I and R as a first step. The second step for B will be easy once we have the estimate for I by Eq. (1.4) . We multiply Eq.(1.1) by S and integrate over Ω to obtain

where at the final step, we have used Cauchy-Schwarz's inequality with a small parameter ε > 0. Similarly, we mutiply Eq.(1.2) by I and integrate of Ω to obtain

where we have used the fact 0 ≤ h(B) ≤ 1. We perform the same calculation for Eq.(1.3) to obtain

Now we choose ε = 1 and add up the above estimates to obtain

We see

The above estimate is good enough to derive the L ∞ -bound in Q T for any fixed T > 0 by using an iteration technique and Sobolev embedding. Indeed, we use the same technique as in Lemma 3.5 in section 3 to obtain that R x i ∈ W 2,1 p 0 (Q T ) for all i = 1, · · · , n, with p 0 > 2 and

where C depends only on known data and the upper bound of T .

By the Sobolev embedding, we see R(x, t) ∈ L q 0 (Q T with q 0 = (n+2)p 0 n+2−4p 0 and

Hence, the energy estimate yields that S(x, t) ∈ L q 0 (Q T ) and

||S|| q 0 ,Q T ≤ C.

Next, we take derivative with respect to x i for Eq.(1.1)-(1.2) and follow the same calculations as for R, we are able to obtain I i ∈ L p 0 (Q T ) for all i and

where C depends only on known data and T . Since all coefficients d i and b(x, t), Z(, 0) are smooth, after a finite number of l-steps, we conclude that R ∈ W 2l,l p (Q T ) and

where C depends only on known data.

Consequently, if l ≥ n+2 2 Sobolev's embedding yields

where C depends only on known data and T . With the above L ∞ (Q T )-bound for R(x, t), we immediately obtain an L ∞ (Q T ) bound for S(x, t) from Eq.(1.1) by applying the maximum principle. Then from Eq.(1.2), the same maximum principle yields an L ∞ (Q T )-bound for I. Finally, from Eq.(1.4) we obtain an L ∞ (Q T ) of B(x, t) by applying a comparison principle. Now for we use Lemma 2.6 to obtain

provided that n ≤ 4, where C depends only on known data. The uniqueness of the weak solution can be proved easily since the weak solution is bounded.

Q.E.D. With the global bound for Z(x, t) we are ready to prove the asymptotic behavior of the solution for the system (1.1)-(1.6).

Proof of Theorem 2.4: First of all, for we use Lemma 2.6 to obtain

provided that n ≤ 4, where C depends only on known data.

Let S ∞ (x) be the following steady-state solution of the elliptic equation:

Obviously, the elliptic problem (4.1)-(4.2) has a unique solution ( [10] ). Moreover, since d is positive, we can apply the maximum principle to see that S ∞ (x) is uniformly bounded in Ω. Furthermore, by Campanato estimate (see [29] ) we have

where C depends only on known data.

S * (x, t) = S(x, t) − S ∞ (x), (I * (x, t), R * (x, t), B * (x, t)) = (I(x, t), R(x, t), B(x, t)), (x, t) ∈ Q.

We also define

Define

For I * , R * and B * , they satisfy the same equations as I, R, B. We multiply Eq.(4.3) by S * and integrate over Ω to obtain

where m 0 = sup Ω |∇S ∞ (x)|. We use Cauchy-Schwarz's inequality with a small parameter ε to obtain

By choosing ε = d 0 2m 0 , we see

where ε 0 > 0 is any small number and C(ε 0 ) depends only on known data and ε 0 . On the other hand, note that I * (x, t), R * (x, t) and B * (x, t) satisfy the same equations as I, R and B. If we perform the same energy estimate, then we have the same estimate as I, R and B except additional terms involving d * k : d * 2 (∇I * · ∇I), d * 3 (∇R * · ∇R).

we follow exactly the same method as for J 1 (t) to obtain the estimates similar to the one in the proof of Theorem 2.3. For these additional terms, we use the same argument as for S * to obtain Ω |d * 2 ∇I * · ∇I|dx ≤ ε Ω |∇I * | 2 sdxx + C(ε)

where ε > 0 is an arbitrary small constant. We choose ε = d 0 2 . Let

We conclude

where g 0 = σ + β 1 + β 2 + γ 4 and C depends only on known data.

We choose ε 0 = d − g 0 2 > 0.

Then |d * k (x, τ )| 2 + |b * (x, τ )| 2 ]dx.

By the assumption H(2.5), we obtain lim t→∞ J(t) = 0.

Once we know I(x, t) goes to 0 in L 2 (Ω)-sense as t → ∞, we can use the same argument for J 4 (t) (see [22] ) to obtain lim t→∞ J 4 (t) = 0.

By the definition of a global attractor (see [22] ), we see that Z ∞ (x) = (S ∞ (x), 0, 0, 0) is a global attractor. Thus, the proof of Theorem 2.4 is completed. Q.E.D.

In this paper we studied a mathematical model for the Cholera epidemic without life-time immunity. The model equations are governed by a coupled reaction-diffusion system with different diffusion coefficients for each species. We established the global well-posedness for the coupled reaction-diffusion system under a certain condition on known data. Moreover, the long-time behavior of the solution is obtained for any space dimension n. Particularly, we prove that there is a global attractor for the system under appropriate conditions on known data. These results justify the mathematical model and provide scientists a deeper understanding of the dynamics of interaction between bacteria and susceptible, infected and recovered human hosts. The mathematical model with the help of real data analysis provides a scientific foundation for policy-makers to make better decisions for the general public in health and medical sciences. The main tools used in this paper come from some delicate theories for elliptic and parabolic equations. The method developed in this paper can be used to study other models such as the avian influenza for birds.

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Population biology of infectious disease II

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Nonlinear elliptic and parabolic equations involving measure data

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Global regularity of solutions to systems of reaction-diffusion with sub-quadratic growth in any space dimension

Reproduction numbers and sub-threshold endemic equilibra for compartmental models of disease transmission

A Cholera model in a patchy environment with water and human movement

Partial Differential Equations

The mathematics of infectious diseases

Global dynamics of a SEIR model with varying total population size

Stability analysis and application of a mathematical cholera model

Second-Order Parabolic Differential Equations

Diffusion, self-diffusion and cross-diffusion

A Contribution to the Mathematical Theory of Epidemics

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Linear and Quasilinear Equations of Parabolic Type

Global stability of infectious disease models using lyapunov functions

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This paper was motivated by some open questions raised by Professor K. Yamazaki in a seminar at Washington State University. The author would like to thank Professor Yamazaki and Professor X. Wang for some interesting discussions. Many thanks also go to Mr. Brian Yin, Esq., from Law Firm Clifford Chance US LLP, who helped to edit the original paper.