# Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission

Authors: Ming-Tao Li; Gui-Quan Sun; Juan Zhang; Zhen Jin
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

For a multigroup cholera model with indirect transmission, the infection for
a susceptible person is almost invariably transmitted by drinking contaminated
water in which pathogens,V. cholerae, are present. The basic reproduction
number ℛ0 is identified and global dynamics are completely determined by ℛ0.
It shows that ℛ0 is a globally threshold parameter in the sense that if it is less
than one, the disease-free equilibrium is globally asymptotically stable; whereas
if it is larger than one, there is a unique endemic equilibrium which is global
asymptotically stable. For the proof of global stability with the disease-free
equilibrium, we use the comparison principle; and for the endemic equilibrium
we use the classical method of Lyapunov function and the graph-theoretic approach.


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## Body

## 1. Introduction

Cholera, a waterborne gastroenteric infection, remains a significant threat to public health in the developing world. Outbreaks of cholera occur cyclically, usually twice per year in endemic areas, and the intensity of these outbreaks varies over longer periods [1]. Hence, in the last few decades, enormous attention is being paid to the cholera disease and several mathematical dynamic models have been developed to study the transmission of cholera [1–7]. In these papers, they consider the population is uniformly mixed, but many factors can lead to heterogeneity in a host population. So in this paper we divide different population into different groups, which can be divided geographically into communities, cities, and countries, to incorporate differential infectivity of multiple strains of the disease agent.In the case of cholera, the transmission usually occurs through ingestion of contaminated water or feces rather than through casual human-human contact [1]. Therefore, direct contact of healthy person with an infected person is not a risk for contracting infection, whereas a healthy person may contract infection by drinking contaminated water in which pathogens, V. cholerae, are present [2]. The members of this bacterial genus (V. cholerae) naturally colonize in lakes, rivers, and estuaries. Therefore we consider that cholera transmits to other individuals via bacteria in the aquatic environment and formulates a multi-group epidemic model for cholera. LetNbe the total population which is divided into four epidemiological compartments, susceptible compartmentS, infectious compartmentI, recovered compartment R, and vaccinated compartmentV. LetWbe the density of V. cholerae in the aquatic environment. As a consequence of the increase in the density of virulent V. cholerae in the aquatic environment, humans become infected and begin to shed increasing numbers of bacteria into the aquatic environment, further elevating bacterial density and exacerbating the outbreak [1]. The growth rate of density of bacteria in the aquatic environment is assumed to be proportional to the number of infectious individuals. We assume that the immunity induced by vaccination is perfect; therefore individuals in vaccinated individualsVcannot be infected. The model is called a multi-group cholera SIRVW epidemic model.In recent years, multi-group epidemic models have been used to describe the transmission dynamics of many infectious disease in heterogeneous individuals, such as HIV/AIDS [8], dengue [9], West-Nile virus [10], sexually transmitted diseases [11]; and so on. It is well known that global dynamics of multi-group models with higher dimensions, especially the global stability of the endemic equilibrium, is a very challenging problem. Lajmanovich and York [12] proved global stability of the unique endemic equilibrium by using a quadratic global Lyapunov function on a class of n-group SIS models for gonorrhea; Hethcote [13] proved global stability of the endemic equilibrium for multi-group SIR model without vital dynamics; Thieme [14] proved global stability of the endemic equilibrium of multi-group SEIRS model under certain restrictions. However, they only proved global stability of the endemic equilibrium for multi-group model under some special conditions. In 2006, Guo et al. [15] have first succeeded to establish the complete global dynamics for a multi-group SIR model, by making use of the theory of non-negative matrices, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory. By using the results or ideas of [15], the papers [16, 17] proved the global stability of the endemic equilibrium for multi-group model with nonlinear incidence rates and the papers [18, 19] proved the global stability of the endemic equilibrium for multi-group model with distributed delays.Distinguishing from these multi-group models with direct transmission from person to person, a multi-group cholera model with indirect transmission from the bacteria of the aquatic environment to person is proposed in this paper. We prove that the disease-free equilibrium is globally asymptotically stable ifℛ0<1, while an endemic equilibrium exists uniquely and is globally asymptotically stable if ℛ0>1.The organization of this paper is as follows. In Section2, we construct a multi-group cholera epidemiological and give some dynamic analysis on the disease-free equilibrium and the endemic equilibrium. An example is given in Section 3 and some conclusions are included in Section 4.


## 2. Mathematical Modeling and Analysis

For a multi-group epidemic model with cholera, the population of human is divided intondiscrete groups, wheren∈ℕ. Let Si(t), Ii(t), Ri(t), and Vi(t) be the numbers of susceptible, infectious, recovered, and vaccinated individuals in group i=1,2,…,nat timet, respectively. LetWi(t)be the density of bacteria in the aquatic environment in groupi=1,2,…,nat timet. Based on the assumptions in Section 1, the disease transmission rate of cholera between compartmentsSiandWjis denoted byβij, which means the susceptible individuals in theith group can contact the bacteria of the aquatic environment in thejth (j=1,2,…,n) group. So the new infection that occurred in theith group is given by∑j=1nβijSiWj. The recruitment rate of individuals intoSi(t)compartment with theith group is given by a constantAi. Within theith group, it is assumed that natural death of human isdi. A simple immunization policy is considered where the vaccination rate inSi(t)compartment is given by a constantγi and the losing immunity rate from vaccination individuals isλi. We assume that individuals inIi(t)compartment recover with a rate constantri. InWi(t)compartment, the brucella shedding rate fromIi(t)compartment iski, and the decaying rate of brucella isδi. So a general multi-group SIRVW epidemic model is described by the following system of differential equations:
(1)dSidt=Ai-(di+γi)Si+λiVi-∑j=1nβijSiWj,dIidt=∑j=1nβijSiWj,-(di+ri)Ii,dRidt=riIi-diRi,dVidt=γiSi-(λi+di)Vi,dWidt=kiIi-δiWi,i=1,2,…,n.The parametersAi, di, λi, γi, ki, andδiare positive for alli=1,2,…,n, which is made for the biological justification. And we assume thatβijis nonnegative for alli,j=1,2,…,nandn-square matrixB=(βij)1≤i,j≤nis irreducible, which implies that every pair of groups is joined by an infectious path so that the presence of an infectious individual in the first group can cause infection in the second group.Observe that the variableRidoes not appear in the first and last two equations of system (1); this allows us to consider the following reduced system:
(2)dSidt=Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj,dIidt=∑j=1n‍βijSiWj-(di+ri)Ii,dVidt=γiSi-(λi+di)Vi,dWidt=kiIi-δiWi,i=1,2,…,n.For each groupi, adding the four equations in system (2) gives
(3)d(Si+Ii+Vi)dt=Ai-di(Si+Ii+Vi)-riIid(Si+Ii+Vi)dt≤(Ai-di(Si+Ii+Vi)),then it follows that(4)limt→∞sup(Si+Ii+Vi)≤Aidi,limt→∞supWi≤kiAidiδi.Therefore, omega limit sets of system (2) are contained in the following bounded region in the nonnegative cone ofℝ4n:
(5)X={(Si,Ii,Vi,Wi)∣Si,Ii,Vi,Wi≥0,0≤(Si+Ii+Vi)kiAidiδi≤Aidi,Wi≤kiAidiδi,i=1,2,…,n}.It can be verified that regionXis positively invariant with respect to system (2). System (2) always has a disease-free equilibrium
(6)P0=(S10,0,V10,0,…,Si0,0,Vi0,0,…,Sn0,0,Vn0,0)on the boundary ofX, where
(7)Si0=Ai(λi+di)di(λi+di+γi),Vi0=Aiγidi(λi+di+γi).


### 2.1. The Basic Reproduction Number

According to the next generation matrix formulated in papers [20–22], we define the basic reproduction numberℛ0of system (2). In order to formulateℛ0, we order the infected variables first by disease state and then by group, that is,
(8)I1,W1,I2,W2,…,In,Wn.Consider the following auxiliary system:(9)dIidt=∑j=1n‍βijSiWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.Follow the recipe from van den Driessche and Watmough [21] to obtain
(10)F=(0β11S100β12S10⋯0β1nS100000⋯000β21S200β22S20⋯0β2nS200000⋯00⋮⋮⋮⋮⋮⋮0βn1Sn00βn2Sn0⋯0βnnSn00000⋯00)2n×2n,V=(d1+r1000⋯00-k1δ100⋯0000d2+r20⋯0000-k2δ2⋯00⋮⋮⋮⋮⋮⋮0000⋯dn+rn00000⋯-k1δn)2n×2n.We can get the inverse ofV, which equals(11)V-1=(1d1+r1000⋯00k1δ1(d1+r1)1δ100⋯00001d2+r20⋯0000k2δ2(d2+r2)1δ2⋯00⋮⋮⋮⋮⋮⋮0000⋯1dn+rn00000⋯knδn(dn+rn)1δn)2n×2n.Thus, the next generation matrix isFV-1,
(12)FV-1=(A11⋯A1nB11⋯B1n⋮⋮⋮⋮An1⋯AnnBn1⋯Bnn0⋯00⋯0⋮⋮⋮⋮0⋯00⋯0)2n×2n,A=(A11A12⋯A1nA21A22⋯A2n⋮⋮⋮An1An2⋯Ann)n×n.So we can calculate the basic reproduction number of system (2),
(13)ℛ0=ρ(FV-1)=ρ(A),where(14)Aij=βijkjSi0δj(dj+rj),Si0=Ai(λi+di)di(λi+di+γi),i=1,2,…,n,andρdenotes the spectral radius. As we will show,ℛ0is the key threshold parameters whose values completely characterize the global dynamics of system (2).


### 2.2.  Global Stability of the Disease-Free Equilibrium of System (2)

For the disease-free equilibriumP0of system (2), we have the following property.Theorem 1.
Ifℛ0<1, the disease-free equilibriumP0of system (2) is globally asymptotically stable in the regionX.Proof.
LetM=F-V, and defines(M)=max{Reλ:λis an eigenvalue ofM}, sos(M)is a simple eigenvalue ofMwith a positive eigenvector [23]. By Theorem  2 in [21], there hold two equivalences:
(15)ℛ0>1⟺s(M)>0,ℛ0<1⟺s(M)<0.
To prove the locally stability of disease-free equilibrium, we check the hypotheses (A1)–(A5) in [21]. Hypotheses (A1)–(A4) are easily verified, while (A5) is satisfied if all eigenvalues of the4n×4nmatrix
(16)J|P0=(M0J3J4)4n×4n
have negative real parts, whereJ3=-F,(17)J4=(-(d1+γ1)λ100⋯00γ1-(d1+λ1)00⋯0000-(d2+γ2)λ2⋯0000γ2-(d2+λ2)⋯00⋮⋮⋮⋮⋮⋮0000⋯-(dn+γn)λn0000⋯γn-(dn+λn))2n×2n.
Calculate the eigenvalues ofJ4:
(18)s(J4)=max{-d1,…,-dn,-(d1+λ1+γ1),…,-(dn+λn+γn)}<0.
Ifℛ0<1, thens(M)<0ands(J|P0)<0, and the disease-free equilibriumP0of (2) is locally asymptotically stable.
Now we will prove that the disease-free equilibriumP0of system (2) is globally attractive whenℛ0<1. From the third equation of system (2), we have
(19)dVidt=γiSi-(λi+di)VidVidt=γi(Ni-(Ii+Vi))-(λi+di)VidVidt≤γiAidi-(λi+γi+di)Vi.
So we can have that, for a small enough positive numberϵ1, there exists ti>0, i=1,2,…,n such that for allt>ti,
(20)Vi≤Aiγidi(λi+γi+di)+ϵ1=Vi0+ϵ1.
Also from the equations of system (2), we have
(21)dSidt=Ai+λiVi-(γi+di)Si-∑j=1n‍βijSiWjdSidt≤Ai+λi(Vi0+ϵ1)-(di+γi)So.
Then(22)limt→∞supSi=Ai+λi(Vi0+ϵ1)di+γi=Si0+ϵ2,(ϵ2=λiϵ1di+γi).
From system (9) andSi≤Si0+ϵ2with allt>ti. Thus, whent>ti, we derive
(23)dIidt=(Si0+ϵ2)∑j=1n‍βijWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.
Consider the following auxiliary system(24)dIi′dt=(Si0+ϵ2)∑j=1n‍βijWj′-(di+ri)Ii′,dWi′dt=kiIi′-δiWi′,i=1,2,…,n.
LetM0be the matrix defined by
(25)M0=(0β110β12⋯0β1n0000⋯000β210β22⋯0β2n0000⋯00⋮⋮⋮⋮⋮⋮0βn10βn2⋯0βnn0000⋯00)2n×2n,
and setM1=M+ϵ2M0. It follows from Theorem  2 in [21] thatℛ0<1if and only ifs(M)<0. Thus, there exists anϵ2>0small enough such thats(M1)<0. Using the Perron-Frobenius theorem, all eigenvalues of the matrixM1have negative real parts whens(M1)<0. Therefore it has
(26)(I1′(t),W1′(t),I2′(t),W2′(t),…,In′(t),Wn′(t))⟶(0,0,0,0,…,0,0),t⟶∞,
which implies that the zero solution of system (24) is globally asymptotically stable. Using the comparison principle of Smith and Waltman [23], we know that
(27)(I1(t),W1(t),I2(t),W2(t),…,In(t),Wn(t))⟶(0,0,0,0,…,0,0),t⟶∞.
By the theory of asymptotic autonomous system of Thieme [24], it is also known that
(28)(S1(t),V1(t),…,Sn(t),Vn(t))⟶(S1(0),V1(0),…,Sn(0),Vn(0)),t⟶∞.
SoP0is globally attractive whenℛ0<1. It follows that the disease-free equilibriumP0of (2) is globally asymptotically stable whenℛ0<1. This completes the proof.


### 2.3. The Uniform Persistence and Unique Positive Solution of System (2)

In this section, we give the proof of the uniform persistence and the unique positive solution of system (2). Define
(29)X0={(Si,Ii,Vi,Wi)∈X∣Ii,Wi>0,i=1,2,…,n},∂X0=X∣X0.Theorem 2.
Whenℛ0>1, there exists a positive constantε1such that when∥Ii(0)∥<ε1,∥Wi(0)∥<ε1for(Si(0),Ii(0),Vi(0),Wi(0))∈X0(30)limsupt→∞max{Ii(t),Wi(t)}>ε1,i=1,2,…,n.Proof.
Consider the following system:(31)dSidt=Ai+λiVi-(γi+di)Si,dVidt=γiSi-(λi+di)Vi,i=1,2,…,n.
Using Corollary  3.2 in Zhao and Jing [25], it then follows that system (31) has a unique positive equilibrium(S10,V10,…,Sn0,Vn0) which is globally asymptotically stable.
As toℛ0>1⇔s(M)>0, chooseε>0small enough such thats(M2)>0, whereM2=M-εM0. Let us consider a perturbed system
(32)dSidt=Ai-(di+γi)Si+λiVi-ε1Si∑j=1n‍βij,dVidt=γiSi-(λi+di)Vi,i=1,2,…,n.
From our previous analysis of system (32), we can restrictε1>0small enough such that (32) admits a unique positive equilibrium(Si0(ε1),Vi0(ε1),i=1,2,…,n)which is globally asymptotically stable.Si0(ε1)is continuous inε1, so we can further restrictε1small enough such thatSi0(ε1)>Si0-ε, i=1,2,…,n.
For the sake of contradiction of Theorem2, there is aT>0such thatIi(t)<ε1, Wi(t)<ε1, i=1,2,…,n, for allt≥T. Then fort≥T, we have
(33)dSidt≥Ai-(di+γi)Si+λiRi-ε1Si∑j=1n‍βij,dRidt=γiSi-(λi+di)Ri,i=1,2,…,n.
Since the equilibrium (Si0(ε1), Vi0(ε1), i=1,2,…,n) of (32) is globally asymptotically stable andSi0(ε1)>Si0-ε, i=1,2,…,n. There exists aT1>T>0such thatSi(t)>Si0-ε, i=1,2,…,n for t>T1. As a consequence, for t>T1, there holds
(34)dIidt≥(Si0-ε)∑j=1n‍βijWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.
Consider the following system(35)dIi′dt=(Si0-ε)∑j=1n‍βijWj′-(di+ri)Ii′,dWi′dt=kiIi′-δiWi′,i=1,2,…,n.
Since the matrixM2has positive eigenvalues(M2)with a positive eigenvector. It is easy to see that
(36)(I1′(t),W1′(t),I2′(t),W2′(t),…,In′(t),Wn′(t))⟶(∞,∞,∞,∞,…,∞,∞),t⟶∞.
Using the comparison principle of Smith and Waltman [23], we also know that
(37)(I1(t),W1(t),I2(t),W2(t),…,In(t),Wn(t))⟶(∞,∞,∞,∞,…,∞,∞),t⟶∞,
which leads to a contradiction, therefore we claim that(38)limsupt→∞max{Ii(t),Wi(t)}>ε1,i=1,2,…,n.
This completes the proof.We also have the following result of system (2).Theorem 3.
Ifℛ0>1, then system (2) admits at least one positive equilibrium, and there is a positive constantεsuch that every solution(Si(t),Ii(t),Vi(t),Wi(t))of the system (2) with(Si(0),Ii(0),Vi(0),Wi(0))∈X0satisfies
(39)min{liminft→∞Ii(t),liminft→∞Wi(t)}≥ε,i=1,2,…,n,
which implies that system (2) is uniformly persistent.Proof.
Now we prove that system (2) is uniformly persistent with respect to(X0,∂X0). By the form of (2), it is easy to see that bothXandX0are positively invariant and∂X0is relatively closed inX. Furthermore system (2) is point dissipative. Let
(40)M∂={(Si(0),Ii(0),Vi(0),Wi(0))∣(Si(t),Ii(t),Vi(t),Wi(t))∈∂X0,∀t≥0,i=1,2,…,n}.
It is easy to show that(41)M∂={(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}.
Noting that{(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}⊆M∂. We only need to proveM∂⊆{(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}. Assume(Si(0),Ii(0),Vi(0),Wi(0),i=1,2,…,n)∈M∂. It suffices to show thatIi(t)=0, Wi(t)=0, for all t≥0, i=1,2,…,n. Suppose not. Then there exist an i0, 1≤i0≤n, andt0≥0such thatIi0(t0)>0, Wi0(t0)>0 and we partition{1,2,…,n}into two setsQ1andQ2such that
(42)(Ii(t0),Wi(t0))T=0,∀i∈Q1,(Ii(t0),Wi(t0))T>0,∀i∈Q2.
Q
1is nonempty due to the definition ofM∂.Q2is non-empty sinceIi0(t0)>0,Wi0(t0)>0. For anyi∈Q2 and we have that
(43)dWi(t0)dt0=kiIi(t0)-δiWi(t0)>kiIi(t0),i∈Q2.
It follows that there is anη>0such thatIi(t)>0, fort0<t<t0+η, i∈Q2.This means that(Si(t),Ii(t),Vi(t),Wi(t),i=1,2,…,n)does not belong to ∂X0fort0<t<t0+η, which contradicts the assumption that (Si(0),Ii(0),Vi(0),Wi(0),i=1,2,…,n)∈M∂. This proves the system (41).
P
0is globally asymptotically stable for system (2). It is clear that there is only an equilibriaumP0inM∂ and by aforementioned claim, it then follows thatP0is isolated invariant set inX,Ws(P0)∩X0=∅. Clearly, every orbit inM∂converges toP0,P0is acyclic inM∂. Using Theorem  4.6 in Thieme [26], we conclude that the system (2) is uniformly persistent with respect to(X0,∂X0). By the result of [27, 28], system (2) has an equilibrium(S1*,I1*,V1*,W1*,…,Sn*,In*,Vn*,Wn*)∈X0. We further claim thatSi*,Vi*>0, i=1,2,…,n. Suppose thatSi*=Vi*=0, i=1,2,…,n; from of (2), we can getIi*=Wi*=0, i=1,2,…,n. It is a contradiction. Then (S1*,I1*,V1*,W1*,…,Sn*,In*,Vn*,Wn*)∈X0 is a componentwise positive equilibrium of system (2). This completes the proof.The following theorem shows that there exists a unique positive solution for system (2) whenℛ0>1.Theorem 4.
Ifℛ0>1, then there only exists a unique positive equilibrium P* for system (2).Proof.
Consider the following system:(44)Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj=0,∑j=1n‍βijSiWj-(di+ri)Ii=0,γiSi-(λi+di)Vi=0,kiIi-δiWi=0,i=1,2,…,n.
We have that(45)Si=di+λidi(di+λi+γi)(Ai-(di+ri)Ii),Wi=kiIiδi,Vi=γiSidi+λi,i=1,2,…,n.
Hence, the equilibrium of system (2) is equal to the following system:
(46)Bi(Ai-niIi)∑j=1n‍βijIj-niIi=0,i=1,2,…,n,
where(47)Bi=ki(di+λi)diδi(di+λi+γi),ni=di+ri,i=1,2,…,n.
Therefore, we only need to prove that (46) has a unique positive equilibrium whenℛ0>1. Use the method in [12] to demonstrate the unique positive equilibrium of (46). First we prove that Ii*=h, i=1,2,…,n, is the only positive solution of (46). Assume that Ii*=handIi*=kare two positive solutions of (46), both nonzero. Ifh≠k, thenhi≠kifor some i (i=1,2,…,n). Assume without loss of generality thath1>k1 and moreover thath1/k1≥hi/kifor all i (i=1,2,…,n). Sincehandkare positive solutions of (46), we substitute them into (46). We obtain
(48)0=B1(A1-n1h1)∑j=1n‍β1jhj-n1h10=B1(A1-n1k1)∑j=1n‍β1jkj-n1k1,
so(49)0=B1(A1-n1h1)∑j=1n‍β1jhjk1h1-n1k10=B1(A1-n1k1)∑j=1n‍β1jkj-n1k1,B1(A1-n1h1)∑j=1n‍β1jhjk1h1=B1(A1-n1k1)∑j=1n‍β1jkj.
But(hi/h1)k1≤kiandB1(A1-n1h1)<B1(A1-n1k1); thus from the above equalities we get
(50)B1(A1-n1h1)∑j=1n‍β1jhjk1h1≤B1(A1-n1h1)∑j=1n‍β1jkj<B1(A1-n1k1)∑j=1n‍β1jkj.
This is a contradiction, so there is only one positive solutionIi*=h, i=1,2,…,n, of (46). So when ℛ0>1, there only exists a unique positive equilibrium for system (2).


### 2.4. Global Stability of the Unique Endemic Solution of System (2)

In this section, we prove that the unique endemic equilibrium of system (2) is globally asymptotically stable inX0. In order to prove global stability of the endemic equilibrium, the Lyapunov function will be used. In the following, we also use a Lyapunov function to prove global stability of the endemic equilibrium.Theorem 5.
Ifℛ0>1, the unique positive equilibriumP*of system (2) is globally asymptotically stable inX0.Proof.
Following [15] we define
(51)ξij=βijSi*Wj*,1≤i,j≤n,n≥2,(52)B=(∑j≠1n‍ξ1j-ξ21⋯-ξn1-ξ12∑j≠2n‍ξ2j⋯-ξn2⋮⋮⋱⋮-ξ1n-ξ2n⋯∑j≠nn‍ξnj)n×n,
which is a Laplacian matrix whose column sums are zero and which is irreducible. Therefore, it follows from Lemma  2.1 of [15] that the solution space of linear system
(53)Bζ=0,
has dimension 1 with a basis(54)ζ=(ζ1,ζ2,…,ζn)T=(c1,c2,…,cn)T,
wherecidenotes the cofactor of theith diagonal entry ofB. Note that from (53) we have that
(55)∑j=1n‍ζiξij=∑j=1n‍ζjξji,i=1,2,…,n.
For suchζ=(ζ1,ζ2,…,ζn)we define a Lyapunov function
(56)L(S,I,V,W)=∑i=1n‍ζi(Si-Si*-Si*lnSi*Si+Ii-Ii*-Ii*lnIi*Ii+Vi-Vi*-Vi*lnVi*Vi+di+riki(Wi-Wi*-Wi*lnWi*Wi)),
whereS=(S1,S2,…,Sn), I=(I1,I2,…,In), V=(V1,V2,…,Vn), andW=(W1,W2,…,Wn). It is easy to see thatL(S,I,V,W)≥0for all(S,I,V,W)≥0and the equalityL(S,I,V,W)=0holds if and only if(S,I,V,W)=(S*,I*,V*,W*). The derivative along the trajectories of system (2) is
(57)L′(S,I,V,W)=∑i=1n‍ζi(Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj-Si*Si(Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj)+∑j=1n‍βijSiWj-(di+ri)Ii-Ii*Ii(∑j=1n‍βijSiWj-(di+ri)Ii)+γiSi-(λi+di)Vi-Vi*Vi(γiSi-(λi+di)Vi)∑j=1n‍+di+riki(kiIi-δiWi-Wi*Wi(kiIi-δiWi)))=L1+L2+L3.
From system (44), we have
(58)Ai=(di+γi)Si*-λiVi*+∑j=1n‍βijSi*Wj*,(59)∑j=1n‍βijSi*Wj*=(di+ri)Ii*=δi(di+ri)Wi*ki.
So(60)L1=∑i=1n‍ζi(∑j=1n‍βijSi*Wj-δi(di+ri)Wiki),L2=∑i=1n‍ζi((di+γi)Si*-λiVi*-(di+γi)Si+λiViSi*Si+Si*Si((di+γi)Si*-λiV*-(di+γi)Si+λiVi)+γiSi-(λi+di)Vi+Vi*Vi(γiSi-(λi+di)Vi))=∑i=1n‍ζi(diSi*(2-SiSi*-Si*Si)+λiVi*(2-SiVi*Si*Vi-Si*ViSiVi*)+diVi*(3-ViVi*-Si*Si-SiVi*Si*Vi))≤0,L3=∑i=1n‍ζi(3∑j=1n‍βijSi*Wj*-∑j=1n‍βijSi*Wj*Si*SiL3=-∑j=1n‍βijSiWjIi*Ii-(di+ri)IiWi*Wi).
Now we claim that(61)∑i=1n‍ζi∑j=1n‍βijSi*Wj=∑i=1n‍ζiδi(di+ri)Wiki.
Appealing to (51), (55) and (59),
(62)∑i=1n‍∑j=1n‍ζiβijSi*Wj=∑i=1n‍∑j=1n‍ζjβjiSj*Wi=∑i=1n‍∑j=1n‍WiWi*ζjβjiSj*Wi*=∑i=1n‍WiWi*∑j=1n‍ζjξji=∑i=1n‍WiWi*∑j=1n‍ζiξij=∑i=1n‍ζiδi(di+ri)Wiki.
From (61) we have
(63)L′(S,I,V,W)≤∑i=1n‍ζi(3∑j=1n‍βijSi*Wj*-∑j=1n‍βijSi*Wj*Si*Si-∑j=1n‍βijSiWjIi*Ii-(di+ri)IiWi*Wi)=∑i,j=1n‍ζiξij(3-Si*Si-SiWjIi*Si*Wj*Ii-Wi*IiWiIi*)=:Hn(S1,I1,W1,…,Sn,In,Wn).
Next we show thatHn≤0for all(S1,I1,W1,…,Sn,In,Wn)∈X0by applying the graph-theoretic approach developed in [29–31]. As in [29],L=G(B)denotes the directed graph associated with matrixB,Qpresents a subgraph ofL,CQdenotes the unique elementary cycle ofQ,E(CQ)presents the set of directed arcs in CQ, andl=l(Q)denotes the number of arcs inCQ. ThenHncan be rewritten as
(64)Hn=∑Q‍Hn,Q,
where(65)Hn,Q=∏(r,m)∈E(Q)‍ξrmHn,Q=×(3l-∑(i,j)∈E(CQ)‍(Si*Si+SiWjIi*Si*Wj*Ii+Wi*IiWiIi*)).
For instance,(66)H1=H1(S1,I1,W1)H1=∑i=j=1‍ζ1ξ11(3-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*)≤0,H2=H2(S1,I1,W1,S2,I2,W2)H2=∑i,j=12‍ζiξij(3-Si*Si-SiWjIi*Si*Wj*Ii-Wi*IiWjIi*)H2=ξ11ξ21(3-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*)H2=+ξ22ξ12(3-S2*S2-S2W2I2*S2*W2*I2-W2*I2W2I2*)H2=+ξ12ξ21(6-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*-S2*S2H2=+ξ12ξ21F-S2W2I2*S2*W2*I2-W2*I2W2I2*)≤0.
Note that for each unicycle graphQ, it is easy to see that
(67)∏(i,j)∈E(CQ)‍Si*Si·SiWjIi*Si*Wj*Ii·Wi*IiWiIi*=∏(i,j)∈E(CQ)‍Wi*WjWiWj*=1.
Therefore,(68)∑(i,j)∈E(CQ)‍(Si*Si+SiWjIi*Si*Wj*Ii+Wi*IiWiIi*)≥3l,
and henceHn,Q≤0for eachQ, andHn,Q=0if and only if
(69)Si*Si=SiWjIi*Si*Wj*Ii=Wi*IiWiIi*,(i,j)∈E(CQ).
Thus(70)L′(S,I,V,W)≤Hn≤0.
The equalityL′(S,I,V,W)=0 holds if and only if Si=Si*, Ii=Ii*, Vi=Vi*, and Wi=Wi* for alli=1,2,…,n. Therefore, following from LaSalle’s Invariance Principle [32], the unique endemic equilibriumP*of system (2) is globally asymptotically stable. This completes the proof.


## 2.1. The Basic Reproduction Number

According to the next generation matrix formulated in papers [20–22], we define the basic reproduction numberℛ0of system (2). In order to formulateℛ0, we order the infected variables first by disease state and then by group, that is,
(8)I1,W1,I2,W2,…,In,Wn.Consider the following auxiliary system:(9)dIidt=∑j=1n‍βijSiWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.Follow the recipe from van den Driessche and Watmough [21] to obtain
(10)F=(0β11S100β12S10⋯0β1nS100000⋯000β21S200β22S20⋯0β2nS200000⋯00⋮⋮⋮⋮⋮⋮0βn1Sn00βn2Sn0⋯0βnnSn00000⋯00)2n×2n,V=(d1+r1000⋯00-k1δ100⋯0000d2+r20⋯0000-k2δ2⋯00⋮⋮⋮⋮⋮⋮0000⋯dn+rn00000⋯-k1δn)2n×2n.We can get the inverse ofV, which equals(11)V-1=(1d1+r1000⋯00k1δ1(d1+r1)1δ100⋯00001d2+r20⋯0000k2δ2(d2+r2)1δ2⋯00⋮⋮⋮⋮⋮⋮0000⋯1dn+rn00000⋯knδn(dn+rn)1δn)2n×2n.Thus, the next generation matrix isFV-1,
(12)FV-1=(A11⋯A1nB11⋯B1n⋮⋮⋮⋮An1⋯AnnBn1⋯Bnn0⋯00⋯0⋮⋮⋮⋮0⋯00⋯0)2n×2n,A=(A11A12⋯A1nA21A22⋯A2n⋮⋮⋮An1An2⋯Ann)n×n.So we can calculate the basic reproduction number of system (2),
(13)ℛ0=ρ(FV-1)=ρ(A),where(14)Aij=βijkjSi0δj(dj+rj),Si0=Ai(λi+di)di(λi+di+γi),i=1,2,…,n,andρdenotes the spectral radius. As we will show,ℛ0is the key threshold parameters whose values completely characterize the global dynamics of system (2).


## 2.2.  Global Stability of the Disease-Free Equilibrium of System (2)

For the disease-free equilibriumP0of system (2), we have the following property.Theorem 1.
Ifℛ0<1, the disease-free equilibriumP0of system (2) is globally asymptotically stable in the regionX.Proof.
LetM=F-V, and defines(M)=max{Reλ:λis an eigenvalue ofM}, sos(M)is a simple eigenvalue ofMwith a positive eigenvector [23]. By Theorem  2 in [21], there hold two equivalences:
(15)ℛ0>1⟺s(M)>0,ℛ0<1⟺s(M)<0.
To prove the locally stability of disease-free equilibrium, we check the hypotheses (A1)–(A5) in [21]. Hypotheses (A1)–(A4) are easily verified, while (A5) is satisfied if all eigenvalues of the4n×4nmatrix
(16)J|P0=(M0J3J4)4n×4n
have negative real parts, whereJ3=-F,(17)J4=(-(d1+γ1)λ100⋯00γ1-(d1+λ1)00⋯0000-(d2+γ2)λ2⋯0000γ2-(d2+λ2)⋯00⋮⋮⋮⋮⋮⋮0000⋯-(dn+γn)λn0000⋯γn-(dn+λn))2n×2n.
Calculate the eigenvalues ofJ4:
(18)s(J4)=max{-d1,…,-dn,-(d1+λ1+γ1),…,-(dn+λn+γn)}<0.
Ifℛ0<1, thens(M)<0ands(J|P0)<0, and the disease-free equilibriumP0of (2) is locally asymptotically stable.
Now we will prove that the disease-free equilibriumP0of system (2) is globally attractive whenℛ0<1. From the third equation of system (2), we have
(19)dVidt=γiSi-(λi+di)VidVidt=γi(Ni-(Ii+Vi))-(λi+di)VidVidt≤γiAidi-(λi+γi+di)Vi.
So we can have that, for a small enough positive numberϵ1, there exists ti>0, i=1,2,…,n such that for allt>ti,
(20)Vi≤Aiγidi(λi+γi+di)+ϵ1=Vi0+ϵ1.
Also from the equations of system (2), we have
(21)dSidt=Ai+λiVi-(γi+di)Si-∑j=1n‍βijSiWjdSidt≤Ai+λi(Vi0+ϵ1)-(di+γi)So.
Then(22)limt→∞supSi=Ai+λi(Vi0+ϵ1)di+γi=Si0+ϵ2,(ϵ2=λiϵ1di+γi).
From system (9) andSi≤Si0+ϵ2with allt>ti. Thus, whent>ti, we derive
(23)dIidt=(Si0+ϵ2)∑j=1n‍βijWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.
Consider the following auxiliary system(24)dIi′dt=(Si0+ϵ2)∑j=1n‍βijWj′-(di+ri)Ii′,dWi′dt=kiIi′-δiWi′,i=1,2,…,n.
LetM0be the matrix defined by
(25)M0=(0β110β12⋯0β1n0000⋯000β210β22⋯0β2n0000⋯00⋮⋮⋮⋮⋮⋮0βn10βn2⋯0βnn0000⋯00)2n×2n,
and setM1=M+ϵ2M0. It follows from Theorem  2 in [21] thatℛ0<1if and only ifs(M)<0. Thus, there exists anϵ2>0small enough such thats(M1)<0. Using the Perron-Frobenius theorem, all eigenvalues of the matrixM1have negative real parts whens(M1)<0. Therefore it has
(26)(I1′(t),W1′(t),I2′(t),W2′(t),…,In′(t),Wn′(t))⟶(0,0,0,0,…,0,0),t⟶∞,
which implies that the zero solution of system (24) is globally asymptotically stable. Using the comparison principle of Smith and Waltman [23], we know that
(27)(I1(t),W1(t),I2(t),W2(t),…,In(t),Wn(t))⟶(0,0,0,0,…,0,0),t⟶∞.
By the theory of asymptotic autonomous system of Thieme [24], it is also known that
(28)(S1(t),V1(t),…,Sn(t),Vn(t))⟶(S1(0),V1(0),…,Sn(0),Vn(0)),t⟶∞.
SoP0is globally attractive whenℛ0<1. It follows that the disease-free equilibriumP0of (2) is globally asymptotically stable whenℛ0<1. This completes the proof.


## 2.3. The Uniform Persistence and Unique Positive Solution of System (2)

In this section, we give the proof of the uniform persistence and the unique positive solution of system (2). Define
(29)X0={(Si,Ii,Vi,Wi)∈X∣Ii,Wi>0,i=1,2,…,n},∂X0=X∣X0.Theorem 2.
Whenℛ0>1, there exists a positive constantε1such that when∥Ii(0)∥<ε1,∥Wi(0)∥<ε1for(Si(0),Ii(0),Vi(0),Wi(0))∈X0(30)limsupt→∞max{Ii(t),Wi(t)}>ε1,i=1,2,…,n.Proof.
Consider the following system:(31)dSidt=Ai+λiVi-(γi+di)Si,dVidt=γiSi-(λi+di)Vi,i=1,2,…,n.
Using Corollary  3.2 in Zhao and Jing [25], it then follows that system (31) has a unique positive equilibrium(S10,V10,…,Sn0,Vn0) which is globally asymptotically stable.
As toℛ0>1⇔s(M)>0, chooseε>0small enough such thats(M2)>0, whereM2=M-εM0. Let us consider a perturbed system
(32)dSidt=Ai-(di+γi)Si+λiVi-ε1Si∑j=1n‍βij,dVidt=γiSi-(λi+di)Vi,i=1,2,…,n.
From our previous analysis of system (32), we can restrictε1>0small enough such that (32) admits a unique positive equilibrium(Si0(ε1),Vi0(ε1),i=1,2,…,n)which is globally asymptotically stable.Si0(ε1)is continuous inε1, so we can further restrictε1small enough such thatSi0(ε1)>Si0-ε, i=1,2,…,n.
For the sake of contradiction of Theorem2, there is aT>0such thatIi(t)<ε1, Wi(t)<ε1, i=1,2,…,n, for allt≥T. Then fort≥T, we have
(33)dSidt≥Ai-(di+γi)Si+λiRi-ε1Si∑j=1n‍βij,dRidt=γiSi-(λi+di)Ri,i=1,2,…,n.
Since the equilibrium (Si0(ε1), Vi0(ε1), i=1,2,…,n) of (32) is globally asymptotically stable andSi0(ε1)>Si0-ε, i=1,2,…,n. There exists aT1>T>0such thatSi(t)>Si0-ε, i=1,2,…,n for t>T1. As a consequence, for t>T1, there holds
(34)dIidt≥(Si0-ε)∑j=1n‍βijWj-(di+ri)Ii,dWidt=kiIi-δiWi,i=1,2,…,n.
Consider the following system(35)dIi′dt=(Si0-ε)∑j=1n‍βijWj′-(di+ri)Ii′,dWi′dt=kiIi′-δiWi′,i=1,2,…,n.
Since the matrixM2has positive eigenvalues(M2)with a positive eigenvector. It is easy to see that
(36)(I1′(t),W1′(t),I2′(t),W2′(t),…,In′(t),Wn′(t))⟶(∞,∞,∞,∞,…,∞,∞),t⟶∞.
Using the comparison principle of Smith and Waltman [23], we also know that
(37)(I1(t),W1(t),I2(t),W2(t),…,In(t),Wn(t))⟶(∞,∞,∞,∞,…,∞,∞),t⟶∞,
which leads to a contradiction, therefore we claim that(38)limsupt→∞max{Ii(t),Wi(t)}>ε1,i=1,2,…,n.
This completes the proof.We also have the following result of system (2).Theorem 3.
Ifℛ0>1, then system (2) admits at least one positive equilibrium, and there is a positive constantεsuch that every solution(Si(t),Ii(t),Vi(t),Wi(t))of the system (2) with(Si(0),Ii(0),Vi(0),Wi(0))∈X0satisfies
(39)min{liminft→∞Ii(t),liminft→∞Wi(t)}≥ε,i=1,2,…,n,
which implies that system (2) is uniformly persistent.Proof.
Now we prove that system (2) is uniformly persistent with respect to(X0,∂X0). By the form of (2), it is easy to see that bothXandX0are positively invariant and∂X0is relatively closed inX. Furthermore system (2) is point dissipative. Let
(40)M∂={(Si(0),Ii(0),Vi(0),Wi(0))∣(Si(t),Ii(t),Vi(t),Wi(t))∈∂X0,∀t≥0,i=1,2,…,n}.
It is easy to show that(41)M∂={(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}.
Noting that{(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}⊆M∂. We only need to proveM∂⊆{(Si(t),0,Vi(t),0)∣Si(t),Vi(t)≥0,i=1,2,…,n}. Assume(Si(0),Ii(0),Vi(0),Wi(0),i=1,2,…,n)∈M∂. It suffices to show thatIi(t)=0, Wi(t)=0, for all t≥0, i=1,2,…,n. Suppose not. Then there exist an i0, 1≤i0≤n, andt0≥0such thatIi0(t0)>0, Wi0(t0)>0 and we partition{1,2,…,n}into two setsQ1andQ2such that
(42)(Ii(t0),Wi(t0))T=0,∀i∈Q1,(Ii(t0),Wi(t0))T>0,∀i∈Q2.
Q
1is nonempty due to the definition ofM∂.Q2is non-empty sinceIi0(t0)>0,Wi0(t0)>0. For anyi∈Q2 and we have that
(43)dWi(t0)dt0=kiIi(t0)-δiWi(t0)>kiIi(t0),i∈Q2.
It follows that there is anη>0such thatIi(t)>0, fort0<t<t0+η, i∈Q2.This means that(Si(t),Ii(t),Vi(t),Wi(t),i=1,2,…,n)does not belong to ∂X0fort0<t<t0+η, which contradicts the assumption that (Si(0),Ii(0),Vi(0),Wi(0),i=1,2,…,n)∈M∂. This proves the system (41).
P
0is globally asymptotically stable for system (2). It is clear that there is only an equilibriaumP0inM∂ and by aforementioned claim, it then follows thatP0is isolated invariant set inX,Ws(P0)∩X0=∅. Clearly, every orbit inM∂converges toP0,P0is acyclic inM∂. Using Theorem  4.6 in Thieme [26], we conclude that the system (2) is uniformly persistent with respect to(X0,∂X0). By the result of [27, 28], system (2) has an equilibrium(S1*,I1*,V1*,W1*,…,Sn*,In*,Vn*,Wn*)∈X0. We further claim thatSi*,Vi*>0, i=1,2,…,n. Suppose thatSi*=Vi*=0, i=1,2,…,n; from of (2), we can getIi*=Wi*=0, i=1,2,…,n. It is a contradiction. Then (S1*,I1*,V1*,W1*,…,Sn*,In*,Vn*,Wn*)∈X0 is a componentwise positive equilibrium of system (2). This completes the proof.The following theorem shows that there exists a unique positive solution for system (2) whenℛ0>1.Theorem 4.
Ifℛ0>1, then there only exists a unique positive equilibrium P* for system (2).Proof.
Consider the following system:(44)Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj=0,∑j=1n‍βijSiWj-(di+ri)Ii=0,γiSi-(λi+di)Vi=0,kiIi-δiWi=0,i=1,2,…,n.
We have that(45)Si=di+λidi(di+λi+γi)(Ai-(di+ri)Ii),Wi=kiIiδi,Vi=γiSidi+λi,i=1,2,…,n.
Hence, the equilibrium of system (2) is equal to the following system:
(46)Bi(Ai-niIi)∑j=1n‍βijIj-niIi=0,i=1,2,…,n,
where(47)Bi=ki(di+λi)diδi(di+λi+γi),ni=di+ri,i=1,2,…,n.
Therefore, we only need to prove that (46) has a unique positive equilibrium whenℛ0>1. Use the method in [12] to demonstrate the unique positive equilibrium of (46). First we prove that Ii*=h, i=1,2,…,n, is the only positive solution of (46). Assume that Ii*=handIi*=kare two positive solutions of (46), both nonzero. Ifh≠k, thenhi≠kifor some i (i=1,2,…,n). Assume without loss of generality thath1>k1 and moreover thath1/k1≥hi/kifor all i (i=1,2,…,n). Sincehandkare positive solutions of (46), we substitute them into (46). We obtain
(48)0=B1(A1-n1h1)∑j=1n‍β1jhj-n1h10=B1(A1-n1k1)∑j=1n‍β1jkj-n1k1,
so(49)0=B1(A1-n1h1)∑j=1n‍β1jhjk1h1-n1k10=B1(A1-n1k1)∑j=1n‍β1jkj-n1k1,B1(A1-n1h1)∑j=1n‍β1jhjk1h1=B1(A1-n1k1)∑j=1n‍β1jkj.
But(hi/h1)k1≤kiandB1(A1-n1h1)<B1(A1-n1k1); thus from the above equalities we get
(50)B1(A1-n1h1)∑j=1n‍β1jhjk1h1≤B1(A1-n1h1)∑j=1n‍β1jkj<B1(A1-n1k1)∑j=1n‍β1jkj.
This is a contradiction, so there is only one positive solutionIi*=h, i=1,2,…,n, of (46). So when ℛ0>1, there only exists a unique positive equilibrium for system (2).


## 2.4. Global Stability of the Unique Endemic Solution of System (2)

In this section, we prove that the unique endemic equilibrium of system (2) is globally asymptotically stable inX0. In order to prove global stability of the endemic equilibrium, the Lyapunov function will be used. In the following, we also use a Lyapunov function to prove global stability of the endemic equilibrium.Theorem 5.
Ifℛ0>1, the unique positive equilibriumP*of system (2) is globally asymptotically stable inX0.Proof.
Following [15] we define
(51)ξij=βijSi*Wj*,1≤i,j≤n,n≥2,(52)B=(∑j≠1n‍ξ1j-ξ21⋯-ξn1-ξ12∑j≠2n‍ξ2j⋯-ξn2⋮⋮⋱⋮-ξ1n-ξ2n⋯∑j≠nn‍ξnj)n×n,
which is a Laplacian matrix whose column sums are zero and which is irreducible. Therefore, it follows from Lemma  2.1 of [15] that the solution space of linear system
(53)Bζ=0,
has dimension 1 with a basis(54)ζ=(ζ1,ζ2,…,ζn)T=(c1,c2,…,cn)T,
wherecidenotes the cofactor of theith diagonal entry ofB. Note that from (53) we have that
(55)∑j=1n‍ζiξij=∑j=1n‍ζjξji,i=1,2,…,n.
For suchζ=(ζ1,ζ2,…,ζn)we define a Lyapunov function
(56)L(S,I,V,W)=∑i=1n‍ζi(Si-Si*-Si*lnSi*Si+Ii-Ii*-Ii*lnIi*Ii+Vi-Vi*-Vi*lnVi*Vi+di+riki(Wi-Wi*-Wi*lnWi*Wi)),
whereS=(S1,S2,…,Sn), I=(I1,I2,…,In), V=(V1,V2,…,Vn), andW=(W1,W2,…,Wn). It is easy to see thatL(S,I,V,W)≥0for all(S,I,V,W)≥0and the equalityL(S,I,V,W)=0holds if and only if(S,I,V,W)=(S*,I*,V*,W*). The derivative along the trajectories of system (2) is
(57)L′(S,I,V,W)=∑i=1n‍ζi(Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj-Si*Si(Ai-(di+γi)Si+λiVi-∑j=1n‍βijSiWj)+∑j=1n‍βijSiWj-(di+ri)Ii-Ii*Ii(∑j=1n‍βijSiWj-(di+ri)Ii)+γiSi-(λi+di)Vi-Vi*Vi(γiSi-(λi+di)Vi)∑j=1n‍+di+riki(kiIi-δiWi-Wi*Wi(kiIi-δiWi)))=L1+L2+L3.
From system (44), we have
(58)Ai=(di+γi)Si*-λiVi*+∑j=1n‍βijSi*Wj*,(59)∑j=1n‍βijSi*Wj*=(di+ri)Ii*=δi(di+ri)Wi*ki.
So(60)L1=∑i=1n‍ζi(∑j=1n‍βijSi*Wj-δi(di+ri)Wiki),L2=∑i=1n‍ζi((di+γi)Si*-λiVi*-(di+γi)Si+λiViSi*Si+Si*Si((di+γi)Si*-λiV*-(di+γi)Si+λiVi)+γiSi-(λi+di)Vi+Vi*Vi(γiSi-(λi+di)Vi))=∑i=1n‍ζi(diSi*(2-SiSi*-Si*Si)+λiVi*(2-SiVi*Si*Vi-Si*ViSiVi*)+diVi*(3-ViVi*-Si*Si-SiVi*Si*Vi))≤0,L3=∑i=1n‍ζi(3∑j=1n‍βijSi*Wj*-∑j=1n‍βijSi*Wj*Si*SiL3=-∑j=1n‍βijSiWjIi*Ii-(di+ri)IiWi*Wi).
Now we claim that(61)∑i=1n‍ζi∑j=1n‍βijSi*Wj=∑i=1n‍ζiδi(di+ri)Wiki.
Appealing to (51), (55) and (59),
(62)∑i=1n‍∑j=1n‍ζiβijSi*Wj=∑i=1n‍∑j=1n‍ζjβjiSj*Wi=∑i=1n‍∑j=1n‍WiWi*ζjβjiSj*Wi*=∑i=1n‍WiWi*∑j=1n‍ζjξji=∑i=1n‍WiWi*∑j=1n‍ζiξij=∑i=1n‍ζiδi(di+ri)Wiki.
From (61) we have
(63)L′(S,I,V,W)≤∑i=1n‍ζi(3∑j=1n‍βijSi*Wj*-∑j=1n‍βijSi*Wj*Si*Si-∑j=1n‍βijSiWjIi*Ii-(di+ri)IiWi*Wi)=∑i,j=1n‍ζiξij(3-Si*Si-SiWjIi*Si*Wj*Ii-Wi*IiWiIi*)=:Hn(S1,I1,W1,…,Sn,In,Wn).
Next we show thatHn≤0for all(S1,I1,W1,…,Sn,In,Wn)∈X0by applying the graph-theoretic approach developed in [29–31]. As in [29],L=G(B)denotes the directed graph associated with matrixB,Qpresents a subgraph ofL,CQdenotes the unique elementary cycle ofQ,E(CQ)presents the set of directed arcs in CQ, andl=l(Q)denotes the number of arcs inCQ. ThenHncan be rewritten as
(64)Hn=∑Q‍Hn,Q,
where(65)Hn,Q=∏(r,m)∈E(Q)‍ξrmHn,Q=×(3l-∑(i,j)∈E(CQ)‍(Si*Si+SiWjIi*Si*Wj*Ii+Wi*IiWiIi*)).
For instance,(66)H1=H1(S1,I1,W1)H1=∑i=j=1‍ζ1ξ11(3-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*)≤0,H2=H2(S1,I1,W1,S2,I2,W2)H2=∑i,j=12‍ζiξij(3-Si*Si-SiWjIi*Si*Wj*Ii-Wi*IiWjIi*)H2=ξ11ξ21(3-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*)H2=+ξ22ξ12(3-S2*S2-S2W2I2*S2*W2*I2-W2*I2W2I2*)H2=+ξ12ξ21(6-S1*S1-S1W1I1*S1*W1*I1-W1*I1W1I1*-S2*S2H2=+ξ12ξ21F-S2W2I2*S2*W2*I2-W2*I2W2I2*)≤0.
Note that for each unicycle graphQ, it is easy to see that
(67)∏(i,j)∈E(CQ)‍Si*Si·SiWjIi*Si*Wj*Ii·Wi*IiWiIi*=∏(i,j)∈E(CQ)‍Wi*WjWiWj*=1.
Therefore,(68)∑(i,j)∈E(CQ)‍(Si*Si+SiWjIi*Si*Wj*Ii+Wi*IiWiIi*)≥3l,
and henceHn,Q≤0for eachQ, andHn,Q=0if and only if
(69)Si*Si=SiWjIi*Si*Wj*Ii=Wi*IiWiIi*,(i,j)∈E(CQ).
Thus(70)L′(S,I,V,W)≤Hn≤0.
The equalityL′(S,I,V,W)=0 holds if and only if Si=Si*, Ii=Ii*, Vi=Vi*, and Wi=Wi* for alli=1,2,…,n. Therefore, following from LaSalle’s Invariance Principle [32], the unique endemic equilibriumP*of system (2) is globally asymptotically stable. This completes the proof.


## 3. A Numerical Example

Consider the system (1), wheni=2, one has the two-group model as follows:
(71)dS1dt=A1-(d1+γ1)S1+λ1V1-(β11S1W1+β12S1W2),dI1dt=β11S1W1+β12S1W2-(d1+r1)I1,dR1dt=r1Ii-d1R1,dV1dt=γ1S1-(λ1+d1)V1,dW1dt=k1I1-δ1W1,dS2dt=A2-(d2+γ2)S2+λ2V2-(β21S2W1+β22S2W2),dI2dt=β21S2W1+β22S2W2-(d2+r2)I2,dR2dt=r2I2-d2R2,dV2dt=γ2S2-(λ2+d2)V2,dW2dt=k2I2-δ2W2.We can give the basic reproduction number of system (71), which is
(72)ℛ0′=A11+A22+(A11-A22)2+4A12A212,where(73)Aij=βijkjSi0δj(dj+rj),Si0=Ai(λi+di)di(λi+di+γi),i=1,2.TakingA1=150, A2=220, d1=0.1, d2=0.1, λ1=0.4, λ2=0.6, λ1=0.5, λ2=0.5, r1=1, r2=1, k1=10, k2=10, δ1=8, δ2=8 and using Matlab ODE solver, we run numerical simulations for two cases.Ifβ11=0.00048, β12=0.0004, β21=0.0004, and β22=0.00045, we have ℛ0′≈0.9804<1. Hence the disease-free equilibrium of system (71) is globally asymptotically stable (see Figure 1(a)). Ifβ11=0.0025, β12=0.001, β21=0.001, and β22=0.0020, we have ℛ0′≈3.6594>1. Hence the endemic equilibrium of system (71) is globally asymptotically stable (see Figure 1(b)).(a) The disease dies out in both groups. (b) The disease persists in both groups. Initial conditions are:S1(0)=280, I1(0)=40, R1(0)=10, V1(0)=130, W1(0)=250, S2(0)=260, I2(0)=20, R2(0)=10, V2(0)=130, W2(0)=300.
(a)
(b)


## 4. Conclusion

Cholera epidemic has become a major health problem for many developing countries. From good understanding of the transmission dynamics of cholera in many emergent epidemic regions, the heterogeneous host population can be divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions. Hence, in this paper, we proposed a multi-group cholera SIRVW epidemiological model. In order to distinguish many multi-group models with direct transmission from person to person, we only considered this multi-group cholera model with indirect transmission from the bacteria of the aquatic environment to person. Firstly, the basic reproduction numberℛ0of this model is given. Then, it is found that the model has two non-negative equilibria, the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium exists without any condition whereas the endemic equilibrium exists providedℛ0>1. Finally, through the analysis of the model it has been found that the global asymptotic behavior of multi-group SIRVW model is completely determined by the size ofℛ0. That is, the disease-free equilibrium is globally asymptotically stable ifℛ0<1while an endemic equilibrium exists uniquely and is globally asymptotically stable ifℛ0>1. By running numerical simulations for the cases of two-groups model, we can see that the disease-free equilibrium of system (71) is globally stable whenℛ0′<1and the unique endemic equilibrium of system (71) is globally stable whenℛ0′>1.



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*Source: 703826-2013-12-22.xml*