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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 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.
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 . 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) .
Many mathematical models were proposed to understand the propagation mechanism of cholera, the earlier one of which was established by Capasso and Paveri-Fontana  to study the 1973 cholera epidemic in the Mediterranean region as follows: where and denote the concentrations of the pathogen and the infective populations, respectively. In addition, Codeço  investigated the role of the aquatic pathogen in dynamics of cholera through the following susceptible-infective-pathogen mode: where 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 . Cholera bacteria live in rivers and interact with the plankton on the surface of the water . When individuals drink contaminated water and are infected, then they will release cholera bacteria through excretion . 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.  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 , Xu and Zhao , Jin and Zhao , and Hsu and Yang  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 and denote the concentrations of susceptible and infected individuals, respectively, and is the concentration of the infectious agents. is the disease-related death rate, denotes the contribution of each infected person to the concentration of cholera, and is the net death rate of the vibrio. is the environment-to-human transmission incidence. Similar to , we assume that satisfies(A1), , , for .From hypothesis (A1), we have .
In this paper, we study the traveling wave solutions of model (3). The formula for minimal wave speed is given. To prove the existence of traveling wave solutions for , 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 , the cone in our paper is bounded. Motivated by [34–37], we prove the nonexistence of traveling wave solutions for by two-sided Laplace transform, which was firstly introduced to prove the nonexistence of traveling wave solutions by Carr and Chmaj  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  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 . 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, also satisfies assumption (A1) with 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 .
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 . Then, there exists a positive constant which is the only positive root of (9). For any , system (5) has a traveling wave solution satisfying boundary condition (8) such that is nonincreasing in . Furthermore, one has that
Proof. Assume is a traveling wave solution of system (13) satisfying boundary condition (14). Obviously, is a solution of system (5). To prove the necessity, it is enough to show that . 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 .
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 and obtain Substituting into (19), we get the characteristic equation that is, where , , and . To investigate distribution of roots of (21), denote and introduce the following lemma .
Then, we get the following lemma about the distribution of eigenvalues.
Lemma 4. Assume . Then, there exists a constant which is the only positive root of (9) such that(a)if , (21) has a negative real root and two nonreal complex conjugate roots with positive real parts;(b)if , (21) has a negative real root and a positive real multiple root;(c)if , (21) has a negative real root and two different positive real roots;(d)one can assume and let be the two positive roots of (21); then if .
Proof. Obviously, , , and . By Descartes’ rule of signs, has only one positive root such that for and for . Direct calculations show that . Since and , 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 is a cubic polynomial.
In this section, we always suppose and unless other conditions are specified. Denote to be the two positive roots of (21) and define where , .
Lemma 5. The function satisfies inequality for any .
Proof. Firstly, assume and, therefore, . Since satisfies (19) and for any , we have
Secondly, set , which implies . We have that The proof is completed.
Lemma 6. For sufficiently small and sufficiently large, the function satisfies for any .
Proof. Let be sufficiently large to ensure . When , then and the lemma is obviously true. Now, suppose . Then, and where . Let . Since we can find sufficiently small and sufficiently large such that Thus, the proof is completed.
Lemma 7. Let . Then, for large enough, the function satisfies for any .
Proof. It is clear that if and only if , that if and only if , and that if and only if . Let . When , then , , and Lemma 7 holds.
In this paragraph, assume . Then, , , and . To prove this lemma, it is enough to show where . Since we only need to show Since , then Since , inequality (34) satisfies if The proof is completed.
To apply Schauder’s fixed point theorem, we will introduce a topology in . Let be a positive constant which will be specified in the following. For , define We will find traveling wave solutions in the following profile set: Obviously, is closed and convex in . Firstly, we change system (18) into the following form: where , Suppose are the two roots of equation . Furthermore, define by In the remainder of this paper, it is always assumed that .
Lemma 8. Consider .
Proof. Let ; 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 . Then, Therefore, we have proved .
Now, we study . If , then , which implies that since , . Assume . From Lemma 7 and system (39), it is clear that which implies where the final inequality is due to and . In conclusion, for any .
Similarly, we can show and the proof is completed.
Lemma 9. Map is continuous with respect to the norm in .
Proof. For , we have
where , and is between and . Therefore, we have
Thus, when , we get
When , it follows that Consequently, we have proved that for any , where That is, In conclusion, is continuous with respect to the norm in .
In addition, consider . Firstly, we have where . Therefore, If , it holds that If , we have Consequently, we conclude that where Thus, is continuous with respect to the norm in . The proof is completed.
Lemma 10. Map is compact with respect to the norm in .
Proof. For any , it is clear that
Since , we have
which implies that . Since , we get
which implies that. Consequently, and are bounded, which shows that is uniformly bounded and equicontinuous with respect to the norm in .
Furthermore, for any positive integer , define Obviously, for fixed , is uniformly bounded and equicontinuous with respect to the norm in , which implies that is a compact operator. Since we have that Similarly, we can prove that when . Thus, when . By Proposition 2.1 in Zeilder , we have that converges to in with respect to the norm . Consequently, is compact with respect to the norm . The proof is completed.