For a multi-group epidemic model with cholera, the population of human is divided into n discrete groups, where n∈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,…,n at time t, respectively. Let Wi(t) be the density of bacteria in the aquatic environment in group i=1,2,…,n at time t. Based on the assumptions in Section 1, the disease transmission rate of cholera between compartments Si and Wj is denoted by βij, which means the susceptible individuals in the ith group can contact the bacteria of the aquatic environment in the jth (j=1,2,…,n) group. So the new infection that occurred in the ith group is given by ∑j=1nβijSiWj. The recruitment rate of individuals into Si(t) compartment with the ith group is given by a constant Ai. Within the ith group, it is assumed that natural death of human is di. A simple immunization policy is considered where the vaccination rate in Si(t) compartment is given by a constant γi and the losing immunity rate from vaccination individuals is λi. We assume that individuals in Ii(t) compartment recover with a rate constant ri. In Wi(t) compartment, the brucella shedding rate from Ii(t) compartment is ki, and the decaying rate of brucella is δi. 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.
2.1. The Basic Reproduction Number
According to the next generation matrix formulated in papers [20–22], we define the basic reproduction number ℛ0 of system (2). In order to formulate ℛ0, 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 of V, 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 is FV-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,
and ρ denotes the spectral radius. As we will show, ℛ0 is 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 equilibrium P0 of system (2), we have the following property.
Theorem 1.
If ℛ0<1, the disease-free equilibrium P0 of system (2) is globally asymptotically stable in the region X.
Proof.
Let M=F-V, and define s(M)=max{Reλ:λ is an eigenvalue of M}, so s(M) is a simple eigenvalue of M with 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 the 4n×4n matrix
(16)J|P0=(M0J3J4)4n×4n
have negative real parts, where J3=-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 of J4:
(18)s(J4)=max{-d1,…,-dn,-(d1+λ1+γ1),…, -(dn+λn+γn)}<0.
If ℛ0<1, then s(M)<0 and s(J|P0)<0, and the disease-free equilibrium P0 of (2) is locally asymptotically stable.
Now we will prove that the disease-free equilibrium P0 of system (2) is globally attractive when ℛ0<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 number ϵ1, there exists ti>0, i=1,2,…,n such that for all t>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) and Si≤Si0+ϵ2 with all t>ti. Thus, when t>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.
Let M0 be the matrix defined by
(25)M0=(0β110β12⋯0β1n0000⋯000β210β22⋯0β2n0000⋯00⋮⋮⋮⋮ ⋮⋮0βn10βn2⋯0βnn0000⋯00)2n×2n,
and set M1=M+ϵ2M0. It follows from Theorem 2 in [21] that ℛ0<1 if and only if s(M)<0. Thus, there exists an ϵ2>0 small enough such that s(M1)<0. Using the Perron-Frobenius theorem, all eigenvalues of the matrix M1 have negative real parts when s(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⟶∞.
So P0 is globally attractive when ℛ0<1. It follows that the disease-free equilibrium P0 of (2) is globally asymptotically stable when ℛ0<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.
When ℛ0>1, there exists a positive constant ε1 such that when ∥Ii(0)∥<ε1,∥Wi(0)∥<ε1 for (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 to ℛ0>1⇔s(M)>0, choose ε>0 small enough such that s(M2)>0, where M2=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 restrict ε1>0 small 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 in ε1, so we can further restrict ε1 small enough such that Si0(ε1)>Si0-ε, i=1,2,…,n.
For the sake of contradiction of Theorem 2, there is a T>0 such that Ii(t)<ε1, Wi(t)<ε1, i=1,2,…,n, for all t≥T. Then for t≥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 and Si0(ε1)>Si0-ε, i=1,2,…,n. There exists a T1>T>0 such that Si(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 matrix M2 has positive eigenvalue s(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.
If ℛ0>1, then system (2) admits at least one positive equilibrium, and there is a positive constant ε such that every solution (Si(t),Ii(t),Vi(t),Wi(t)) of the system (2) with (Si(0),Ii(0),Vi(0),Wi(0)) ∈X0 satisfies
(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 both X and X0 are positively invariant and ∂X0 is relatively closed in X. 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 prove M∂⊆{(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 that Ii(t)=0, Wi(t)=0, for all t≥0, i=1,2,…,n. Suppose not. Then there exist an i0, 1≤i0≤n, and t0≥0 such that Ii0(t0)>0, Wi0(t0)>0 and we partition {1,2,…,n} into two sets Q1 and Q2 such that
(42)(Ii(t0),Wi(t0))T=0, ∀i∈Q1,(Ii(t0),Wi(t0))T>0, ∀i∈Q2.
Q
1
is nonempty due to the definition of M∂. Q2 is non-empty since Ii0(t0)>0,Wi0(t0)>0. For any i∈Q2 and we have that
(43)dWi(t0)dt0=kiIi(t0)-δiWi(t0)>kiIi(t0), i∈Q2.
It follows that there is an η>0 such that Ii(t)>0, for t0<t<t0+η, i∈Q2. This means that (Si(t),Ii(t),Vi(t),Wi(t), i=1,2,…,n) does not belong to ∂X0 for t0<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
0
is globally asymptotically stable for system (2). It is clear that there is only an equilibriaum P0 in M∂ and by aforementioned claim, it then follows that P0 is isolated invariant set in X, Ws(P0)∩X0=∅. Clearly, every orbit in M∂ converges to P0, P0 is acyclic in M∂. 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 that Si*,Vi*>0, i=1,2,…,n. Suppose that Si*=Vi*=0, i=1,2,…,n; from of (2), we can get Ii*=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) when ℛ0>1.
Theorem 4.
If ℛ0>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 when ℛ0>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*=h and Ii*=k are two positive solutions of (46), both nonzero. If h≠k, then hi≠ki for some i (i=1,2,…,n). Assume without loss of generality that h1>k1 and moreover that h1/k1≥hi/ki for all i (i=1,2,…,n). Since h and k are 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≤ki and B1(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 solution Ii*=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 in X0. 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.
If ℛ0>1, the unique positive equilibrium P* of system (2) is globally asymptotically stable in X0.
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,
where ci denotes the cofactor of the ith diagonal entry of B. Note that from (53) we have that
(55)∑j=1nζiξij=∑j=1nζjξji, i=1,2,…,n.
For such ζ=(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)),
where S=(S1,S2,…,Sn), I=(I1,I2,…,In), V=(V1,V2,…,Vn), and W=(W1,W2,…,Wn). It is easy to see that L(S,I,V,W)≥0 for all (S,I,V,W)≥0 and the equality L(S,I,V,W)=0 holds 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 that Hn≤0 for all (S1,I1,W1,…,Sn,In,Wn)∈X0 by applying the graph-theoretic approach developed in [29–31]. As in [29], L=G(B) denotes the directed graph associated with matrix B, Q presents a subgraph of L, CQ denotes the unique elementary cycle of Q, E(CQ) presents the set of directed arcs in CQ, and l=l(Q) denotes the number of arcs in CQ. Then Hn can 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 graph Q, 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 hence Hn,Q≤0 for each Q, and Hn,Q=0 if 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 equality L′(S,I,V,W)=0 holds if and only if Si=Si*, Ii=Ii*, Vi=Vi*, and Wi=Wi* for all i=1,2,…,n. Therefore, following from LaSalle’s Invariance Principle [32], the unique endemic equilibrium P* of system (2) is globally asymptotically stable. This completes the proof.