Competitive exclusion in a vector–host epidemic model with distributed delay^†

Abstract

A multi-strain model of a vector-borne disease with distributed delay in the vector and the host is investigated. It is shown that if the reproduction number of the model ℛ₀<1, the unique disease-free equilibrium is globally asymptotically stable. Without loss of generality, strain one is assumed to have the largest reproduction number. In this case, the dominance equilibrium of strain one is shown to be locally stable. The basic reproduction number for a strain i () is written as a product of the reproduction number of the vector and the reproduction number of the host , i.e. . The competitive exclusion principle is derived under the somewhat stronger condition that if strain one maximizes both the reproduction number of the host and the reproduction number of the vector , strain one dominance equilibrium is globally asymptotically stable.

Keywords:

• vector–host
• competitive exclusion
• distributed delay
• global stability
• Lyapunov function

AMS Subject Classification::

• 92D30

1. Introduction

^†Vector-borne diseases are among the most complex infectious diseases to understand, model and control. Their complexity stems from the multi-staged life cycle of the vector, often influenced by the climate, the intricate evolution of the parasite, developing both in the vector and the host, and the fact that most vector-borne diseases have humans as well as non-human species as hosts. Many factors make controlling vector-borne diseases particularly challenging. This article focuses on one particular aspect, namely the evolution of the parasite. Parasites of vector-borne diseases exist in several species or serotypes. For instance, four species of the Plasmodium parasite cause malaria: P. falciparum, P. ovale, P. vivax, and P.malariae. Four serotypes of the dengue fever virus cause dengue. However, within these large classification groups, each pathogen can be antigenically diverse and may exist in multiple variants, called strains. Multi-strain models for vector-borne diseases have been considered before (see 689 and the references therein), but in most cases strains are assumed to interact through cross-immunity or other mechanism which generates coexistence.

For directly transmitted infections, it is well known that in the simplest case when no trade off mechanism is present 22, the only possible outcome of the competition of n strains is competitive exclusion. In this case, the strain with the largest reproduction number outcompetes and eliminates the remaining strains, a result first established by Bremermann and Thieme 2. In the present article, we address the question whether this principle extends to vector-borne diseases. Because the dynamics of the vector is subjected to a significant influence from the incubation period of the pathogen within the vector, called extrinsic incubation period, and the incubation period within the host, called intrinsic incubation period, we incorporate distributed delay both in the vector and the host to account for these delays. Natural delays occurring in the dynamics of vector-borne diseases are typically incorporated as discrete delays or age-structure. As such they lead to backward bifurcation, oscillations 17,16–18,2934 and even chaos 23.

In this article, we consider the outcome of the competition in the resulting multi-strain vector-borne model with distributed delay. Originally our hypothesis was that, analogously to directly transmitted diseases, the strain with the maximal reproduction number eliminates all the rest. However, the analysis revealed that we need a stronger condition for competitive dominance in the case of vector-borne diseases. In particular, we notice that the basic reproduction number for a strain i ( is written as a product of the reproduction number of the vector and the reproduction number of the host , i.e. . We find that for vector-borne diseases the competitive exclusion principle can be formulated as follows: The strain that maximizes both the vector and the human reproduction numbers dominates in the population. Such a strain will also have an overall maximal reproduction number. We should mention, however, that our competitive exclusion principle for vector-borne diseases is not complete. It would not give the outcome of the competition of the strains when different strains maximize the vector and the human reproduction numbers. For instance, if strain one maximizes the human reproduction number, that is, for all i≠1, but strain two maximizes the vector reproduction number, for all i≠2, then our competitive exclusion principle does not imply which strain will dominate in the population. We hypothesize that the strain with the maximal basic reproduction number will dominate but with a different approach than the one we have applied here may be necessary to establish this result.

This paper is structured as follows. In Section 2, we introduce a multi-strain version of the vector-borne disease model with two distributed delays. In Section 3, we discuss the equilibria and establish their local stabilities. In Section 4, we establish global stability of the disease-free equilibrium. In Section 5, we verify competitive exclusion principle under the assumption that strain one maximizes both the human and the vector reproduction numbers. Section 6 summarizes our results.

2. Model formulation

As in the introduction, we assume the pathogen causes the vector–host transmission disease existence through multiple strains. The multi-strain model of the vector–host interaction divides the vector population under consideration into the following groups: susceptible vectors, denoted by S_v, and infected vectors, which are divided into n subgroups: those infected with a strain i, denoted by ). If we let N_v(t) be the total number of vectors at time t, we have . Let N_h(t) be the density of the total host population at time t, which is composed of the number of susceptible individuals S_h(t), the number of infective individuals with a strain i, (i=1, 2, …, n ), and the number of recovered or immune individuals R_h(t). Thus, . For the disease transmission, it is assumed that a susceptible host can receive the infection through contact with a strain i infective vectors. Similarly, a susceptible vector can receive the infection through contact with strain i infectious human. Furthermore, it is assumed that a susceptible host and susceptible mosquito become exposed when they receive the infection from an infective individual and remain exposed for some period before becoming infectious.

Let Λ_h and Λ_v be the birth rate of the host population and the vectors, respectively. b is the average rate of biting on hosts by a single mosquito (number of bites per unit time), and the hosts are always sufficient in abundance; so that it is reasonable to assume that the biting rate b is constant. Thus, the number of bites on hosts per unit time per host is b/N_h(t). If is probability of infected bites with strain i on host individuals that produce an infection, is the infection force of the interaction between the infected mosquitos with strain i infected vectors and the susceptible host S_h(t). As pointed out in the introduction, there exists extrinsic and intrinsic incubation period in mosquitos and hosts, respectively. Let τ be the extrinsic incubation period of the parasite in the vector population. Since only a proportion of vectors survive this incubation period, the force of infection on host individuals at the present time t is , where is the death rate of the vectors infected with strain i. In reality, however, the incubation period is not a number but an interval during which the maturation of the parasite occurs in different hosts and vectors. Hence, we assume that τ is a distributed parameter (see 326). As a result, the force of infection becomes where the kernel function represents the distribution of the infectivity of the parasites in host population where the time taken to become infectious is τ.

Similarly, let μ_h be the rate of host individuals, and be the recovery rate of infected host individuals with strain i. is the probability that a susceptible vector bites on the infective host individual with strain i that produces an infection. Based on the above facts and that the average total rate of contacts between host and vectors must be conserved [28, P₆₆₇], gives the incidence of new cases of infection for the vector at the present time t. The kernel function expresses the infectivity on susceptible vectors during the intrinsic incubation period. Here, τ_v, τ_h are, respectively, the upper limits of the parasites incubation periods in vectors and hosts. The terms account for the probability of survival as infectious of host individuals during the intrinsic incubation period.

In a more general sense, there would be a possibility that the removed population is alive and bitten again by vectors, or multiple infectious bites on infected people may affect the disease progression. For simplicity, however, we assume that the recovered population will no longer be involved in the transmission process, and multiple infectious bites on infected hosts do not play a role in the progression of the disease. Under the above assumption, our multi-strain model with extrinsic and intrinsic distributed incubation periods can be formulated as follows: Model (1) is equipped with the following initial conditions: All parameters in model (1) are non-negative. We define the following space of functions where, the Banach space C([−h, 0], R) of continuous functions mapping the interval [−h, 0] into R is equipped with the sup-norm ||ψ||=sup _{−h≤θ≤0}|ψ (θ)|. By the standard theory of functional differential equations 12, it can be verified that solutions of (1) with non-negative initial conditions (2) has a unique solution which remains non-negative for all t≥0. Moreover, we can show the solutions of system (1) are ultimately uniformly bounded in X. In fact, it follows from the total vector population size N_v(t) satisfying N_v′(t)=Λ_v−μ_v N_v(t) that Similarly, we have So, it is assumed that the total vector and host population sizes N_v(t), N_h(t) are asymptotically constant.

Let It is easy to see that the set Ω is positively invariant for system (1). We note that R_h(t) can be removed from the equations of system (1), it is sufficient to analyse the dynamical behaviour of solutions to system (1) without the equation of R_h(t). For simplicity, we introduce the following notation. Furthermore, we impose the following assumptions:

Assumption 1

• (1) It is assumed that and are continuous on [0, h , i=1, 2, …, n;

• (2)  and satisfy

• (3)  for 0≤τ≤h, i=1, 2, …, n. Here h=max{τ_h, τ_v}.

In the next section, we investigate the dynamics of the following system

From the above discussion, we have always the following proposition for system (3)

Proposition 2.1

There exists a unique solution of system (3) with initial conditions (2). This solution is non-negative for all t≥0. Moreover, all the solutions of system (3) are ultimately uniformly bounded in Ω.

3. Equilibria and local stability

System (3) always has a unique disease-free equilibrium ℰ₀, which is given by where 0=(0, …, 0) is an n-dimensional vector of zeroes.

According to the definition of the basic reproduction number 4, which is the expected number of secondary cases produced in an entirely susceptible population by a typical infected individual during its entire infectious period, we define the reproduction number of strain i:

We notice that the disease cycle in vector-borne disease consists of two transmission pathways: human-to-vector transmission and vector-to-human transmission. Each of these pathways is characterized with its own reproductive number. The reproductive number of human-to-vector transmission of strain i is given by and gives the number of secondary infected vectors that one infectious host will produce in an entirely susceptible vector population. Similarly, the reproduction number of the vector-to-human pathway of strain i is given by Obviously, the reproduction number for a strain i in system (3) is a product of the human and vector reproduction numbers:

We define a reproduction number of the whole system (3)

By direct calculation, we show that for each i, in system (3), there is a corresponding strain-i equilibrium ℰ_i given by where the non-zero components and are in position i+1 and n+j+1, respectively. Moreover, equilibrium ℰ_i exists if and only if . The non-zero components of the equilibrium ℰ_i are given by

Now we investigate the stability of the equilibria in system (3), let us first linearize system (3) at disease-free equilibrium ℰ₀. In particular, let S_v(t)=Λ_v/μ_v+x_v(t), , S_h(t)=Λ_h/μ_h+x_h(t), . We look for exponential solutions of the time-dependent system for the perturbations. Thus, we obtain the following eigenvalue problems: Solving the linear system (4), it is easy to obtain that −μ_v,−μ_h are two negative characteristic roots of system (3) and other characteristic roots (n−2) are determined by the following equation: We now show that all characteristic roots in Equation (5) have a negative real part. Note that Equation (5) is continuous functions of τ, and there is a continuity in the eigenvalues for τ≥0. Using Rouche's Theorem [5, Theorem 9.17.4], we know that given the eigenvalues were all negative in the non-delay case, introducing a delay can cause a change in stability if and only if for some value of τ, there exists a purely imaginary root of Equation (5), i.e. a pair of eigenvalues must cross the imaginary axis if they are to become positive. For the Equation (5), we have the following lemma:

Lemma 3.1

If the characteristic equation (5) has no purely imaginary roots.

Proof

Assume the contrary. Set To proceed, we first consider Equation (6) with τ=0. It is easy to verify that all roots in Equation (6) have negative real parts for and τ=0. For τ>0, we show that Q(λ, τ)=0 has no purely imaginary roots. Set Let λ=i ω (ω>0) be a root of Equation (6). Thus,we have Thus, we have . From Equation (6), we have F₂(i ω)−F₃F₁(i ω, τ)=0. Therefore, we have That is, Thus, we have It follows from Equation (7) that we obtain

Obviously, if , it is impossible for Equation (8) to hold. Hence, the linearized system (5) cannot have any roots with a positive real part.   ▪

Therefore, from the above discussion, we have established the following result:

Theorem 3.1

If then the disease-free equilibrium ℰ₀ of system (3) is locally asymptotically stable. If ℛ₀>1, it is unstable.

Now we shall establish the local stability of the single-strain equilibrium ℰ_i for a fixed i. We have the following result:

Theorem 3.2

Assume for a fixed i, If for all j≠i, then the strain-i equilibrium ℰ_i is locally asymptotically stable. Otherwise, if there exists k₀ such then the single-strain equilibrium ℰ_i is unstable.

Proof

To simplify the presentation, without loss of generality, we shall assume that i=1, i.e. , and , for i=2, …, n. We linearize system (3) around the strain-one equilibrium ℰ₁. Let , , , , , , for i=2, …, n. We obtain the following linearized system:

We notice that the linearized equations for strains k=2, 3, …, n can separate from the whole system. Thus, by direct calculation, we obtain the following characteristic equations: Similar to proof in Lemma 3.1, it is easy to show that if , Equation (10) has no purely imaginary roots and all eigenvalues of these equations have negative real parts.

Therefore, the stability of ℰ₁ depends on the eigenvalues of the following system: By looking for exponential solutions of (11), we obtain the following the characteristic equation Now we show that Equation (12) has no purely imaginary roots. In fact, we divide both sides by (λ+μ_h), then we set If λ is a root with ℜλ≥0, it follows from Equation (13) that Using the equilibrium point satisfies the corresponding equation, we obtain It is a contradiction. Hence, using Rouche's Theorem [5, Theorem 9.17.4], we only show that Equation (12) cannot have any roots with a non-negative real part. Therefore, the strain-one equilibrium ℰ₁ is locally asymptotically stable for and . Otherwise ℰ₁ is unstable.   ▪

4. Global stability of the disease-free equilibrium

In the previous section, we have shown that equilibria are locally stable, i.e. given the conditions on the parameters, if the initial conditions are close enough to the equilibrium, the solution will converge to that equilibrium. In this section, our objective is to extend these results to global results. Now we first establish the global stability of the disease-free equilibrium. We have the following result:

Theorem 4.1

If the disease-free equilibrium ℰ₀ is globally asymptotical stable.

Proof

From the first and the third equations of system (3), we obtain

It follows from the above inequalities that Thus, from system (3) and inequalities (15), we have

From Equation (16), we obtain that Thus, from the above expression, for i=1, …, n, we have Similarly, for i=1, …, n we also have From Equations (17) and (18), we have Since ℛ₀<1 and is bounded, it follows from the above expression that Similarly, from Equation (18), we have Therefore, as t→∞. From the above discussion and Theorem 3.1, we can conclude that the disease-free equilibrium ℰ₀ is globally asymptotically stable for ℛ₀<1. This completes the proof of Theorem 4.1.   ▪

5. Principle of competitive exclusion

For directly transmitted diseases, Bremermann and Thieme 2 establish that in the simplest scenario when competitive exclusion is the only outcome, the strain with the largest reproduction number persists, while the remaining strains die out. While extending the Bremermann and Thieme 2 result to vector-borne diseases remains an open problem (in the case of vector-borne diseases, we seem to need a stronger condition for competitive exclusion), we were able to show competitive exclusion under the somewhat stronger condition that the winning strain maximizes both the human and the vector reproduction numbers, not just the overall reproduction number.

Mathematically speaking, establishing the competitive exclusion principle means establishing the global stability of the strain one equilibrium ℰ₁. From Theorem 3.2, we obtain that under some conditions the equilibrium ℰ₁ is locally asymptotically stable. In this section, we are ready to establish the global stability of the strain one equilibrium ℰ₁ by constructing the Lyapunov functional, similar to the one used in 10111415,19–21,24253132. We note that for constructing the Lyapunov function, such as f(x)=x−1−ln x, here f(x) is not defined if x=0. Thus, in order to construct our Lyapunov function, we must first show that when ℛ₀>1, the disease persists in the form of strain one. Now we apply the persistence theory by Hale and Waltmann 13 for infinite-dimensional systems to show the uniform persistence of strain one in system (3). The methods and techniques have been recently employed by other authors (see 273033).

To proceed, we introduce the following notation and terminology. Let X be a complete metric space with metric d. Suppose that T is a continuous semiflow on X, that is, a continuous mapping T: [0, ∞)×X→X with the following properties: where T_t denotes the mapping from X to X given by T_t (x)=T (t, x). The distance d(x, Y) of a point x∈X to a subset Y of X is defined by

Recall that the positive orbit γ⁺(x) through x is defined as γ⁺(x)=∪_t>0{T (t)x}, and its ω−limit set is . Define by W^s (E) the strong stable set of a compact invariant set E as W^s (E)={x: x∈X, ω(x)≠φ, ω(x)⊂E}. Denote by T(t), t≥0, the family of solution operators corresponding to Equation (3). The following assumption must be satisfied

(H1): Let X₀⊂X, X⁰⊂X, X₀∩X⁰=∅. X₀ is open and dense in X. Moreover, the C₀-semigroup T(t) on X satisfies

Let T_b(t)=T (t)|X₀ and A_b be the global attractor for T_b(t). The following result is discussed in [13, Theorem 4.2]:

Lemma 5.1

Suppose that T(t) satisfies (H1) and the following conditions:

• (i) There is a t₀>0 such that T(t) is compact for t>t₀;

• (ii) T(t) is point dissipative in X;

• (iii)  is isolated and has an acyclic covering

    M˜, where

• (iv) W^s (M_i)∩X₀=∅ for i=1, 2, …, n.

Then T(t) is a uniform repeller with respect to X₀, i.e., there is an η>0 such that for any x∈X⁰, lim inf;_t→+∞ d (T(t), X₀)≥η.

We use the above lemma to establish the following result:

Theorem 5.1

Assume that ℛ₀>1. Then the disease persists in the population. If and strains from the 2 to n approach zero, that is Then, for all initial conditions that belong to Ω₀, strain one can persist. In particular, there exists a constant γ>0 such that

Proof

Under the assumptions of Theorem 5.1, system has two non-negative equilibrium ℰ₀ and ℰ₁. Let φ be a vector of initial conditions for system (3). We assume the functions in the vector φ are continuous. Let where, the Banach space C([−h, 0], ℝ⁺) of continuous functions mapping the interval [−h, 0] into ℝ⁺ equipped with the sup-norm ||ψ||=sup _{−h≤θ≤0}|ψ (θ)|. Set Obviously, we have X=X⁰∪X₀. It suffices to show that there exists an η>0 such that for any solution u_t of system (3) initiating from X⁰, lim inf;_t→∞ d (U_t, X₀)≥η. To this end, we have to verity below that the conditions of Lemma 5.1 are satisfied.

First, it follows that both X and X⁰ are positively invariant. Clearly, ∂ X₀ is relatively closed in X. It is easy to verify that system (3) is point dissipative. Set We now claim that M_∂={(Λ_v/μ_v, 0, Λ_h/μ_h, 0)}, where 0=(0, …, 0) is an n−dimensional vector of zeroes. Assuming φ∈M_∂ it suffices to show that Assume that on the contrary, there exists t₀>0 such that case (i) for some strain-i₀, we have while or case (ii) for some strain-j₀, we have while

In case (i), from the third equation of (3), we have Hence, there is a sufficiently small constant ϵ₀ such that . On the other hand, from , we obtain a positive ϵ₁ (0<ϵ₁<ϵ₀) such that . Thus, we obtain . This is in contradiction with the assumption that Similarly, we can show the case (ii) does not hold.

Let , where Y₂ is the global attractor of T(t) restricted to ∂ X. We now show that Ω₂={ℰ₀}. In fact, it follows from Ω₂⊆ M_∂ and the first and third equations of (3). we have lim _t→+∞ S_v(t)=Λ_v/μ_v, lim _t→+∞ S_v(t)=Λ_v/μ_v. Thus, {ℰ₀} is the isolated invariant set in X.

Finally, we need to show that W^s(ℰ₀)∩X⁰=∅.

Assume on the contrary that there exists a solution U_t∈X⁰ such that Hence, for each ϵ>0 there exists a time T₀ such that for each t>T₀ we have By shifting the dynamical system, we may assume that the above inequalities are satisfied for t≥0. From the first and third equations of (3) and taking into account the above inequalities, we have Therefore, from Equation (20), we have

Thus, there exists a T₁>T₀ such that, for any t>T₁, and using the second equation of system (3) and the inequality above, we have Similarly, there exists a T₂>T₀, such that for any t≥T₂, we have By the mean value theorem for integrals, we know that for any time t, there is a ξ_t such that

Therefore, from Equations (21)–(23), we obtain Notice that . Thus, we may assume we have chosen sufficiently small ϵ in such a way that we have

Since as t→∞, then by a standard comparison argument, the solution of the following equation with initial conditions has to converge to (0, 0) as well for inequality (25). On the other hand, for i=1, …, n, if we set, Obviously, it follows from the solution that W_i(t)→0 as t→∞. In particular, we have W₁(t)→0 as t→∞. However, by directly calculating the derivative in Equation (27), we obtain Thus, for the above expression (28), if we consider i=1 and use Equation (25), then we have W₁(t)→∞ as t→∞. This is a contradiction.

Therefore, we have W^s(ℰ₀)∩X⁰=∅. This completes the proof of Theorem 5.1.   ▪

Now we are able to state our main result.

Theorem 5.2

Assume that and for i=2, …, n. Then, equilibrium ℰ₁ is globally asymptotically stable.

Proof

Because of the complexity of the expressions, we define the Lyapunov functional in components and take the derivative of each component separately. Set f(x)=x−1−ln x, x>0. The components of the Lyapunov's functional are defined as follows: where, .

Calculating the derivative of the above expressions in Equation (29) along with system (3), we obtain The above equations can be reduced to the following:

Differentiating the time derivative of with respective to t, we obtain Notice that and is bounded. It follows that . Moreover, Hence, we have Similarly, we have Set Thus, by Equations (30)–(33), we obtain Similarly, set where, .

Calculating the derivative of the expressions in Equation (36), respectively, along system (3), we obtain Set By Equations (37)–(39) and the system (3), we obtain Set From Equations (35) and (40), we obtain It follows from Equation (41) that Using the equality , we obtain Using f(x)=x−1−ln x, x>0 and the expression of and , from Equation (42), we obtain Since f(x)≥0 for x>0, and we have d W(t)/d t≤0. Let

We want to show that the largest invariant set in ¯ Ω is the singleton {ℰ₁}. In fact, from Equation (43), d W(t)/d t=0, and using the fact that 1−x+ln x≤0 for all x>0 with equality holding iff x=1, imply and for all τ∈[0, h) and t>0. Furthermore, for each element of ¯ Ω, we have and, since ¯ Ω is invariant, d S_v/d t=0, d S_h/d t=0. Using Equations (44) and (45), we obtain Rearranging gives Using the equilibrium ℰ₁ satisfies the relation , we easily obtain for t≥0. Similarly, it is easy to prove that for t≥0. Thus, we show d W(t)/d t=0 if and only if , for t≥0. Hence, the largest invariant set in ¯ Ω is the singleton {ℰ₁}. By the LaSalle invariance principle and Theorem 3.2, we show that equilibrium ℰ₁ is globally asymptotically stable.   ▪

6. Discussion

In this paper, we introduce a multi-strain model of a vector-borne disease with distributed delay in the vector and the host. We define the basic reproduction number of the disease as the maximum of the reproduction numbers of each strain. We show that if ℛ₀<1 the disease-free equilibrium is locally and globally stable, that is the number of infected with each strain goes to zero. Furthermore, we show that if ℛ₀>1, then the disease persists.

Our main objective with the article was to extend the competitive exclusion result established by Bremermann and Thieme 2. This principle states that in the basic ODE model when multiple strains compete only the strain with the largest reproduction number persists, if its reproduction number is larger than one, and the other strains die out. In this case, coexistence is not possible outside of the trivial scenario when the reproduction numbers of two different strains are equal.

To prove the competitive exclusion result for the multi-strain vector-borne model with distributed delay, we need to prove global stability of the a single-strain equilibrium. We approached the problem using a Lyapunov function. However, we were able to establish only the following weaker result: It is well known that the reproduction number in vector-borne diseases splits into a vector reproduction number and host reproduction number, namely . Our result states that the strain that maximizes both the vector reproduction number and the human reproduction number will dominate in the population and eliminate the rest. In particular, if we assume that and for i=2, ˙s, n then strain one will dominate and eliminate the rest. In this case, coexistence does not occur. In fact, it can be shown that model (3) does not have a coexistence equilibrium if all strain reproduction numbers are different.

Our result does not resolve the competition outcome in the case when different strains maximize the human and the vector reproduction numbers. In this sense, the question whether complete competitive exclusion holds in the vector–host model is still an open question. We surmise that because coexistence equilibrium is not present, competitive exclusion in this case also occurs with the strain with maximal reproduction number eliminating the rest. Establishing this result, however, may need a different approach.

From the perspective of public health, public health efforts will work best if directed to monitoring the vector and host reproduction numbers of the multiple strains in vector-borne diseases. Control measures may need to be applied to the strain(s) that maximize the vector and the host reproduction numbers.

At last, we should mention that, similar to paper 25, our analysed approaches in this paper can be applied for a multi-strain model of a vector-borne disease with discrete delays. The obtained results in this paper also hold for the multi-strain model with discrete delays.

Acknowledgements

The authors thank J.M.Cushing (Editor-in-Chief), the handling editor and the anonymous referees for their useful feedback and helpful comments and suggestions which have improved the paper. This work was partially written when the first author was visiting the Department of Mathematics, University of Florida in 2012; he thanks them for their hospitality. This work was carried out with the support from the NSF under grant DMS-0817789, the NSF of China grants 11271314 and 10911120387 and the University Key Teacher Foundation of He'nan Province (2009GGJS-076).

Notes

^†This article is dedicated to Mimmo Iannelli on the occasion of his 65th birthday.