Global Stability of Multigroup Dengue Disease Transmission Model

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.


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 graphtheoretic 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 H i t , I H i t , R H i t , S V i t , I V i t , respectively.The model to be studied takes the following form: Here A H i and A V i represent the recruitment rate of the humans and the mosquitoes in the ith group, β H ij represents the contact rate between susceptible humans S H i and infectious mosquitoes I V j , β V ij is the contact rate between infected people I H j and susceptible mosquitoes S V i , μ H i and μ V i represent the death rate of the humans and the mosquitoes in the ith group, and γ H i represents the recovery rate of the humans in the ith group.All parameter values are assumed to be nonnegative and 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.

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

Mathematical Analysis
From the first and the fourth equation in 1.1 , we know lim sup For each i, adding the five equations in 1.1 , we obtain

Journal of Applied Mathematics
Before going into any detail, we simplify the system.For each i-group, since the variable R H i dose not appear in the first two and the last two equations of 1.1 , it suffices to consider the following reduced system:

3.4
where i 1, 2, . . ., n, in the feasible region It can be verified that D is positively invariant with respect to system 3.4 .Behaviors of R H i 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 satisfies where i 1, 2, . . ., n.It is easy to see that the disease-free equilibrium denoted by E 0 S 0 , I 0 exists for all positive parameter values, where S 0 where We know that for all S ∈ D, S ≤ S 0 , so for all S ∈ D, M S ≤ M 0 .We define the basic reproduction number R 0 as the spectral radius of M 0 ; that is R 0 ρ M 0 .We set Theorem 3.1.Assume that B H , B V , and B M are irreducible.
1 If R 0 ≤ 1, then the disease-free equilibrium E 0 of system 3.4 is globally asymptotically stable in D.
Proof.Since B M is irreducible and nonnegative, we know that M S and M 0 are irreducible and nonnegative.Therefore, by Proposition 2.3 ii , there exists a left eigenvector ω ω H , ω V > 0 of M 0 corresponding to ρ M 0 , where

3.11
Denote the transpose of I as I T .Differentiating L along the solution of system 3.4 , we obtain

3.12
Therefore, we obtain ii if R 0 1, L 0 ⇔ S S 0 or I 0.
Thus, we know that the singleton {E 0 } is the only compact invariant subset of {L 0}.By LaSalle's Invariance Principle 17 , E 0 is globally asymptotically stable in D, if R 0 ≤ 1.
If R 0 > 1 and I > 0, it is easy to see that ω M 0 I T − I T ρ M 0 − 1 ωI T > 0.

3.13
Then, according to continuity, there exists a neighborhood B E 0 of E 0 , B E 0 ⊆ D, such that for all S, I ∈ B E 0 L ω M S I T − I T > 0.

3.14
This implies that E 0 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 R 0 > 1, the instability of E 0 implies the uniform persistence of 3.4 .The proof is complete.
Uniform persistence of 3.4 , together with uniform boundedness of solutions in Proof.The uniqueness of endemic equilibrium is obvious in

3.15
It is easy to see that

3.16
Since B H is irreducible and nonnegative, we get n k / j β H kj / 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 v 1 , v 2 , . . ., v n , where v i C ii > 0 for i 1, 2, . . ., n. Define a Lyapunov function as follows: 3.17 Together with 3.6 , we get the derivative of V along the solution of system 3.4

3.18
According to x 1 /x 2 x 2 /x 1 ≥ 2 for each x 1 , x 2 > 0, with equality holding if and only if x 1 x 2 , we have where i 1, 2, . . ., n and equalities hold, respectively, if and only if

3.21
We first show K 1 ≡ 0 for all S, I ∈

3.28
Since the right-hand side of 3.28 is strictly decreasing in η, by 3.

Discussion
Taking the basic reproduction number R 0 as a sharp threshold parameter, we establish the global dynamics of system 3.4 .Our result implies that, if R 0 ≤ 1, then the dengue disease always dies out in all groups; if R 0 > 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 V j can infect ones in individuals S H i directly or indirectly; individuals in I H j can infect ones in mosquitoes S V i directly or indirectly, and individuals in I H j can infect ones in S H i by mosquitoes indirectly, respectively.
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.
One has the following result on the endemic equilibrium E * .Assume that B H , B V , and B M are irreducible.If R 0 > 1, then the endemic equilibrium E * of system 3.4 is globally asymptotically stable in • D 20, 21 .Corollary 3.2.Assume B H , B V , and B M are irreducible.If R 0 > 1, then 3.4 has at least one endemic equilibrium.Denote the endemic equilibrium by E * S * , I * , where S * • D.
B has vertices {1, 2, . . ., n} with a directed arc i, j from i to j if and only if n k / j β V ij β H kj / 0. Since B is irreducible, by a similar argument in 2 , we obtain K 2 ≤ 0 for all S, I ∈ .25 Therefore, K 1 ≡ 0 for all S, I ∈ • D.Γ

Corollary 3.4. Assume
6 , we get that 3.28 holds if and only if η 1, namely, at E * .By LaSalle's Invariance Principle, E 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.that B V and B M are irreducible and n k / j β H kj / 0 or B H , B M are irreducible and n k / j β V kj / 0 , j 1, 2, . . ., n.If R 0 > 1, then the endemic equilibrium E * is globally asymptotically stable in • D. * of system 3.4 is globally asymptotically stable in • D.