Global stability of the dengue disease transmission models

In this paper, we further investigate the global stability of the dengue transmission 
 models. By using persistence theory, it is showed that the disease of system uniformly persists when the basic reproduction number is larger than unity. 
 By constructing suitable Lyapunov 
 function methods and LaSalle Invariance Principle, 
 we show that the unique endemic equilibrium of the model is always 
globally asymptotically stable as long as it exists.


1.
Introduction. Dengue fever is a mosquito-borne tropical disease caused by dengue virus. It has been one of the most important public health problems in the tropical and subtropical developing countries and regions. The number of cases of dengue fever has increased dramatically since 1960s, with between 50 and 528 million people infected yearly [6,1]. There are currently no licensed vaccines or specific therapeutics, and substantial vector control efforts have not stopped its rapid emergence and global spread [18]. The contemporary worldwide distribution of the risk of dengue virus infection and its global dynamics is still poorly known [10].
Mathematical models have been developed in the literatures [7,8,17,20,2,15] to gain insights into the transmission dynamics of dengue in a community. Recently, a deterministic model for the transmission dynamics of a strain of dengue disease has been investigated by Garba, et al. [9]. The basic reproduction number practical importance in biology [15], which can determine the long-term dynamics exhibited by a given nonlinear dynamical system. It is also generally believed that when the nontrivial equilibrium of the model is unique in the feasible region, it is globally asymptotically stable. To demonstrate global stability of the equilibria in some nonlinear dynamical system, some methods are applied and developed in literatures [16,14,12,13,3,4,5]. Especially, it is more suitable to construct Lyapunov function (or functionals) for higher-dimensional systems to show that the equilibrium of the model is globally asymptotically stable.
Motivated by the works of Li et al. [14] and Kalajdzievska [12], in the present note, using Lyapunov function methods, we show that the endemic equilibrium for the dengue epidemic model with the mass action formulation is globally asymptotically stable if 0 ≥ 1 in the paper [9]. The rest of this paper is organized as follows. Section 2 states the global stability of equilibrium for the dengue model without vaccination. The global stability of equilibrium for the vaccination dengue model is given in Section 3. Finally, concluding remarks are addressed in Section 4.

2.
The global stability for the dengue model without vaccination. In this section, we mainly investigate the global stability for the existing endemic equilibrium of the dengue model without vaccination in the literature [9].
Let S H (t), E H (t), I H (t), R H (t), respectively, denote susceptible humans, exposed humans, infectious humans and recovered humans. Let S V (t), E V (t), I V (t), respectively, denote susceptible mosquitoes, exposed mosquitoes, infectious mosquitoes. Since the variable equation for the recovered humans in the dengue model does not appear explicitly in other variable equations, we only consider the following the dengue model without vaccination in paper [9]. (2. 2) The parameters Π H , µ H , σ H , δ H , τ H , Π V , µ V , σ V , δ V are poitive constants and have the same description as those of paper [9]. By using the next generation matrix method, the basic reproduction number m 0 of the system (2.1) is given by where, By directly calculating, we obtain the the following result (These results also can be obtained from paper [9]).
Theorem 2.1. The dengue model with mass action incidence, given by (2.1) with (2.2), has the disease-free equilibrium E 0 if m 0 ≤ 1, and has a unique endemic Now we first prove that the disease can persist by showing that the system (2.1) is uniformly persistent. Some methods and techniques have been recently employed by other authors [11,19] to show the nonlinear system is uniformly persistent. We investigate our system (2.1) with terminology used in Hofbauer and So [11]. Let (X, d) be metric space and f : X → X be continuous with a closed subspace Y such that X/Y is forward invariant under f . It is assumed that X has a global attractor A. Let M be the maximal compact invariant set in Y . Then f is uniformly persistent (with respect to Y ) i.e. there exists a positive constant number ε > 0 such that lim inf t→∞ d(f (x(t)), Y ) > ε for all x ∈ X/Y if and if is isolated in A and W s (M ) = {x ∈ X : f (x(t)) → M as t → ∞} [11]. Let X = R 6 + and Y = ∂R 6 + , the boundary of X.
Theorem 2.2 If m 0 > 1, then system (2.1) is uniformly persistent. Proof. For any solution of system (2.1) X = (S H , E H , I H , S V , E V , I V ) ∈ R 6 + , let T define the map induced by system (2.1). It is easy to follow that if the initial conditions in system (2.1) is positive,then the solutions (S H (t), E H (t), I H (t), S V (t), E V (t), I V (t)) > 0 for t > 0. Therefore, X ∈ X/Y is positively invariant for the system (2.1). Obviously, it is easy to obtain that system (2.1) has a global attractor and Suppose on the contrary that there exists a solution ( with the positive initial conditions such that Then, for any sufficiently small positive number ε > 0, there exists t 0 > 0, such that From the equations of (2.1), for any t > t 0 , we have Consider the following matrix defined by Consider the following linear system T is denoted the map induced by system (2.6). Noticing that each entry of T is positive and it follows from m 0 > 1 that the special radius of T is larger than unity. Let be a solution of (2.6) through the initial condition (E H0 , I H0 , Hence solutions of (2.6) are unbounded. As a result, E i (t), i = H, V and I i (t), i = H, V also become unbounded large as t → ∞. We obtain a contradiction and conclude that W s (E 0 ) ∩ X 0 = Ø. Therefore,system (2.1) is uniformly persistent with respect to Y by Theorem 4.1 of paper [11].i.e. there exists ε 0 > 0 such that Now, using Lyapunov function methods, we investigate the global stability of the unique endemic equilibrium E * for system (2.1). We firstly establish the following result: It is obvious that the endemic equilibrium E * of (2.1) corresponds to the positive equilibrium E * (1; 1; 1; 1; 1; 1) of (2.7), and that the global stability of E * is the same as that of E * (1; 1; 1; 1; 1; 1) , so we will discuss the global stability of the equilibrium E * (1; 1; 1; 1; 1; 1) of system (2.7) instead of E * .
Consider the following Lyapunov function: where a 1 , a 2 , a 3 , a 4 , a 5 > 0 are constants to be determined. Noticing that W 1 (t) has a global minimum at E * . By directly calculating the derivation of W 1 (t) along the solutions of system (2.1), we have (2.9) Positive constants a 1 , a 2 , a 3 , a 4 and a 5 are chosen as It can be verified that a 1 , a 2 , a 3 , a 4 , a 5 satisfy the following relations (2.10) It is easy to verify that system (2.10) is compatible. Thus, system (2.9) can be reduced the following (2.11) Eq.(2.11) can be rewritten as the following forms (2.12) where the parameters b 1 , b 2 , b 3 , b 4 , b 5 , b 6 ≥ 0 are constants to be determined.
Let the coefficients for the same terms between Eq.(2.11) and Eq.(2.12) be equal, which yields the following equations (2.13) Incorporating the following equality, (2.14) To assure that b 3 , b 5 and b 6 are all nonnegative, b 4 satisfy the following inequality: In fact, b 4 always exists. For example, we may choose Thus, we can assure that the existence of b i ≥ 0, (i = 1, 2, 3, 5, 5, 6) in (2.12). Using the arithmetic mean is greater than or equal to the geometric mean, thus, obviously, each term in brackets of Eq.(2.12) is negative definite. Therefore, it follows from (2. From Theorem 2.2 and the characteristic of the system (2.1), we can see that the maximum invariant set of (2.1) on the set D is the singleton {E * }. Therefore, the endemic equilibrium E * of system (2.1) is globally stable in D by LaSalle Invariable Principle. Similar analysis in Theorem 2.3, we can show that the disease free equilibrium E 0 of the dengue model (2.1) is globally stable. In fact, by constructing the following Lyapunov function: Thus, we establish the following result 3. The vaccination dengue model. In this section, we investigate the global stability of the endemic equilibrium for the dengue model with vaccination in paper [9]. We still denote S H (t), E H (t), I H (t), P H (t), respectively, susceptible humans, exposed humans, infectious humans and vaccinated humans. Let S V (t), E V (t), I V (t), respectively, denote susceptible mosquitoes, exposed mosquitoes, infectious mosquitoes. This extended model include a population of vaccinated individuals. Since it is assumed that the vaccine is imperfect, vaccinated individuals acquire infection at a rate λ m H (1 − ε), where ε is vaccine efficiency. It is assumed that the vaccine wanes at a rate ω and susceptible individuals is vaccinated at a rate ξ. We consider the following vaccination dengue model Firstly, the vaccination reproduction number of the model (3.1), denoted by m vac , is given by where, .
It is easy to verify that the region Γ vac is positively invariant and attracting.
Using the same method as proving the persistence of (2.1). Then we firstly give the following result.
Eq.(3.3), system (3.1) can be written as follows (3.4) Thus, the endemic equilibrium E * vac of (3.1) corresponds to the positive equilibrium E * vac (1; 1; 1; 1; 1; 1; 1) of (3.4), and that the global stability of E * vac is the same as that of E * vac (1; 1; 1; 1; 1; 1; 1) , so we will discuss the global stability of the equilibrium E * vac (1; 1; 1; 1; 1; 1; 1) of system (3.4) instead of E * vac . Considering the following Lyapunov function: 5) where a 1 , a 2 , a 3 , a 4 , a 5 , a 6 > 0 are constants to be determined. Note that W 3 (t) is a Lyapunov function and has a global minimum at E * vac . By directly calculating the derivation of W 3 (t) along the solution of (3.4), we have 4. Concluding remarks. In this paper, by constructing Lyapunov function methods and LaSalle Invariance Principle, we show the global stability of the endemic equilibrium in two dengue transmission models in paper [9]. Our results show that the unique endemic equilibrium of the models with mass incidence is always globally asymptotically stable as long as it exists. These results can further enrich the dynamics of dengue virus transmission. In contrast, authors in paper [9] have showed that the dengue model with standard incidence rate exhibits the backward bifurcation. Therefore,these obtained results show that the incidence rate (the rate of new infection) have played an important role in analyzing the disease transmission dynamical behavior. In this paper, it is also showed that it is important to construct Lyapunov function (or functionals) methods and LaSalle Invariance Principle in analyzing the global asymptotic stability of some higher-dimensional dynamical systems.