# Global Stability of Multigroup Dengue Disease Transmission Model

Authors: Deqiong Ding; Xueping Wang; Xiaohua Ding
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We investigate a class of multigroup dengue epidemic
model. We show that the global dynamics are determined by the basic reproductive
numberR0. 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.


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## Body

## 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 SHi(t), IHi(t), RHi(t), SVi(t), IVi(t), respectively. The model to be studied takes the following form:(1.1)SHi′=AHi-∑j=1nβHijSHiIVj-μHiSHi,IHi′=∑j=1nβHijSHiIVj-(μHi+γHi)IHi,RHi′=γHiIHi-μHiRHi,SVi′=AVi-∑j=1nβVijSViIHj-μViSVi,IVi′=∑j=1nβVijSViIHj-μViIVi,
where i=1,2,…,n. Here AHi and AVi represent the recruitment rate of the humans and the mosquitoes in the ith group, βHij represents the contact rate between susceptible humans SHi and infectious mosquitoes IVj, βVij is the contact rate between infected people IHj and susceptible mosquitoes SVi, μHi and μVi represent the death rate of the humans and the mosquitoes in the ith group, and γHi represents the recovery rate of the humans in the ith group. All parameter values are assumed to be nonnegative and AHi,AVi,μHi,μVi>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 Section2, 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]).
LetU=(uij)n×n. We say thatU≥0(Uis nonnegative), if all its entriesuijare real and nonnegative.
IfU=(uij)n×n and W=(wij)n×n are both nonnegative, we write U≥W if uij≥wij for all i and j, and U>W if uij≥wij and U≠W.Definition 2.2 (see [15]).
A matrixU=(uij)n×nis said to be reducible if either(i)
n=1 and U=0; or(ii)
n≥2, there is a permutation matrix P,(2.1)PUPT=(U10U2U3),
where U1 and U3 are square matrices. Otherwise, U is irreducible.LetΓ(U) denote the directed graph of (uij)n×n. We have the following proposition.Proposition 2.3 (see [16]).
For matrixU, one has (i)
IfU is nonnegative, then the spectral radius ρ(U) of U is an eigenvalue, and U has a nonnegative eigenvector corresponding to ρ(U).(ii)
IfU is nonnegative and irreducible, then ρ(U) is a simple eigenvalue, and U has a positive eigenvector x corresponding to ρ(U).(iii)
If0<W<U, then ρ(W)≤ρ(U). Moreover, if 0<W<U and W+U is irreducible, then ρ(W)<ρ(U).(iv)
IfU is nonnegative and irreducible, and W is diagonal and positive (namely, all of its entries are positive), then UW is irreducible.(v)
MatrixU 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 know(3.1)lim⁡⁡sup⁡t⟶∞SHi≤AHiμHi,lim⁡⁡sup⁡t⟶∞SVi≤AViμVi.
For each i, adding the five equations in (1.1), we obtain(3.2)(SHi+IHi+RHi+SVi+IVi)′=AHi+AVi-μHi(SHi+IHi+RHi)-μVi(SVi+IVi)≤AHi+AVi-μi*(SHi+IHi+RHi+SVi+IVi),
where μi*=min{μHi,μVi}. Thus,(3.3)lim⁡⁡sup⁡t⟶∞(SHi+IHi+RHi+SVi+IVi)≤AHi+AViμi*.
Before going into any detail, we simplify the system. For each i-group, since the variable RHi 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)SHi′=AHi-∑j=1nβHijSHiIVj-μHiSHi,IHi′=∑j=1nβHijSHiIVj-(μHi+γHi)IHi,SVi′=AVi-∑j=1nβVijSViIHj-μViSVi,IVi′=∑j=1nβVijSViIHj-μViIVi,
where i=1,2,…,n, in the feasible region(3.5)D={(S,I)∈R+4n∣SHi≤AHiμHi,SVi≤AViμVi,SHi+IHi+SVi+IVi≤AHi+AViμi*,i=1,2,…,n},
where S=(SH,SV), I=(IH,IV), SH=(SH1,…,SHn), SV=(SV1,…,SVn), IH=(IH1,…,IHn) and IV=(IV1,…,IVn). It can be verified that D is positively invariant with respect to system (3.4). Behaviors of RHi 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(3.6)AHi-∑j=1nβHijSHiIVj-μHiSHi=0,∑j=1nβHijSHiIVj-(μHi+γHi)IHi=0,AVi-∑j=1nβVijSViIHj-μViSVi=0,∑j=1nβVijSViIHj-μViIVi=0,
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 SHi0=AHi/μHi, SVi0=AVi/μVi and IHi0=IVi0=0, i=1,2,…,n.Denote(3.7)M(S)=(0MH(S)MV(S)0),
where MH(S)=(βHijSHi/μHi+γHi)n×n, MV(S)=(βVijSVi/μVi)n×n. We also denote MH(SH0)=MH0 and MV(SV0)=MV0(3.8)M0=(0MH0MV00).
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 set(3.9)BH=(βH11βH21⋯βHn1βH12βH22⋯βHn2⋮⋮⋱⋮βH1nβH2n⋯βHnn),(3.10)BV=(βV11βV21⋯βVn1βV12βV22⋯βVn2⋮⋮⋱⋮βV1nβV2n⋯βVnn),BM=(0BHBV0).Theorem 3.1.
Assume thatBH,  BV,  BM are irreducible.
(1)
IfR0≤1, then the disease-free equilibrium E0 of system (3.4) is globally asymptotically stable in D.(2)
IfR0>1, then E0 is unstable and system (3.4) is uniformly persistent in D°.Proof.
SinceBM 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=(ωH,ωV)>0 of M0 corresponding to ρ(M0), where ωH=(ωH1,ωH2,…,ωHn), ωV=(ωV1,ωV2,…,ωVn); that is, ωρ(M0)=ωM0. Define
(3.11)L=∑i=1n(ωHiμHi+γHiIHi+ωViμViIVi).
Denote the transpose of I as IT. Differentiating L along the solution of system (3.4), we obtain
(3.12)L′=∑i=1n{ωHiμHi+γHi[∑j=1nβHijSHiIVj-(μHi+γHi)IHi]+ωViμVi(∑j=1nβVijSViIHj-μViIVi)}=∑i=1n[ωHi(∑j=1nβHijSHiIVjμHi+γHi-IHi)+ωVi(∑j=1nβVijSViIHjμVi-IVi)]=ω(M(S)IT-IT)≤ω(M0IT-IT)=(ρ(M0)-1)ωIT≤0.
Therefore, we obtain(i)
ifR0<1, L′=0⇔I=0;(ii)
ifR0=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
(3.13)ω(M0IT-IT)=(ρ(M0)-1)ωIT>0.
Then, according to continuity, there exists a neighborhood 𝔅(E0) of E0, 𝔅(E0)⊆D, such that for all (S,I)∈𝔅(E0)(3.14)L′=ω(M(S)IT-IT)>0.
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.
AssumeBH,  BV, and  BM are irreducible. If  R0>1, then (3.4) has at least one endemic equilibrium.Denote the endemic equilibrium byE*=(S*,I*), where SHi*,SVi*,IHi*,IVi*>0, i=1,2,…,n. One has the following result on the endemic equilibrium E*.Theorem 3.3.
Assume thatBH,  BV, and  BM 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 inD°, if we prove that the endemic equilibrium E* is globally stable when R0>1. We denote β̅Hij=βHijSHi*IVj*, β̅Vij=βVijSVi*IHj*(3.15)B̅=(∑k,j≠1nβ̅V1jβ̅H1k-∑k≠1nβ̅V21β̅Hk1⋯-∑k≠1nβ̅Vn1β̅Hk1-∑k≠2nβ̅V12β̅Hk2∑k,j≠2nβ̅V2jβ̅H2k⋯-∑k≠2nβ̅Vn2β̅Hk2⋮⋮⋱⋮-∑k≠nnβ̅V1nβ̅Hkn-∑k≠nnβ̅V2nβ̅Hk2⋯∑k,j≠nnβ̅Vnjβ̅Hnk).
It is easy to see that
(3.16)B̅=(∑k,j≠1nβ̅V1jβ̅H1k0⋯00∑k,j≠2nβ̅V2jβ̅H2k⋯0⋮⋮⋱⋮00⋯∑k,j≠nnβ̅Vnjβ̅Hnk)-(∑k≠1nβ̅Hk10⋯00∑k≠2nβ̅Hk2⋯0⋮⋮⋱⋮00⋯∑k≠nnβ̅Hkn)(0β̅V21⋯β̅Vn1β̅V120⋯β̅Vn2⋮⋮⋱⋮β̅V1nβ̅V2n⋯0).
Since BH is irreducible and nonnegative, we get ∑k≠jnβ̅Hkj≠0, j=1,2,…,n. Together with BV being irreducible and nonnegative, by Proposition 2.3(iv), we know that B̅ is irreducible. Let Cij denote the cofactor of the (i,j) entry of B̅. According to Lemma  2.1 in [2], we have that the equation B̅v=0 has a positive solution v=(v1,v2,…,vn), where vi=Cii>0 for i=1,2,…,n. Define a Lyapunov function as follows:
(3.17)V=∑i=1nvi[∑k=1nβ̅Vik(SHi-SHi*ln⁡SHi+IHi-IHi*ln⁡IHi)+∑k=1nβ̅Hik(SVi-SVi*ln⁡SVi+IVi-IVi*ln⁡IVi)].
Together with (3.6), we get the derivative of V along the solution of system (3.4)
(3.18)V′=∑i=1nvi{∑k=1nβ̅Vik[AHi-μHiSHi-SHi*SHi(AHi-∑j=1nβHijSHiIVj-μHiSHi)-(μHi+γHi)IHi-IHi*IHi(∑j=1nβHijSHiIVj-(μHi+γHi)IHi)]+∑k=1nβ̅Hik[AVi-μViSVi-SVi*SVi(AVi-∑j=1nβVijSViIHj-μViSVi)-μViIVi-IVi*IVi(∑j=1nβVijSViIHj-μViIVi)]},=∑i,k=1nvi[β̅VikμHiSHi*(2-SHiSHi*-SHi*SHi)+β̅HikμViSVi*(2-SViSVi*-SVi*SVi)]+∑i,k=1nviβ̅Vik(2∑j=1nβ̅Hij-SHi*SHi∑j=1nβ̅Hij+∑j=1nβ̅HijIVjIVj*-IHiIHi*∑j=1nβ̅Hij-IHi*IHi∑j=1nβHijSHiIVj)+∑i,k=1nviβ̅Hik(2∑j=1nβ̅Vij-SVi*SVi∑j=1nβ̅Vij+∑j=1nβ̅VijIHjIHj*-IViIVi*∑j=1nβ̅Vij-IVi*IVi∑j=1nβVijSViIHj).
According to (x1/x2)+(x2/x1)≥2 for each x1,x2>0, with equality holding if and only if x1=x2, we have
(3.19)μHiSHi*(2-SHi*SHi-SHiSHi*)≤0,μViSVi*(2-SVi*SVi-SViSVi*)≤0,
where i=1,2,…,n and equalities hold, respectively, if and only if
(3.20)SHi=SHi*,SVi=SVi*,i=1,2,…,n.
Hence,
(3.21)V′≤∑i,k=1nviβ̅Vik(2∑j=1nβ̅Hij-SHi*SHi∑j=1nβ̅Hij+∑j=1nβ̅HijIVjIVj*-IHiIHi*∑j=1nβ̅Hij-IHi*IHi∑j=1nβHijSHiIVj)+∑i,k=1nviβ̅Hik(2∑j=1nβ̅Vij-SVi*SVi∑j=1nβ̅Vij+∑j=1nβ̅VijIHjIHj*-IViIVi*∑j=1nβ̅Vij-IVi*IVi∑j=1nβVijSViIHj)=∑i=1nvi[∑k=1nβ̅Vik(∑j=1nβ̅HijIVjIVj*-IHiIHi*∑j=1nβ̅Hij)+∑k=1nβ̅Hik(∑j=1nβ̅VijIHjIHj*-IViIVi*∑j=1nβ̅Vij)]+∑i,j,k=1nviβ̅Vikβ̅Hij(4-SHi*SHi-SVi*SVi-IHi*SHiIVjIHiSHi*IVj*-IVi*SViIHjIViSVi*IHj*)=:K1+K2.
We first show K1≡0 for all (S,I)∈D°. It follows from B̅v=0 that
(3.22)∑k,j=1nβ̅Vijβ̅Hikvi=∑k,j=1nβ̅Vjiβ̅Hkivj,i=1,2,…,n. This implies that
(3.23)∑i=1nvi∑k=1nβ̅Vik∑j=1nβ̅HijIVjIVj*=∑j=1nIVjIVj*∑k=1n∑i=1nviβ̅Vikβ̅Hij=∑j=1nIVjIVj*vj∑k,i=1nβ̅Vjkβ̅Hji.
Thus,
(3.24)∑i=1nvi∑k=1nβ̅Vik∑j=1nβ̅HijIVjIVj*-∑i=1nvi∑k=1nβ̅HikIViIVi*∑j=1nβ̅Vij=0.
Similarly, we produce
(3.25)∑i=1nvi∑k=1nβ̅Hik∑j=1nβ̅VijIHjIHj*-∑i=1nvi∑k=1nβ̅VikIHiIHi*∑j=1nβ̅Hij=0.
Therefore, K1≡0 for all (S,I)∈D°.
Γ(B̅) has vertices {1,2,…,n} with a directed arc (i,j) from i to j if and only if ∑k≠jnβ̅Vijβ̅Hkj≠0. Since B̅ is irreducible, by a similar argument in [2], we obtain K2≤0 for all (S,I)∈D°. Furthermore, we produce that
(3.26)V′≤0.
If (3.20) holds, we have
(3.27)K2=0⟺IHi=ηIHi*,IVi=ηIVi*,i=1,2,…,n,
where η is arbitrary positive numbers.
According to (3.20) and (3.27), we know that V′=0⇔SHi=SHi*,SVi=SVi*,IHi=ηIHi*,IVi=ηIVi*,  i=1,2,…,n. Substituting (3.20) and (3.27) into system (3.4), we obtain
(3.28)0=AHi-η∑j=1nβHijSHi*IVj*-μHiSHi*,0=AVi-η∑j=1nβViSVi*IHj*-μViSVi*.
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 Theorem3.3 and the definition of matrix B̅, it is easy to get a corollary as follows.Corollary 3.4.
Assume thatBV,  BM are irreducible and  ∑k≠jnβHkj≠0(or  BH,  BM are irreducible and  ∑k≠jnβVkj≠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 numberR0 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 Theorem3.3 and Corollary 3.4 mean that mosquitoes in IVj can infect ones in individuals SHi directly or indirectly; individuals in IHj can infect ones in mosquitoes SVi directly or indirectly, and individuals in IHj can infect ones in SHi by mosquitoes indirectly, respectively.



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