Traveling wave solutions for epidemic cholera model with disease-related death.

Based on Codeço's cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speed c (∗) is given. To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder's fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.

1. Introduction

In the past and at present, cholera has been a serious threat to human health, which is an acute, diarrheal illness caused by infection of the intestine with the bacterium Vibrio cholera. An estimated 3–5 million cases and over 100,000 deaths occur each year around the world [1]. The cholera bacterium is usually found in water or food sources that have been contaminated by feces from a person infected with cholera. Cholera is most likely to be found and spread in places with inadequate water treatment, poor sanitation, and inadequate hygiene. Therefore, cholera outbreaks have been occurring in developing countries—for example, Iraq (2007-2008), Guinea Bissau (2008), Zimbabwe (2008-2009), Haiti (2010), Democratic Republic of Congo (2011-2012), and Sierra Leone (2012) [2].

Many mathematical models were proposed to understand the propagation mechanism of cholera, the earlier one of which was established by Capasso and Paveri-Fontana [3] to study the 1973 cholera epidemic in the Mediterranean region as follows: where B(t) and I(t) denote the concentrations of the pathogen and the infective populations, respectively. In addition, Codeço [4] investigated the role of the aquatic pathogen in dynamics of cholera through the following susceptible-infective-pathogen mode: where S(t) is the susceptible individuals. In this model, human is divided into two groups: the susceptible and the infective. As pointed out in [4-8], bacterium Vibrio cholera can spread by direct human-to-human and indirect environment-to-human modes. To understand the complex dynamics of cholera, model (2) is extended by [6, 9–15], and so forth.

In all previous models, the influences of space distribution of human on the transmission of cholera are omitted. Cholera usually spreads in spacial wave [16]. Cholera bacteria live in rivers and interact with the plankton on the surface of the water [17]. When individuals drink contaminated water and are infected, then they will release cholera bacteria through excretion [18]. Capasso et al. [19-22] developed model (1) by incorporating the bacterium diffusion in a bounded area and studied the existence and stability of solutions. To deeply investigate the interaction of transmission modes and bacterium diffusion, Bertuzzo et al. [23, 24] incorporated patchy structure into model (2) and supposed that the pathogen in water could diffuse among these patches. Furthermore, Mari et al. [25] studied the influence of diffusion of both human and pathogen on cholera dynamics through a patchy model.

Infectious case usually is found firstly at some location and then spreads to other areas. Consequently, the most important question for cholera is as follows: what is the spreading speed of cholera? However, the above spacial models mainly focus on the stability of solutions, not the spreading speed. Traveling wave solution is an important tool used to study the spreading speed of infectious diseases [26-28]. Based on Capasso's model (1), Zhao and Wang [29], Xu and Zhao [30], Jin and Zhao [31], and Hsu and Yang [32] studied the influences of pathogen diffusion on the spread speed of cholera.

In above diffusive cholera models, diffusion of aquatic pathogen is neglected. In this paper, we investigate the effects of the disease-related death and aquatic pathogen in cholera epidemic dynamics by developing model (2). Based on model (2) and ignoring natural birth and death, a general diffusive epidemic cholera model incorporating the disease-related death and aquatic pathogen dynamics can be formulated as the following reaction-diffusion system: where S = S(x, t) and I = I(x, t) denote the concentrations of susceptible and infected individuals, respectively, and B = B(x, t) is the concentration of the infectious agents. δ is the disease-related death rate, γ denotes the contribution of each infected person to the concentration of cholera, and m is the net death rate of the vibrio. f(B) is the environment-to-human transmission incidence. Similar to [15], we assume that f(B) satisfiesFrom hypothesis (A1), we have f(B) ≤ f′(0)B.

f(0) = 0, lim⁡ f(B) < +∞, f′(B) > 0, −M 0 ≤ f′′(B) ≤ 0 for B ≥ 0.

In this paper, we study the traveling wave solutions of model (3). The formula for minimal wave speed c* is given. To prove the existence of traveling wave solutions for c > c*, an invariant cone is constructed and Schauder's fixed point theorem is introduced. Schauder's fixed point theorem is applied widely to prove the existence of traveling wave solutions (e.g., [26, 33, 34]). However, unlike Wang and Wu [34], the cone in our paper is bounded. Motivated by [34-37], we prove the nonexistence of traveling wave solutions for c < c* by two-sided Laplace transform, which was firstly introduced to prove the nonexistence of traveling wave solutions by Carr and Chmaj [37] and then was applied by [34-36]. To apply two-sided Laplace transform, the exponential decrease of traveling wave solutions is needed, which is proved in [34] by analysis method. However, it cannot be applied to our model due to the nonlinearity of cholera incidence. Therefore, in this paper, a new method is proposed to get the exponential decrease of traveling wave solutions, which is inspired by the proof of Stable Manifold Theorem in [38]. In addition, our method can be applied to reaction-diffusion systems consisting of more than three equations.

This paper is organized as follows. Section 2 is focused on the existence of traveling wave solutions. Firstly, the existence of traveling wave solutions for original system is proved to be equivalent to that of a new simple system. Then, two pairs of upper and lower solutions are constructed to get an invariant cone and Schauder's fixed point theorem is applied for new system. Section 3 is devoted to the nonexistence of traveling wave solutions. For this aim, a new method is proposed to show the exponential decrease of traveling wave solutions and two-sided Laplace transform is used.

2. Existence of Traveling Wave Solutions

For convenience in discussing the model, we introduce dimensionless variables and parameters. Setting we obtain where Obviously, g(u 3) also satisfies assumption (A1) with M 0 being replaced by a new constant .

A traveling wave solution of system (5) is a nonnegative nontrivial solution of the form satisfying boundary condition where U 0 > U 1 ≥ 0.

Before giving the main theorem, we introduce the equation for minimal wave speed: where The main result of this section is given as follows.

Theorem 1

Suppose U 0 > 1/g′(0). Then, there exists a positive constant c* which is the only positive root of (9). For any >c*, system (5) has a traveling wave solution (U(y + cτ), W(y + cτ), V(y + cτ)) satisfying boundary condition (8) such that U(ξ) is nonincreasing in R. Furthermore, one has that

To study the existence of traveling wave solutions, using constant variation method in the second equation of (5) gives Then, system (5) changes into

Lemma 2

(U(y + cτ), V(y + cτ)) is a traveling wave solution of system (13) satisfying boundary condition if and only if (U(y + cτ), W(y + cτ), V(y + cτ)) is a traveling wave solution of system (5) satisfying boundary condition (8), where

Proof

Assume (U(y + cτ), V(y + cτ)) is a traveling wave solution of system (13) satisfying boundary condition (14). Obviously, (U(y + cτ), W(y + cτ), V(y + cτ)) is a solution of system (5). To prove the necessity, it is enough to show that W(+∞) = W(−∞) = 0. Consider where the third equality is due to L'Hopital principal. The sufficiency is clear and is omitted. The proof is completed.

From Lemma 2, we only need to study traveling wave solutions of (13) satisfying boundary condition (14). Substituting traveling profile into system (13) yields the following equations: where ′ denotes the derivative with respect to ξ.

In the following, we will use Schauder's fixed point theorem to prove the existence of traveling wave solutions. To achieve this goal, we firstly linearize the second equation of (18) at (U 0, 0) and obtain Substituting ϕ(ξ) = e v into (19), we get the characteristic equation that is, where a 0 = κ[g′(0)U 0 − 1]/c, a 1 = −(1 + κ) < 0, and a 2 = (κ − c 2)/c. To investigate distribution of roots of (21), denote and introduce the following lemma [39].

Lemma 3

(a) If Δ0 > 0, (21) has one real root and two nonreal complex conjugate roots.

(b) If Δ0 = 0, (21) has a multiple root and all its roots are real.

(c) If Δ0 < 0, (21) has three distinct real roots.

Then, we get the following lemma about the distribution of eigenvalues.

Lemma 4

Assume U 0 > 1/g′(0). Then, there exists a constant c* > 0 which is the only positive root of (9) such that

if 0 < c < c*, (21) has a negative real root and two nonreal complex conjugate roots with positive real parts;

if c = c*, (21) has a negative real root and a positive real multiple root;

if c > c*, (21) has a negative real root and two different positive real roots;

one can assume c > c* and let λ 1 < λ 2 be the two positive roots of (21); then H(λ 1 + ε) < 0 if 0 < ε < λ 2 − λ 1.

Obviously, b 3 < 0, b 2 < 0, and b 0 > 0. By Descartes' rule of signs, Δ(c) = 0 has only one positive root c* such that Δ < 0 for c > c* and Δ > 0 for 0 < c < c*. Direct calculations show that Δ0 = Δ/(108c 4). Since a 0 > 0 and a 1 < 0, Descartes' rule of signs shows that (21) has only one negative real root and Routh-Hurwitz criterion indicates that (21) has roots with positive real parts. Then, the combination of Lemma 3 and above analysis completes the proof of (a)–(c). (d) is obviously true, since H(λ) is a cubic polynomial.

In this section, we always suppose U 0 > 1/g′(0) and c > c* unless other conditions are specified. Denote λ 1 < λ 2 to be the two positive roots of (21) and define where V 0 = JU 0, J = lim⁡ g(V).

Lemma 5

The function satisfies inequality for any ξ ≠ ln⁡V 0/λ 1.

Firstly, assume ξ < ln⁡V 0/λ 1 and, therefore, . Since satisfies (19) and g′′(V) ≤ 0 for any V ≥ 0, we have where 0 < θ < 1.

Secondly, set ξ > ln⁡V 0/λ 1, which implies . We have that The proof is completed.

Lemma 6

For α > 0 sufficiently small and σ > 0 sufficiently large, the function satisfies for any ξ ≠ 1/αln⁡(U 0/σ).

Let σ > 0 be sufficiently large to ensure 1/αln⁡(U 0/σ) < ln⁡V 0/λ 1. When ξ > 1/αln⁡(U 0/σ), then and the lemma is obviously true. Now, suppose ξ < 1/αln⁡(U 0/σ). Then, and where 0 < θ < 1. Let σ = 1/α. Since we can find α > 0 sufficiently small and σ > 0 sufficiently large such that Thus, the proof is completed.

Lemma 7

Let ε = min⁡{α, λ 1, λ 2 − λ 1}/2. Then, for M > 0 large enough, the function satisfies for any ξ ≠ 1/εln⁡(1/M).

It is clear that if and only if ξ = 1/αln⁡(U 0/σ), that if and only if ξ = 1/εln⁡(1/M), and that 1/εln⁡(1/M) < 1/αln⁡(U 0/σ) if and only if M > (σ/U 0)(. Let M > (σ/U 0)(. When ξ > 1/εln⁡(1/M), then e (1 − Me ) < 0, , and Lemma 7 holds.

In this paragraph, assume ξ < 1/εln⁡(1/M). Then, ξ < 1/αln⁡(U 0/σ), , and . To prove this lemma, it is enough to show where 0 < θ < 1. Since we only need to show Since ξ < 1/αln⁡(U 0/σ), then Since H(λ 1 + ε) < 0, inequality (34) satisfies if The proof is completed.

To apply Schauder's fixed point theorem, we will introduce a topology in C(R, R 2). Let μ be a positive constant which will be specified in the following. For Φ(ξ) = (ϕ 1(ξ), ϕ 2(ξ)), define We will find traveling wave solutions in the following profile set: Obviously, Γ is closed and convex in C(R, R 2). Firstly, we change system (18) into the following form: where β 1 > lim⁡ g(V), Suppose Λ1 < 0 < Λ2 are the two roots of equation Λ2 − cΛ − 1 = 0. Furthermore, define F = (F 1, F 2) : Γ → C(R, R 2) by In the remainder of this paper, it is always assumed that μ < min⁡{β 1/c, κ/c, −Λ1, Λ2}.

Lemma 8

Consider F(Γ) ⊂ Γ.

Let (U(·), V(·)) ∈ Γ; that is , . Then, we need to prove that

First of all, we have From Lemma 6 and system (39), we get where the second inequality is due to β 1 > lim⁡ g(V). Then, Therefore, we have proved .

Now, we study F 2(U(·), V(·))(ξ). If ξ ≥ ξ 0≜1/εln⁡(1/M), then , which implies that since , . Assume ξ < ξ 0. From Lemma 7 and system (39), it is clear that which implies where the final inequality is due to and . In conclusion, for any ξ ∈ R.

Similarly, we can show and the proof is completed.

Lemma 9

Map F = (F 1, F 2) : Γ → C(R, R 2) is continuous with respect to the norm |·| in B (R, R 2).

For Φ1(·) = (U 1(·), V 1(·)), Φ2(·) = (U 2(·), V 2(·)) ∈ Γ, we have where M 1 = β 1 + J + U 0 g′(0), and V* is between V 1(ξ) and V 2(ξ). Therefore, we have Then, Thus, when ξ ≤ 0, we get

When ξ > 0, it follows that Consequently, we have proved that for any ξ ∈ R, where That is, In conclusion, F 1 : Γ → C(R, R) is continuous with respect to the norm |·| in B (R, R).

In addition, consider F 2(U(ξ), V(ξ)). Firstly, we have where M 3 = (J + U 0 g′(0))κ/(κ − cμ). Therefore, If ξ < 0, it holds that If ξ ≥ 0, we have Consequently, we conclude that where Thus, F 2 : Γ → C(R, R) is continuous with respect to the norm |·| in B (R, R). The proof is completed.

Lemma 10

Map F = (F 1, F 2) : Γ → Γ is compact with respect to the norm |·| in B (R, R 2).

For any Φ(ξ) = (U(ξ), V(ξ)) ∈ Γ, it is clear that Since |H 1(U, V)(ξ)| = |[β 1 − g(V(ξ))]U(ξ)| ≤ β 1 U 0, we have which implies that  |(d/dξ)F 1(Φ(·))(·)| < 2β 1 U 0/c. Since Φ(·) = (U(·), V(·)) ∈ Γ, we get Then, which implies that|(d/dξ)F 2(Φ(·))(·)| < 2(β 1 U 0 + V 0)/(Λ2 − Λ1). Consequently, |(d/dξ)F 1(Φ(·))(·)| and |(d/dξ)F 2(Φ(·))(·)| are bounded, which shows that F(Γ) is uniformly bounded and equicontinuous with respect to the norm |·| in B (R, R 2).

Furthermore, for any positive integer n, define Obviously, for fixed n, F (Γ) is uniformly bounded and equicontinuous with respect to the norm |·| in B (R, R 2), which implies that F : Γ → Γ is a compact operator. Since we have that Similarly, we can prove that |F 2 (U(·),V(·))(·)−F 2(U(·),V(·))(·)| → 0 when n → +∞. Thus, |F (U(·),V(·))(·)−F(U(·),V(·))(·)| → 0 when n → +∞. By Proposition 2.1 in Zeilder [40], we have that F converges to F in Γ with respect to the norm |·|. Consequently, F : Γ → Γ is compact with respect to the norm |·|. The proof is completed.

Proof of Theorem 1

Combination of Schauder's fixed point theorem and Lemmas 8, 9, and 10 shows that there exists a nonnegative traveling wave solution (U (·), V (·)) ∈ Γ such that (U (ξ), V (ξ)) → (U 0, 0) when ξ → −∞. By the definition of and , there is a ξ 0 < 0 such that and when ξ < ξ 0. Therefore, if ξ < ξ 0, we have that cU ′ = −g(V (ξ))U(ξ) < 0 which implies that U 0 > U 1 ≥ 0. It is clear that U (ξ) is nonincreasing in R.

From Lemma 2 and system (18), it follows that By the first and second equation of (69), we have Since the traveling wave solution (U (·), V (·)) ∈ Γ and is the fixed point of the operator F, thus (U (ξ), V (ξ)) = F(U (·), V (·))(ξ). The proof of Lemma 10 shows that V ′(ξ) = F 2′(U (·), V (·))(ξ) is bounded. Using L'Hopital principal shows that Integrating the third equation of (69) from −∞ to ξ yields Then, the boundedness of V (ξ) and V ′(ξ) implies ∫− + V (η)dη < +∞, which shows that lim⁡ V (ξ) = 0 and ∫− + W (η)dη = ∫− + V (η)dη = c(U 0 − U 1).

Next, we prove 0 ≤ V (ξ) ≤ U 0 − U 1, which is motivated by Wang and Wu [34]. Define It is clear that R(ξ) satisfies R(−∞) = 0, and R(+∞) = U 0 − U 1. Denote N(ξ) = V (ξ) + R(ξ), which satisfies Due to N′(+∞) = 0, it follows that which implies that N(ξ) is nondecreasing in R. Since N(−∞) = 0 and N(+∞) = U 0 − U 1, we have 0 ≤ V (ξ) ≤ U 0 − U 1 for all ξ ∈ R. The proof is completed.

3. Nonexistence of Traveling Wave Solutions

In this section, we give the conditions on which system (5) has no traveling wave solutions.

Theorem 11

(I) Assume U 0 > 1/g′(0). Then, for any 0 < c < c*, system (5) has no nonnegative nontrivial traveling wave solutions (U(y + cτ), W(y + cτ), V(y + cτ)) satisfying boundary condition (8).

(II) Suppose 0 < U 0 ≤ 1/g′(0). Then, for any c > 0, system (5) has no traveling wave solutions (U(y + cτ), W(y + cτ), V(y + cτ)) satisfying boundary condition (8).

To prove (I) by two-sided Laplace transform, the following lemma about the exponential decrease of traveling wave solutions is needed.

Lemma 12

Assume U 0 > 1/g′(0). If (U(y + cτ), W(y + cτ), V(y + cτ)) is a nonnegative nontrivial traveling wave solution of (5) satisfying boundary condition (8), there exists a positive constant α such that

Substituting traveling wave profile into the second and third equations of (5), it follows that Setting V′ = Z, we have Furthermore, system (80) can be written as where and . Obviously, . Since and g′(V) is bounded in [0, +∞), for any small constant ε > 0, there exists ξ 0 ∈ R such that for all ξ ≤ ξ 0.

Since (U(ξ), W(ξ), V(ξ)) is the solution of (79) if and only if (U(ξ − η), W(ξ − η), V(ξ − η)) is the solution of (79) for any η ∈ R and they satisfy the same boundary condition at ±∞, we can select large η such that (84) holds with ξ 0 = 0 and (U(ξ), W(ξ), V(ξ)) being replaced by (U(ξ − η), W(ξ − η), V(ξ − η)). Therefore, we suppose ξ 0 = 0.

Calculations show that the eigenfunction of A is H(λ) defined by (21). Then, by Lemma 4, there exists constant matrix C such that where λ 0 < 0 and the eigenvalues λ 1, λ 2 of 2 × 2 matrix Q have positive real parts. Setting ψ = Cφ, the system (81) then has the form Obviously, lim⁡ φ(ξ) = 0 and G(ξ, φ) = C −1 f(ξ, Cψ) satisfies (84). Denote Then, Φ1′ = BΦ1, Φ2′ = BΦ2, and It is not difficult to see that we can choose positive constants α and L such that

It is clear that (86) is equivalent to where Since lim⁡ u(ξ) = 0, then multiplying the first equality of (90) by e − and setting ξ → −∞ yield or Thus, (90) has the form where u 0 = (0,u 2(0)).

To study the properties of u(ξ), we need to construct a functional sequence. Define for ξ ≤ 0. Obviously, we have for any ξ ≤ 0. Assume that the induction hypothesis holds for j = 1,2,…, m and ξ ≤ 0. Then, using the condition (84) satisfied by the function G, it follows from the induction hypothesis that for ξ ≤ 0 we have Setting ε ≤ 3α/(8L) yields It then follows by induction that (97) holds for all j = 1,2, 3,… and ξ < 0. Thus, for any n > m > N and ξ < 0, we have which implies that {u (ξ)} is a Cauchy sequence of continuous functions. It follows that uniformly for all ξ < 0. Taking the limit of both sides of (95), it follows from the uniform convergence that the continuous function satisfies the integral equation (94). Then, the uniqueness of solution of (94) implies . Consequently, (97) and show that for any ξ < 0. Since lim⁡ u(ξ) = 0, it follows ||u(ξ)e −|| < +∞. Thus, and the proof is completed.

Proof of Theorem 11(I)

Assume (5) has a traveling wave solution (U(y + cτ), W(y + cτ), V(y + cτ)) satisfying boundary condition (8). From the first equation of (18), it follows By Lemma 12, there exists a positive constant α′ such that For 0 < λ < α, define two-sided Laplace transform as follows: Obviously, there exists λ* > 0 and L[V(·)](λ) is increasing in [0, λ*) such that λ* < +∞ satisfying lim⁡ L[V(·)](λ) = +∞ or λ* = +∞.

Furthermore, we have The second equality of (18) can be rewritten as where Denote G(V): = g′(0)V − g(V), from which it follows that G′(0) = 0. Since g(V) satisfies hypothesis (A1), we have G(V) ≥ 0 for any V ≥ 0.

Taking two-sided Laplace transform of (109), we have If λ* < +∞, it is clear that L[V(·)](λ*−) = +∞. Since Lemma 12, inequality (106), and G′(0) = 0 imply L[Q(·)](λ*) < +∞. However, Lemma 4 shows that 0 < Θ(λ*)<+∞, which contradicts (111). Hence, we have λ* = +∞. Furthermore, (111) can be rewritten as Since lim⁡Θ(λ) = +∞, it is impossible that the above equality holds, again a contradiction.

Proof of Theorem 11(II)

Suppose (U(y + cτ), W(y + cτ), V(y + cτ)) is a traveling wave solution of system (5) satisfying boundary condition (8). Then, by the proof of Theorem 1, it is clear that ∫− + V(ξ)dξ = ∫− + H 2(U, V)(ξ)dξ. Furthermore, which is a contradiction. The proof is completed.