Global Stability of Multigroup Dengue Disease Transmission Model

Abstract

We investigate a class of multigroup dengue epidemic model. We show that the global dynamics are determined by the basic reproductive number 𝑅₀. We present that when 𝑅₀≤1, there is a unique disease-free equilibrium which is globally asymptotically stable; when 𝑅₀>1, there exists a unique endemic equilibrium and it is globally asymptotically stable proved by a graph-theoretic approach to the method of global Lyapunov function.

1. Introduction

To understand and control the spread of infectious disease in population, mathematical epidemic models have been paid more attention. One essential assumption in most classical epidemic models is that the individuals are homogeneously mixed. However, many infectious diseases, such as measles, mumps, and gonorrhea, occur in heterogeneous host population, so multigroup epidemic models seem more reasonable. One of the earliest multigroup models is analysed by Lajmanovich and Yorke [1] for gonorrhea in a nonhomogeneous population. However, because of the large scale and complexity of multigroup models, progresses in the mathematical analysis of their global dynamics have been slow, particularly, the question of uniqueness and global stability of the endemic equilibrium. Recently, a graph-theoretic approach to the method of global Lyapunov functions in [2, 3] was proposed to resolve the open problem on the uniqueness and global stability of the endemic equilibrium. Subsequently, a series of good results were produced about multigroup epidemic models in [4–8].

In this paper, we study a multigroup dengue disease transmission model by the method in [2, 3]. In the model, the population is divided into 𝑛 groups. Each group is divided into five disjoint classes: susceptible individuals, infective individuals, removed individuals, susceptible mosquitoes, and infective mosquitoes whose numbers of individuals at time 𝑡 are denoted by 𝑆_𝐻𝑖(𝑡), 𝐼_𝐻𝑖(𝑡), 𝑅_𝐻𝑖(𝑡), 𝑆_𝑉𝑖(𝑡), 𝐼_𝑉𝑖(𝑡), respectively. The model to be studied takes the following form:𝑆^′_𝐻𝑖=𝐴_𝐻𝑖−^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖𝑆_𝐻𝑖,𝐼^′_𝐻𝑖=^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖,𝑅^′_𝐻𝑖=𝛾_𝐻𝑖𝐼_𝐻𝑖−𝜇_𝐻𝑖𝑅_𝐻𝑖,𝑆^′_𝑉𝑖=𝐴_𝑉𝑖−^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝑆_𝑉𝑖,𝐼^′_𝑉𝑖=^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝐼_𝑉𝑖, (1.1) where 𝑖=1,2,…,𝑛. Here 𝐴_𝐻𝑖 and 𝐴_𝑉𝑖 represent the recruitment rate of the humans and the mosquitoes in the 𝑖th group, 𝛽_𝐻𝑖𝑗 represents the contact rate between susceptible humans 𝑆_𝐻𝑖 and infectious mosquitoes 𝐼_𝑉𝑗, 𝛽_𝑉𝑖𝑗 is the contact rate between infected people 𝐼_𝐻𝑗 and susceptible mosquitoes 𝑆_𝑉𝑖, 𝜇_𝐻𝑖 and 𝜇_𝑉𝑖 represent the death rate of the humans and the mosquitoes in the 𝑖th group, and 𝛾_𝐻𝑖 represents the recovery rate of the humans in the 𝑖th group. All parameter values are assumed to be nonnegative and 𝐴_𝐻𝑖,𝐴_𝑉𝑖,𝜇_𝐻𝑖,𝜇_𝑉𝑖>0.

Dengue fever (DF) is an acute mosquito-transmitted disease, with a recorded prevalence in 101 countries [9–11]. An estimated 50–100 million people per year are infected, with approximately 25,000 deaths annually [12]. Thus, the study of DF is perceived as signification and receives much attention. When 𝑛=1, the model (1.1) had been studied extensively. For example, the global stability of the equilibria was proved with the results of the theory of competitive systems and stability of periodic orbits in [13]; in [14], the global stability of the equilibria was proved with Lyapunov functions under some conditions.

The organization of this paper is as follows. In Section 2, we quote some results from graph theory which will be used in the proof of our main results. In Section 3, we present a global analysis of the system (1.1). At Section 4, we give a further discussion.

2. Preliminaries

In this section, we will give some previous results which will be useful for our main results.

Definition 2.1 (see [15]). Let 𝑈=(𝑢_𝑖𝑗)_𝑛×𝑛. We say that 𝑈≥0 (𝑈 is nonnegative), if all its entries 𝑢_𝑖𝑗 are real and nonnegative.
If 𝑈=(𝑢_𝑖𝑗)_𝑛×𝑛 and 𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛 are both nonnegative, we write 𝑈≥𝑊 if 𝑢_𝑖𝑗≥𝑤_𝑖𝑗 for all 𝑖 and 𝑗, and 𝑈>𝑊 if 𝑢_𝑖𝑗≥𝑤_𝑖𝑗 and 𝑈≠𝑊.

Definition 2.2 (see [15]). A matrix 𝑈=(𝑢_𝑖𝑗)_𝑛×𝑛 is said to be reducible if either (i)𝑛=1 and 𝑈=0; or(ii)𝑛≥2, there is a permutation matrix 𝑃,𝑃𝑈𝑃^𝑇=𝑈₁0𝑈₂𝑈₃, (2.1) where 𝑈₁ and 𝑈₃ are square matrices. Otherwise, 𝑈 is irreducible.

Let Γ(𝑈) denote the directed graph of (𝑢_𝑖𝑗)_𝑛×𝑛. We have the following proposition.

Proposition 2.3 (see [16]). For matrix 𝑈, one has (i)If 𝑈 is nonnegative, then the spectral radius 𝜌(𝑈) of 𝑈 is an eigenvalue, and 𝑈 has a nonnegative eigenvector corresponding to 𝜌(𝑈). (ii)If 𝑈 is nonnegative and irreducible, then 𝜌(𝑈) is a simple eigenvalue, and 𝑈 has a positive eigenvector 𝑥 corresponding to 𝜌(𝑈).(iii)If 0<𝑊<𝑈, then 𝜌(𝑊)≤𝜌(𝑈). Moreover, if 0<𝑊<𝑈 and 𝑊+𝑈 is irreducible, then 𝜌(𝑊)<𝜌(𝑈). (iv)If 𝑈 is nonnegative and irreducible, and 𝑊 is diagonal and positive (namely, all of its entries are positive), then 𝑈𝑊 is irreducible.(v)Matrix 𝑈 is irreducible if and only if Γ(𝑈) is strongly connected.

3. Mathematical Analysis

From the first and the fourth equation in (1.1), we knowlimsup_𝑡⟶∞𝑆_𝐻𝑖≤𝐴_𝐻𝑖𝜇_𝐻𝑖,limsup_𝑡⟶∞𝑆_𝑉𝑖≤𝐴_𝑉𝑖𝜇_𝑉𝑖. (3.1) For each 𝑖, adding the five equations in (1.1), we obtain𝑆_𝐻𝑖+𝐼_𝐻𝑖+𝑅_𝐻𝑖+𝑆_𝑉𝑖+𝐼_𝑉𝑖′=𝐴_𝐻𝑖+𝐴_𝑉𝑖−𝜇_𝐻𝑖𝑆_𝐻𝑖+𝐼_𝐻𝑖+𝑅_𝐻𝑖−𝜇_𝑉𝑖𝑆_𝑉𝑖+𝐼_𝑉𝑖≤𝐴_𝐻𝑖+𝐴_𝑉𝑖−𝜇^∗_𝑖𝑆_𝐻𝑖+𝐼_𝐻𝑖+𝑅_𝐻𝑖+𝑆_𝑉𝑖+𝐼_𝑉𝑖, (3.2) where 𝜇^∗_𝑖=min{𝜇_𝐻𝑖,𝜇_𝑉𝑖}. Thus,limsup_𝑡⟶∞𝑆_𝐻𝑖+𝐼_𝐻𝑖+𝑅_𝐻𝑖+𝑆_𝑉𝑖+𝐼_𝑉𝑖≤𝐴_𝐻𝑖+𝐴_𝑉𝑖𝜇^∗_𝑖. (3.3) Before going into any detail, we simplify the system. For each 𝑖-group, since the variable 𝑅_𝐻𝑖 dose not appear in the first two and the last two equations of (1.1), it suffices to consider the following reduced system:𝑆^′_𝐻𝑖=𝐴_𝐻𝑖−^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖𝑆_𝐻𝑖,𝐼^′_𝐻𝑖=^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖,𝑆^′_𝑉𝑖=𝐴_𝑉𝑖−^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝑆_𝑉𝑖,𝐼^′_𝑉𝑖=^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝐼_𝑉𝑖, (3.4) where 𝑖=1,2,…,𝑛, in the feasible region𝐷=(𝑆,𝐼)∈𝑅₊^4𝑛∣𝑆_𝐻𝑖≤𝐴_𝐻𝑖𝜇_𝐻𝑖,𝑆_𝑉𝑖≤𝐴_𝑉𝑖𝜇_𝑉𝑖,𝑆_𝐻𝑖+𝐼_𝐻𝑖+𝑆_𝑉𝑖+𝐼_𝑉𝑖≤𝐴_𝐻𝑖+𝐴_𝑉𝑖𝜇^∗_𝑖,𝑖=1,2,…,𝑛, (3.5) where 𝑆=(𝑆_𝐻,𝑆_𝑉), 𝐼=(𝐼_𝐻,𝐼_𝑉), 𝑆_𝐻=(𝑆_𝐻1,…,𝑆_𝐻𝑛), 𝑆_𝑉=(𝑆_𝑉1,…,𝑆_𝑉𝑛), 𝐼_𝐻=(𝐼_𝐻1,…,𝐼_𝐻𝑛) and 𝐼_𝑉=(𝐼_𝑉1,…,𝐼_𝑉𝑛). It can be verified that 𝐷 is positively invariant with respect to system (3.4). Behaviors of 𝑅_𝐻𝑖 can then be determined from the third equation in (1.1). Our results in this paper will be stated for system (3.4) in 𝐷 and can be translated straightforwardly to system (1.1). Let 𝐷^∘ denote the interior of 𝐷.

An equilibrium (𝑆,𝐼) of (3.4) satisfies𝐴_𝐻𝑖−^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖𝑆_𝐻𝑖=0,_𝑛𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖𝐴=0,_𝑉𝑖−^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝑆_𝑉𝑖=0,_𝑛𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝐼_𝑉𝑖=0, (3.6) where 𝑖=1,2,…,𝑛. It is easy to see that the disease-free equilibrium denoted by 𝐸⁰=(𝑆⁰,𝐼⁰) exists for all positive parameter values, where 𝑆⁰_𝐻𝑖=𝐴_𝐻𝑖/𝜇_𝐻𝑖, 𝑆⁰_𝑉𝑖=𝐴_𝑉𝑖/𝜇_𝑉𝑖 and 𝐼⁰_𝐻𝑖=𝐼⁰_𝑉𝑖=0, 𝑖=1,2,…,𝑛.

Denote𝑀(𝑆)=0𝑀_𝐻(𝑆)𝑀_𝑉(𝑆)0, (3.7) where 𝑀_𝐻(𝑆)=(𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖/𝜇_𝐻𝑖+𝛾_𝐻𝑖)_𝑛×𝑛, 𝑀_𝑉(𝑆)=(𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖/𝜇_𝑉𝑖)_𝑛×𝑛. We also denote 𝑀_𝐻(𝑆⁰_𝐻)=𝑀_𝐻0 and 𝑀_𝑉(𝑆⁰_𝑉)=𝑀_𝑉0𝑀₀=0𝑀_𝐻0𝑀_𝑉00. (3.8) We know that for all 𝑆∈𝐷, 𝑆≤𝑆⁰, so for all 𝑆∈𝐷, 𝑀(𝑆)≤𝑀₀. We define the basic reproduction number 𝑅₀ as the spectral radius of 𝑀₀; that is 𝑅₀=𝜌(𝑀₀). We set𝐵_𝐻=𝛽_𝐻11𝛽_𝐻21⋯𝛽_𝐻𝑛1𝛽_𝐻12𝛽_𝐻22⋯𝛽_𝐻𝑛2𝛽⋮⋮⋱⋮_𝐻1𝑛𝛽_𝐻2𝑛⋯𝛽_𝐻𝑛𝑛𝐵_𝑉,=𝛽_𝑉11𝛽_𝑉21⋯𝛽_𝑉𝑛1𝛽_𝑉12𝛽_𝑉22⋯𝛽_𝑉𝑛2𝛽⋮⋮⋱⋮_𝑉1𝑛𝛽_𝑉2𝑛⋯𝛽_𝑉𝑛𝑛,𝐵_𝑀=0𝐵_𝐻𝐵_𝑉0. (3.9)(3.10)

Theorem 3.1. Assume that 𝐵_𝐻, 𝐵_𝑉, 𝐵_𝑀 are irreducible.
(1)If 𝑅₀≤1, then the disease-free equilibrium 𝐸⁰ of system (3.4) is globally asymptotically stable in 𝐷.(2)If 𝑅₀>1, then 𝐸⁰ is unstable and system (3.4) is uniformly persistent in 𝐷^∘.

Proof. Since 𝐵_𝑀 is irreducible and nonnegative, we know that 𝑀(𝑆) and 𝑀₀ are irreducible and nonnegative. Therefore, by Proposition 2.3(ii), there exists a left eigenvector 𝜔=(𝜔_𝐻,𝜔_𝑉)>0 of 𝑀₀ corresponding to 𝜌(𝑀₀), where 𝜔_𝐻=(𝜔_𝐻1,𝜔_𝐻2,…,𝜔_𝐻𝑛), 𝜔_𝑉=(𝜔_𝑉1,𝜔_𝑉2,…,𝜔_𝑉𝑛); that is, 𝜔𝜌(𝑀₀)=𝜔𝑀₀. Define 𝐿=^𝑛_𝑖=1𝜔_𝐻𝑖𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖+𝜔_𝑉𝑖𝜇_𝑉𝑖𝐼_𝑉𝑖. (3.11) Denote the transpose of 𝐼 as 𝐼^𝑇. Differentiating 𝐿 along the solution of system (3.4), we obtain 𝐿^′=^𝑛_𝑖=1𝜔_𝐻𝑖𝜇_𝐻𝑖+𝛾_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖+𝜔_𝑉𝑖𝜇_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝐼_𝑉𝑖=^𝑛_𝑖=1𝜔_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗𝜇_𝐻𝑖+𝛾_𝐻𝑖−𝐼_𝐻𝑖+𝜔_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗𝜇_𝑉𝑖−𝐼_𝑉𝑖=𝜔𝑀(𝑆)𝐼^𝑇−𝐼^𝑇≤𝜔𝑀₀𝐼^𝑇−𝐼^𝑇=𝜌𝑀₀−1𝜔𝐼^𝑇≤0. (3.12) Therefore, we obtain(i)if 𝑅₀<1, 𝐿′=0⇔𝐼=0;(ii)if 𝑅₀=1, 𝐿′=0⇔𝑆=𝑆⁰ or 𝐼=0.Thus, we know that the singleton {𝐸⁰} is the only compact invariant subset of {𝐿′=0}. By LaSalle’s Invariance Principle [17], 𝐸⁰ is globally asymptotically stable in 𝐷, if 𝑅₀≤1. If 𝑅₀>1 and 𝐼>0, it is easy to see that 𝜔𝑀₀𝐼^𝑇−𝐼^𝑇=𝜌𝑀₀−1𝜔𝐼^𝑇>0. (3.13) Then, according to continuity, there exists a neighborhood 𝔅(𝐸⁰) of 𝐸⁰, 𝔅(𝐸⁰)⊆𝐷, such that for all (𝑆,𝐼)∈𝔅(𝐸⁰)𝐿′=𝜔𝑀(𝑆)𝐼^𝑇−𝐼^𝑇>0. (3.14) This implies that 𝐸⁰ is unstable. Using a uniform persistence result from [18] and a similar argument as in the proof of Proposition 3.3 of [19], we know that, when 𝑅₀>1, the instability of 𝐸⁰ implies the uniform persistence of (3.4). The proof is complete.

Uniform persistence of (3.4), together with uniform boundedness of solutions in 𝐷^∘, implies the existence of an equilibrium of system (3.4) in 𝐷^∘ [20, 21].

Corollary 3.2. Assume 𝐵_𝐻, 𝐵_𝑉, and 𝐵_𝑀 are irreducible. If 𝑅₀>1, then (3.4) has at least one endemic equilibrium.

Denote the endemic equilibrium by 𝐸^∗=(𝑆^∗,𝐼^∗), where 𝑆^∗_𝐻𝑖,𝑆^∗_𝑉𝑖,𝐼^∗_𝐻𝑖,𝐼^∗_𝑉𝑖>0, 𝑖=1,2,…,𝑛. One has the following result on the endemic equilibrium 𝐸^∗.

Theorem 3.3. Assume that 𝐵_𝐻, 𝐵_𝑉, and 𝐵_𝑀 are irreducible. If 𝑅₀>1, then the endemic equilibrium 𝐸^∗ of system (3.4) is globally asymptotically stable in 𝐷^∘.

Proof. The uniqueness of endemic equilibrium is obvious in 𝐷^∘, if we prove that the endemic equilibrium 𝐸^∗ is globally stable when 𝑅₀>1. We denote 𝛽_𝐻𝑖𝑗=𝛽_𝐻𝑖𝑗𝑆^∗_𝐻𝑖𝐼^∗_𝑉𝑗, 𝛽_𝑉𝑖𝑗=𝛽_𝑉𝑖𝑗𝑆^∗_𝑉𝑖𝐼^∗_𝐻𝑗𝐵=^{𝑛𝑘,𝑗≠1}𝛽_𝑉1𝑗𝛽_𝐻1𝑘−^𝑛_𝑘≠1𝛽_𝑉21𝛽_𝐻𝑘1⋯−^𝑛_𝑘≠1𝛽_𝑉𝑛1𝛽_𝐻𝑘1−^𝑛_𝑘≠2𝛽_𝑉12𝛽_𝐻𝑘2^𝑛_{𝑘,𝑗≠2}𝛽_𝑉2𝑗𝛽_𝐻2𝑘⋯−^𝑛_𝑘≠2𝛽_𝑉𝑛2𝛽_𝐻𝑘2−⋮⋮⋱⋮_{𝑛𝑘≠𝑛}𝛽_𝑉1𝑛𝛽_𝐻𝑘𝑛−^𝑛_𝑘≠𝑛𝛽_𝑉2𝑛𝛽_𝐻𝑘2⋯^𝑛_{𝑘,𝑗≠𝑛}𝛽_𝑉𝑛𝑗𝛽_𝐻𝑛𝑘. (3.15) It is easy to see that 𝐵=^{𝑛𝑘,𝑗≠1}𝛽_𝑉1𝑗𝛽_𝐻1𝑘00⋯0_{𝑛𝑘,𝑗≠2}𝛽_𝑉2𝑗𝛽_𝐻2𝑘⋯0⋮⋮⋱⋮00⋯^𝑛_{𝑘,𝑗≠𝑛}𝛽_𝑉𝑛𝑗𝛽_𝐻𝑛𝑘−^𝑛𝑘≠1𝛽_𝐻𝑘100⋯0_𝑛𝑘≠2𝛽_𝐻𝑘2⋯0⋮⋮⋱⋮00⋯^𝑛_𝑘≠𝑛𝛽_𝐻𝑘𝑛0𝛽_𝑉21⋯𝛽_𝑉𝑛1𝛽_𝑉120⋯𝛽_𝑉𝑛2⋮⋮⋱⋮𝛽_𝑉1𝑛𝛽_𝑉2𝑛.⋯0 (3.16) Since 𝐵_𝐻 is irreducible and nonnegative, we get ∑^𝑛_𝑘≠𝑗𝛽_𝐻𝑘𝑗≠0, 𝑗=1,2,…,𝑛. Together with 𝐵_𝑉 being irreducible and nonnegative, by Proposition 2.3(iv), we know that 𝐵 is irreducible. Let 𝐶_𝑖𝑗 denote the cofactor of the (𝑖,𝑗) entry of 𝐵. According to Lemma  2.1 in [2], we have that the equation 𝐵𝑣=0 has a positive solution 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛), where 𝑣_𝑖=𝐶_𝑖𝑖>0 for 𝑖=1,2,…,𝑛. Define a Lyapunov function as follows: 𝑉=^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘𝑆_𝐻𝑖−𝑆^∗_𝐻𝑖ln𝑆_𝐻𝑖+𝐼_𝐻𝑖−𝐼^∗_𝐻𝑖ln𝐼_𝐻𝑖+^𝑛_𝑘=1𝛽_𝐻𝑖𝑘𝑆_𝑉𝑖−𝑆^∗_𝑉𝑖ln𝑆_𝑉𝑖+𝐼_𝑉𝑖−𝐼^∗_𝑉𝑖ln𝐼_𝑉𝑖. (3.17) Together with (3.6), we get the derivative of 𝑉 along the solution of system (3.4) 𝑉′=^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘𝐴_𝐻𝑖−𝜇_𝐻𝑖𝑆_𝐻𝑖−𝑆^∗_𝐻𝑖𝑆_𝐻𝑖𝐴_𝐻𝑖−^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖𝑆_𝐻𝑖−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖−𝐼^∗_𝐻𝑖𝐼_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗−𝜇_𝐻𝑖+𝛾_𝐻𝑖𝐼_𝐻𝑖+^𝑛_𝑘=1𝛽_𝐻𝑖𝑘𝐴_𝑉𝑖−𝜇_𝑉𝑖𝑆_𝑉𝑖−𝑆^∗_𝑉𝑖𝑆_𝑉𝑖𝐴_𝑉𝑖−^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝑆_𝑉𝑖−𝜇_𝑉𝑖𝐼_𝑉𝑖−𝐼^∗_𝑉𝑖𝐼_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗−𝜇_𝑉𝑖𝐼_𝑉𝑖,=^𝑛_𝑖,𝑘=1𝑣_𝑖𝛽_𝑉𝑖𝑘𝜇_𝐻𝑖𝑆^∗_𝐻𝑖𝑆2−^𝐻𝑖𝑆^∗_𝐻𝑖−𝑆^∗_𝐻𝑖𝑆_𝐻𝑖+𝛽_𝐻𝑖𝑘𝜇_𝑉𝑖𝑆^∗_𝑉𝑖𝑆2−^𝑉𝑖𝑆^∗_𝑉𝑖−𝑆^∗_𝑉𝑖𝑆_𝑉𝑖+^𝑛_𝑖,𝑘=1𝑣_𝑖𝛽_𝑉𝑖𝑘2^𝑛_𝑗=1𝛽_𝐻𝑖𝑗−𝑆^∗_𝐻𝑖𝑆_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗+^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝐼_𝑉𝑗𝐼^∗_𝑉𝑗−𝐼_𝐻𝑖𝐼^∗_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗−𝐼^∗_𝐻𝑖𝐼_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗+^𝑛_𝑖,𝑘=1𝑣_𝑖𝛽_𝐻𝑖𝑘2^𝑛_𝑗=1𝛽_𝑉𝑖𝑗−𝑆^∗_𝑉𝑖𝑆_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗+^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝐼_𝐻𝑗𝐼^∗_𝐻𝑗−𝐼_𝑉𝑖𝐼^∗_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗−𝐼^∗_𝑉𝑖𝐼_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗. (3.18) According to (𝑥₁/𝑥₂)+(𝑥₂/𝑥₁)≥2 for each 𝑥₁,𝑥₂>0, with equality holding if and only if 𝑥₁=𝑥₂, we have 𝜇_𝐻𝑖𝑆^∗_𝐻𝑖𝑆2−^∗𝐻𝑖𝑆_𝐻𝑖−𝑆_𝐻𝑖𝑆^∗_𝐻𝑖𝜇≤0,_𝑉𝑖𝑆^∗_𝑉𝑖𝑆2−^∗𝑉𝑖𝑆_𝑉𝑖−𝑆_𝑉𝑖𝑆^∗_𝑉𝑖≤0, (3.19) where 𝑖=1,2,…,𝑛 and equalities hold, respectively, if and only if 𝑆_𝐻𝑖=𝑆^∗_𝐻𝑖,𝑆_𝑉𝑖=𝑆^∗_𝑉𝑖,𝑖=1,2,…,𝑛. (3.20) Hence, 𝑉^′≤^𝑛_𝑖,𝑘=1𝑣_𝑖𝛽_𝑉𝑖𝑘2^𝑛_𝑗=1𝛽_𝐻𝑖𝑗−𝑆^∗_𝐻𝑖𝑆_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗+^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝐼_𝑉𝑗𝐼^∗_𝑉𝑗−𝐼_𝐻𝑖𝐼^∗_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗−𝐼^∗_𝐻𝑖𝐼_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆_𝐻𝑖𝐼_𝑉𝑗+^𝑛_𝑖,𝑘=1𝑣_𝑖𝛽_𝐻𝑖𝑘2^𝑛_𝑗=1𝛽_𝑉𝑖𝑗−𝑆^∗_𝑉𝑖𝑆_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗+^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝐼_𝐻𝑗𝐼^∗_𝐻𝑗−𝐼_𝑉𝑖𝐼^∗_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗−𝐼^∗_𝑉𝑖𝐼_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝑆_𝑉𝑖𝐼_𝐻𝑗=^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝐼_𝑉𝑗𝐼^∗_𝑉𝑗−𝐼_𝐻𝑖𝐼^∗_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗+^𝑛_𝑘=1𝛽_𝐻𝑖𝑘^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝐼_𝐻𝑗𝐼^∗_𝐻𝑗−𝐼_𝑉𝑖𝐼^∗_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗+^𝑛_{𝑖,𝑗,𝑘=1}𝑣_𝑖𝛽_𝑉𝑖𝑘𝛽_𝐻𝑖𝑗𝑆4−^∗𝐻𝑖𝑆_𝐻𝑖−𝑆^∗_𝑉𝑖𝑆_𝑉𝑖−𝐼^∗_𝐻𝑖𝑆_𝐻𝑖𝐼_𝑉𝑗𝐼_𝐻𝑖𝑆^∗_𝐻𝑖𝐼^∗_𝑉𝑗−𝐼^∗_𝑉𝑖𝑆_𝑉𝑖𝐼_𝐻𝑗𝐼_𝑉𝑖𝑆^∗_𝑉𝑖𝐼^∗_𝐻𝑗=∶𝐾₁+𝐾₂. (3.21) We first show 𝐾₁≡0 for all (𝑆,𝐼)∈𝐷^∘. It follows from 𝐵𝑣=0 that ^𝑛_𝑘,𝑗=1𝛽_𝑉𝑖𝑗𝛽_𝐻𝑖𝑘𝑣_𝑖=^𝑛_𝑘,𝑗=1𝛽_𝑉𝑗𝑖𝛽_𝐻𝑘𝑖𝑣_𝑗, (3.22)𝑖=1,2,…,𝑛. This implies that ^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝐼_𝑉𝑗𝐼^∗_𝑉𝑗=^𝑛_𝑗=1𝐼_𝑉𝑗𝐼^∗_𝑉𝑗^𝑛𝑛_𝑘=1𝑖=1𝑣_𝑖𝛽_𝑉𝑖𝑘𝛽_𝐻𝑖𝑗=^𝑛_𝑗=1𝐼_𝑉𝑗𝐼^∗_𝑉𝑗𝑣_𝑗^𝑛_𝑘,𝑖=1𝛽_𝑉𝑗𝑘𝛽_𝐻𝑗𝑖. (3.23) Thus, ^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝐼_𝑉𝑗𝐼^∗_𝑉𝑗−^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝐻𝑖𝑘𝐼_𝑉𝑖𝐼^∗_𝑉𝑖^𝑛_𝑗=1𝛽_𝑉𝑖𝑗=0. (3.24) Similarly, we produce ^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝐻𝑖𝑘^𝑛_𝑗=1𝛽_𝑉𝑖𝑗𝐼_𝐻𝑗𝐼^∗_𝐻𝑗−^𝑛_𝑖=1𝑣_𝑖^𝑛_𝑘=1𝛽_𝑉𝑖𝑘𝐼_𝐻𝑖𝐼^∗_𝐻𝑖^𝑛_𝑗=1𝛽_𝐻𝑖𝑗=0. (3.25) Therefore, 𝐾₁≡0 for all (𝑆,𝐼)∈𝐷^∘.
Γ(𝐵) has vertices {1,2,…,𝑛} with a directed arc (𝑖,𝑗) from 𝑖 to 𝑗 if and only if ∑^𝑛_𝑘≠𝑗𝛽_𝑉𝑖𝑗𝛽_𝐻𝑘𝑗≠0. Since 𝐵 is irreducible, by a similar argument in [2], we obtain 𝐾₂≤0 for all (𝑆,𝐼)∈𝐷^∘. Furthermore, we produce that 𝑉′≤0. (3.26) If (3.20) holds, we have 𝐾₂=0⟺𝐼_𝐻𝑖=𝜂𝐼^∗_𝐻𝑖,𝐼_𝑉𝑖=𝜂𝐼^∗_𝑉𝑖,𝑖=1,2,…,𝑛, (3.27) where 𝜂 is arbitrary positive numbers.
According to (3.20) and (3.27), we know that 𝑉^′=0⇔𝑆_𝐻𝑖=𝑆^∗_𝐻𝑖,𝑆_𝑉𝑖=𝑆^∗_𝑉𝑖,𝐼_𝐻𝑖=𝜂𝐼^∗_𝐻𝑖,𝐼_𝑉𝑖=𝜂𝐼^∗_𝑉𝑖, 𝑖=1,2,…,𝑛. Substituting (3.20) and (3.27) into system (3.4), we obtain 0=𝐴_𝐻𝑖−𝜂^𝑛_𝑗=1𝛽_𝐻𝑖𝑗𝑆^∗_𝐻𝑖𝐼^∗_𝑉𝑗−𝜇_𝐻𝑖𝑆^∗_𝐻𝑖,0=𝐴_𝑉𝑖−𝜂^𝑛_𝑗=1𝛽_𝑉𝑖𝑆^∗_𝑉𝑖𝐼^∗_𝐻𝑗−𝜇_𝑉𝑖𝑆^∗_𝑉𝑖. (3.28) Since the right-hand side of (3.28) is strictly decreasing in 𝜂, by (3.6), we get that (3.28) holds if and only if 𝜂=1, namely, at 𝐸^∗. By LaSalle’s Invariance Principle, 𝐸^∗ is globally asymptotically stable in 𝐷^∘. The proof is complete.

From the process of proof of Theorem 3.3 and the definition of matrix 𝐵, it is easy to get a corollary as follows.

Corollary 3.4. Assume that 𝐵_𝑉, 𝐵_𝑀 are irreducible and ∑^𝑛_𝑘≠𝑗𝛽_𝐻𝑘𝑗≠0(or 𝐵_𝐻, 𝐵_𝑀 are irreducible and ∑^𝑛_𝑘≠𝑗𝛽_𝑉𝑘𝑗≠0), 𝑗=1,2,…,𝑛. If 𝑅₀>1, then the endemic equilibrium 𝐸^∗ of system (3.4) is globally asymptotically stable in 𝐷^∘.

4. Discussion

Taking the basic reproduction number 𝑅₀ as a sharp threshold parameter, we establish the global dynamics of system (3.4). Our result implies that, if 𝑅₀≤1, then the dengue disease always dies out in all groups; if 𝑅₀>1, then the dengue disease always persists at the unique endemic equilibrium level in all groups, independent of the initial condition.

Biologically, our assumptions in Theorem 3.3 and Corollary 3.4 mean that mosquitoes in 𝐼_𝑉𝑗 can infect ones in individuals 𝑆_𝐻𝑖 directly or indirectly; individuals in 𝐼_𝐻𝑗 can infect ones in mosquitoes 𝑆_𝑉𝑖 directly or indirectly, and individuals in 𝐼_𝐻𝑗 can infect ones in 𝑆_𝐻𝑖 by mosquitoes indirectly, respectively.