# A New Approach to the Method of Lyapunov Functionals and Its Applications

Authors: Yunguo Jin; Shouming Zhong
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.


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## Body

## 1. Introduction

Graph theory has developed into a substantial body of knowledge. A graph theoretic approach developed in [1, 2] is used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multigroup models in mathematical epidemiology. Using results from graph theory, a systematic approach developed in [3] allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The graph-theoretical approach has been applied to various classes of coupled systems in engineering, ecology, and epidemiology (see, e.g., [1–13]). However, as in [14], while using the same Lyapunov function [3], sometimes graph theory can be replaced by positive operator theory. Furthermore, it seems that all authors use the graph theory under the assumption that the relative matrix is irreducible (see, e.g., [1–13]).Motivated by the above discussion, in this paper, we show some results which can replace graph theory used to construct global Lyapunov functions in some coupled systems of differential equations (see, e.g., [1–13]) and a more general threshold without the assumption that the relative matrix is irreducible. For some coupled systems of differential equations (see, e.g., [1–13]), by a similar approach to the one discussed in this paper, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.Various epidemics continue to pose a public health threat to humans. One of the most important subjects in this study of epidemic models (see, e.g., [1–3, 7–13]) is to obtain a threshold that determines the persistence or extinction of a disease. In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea, and bacterial infections, may have a different ability to transmit the infections in different ages. For example, measles and varicella always occur in juveniles, while it is reasonable to consider the disease transmission in adult population such as typhus and diphtheria. A heterogeneous host population can be divided into several homogeneous groups according to models of transmission, contact patterns, or geographic distributions. Since the time it takes from the moment of new infection to the moment of becoming infectious may differ from individual to individual, it is indeed a random variable. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and show a general threshold to improved existing results.Some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations are shown in the next sections. In Section3, a new approach to the method of Lyapunov functionals is applied to an epidemic model to obtain a more general threshold for the extinction and persistence of a disease.


## 2. Some Results Obtained by Mathematical Induction

In this section, we show some results which can replace graph theory used to construct global Lyapunov functions in some coupled systems of differential equations.Assumption 1.
There existvk>0, such that
(1)∑j=1n‍β¯kjvk-∑j=1n‍β¯jkvj=0,
where β¯kj≥0, k,j=1,2,…,n, n≥2.Theorem 2.
Under Assumption1, the following results hold:
(2)∑k,j=1n‍vkβ¯kj[Fk(xk)-Fj(xj)]=0,
where Fk(xk), k=1,2,…,n, are arbitrary functions. In particular, if Fk(xk)>0, k=1,2,…,n, then
(3)∑k,j=1n‍vkβ¯kj[1-Fk(xk)Fj(xj)]≤0.Proof.
Note that(4)∑k,j=1n‍vkβ¯kj[Fk(xk)-Fj(xj)]=∑n≥k≥j≥1‍(vkβ¯kj-vjβ¯jk)[Fk(xk)-Fj(xj)]=:Zn.
Let
(5)ΔL=∑L≥k≥j≥2‍(vkβ¯kj-vjβ¯jk)[Fk(xk)-Fj(xj)],L=2,3,…,n.
According to Assumption 1, it is easy to see that we only need to prove the following result:
(6)Zn=∑L=2q+1‍(-vn-1β¯n-1,L+⋯+vLβ¯L,q+2-vq+2β¯q+2,LvLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL+vLβ¯L,n-1-vn-1β¯n-1,L+⋯+vLβ¯L,q+2-vq+2β¯q+2,L)×[FL(xL)-F1(x1)]+Δq+1=:Aq+1,q=1,2,…,n-2.
In fact, if the above result holds, then we have
(7)Zn=∑L=1n‍(v2β¯2L-vLβ¯L2)[F2(x2)-F1(x1)]+Δ2.
Using Assumption 1, we have Zn=Δ2=0. First, we show that Zn=An-1. We can rewrite Zn as
(8)Zn=∑L=2n‍{(vnβ¯nL-vLβ¯Ln)[Fn(xn)-FL(xL)]+(vLβ¯L1-v1β¯1L)[FL(xL)-F1(x1)]}+Δn-1.
Then, using the fact that ab+cd=a(b+d)+d(c-a), where a, b, c, and d are arbitrary numbers, we may obtain
(9)Zn=∑L=2n‍{(vnβ¯nL-vLβ¯Ln)[Fn(xn)-F1(x1)]+(vLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL)×[FL(xL)-F1(x1)](vnβ¯nL-vLβ¯Ln)}+Δn-1=∑L=2n‍(vnβ¯nL-vLβ¯Ln)[Fn(xn)-F1(x1)]+(vnβ¯n1-v1β¯1n)[Fn(xn)-F1(x1)]+∑L=2n-1‍(vLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL)×[FL(xL)-F1(x1)]+Δn-1=∑L=1n‍(vnβ¯nL-vLβ¯Ln)[Fn(xn)-F1(x1)]+Δn-1+∑L=2n-1‍(vLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL)×[FL(xL)-F1(x1)].
Using Assumption 1, we have Zn=An-1. Next, we show that Aq+1=Aq, q=1,2,…,n-2. We can rewrite Aq+1 as
(10)Aq+1=∑L=2q+1‍(-vn-1β¯n-1,L+⋯+vLβ¯L,q+2-vq+2β¯q+2,LvLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL+vLβ¯L,n-1-vn-1β¯n-1,L+⋯+vLβ¯L,q+2-vq+2β¯q+2,L)×[FL(xL)-F1(x1)]+Δq+∑j=2q+1‍(vq+1β¯q+1,j-vjβ¯j,q+1)[Fq+1(xq+1)-Fj(xj)].
By a similar argument as for the discussion
(11)∑L=2n‍{(vnβ¯nL-vLβ¯Ln)[Fn(xn)-FL(xL)]+(vLβ¯L1-v1β¯1L)[FL(xL)-F1(x1)]}=∑L=1n‍(vnβ¯nL-vLβ¯Ln)[Fn(xn)-F1(x1)]+∑L=2n-1‍(vLβ¯L1-v1β¯1L+vLβ¯Ln-vnβ¯nL)×[FL(xL)-F1(x1)],
in the proof of Zn=An-1, we can obtain
(12)∑L=2q+1‍(vLβ¯L1-v1β¯1L)[FL(xL)-F1(x1)]+∑j=2q+1‍(vq+1β¯q+1,j-vjβ¯j,q+1)[Fq+1(xq+1)-Fj(xj)]=∑L=1q+1‍(vq+1β¯q+1,L-vLβ¯L,q+1)[Fq+1(xq+1)-F1(x1)]+∑L=2q‍(vLβ¯L1-v1β¯1L+vLβ¯L,q+1-vq+1β¯q+1,L)×[FL(xL)-F1(x1)].
Substituting (12) into (10), we have
(13)Aq+1=∑L=2q+1‍(+⋯+vLβ¯L,q+2-vq+2β¯q+2,LvLβ¯Ln-vnβ¯nL+vLβ¯L,n-1-vn-1β¯n-1,L+⋯+vLβ¯L,q+2-vq+2β¯q+2,L)×[FL(xL)-F1(x1)]+Δq+∑L=1q+1‍(vq+1β¯q+1,L-vLβ¯L,q+1)[Fq+1(xq+1)-F1(x1)]+∑L=2q‍(vLβ¯L1-v1β¯1L+vLβ¯L,q+1-vq+1β¯q+1,L)×[FL(xL)-F1(x1)]=∑L=1n‍(vq+1β¯q+1,L-vLβ¯L,q+1)[Fq+1(xq+1)-F1(x1)]+Aq.
By Assumption 1, we can get Aq+1=Aq. Using 1-(Fk(xk)/Fj(xj))+ln(Fk(xk)/Fj(xj))≤0 and the result above, we can obtain that ∑k,j=1n‍vkβ¯kj(1-(Fk(xk)/Fj(xj)))≤0. This completes the proof.Remark 3.
Some results in [1–13] from graph theory can be obtained by Theorem 2. For example, ∑k,j=1n‍vkβ¯kjln((EkEj*)/(Ek*Ej))=0 in [11] and ∑k,j=1n‍vkβ¯kj[n+2-(Sk*/Sk)-((SkIjyk,1*)/(Sk*Ij*yk,1))-∑i=2n‍((yk,i-1yk,i*)/(yk,i-1*yk,i))-((yk,nIk*)/(yk,n*Ik))≤0] in [10].The following result is one result of Kirchhoff’s Three Theorem (a result in graph theory). In the following, using mathematical induction, we show the result.Theorem 4.
If(β¯kj)n×n, n≥2, is irreducible, then Assumption 1 holds.Proof.
Obviously, the result holds forn=2. We assume n≥3 and can rewrite (1) as
(14)[∑l≠1‍β¯1l-β¯21⋯-β¯n1-β¯12∑l≠2‍β¯2l⋯-β¯n2⋮⋮⋱⋮-β¯1n-β¯2n⋯∑l≠n‍β¯nl](v1v2⋮vn)=(00⋮0).
Note that (14) is equivalent to the following system:
(15)[∑l≠1‍β¯1l-β¯21⋯-β¯n10∑l≠2‍β¯2l-β¯21β¯12∑l≠1β¯1l⋯-β¯n2-β¯n1β¯12∑l≠1β¯1l⋮⋮⋱⋮0-β¯2n-β¯21β¯1n∑l≠1β¯1l⋯∑l≠n‍β¯nl-β¯n1β¯1n∑l≠1β¯1l]×(v1v2⋮vn)=(00⋮0).
Consider the following system:(16)[∑l≠2‍β¯2l-β¯21β¯12∑l≠1β¯1l-β¯32-β¯31β¯12∑l≠1β¯1l⋯-β¯n2-β¯n1β¯12∑l≠1β¯1l-β¯23-β¯21β¯13∑l≠1β¯1l∑l≠3‍β¯3l-β¯31β¯13∑l≠1β¯1l⋯-β¯n3-β¯n1β¯13∑l≠1β¯1l⋮⋮⋱⋮-β¯2n-β¯21β¯1n∑l≠1β¯1l-β¯3n-β¯31β¯1n∑l≠1β¯1l⋯∑l≠n‍β¯nl-β¯n1β¯1n∑l≠1β¯1l]×(v2⋮vn)=(0⋮0).Let(17)[-a11-a21⋯-an-1,1-a12-a22⋯-an-1,2⋮⋮⋱⋮-a1,n-1-a2,n-1⋯-an-1,n-1]=[∑l≠2‍β¯2l-β¯21β¯12∑l≠1β¯1l-β¯32-β¯31β¯12∑l≠1β¯1l⋯-β¯n2-β¯n1β¯12∑l≠1β¯1l-β¯23-β¯21β¯13∑l≠1β¯1l∑l≠3‍β¯3l-β¯31β¯13∑l≠1β¯1l⋯-β¯n3-β¯n1β¯13∑l≠1β¯1l⋮⋮⋱⋮-β¯2n-β¯21β¯1n∑l≠1β¯1l-β¯3n-β¯31β¯1n∑l≠1β¯1l⋯∑l≠n‍β¯nl-β¯n1β¯1n∑l≠1β¯1l]. We can rewrite (16) as
(18)[∑l≠1‍a1l-a21⋯-an-1,1-a12∑l≠2‍a2l⋯-an-1,2⋮⋮⋱⋮-a1,n-1-a2,n-1⋯∑l≠n-1‍an-1,l](v2v3⋮vn)=(00⋮0).
From (17), it is easy to see that (akj)(n-1)×(n-1), n≥3, is also irreducible. Note that ∑l≠1‍β¯1l≠0. According to the discussion above, we can deduce that if the result (if (β¯kj)n×n, n≥2, is irreducible, then Assumption 1 holds) holds for n=L, L≥2, then it holds also for n=L+1. The proof is complete.


## 3. An Example of an Epidemic Model with Stage Structure and Latency Spreading in a Heterogeneous Host Population

In this section, we present an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using the results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.We formulate an epidemic model with latency spreading in a heterogeneous host population. LetSk(1), Sk(2), Ik, and Rk denote the immature susceptible, mature susceptible, infectious, and recovered population in the kth group, respectively. The disease incidence in the kth group can be calculated as
(19)∑i=12‍∑j=1n‍βkj(i)Sk(i)Ij,
where the sum takes into account cross-infections from all groups and βkj(i) is the transmission coefficient between compartments Sk(i) and Ij. Let dk(1) and dk(2) represent death rates of Sk(1) and Sk(2) populations, respectively. Let τ be the latent period of the population. e-dk(i)τβkj(i)Sk(i)(t-τ)Ij(t-τ) represents the individuals surviving in the latent period τ and becoming infective at time t. Let pk(i)(τ):[0,h]→ℝ+ be integrable function with ∫0h‍pk(i)(τ)dτ=1. We assume that τ is distributed according to pk(i)(τ) over the interval [0,h], where h is the upper bound of the latent period. Then, we obtain the following model for a disease with latency:
(20)S˙k(1)=bk-dk(1)Sk(1)-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1),S˙k(2)=akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2),I˙k=∑i=12‍∑j=1n‍∫0h‍pk(i)(τ)e-dk(i)τβkj(i)Sk(i)(t-τ)Ij(t-τ)dτ-(dk+μk+γk)Ik,R˙k=γkIk-dkRk,k=1,2,…,n,
where bk(1) denotes influx of individuals into the immature susceptible class in the kth group. ak is the conversion rate from immature individual to mature individual in group k. dk, μk, and γk are the natural death rate, the disease-related death rate, and the recovery rate in the kth group, respectively. All parameter values are assumed to be nonnegative and bk, h, ak, dk(i), dk>0.Letfk(i)(τ)=pk(i)(τ)e-dk(i)τ, ck(i)=∫0h‍pk(i)(τ)e-dk(i)τdτ, φk(Sk(1))=bk-dk(1)Sk(1), and mk=dk+μk+γk. Since the variables Rk do not appear in the remaining three equations of (20), we can consider the following reduced system:
(21)S˙k(1)=φk(Sk(1))-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1),S˙k(2)=akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2),I˙k=∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)(t-τ)Ij(t-τ)dτ-mkIk,k=1,2,…,n.
The initial conditions for system (21) take the form
(22)Sk(1)(θ)=ϕk(1)(θ)≥0,Sk(2)(θ)=ϕk(2)(θ)≥0,Ik(θ)=ϕk(θ)≥0,ϕk(i)(0)>0,ϕk(0)>0,i=1,2,k=1,2,…,n,θ∈[-h,0],
where (ϕ1(1)(θ),ϕ1(2)(θ),…,ϕn(θ))∈C([-h,0],ℝ+3n), the Banach space of continuous functions mapping the interval [-h,0] into ℝ+3n.We see that system (21) exits a disease-free equilibrium P0=(S10(1),S10(2),…,Sn0(1),Sn0(2),0,0,…,0), where
(23)φk(Sk0(1))=dk(2)Sk0(2),akSk0(1)=dk(2)Sk0(2),k=1,2,…,n.
Let
(24)Akj=∑i=12ck(i)βkj(i)Sk0(i)mk,k,j=1,2,…,n,
and Q=(Akj)n×n. We assume that(H1)
there exist(w1,w2,…,wn)>0 (wk>0, k=1,2,…,n), such that
(25)(w1,w2,…,wn)Q<(w1,w2,…,wn);(H2)
there exist(w1,w2,…,wn)>0, such that
(26)(w1,w2,…,wn)Q=(w1,w2,…,wn);(H3)
there exist(w1,w2,…,wn)>0, such that
(27)(w1,w2,…,wn)Q>(w1,w2,…,wn).Let(i)
R0<1 if and only if (H1) holds;(ii)
R0=1 if and only if (H2) holds;(iii)
R0>1 if and only if (H3) holds.Remark 5.
By an approach as the one in [1–3, 7–13], we define r0. Let r0=ρ(Q), where ρ denotes the spectral radius. If a matrix is irreducible, then, for the eigenvalue of maximum, the associated eigenvector is positive. Note that the authors in [1–3, 7–13] discussed some coupled systems of differential equations under a definition with an approach as the one of definition of r0 and the assumption that the relevant matrix is irreducible. In fact, if r0<1 and the relevant matrix is irreducible, then R0<1; if r0=1 and the relevant matrix is irreducible, then R0=1; if r0>1 and the relevant matrix is irreducible, then R0>1. However, the reverse is not true. For example,
(28)(Akj)2×2=[1012].
Furthermore, let
(29)(Akj)4×4=[1100110000110011].
Obviously, R0>1 holds also for arbitrary (w1,w2,w3,w4)>0. (Akj)4×4 conforms to the conditions of Theorem 8 but is not in accord with the conditions of Corollary 10. The definition of r0 and the assumption that the relative matrix is irreducible in the results of Corollaries 9 and 10 are analogous with relative definition and assumption in [1–3, 7–13].
The equilibria of (21) are the same as those of the associated ODE system:
(30)S˙k(1)=φk(Sk(1))-Sk(1)∑j=1n‍βkj(1)Ij-akSk(1),S˙k(2)=akSk(1)-Sk(2)∑j=1n‍βkj(2)Ij-dk(2)Sk(2),I˙k=∑i=12‍ck(i)Sk(i)∑j=1n‍βkj(i)Ij-mkIk,k=1,2,…,n.
Let
(31)Nk=Sk(1)+Sk(2)+Ik,d_k=min{dk(1),dk(2),dk},k=1,2,…,n.
From (30) and ck(i)≤1, we have
(32)N˙k≤φk(Sk(1))+dk(1)Sk(1)-d_kNk.
We derive from (32) that the region
(33)Γ={φk(0)+dk(1)Sk0(1)d_k(S1(1),S1(2),…,Sn(1),Sn(2),I1,…,In)∈ℝ+3n:Sk(1)≤Sk0(1),Sk(2)≤Sk0(2),Sk(1)+Sk(2)+Ik≤φk(0)+dk(1)Sk0(1)d_k,k=1,2,…,n}
is a forward invariant compact absorbing set with respect to (30). Let Γ∘ denote the interior of Γ.
Note thatΓ is positively invariant with respect to (21). In fact, let
(34)E˙k=∑i=12‍∑j=1n‍βkj(i)Sk(i)Ij-∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)(t-τ)Ij(t-τ)dτ-dkEk,k=1,2,…,n.
Then, we have
(35)N˙k+E˙k≤φk(Sk(1))+dk(1)Sk(1)-d_k(Nk+Ek).
Furthermore, using the fact that Nk≤Nk+Ek, we may obtain that Γ is positively invariant with respect to (21).Lemma 6.
IfR0>1, then P0 of system (30) is unstable in Γ and there exists a positive equilibrium P* in Γ∘.Proof.
LetI=(I1,I2,…,In) and M=∑k=1n‍((ωkIk)/(mk)). Thus
(36)M˙=∑k=1n‍ωk[∑i=12ck(i)∑j=1nβkj(i)Sk(i)Ijmk-Ik]≤(ω1,ω2,…,ωn)(QIT-IT).
If R0>1, by continuity, we obtain M˙>0, in a neighborhood of P0 in Γ∘. This implies that P0 is unstable. Using the uniform persistence result from [15] and by a similar argument to that in the proof in [1], we can show that if R0>1, the instability of P0 implies the uniform persistence of system (30). This, together with the uniform boundedness of solutions of (30) in Γ∘, implies (30) has at least a positive equilibrium P* in Γ∘. The proof is completed.Let(37)P*=(S1(1)*,S1(2)*,…,Sn(1)*,Sn(2)*,I1*,I2*,…,In*);
then the components of P* satisfy
(38)φk(Sk(1)*)=∑i=12‍Sk(i)*∑j=1n‍βkj(i)Ij*+dk(2)Sk(2)*,(39)∑i=12‍ck(i)Sk(i)*∑j=1n‍βkj(i)Ij*=mkIk*,(40)akSk(1)*=dk(2)Sk(2)*+Sk(2)*∑j=1n‍βkj(2)Ij*,k=1,2,…,n.
Next, we will study the global stability of equilibria of system (21).Theorem 7.
Ifck(1)≥ck(2), k=1,2,…,n, and R0≤1, then P0 of system (21) is globally asymptotically stable in Γ.Proof.
LetS0=(S10(1),S10(2),…,Sn0(1),Sn0(2)). Consider a Lyapunov functional L1+L2, where
(41)L1=∑k=1n‍ωkmk[∑i=12‍ck(i)(Sk(i)-Sk0(i)-Sk0(i)lnSk(i)Sk0(i))+Ik],L2=∑k=1n‍ωkmk[∑i=12‍∑j=1n‍βkj(i)∫0h‍fk(i)(τ)∫t-τt‍Sk(i)(x)Ij(x)dxdτ].
Differentiating L1 along the solution of system (21), we obtain
(42)L˙1=∑k=1n‍ωkmk{ck(1)[φk(Sk(1))-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1)]+ck(2)[akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2)]+∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Skj(i)(t-τ)×Ij(t-τ)dτ-mkIk-ck(1)Sk0(1)Sk(1)[φk(Sk(1))-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1)]-ck(2)Sk0(2)Sk(2)[akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2)]}.
Differentiating L2 along the solution of system (21), we obtain
(43)L2=∑k=1n‍ωkmk[∑i=12‍∑j=1n‍ck(i)βkj(i)Sk(i)Ij-∑i=12‍∑j=1n‍βkj(i)∫0h‍fk(i)(τ)Sk(i)(t-τ)Ij(t-τ)dτ].
Therefore
(44)L˙=∑k=1n‍ωkmk{∑i=12‍∑j=1n‍φk(Sk(1))(ck(1)-ck(1)Sk0(1)Sk(1))+dk(2)Sk0(2)×(ck(2)-ck(2)Sk(2)Sk0(2))+akSk0(1)×[ck(1)-ck(1)Sk(1)Sk0(1)+ck(2)Sk(1)Sk0(1)-ck(2)Sk(1)Sk0(2)Sk0(1)Sk(2)]+∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk}=∑k=1n‍ωkmk{φk(Sk(1))(ck(1)-ck(1)Sk0(1)Sk(1))+∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk+dk(2)Sk0(2)[ck(1)+ck(2)-ck(1)Sk(1)Sk0(1)+ck(2)Sk(1)Sk0(1)-ck(2)Sk(1)Sk0(2)Sk0(1)Sk(2)-ck(2)Sk(2)Sk0(2)]}.
From (23), we know that
(45)φk(Sk0(1))(ck(1)-ck(1)Sk0(1)Sk(1))=dk(2)Sk0(2)(ck(1)-ck(1)Sk0(1)Sk(1)),k=1,2,…,n.
By (45), we obtain
(46)L˙=∑k=1n‍ωkmk{[φk(Sk(1))-φk(Sk0(1))](ck(1)-ck(1)Sk0(1)Sk(1))+∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk+dk(2)Sk0(2)[2ck(1)+ck(2)-ck(1)Sk(1)Sk0(1)-ck(1)Sk0(1)Sk(1)+ck(2)Sk(1)Sk0(1)-ck(2)Sk(1)Sk0(2)Sk0(1)Sk(2)-ck(2)Sk(2)Sk0(2)]}.
We can rewrite the equation as
(47)L˙=∑k=1n‍ωkmk{[φk(Sk(1))-φk(Sk0(1))](ck(1)-ck(1)Sk0(1)Sk(1))+∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk+dk(2)Sk0(2)[(Sk0(1)Sk(1)+Sk(1)Sk0(2)Sk0(1)Sk(2)+Sk(2)Sk0(2))2ck(1)+ck(2)-(ck(1)-ck(2))×(Sk(1)Sk0(1)+Sk0(1)Sk(1))-ck(2)(Sk0(1)Sk(1)+Sk(1)Sk0(2)Sk0(1)Sk(2)+Sk(2)Sk0(2))]}.
By the fact that φk is strictly decreasing function and the arithmetic-geometric mean, we have
(48)L˙≤∑k=1n‍ωkmk{∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk+dk(2)Sk0(2)×[2ck(1)+ck(2)-2(ck(1)-ck(2))-3ck(2)]∑i=12‍∑j=1n‍}=∑k=1n‍ωkmk(∑i=12‍∑j=1n‍Sk0(i)ck(i)βkj(i)Ij-mkIk)=:U,
where equality holds if and only if
(49)Sk(1)=Sk0(1),Sk(2)=Sk0(2),k=1,2,…,n.
Thus
(50)U=∑k=1n‍ωk[∑i=12ck(i)∑j=1nβkj(i)Sk0(i)Ijmk-Ik]=(ω1,ω2,…,ωn)(QIT-IT).
If R0<1, then L˙=0 if and only if IT=0. If R0=1, then L˙=0 implies U=0. If R0=1 and U˙=0, then (49) holds. If (49) holds, then, from the first two equations of (21), we may obtain ∑j=1n‍βkj(1)Sk(1)Ij≡0, ∑j=1n‍βkj(2)Sk(2)Ij≡0, k=1,2…,n. Therefore, if R0=1 and U˙=0, then we have Ik=0. Therefore, L˙=0 if and only if I=0 and S=S0. Hence the largest invariant subset of the set where L˙=0 is the singleton {P0}. By LaSalle’s Invariance Principle, P0 is globally attractive. Using the same proof as the one for Corollary 5.3.1 in [16], we can show that P0 is locally stable. Hence, the disease-free equilibrium P0 is globally asymptotically stable in Γ for R0≤1. This completes the proof.Theorem 8.
Under Assumption1, P* of system (21) is globally asymptotically stable in Γ, if ck(1)≥ck(2), k=1,2,…,n, and R0>1.Proof.
Setβ¯kj=∑i=12‍ck(i)βkj(i)Sk(i)*Ij*, 1≤k,j≤n. Consider a Lyapunov functional V=V1+V2, where
(51)V1=∑k=1n‍vk[∑i=12‍ck(i)(Sk(i)-Sk(i)*-Sk(i)*lnSk(i)Sk(i)*)+Ik-Ik*-Ik*lnIkIk*∑i=12‍],V2=∑k=1n‍vk∑i=12‍∑j=1n‍βkj(i)×∫0hfk(i)(τ)‍∫t-τt‍(Sk(i)(x)Ij(x)Sk(i)*Ij*Sk(i)(x)Ij(x)-Sk(i)*Ij*-Sk(i)*Ij*×lnSk(i)(x)Ij(x)Sk(i)*Ij*)dxdτ.
Differentiating V1 along the solution of system (21), we obtain
(52)V˙1=∑k=1n‍vk{ck(1)[φk(Sk(1))-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1)]+ck(2)[akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2)]+∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)(t-τ)Ij(t-τ)dτ-mkIk-ck(1)Sk(1)*Sk(1)[φk(Sk(1))-∑j=1n‍βkj(1)Sk(1)Ij-akSk(1)]-ck(2)Sk(2)*Sk(2)[akSk(1)-∑j=1n‍βkj(2)Sk(2)Ij-dk(2)Sk(2)]-Ik*Ik[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)(t-τ)×Ij(t-τ)dτ-mkIk∑i=12‍∑j=1n‍]}.
Differentiating V2 along the solution of system (21), we obtain
(53)V˙2=∑k=1n‍vk∑i=12‍∑j=1n‍βkj(i)×{Sk(i)IjSk(i)(t-τ)Ij(t-τ)ck(i)Sk(i)Ij-∫0h‍fk(i)(τ)[Sk(i)IjSk(i)(t-τ)Ij(t-τ)Sk(i)(t-τ)Ij(t-τ)+Sk(i)*Ij*×lnSk(i)IjSk(i)(t-τ)Ij(t-τ)]dτ}.
Therefore
(54)V˙=∑k=1n‍vk{×(t-τ)Ij(t-τ)dτ∑i=12‍∑j=1n‍]φk(Sk(1))(ck(1)-ck(1)Sk(1)*Sk(1))+dk(2)Sk(2)*(ck(2)-ck(2)Sk(2)Sk(2)*)+∑i=12‍ck(i)∑j=1n‍βkj(i)Sk(i)*Ij+mkIk*-mkIk+akSk(1)*×[ck(1)-ck(1)Sk(1)Sk(1)*+ck(2)Sk(1)Sk(1)*-ck(2)Sk(1)Sk(2)*Sk(1)*Sk(2)]-Ik*Ik[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)×(t-τ)Ij(t-τ)dτ∑i=12‍∑j=1n‍]-Δ},
where
(55)Δ=∑i=12‍∑j=1n‍βkj(i)∫0h‍fk(i)(τ)Sk(i)*Ij*lnSk(i)IjSk(i)(t-τ)Ij(t-τ)dτ.
From (38), we know that
(56)φk(Sk(1)*)(ck(1)-ck(1)Sk(1)*Sk(1))=(∑i=12‍∑j=1n‍βkj(i)Sk(i)*Ij*+dk(2)Sk(2)*)(ck(1)-ck(1)Sk(1)*Sk(1)).
It follows from (39), (40), and (56) that
(57)V˙=∑k=1n‍vk{[φk(Sk(1))-φk(Sk(1)*)](ck(1)-ck(1)Sk(1)*Sk(1))+dk(2)Sk(2)*(ck(2)-ck(2)Sk(2)Sk(2)*)+∑i=12‍ck(i)βkj(i)Sk(i)*Ij*(IjIj*-IkIk*+1)+Sk(1)*∑j=1n‍βkj(1)Ij*(ck(1)-ck(1)Sk(1)*Sk(1))+(dk(2)Sk(2)*+Sk(2)*∑j=1n‍βkj(2)Ij*)×[2ck(1)-ck(1)Sk(1)*Sk(1)-ck(1)Sk(1)Sk(1)*+ck(2)Sk(1)Sk(1)*-ck(2)Sk(1)Sk(2)*Sk(1)*Sk(2)]-Ik*Ik×[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)(t-τ)Ij(t-τ)dτ]-Δ(ck(1)-ck(1)Sk(1)*Sk(1))}.
By Theorem 2 and the fact that φk is strictly decreasing function, we obtain(58)V˙≤∑k=1n‍vk{[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)dk(2)Sk(2)*(ck(2)-ck(2)Sk(2)Sk(2)*)+∑i=12‍ck(i)βkj(i)Sk(i)*Ij*+Sk(1)*∑j=1n‍βkj(1)Ij*(ck(1)-ck(1)Sk(1)*Sk(1))+(dk(2)Sk(2)*+Sk(2)*∑j=1n‍βkj(2)Ij*)×[2ck(1)-ck(1)Sk(1)*Sk(1)-ck(1)Sk(1)Sk(1)*+ck(2)Sk(1)Sk(1)*-ck(2)Sk(1)Sk(2)*Sk(1)*Sk(2)]-Ik*Ik[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)×(t-τ)Ij(t-τ)dτ∑i=12‍∑j=1n‍]-Δ}=∑k=1n‍vk{dk(2)Sk(2)*(ck(2)+2ck(1)-ck(1)Sk(1)*Sk(1)-ck(1)Sk(1)Sk(1)*+ck(2)Sk(1)Sk(1)*-ck(2)Sk(1)Sk(2)*Sk(1)*Sk(2)-ck(2)Sk(2)Sk(2)*)+Sk(2)*∑j=1n‍βkj(2)Ij*[2ck(1)+ck(2)-ck(1)Sk(1)*Sk(1)-ck(1)Sk(1)Sk(1)*+ck(2)Sk(1)Sk(1)*-ck(2)Sk(1)Sk(2)*Sk(1)*Sk(2)]+Sk(1)*∑j=1n‍βkj(1)Ij*(2ck(1)-ck(1)Sk(1)*Sk(1))-Ik*Ik[∑i=12‍∑j=1n‍∫0h‍fk(i)(τ)βkj(i)Sk(i)×(t-τ)Ij(t-τ)dτ∑i=12‍∑j=1n‍]-Δ}=∑k=1n‍vk{dk(2)Sk(2)*[×(Sk(1)*Sk(1)+Sk(1)Sk(2)*Sk(1)*Sk(2)+Sk(2)Sk(2)*)ck(2)+2ck(1)-(ck(1)-ck(2))×(Sk(1)*Sk(1)+Sk(1)Sk(1)*)-ck(2)(Sk(1)*Sk(1)+Sk(1)Sk(2)*Sk(1)*Sk(2)+Sk(2)Sk(2)*)]+Sk(2)*×∑j=1n‍βkj(2)Ij*[Sk(1)*Sk(1)+Sk(1)Sk(2)*Sk(1)*Sk(2)2ck(1)+ck(2)-(ck(1)-ck(2))×(Sk(1)*Sk(1)+Sk(1)Sk(1)*)-∫0h‍fk(2)(τ)×(Sk(1)*Sk(1)+Sk(1)Sk(2)*Sk(1)*Sk(2)+((Sk(2)Ik*Sk(2)(Sk(2)*Sk(2)IkIj*)-1×(t-τ)Ij(t-τ)Sk(2)Ik*Sk(2))×(Sk(2)*Sk(2)IkIj*)-1)Sk(1)*Sk(1)+Sk(1)Sk(2)*Sk(1)*Sk(2))dτ]+Sk(1)*∑j=1n‍βkj(1)Ij*×[×(Sk(1)*Sk(1)IkIj*)-1)Sk(1)*Sk(1))2ck(1)-∫0h‍fk(1)(τ)×(Sk(1)*Sk(1)+((Sk(1)*Sk(1)IkIj*)-1(Sk(1)Ik*Sk(1)(t-τ)Ij(t-τ))×(Sk(1)*Sk(1)IkIj*)-1)Sk(1)*Sk(1))dτ]-Δ(ck(2)-ck(2)Sk(2)Sk(2)*)}=:B1. By the arithmetic-geometric mean, we easily see that (59)B1≤∑k=1n‍vk{+Sk(2)*∑j=1n‍βkj(2)Ij*[∫0h‍fk(2)(τ)(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)132ck(1)+ck(2)-2(ck(1)-ck(2))dk(2)Sk(2)*[ck(2)+2ck(1)-2(ck(1)-ck(2))-3ck(2)]+Sk(2)*∑j=1n‍βkj(2)Ij*[∫0h‍fk(2)(τ)(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)1/32ck(1)+ck(2)-2(ck(1)-ck(2))-3∫0h‍fk(2)(τ)×(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)1/3dτ]+Sk(1)*∑j=1n‍βkj(1)Ij*×[(Ik*Sk(1)(t-τ)Ij(t-τ)Sk(1)IkIj*)1/22ck(1)-2∫0h‍fk(1)(τ)×(Ik*Sk(1)(t-τ)Ij(t-τ)Sk(1)IkIj*)1/2dτ]-Δ+Sk(2)*∑j=1n‍βkj(2)Ij*[∫0h‍fk(2)(τ)(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)132ck(1)+ck(2)-2(ck(1)-ck(2))}=∑k=1n‍vk{3Sk(2)*∑j=1n‍βkj(2)Ij*×[(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)1/3ck(2)-∫0h‍fk(2)(τ)×(Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*)1/3dτ]+2Sk(1)*×∑j=1n‍βkj(1)Ij*[(Ik*Sk(1)(t-τ)Ij(t-τ)Sk(1)IkIj*)1/2ck(1)-∫0h‍fk(1)(τ)×(Ik*Sk(1)(t-τ)Ij(t-τ)Sk(1)IkIj*)1/2dτ]-Δ∑j=1n‍}=:B2. Let Y(x)=1-x+lnx. We can rewrite B2 as
(60)∑k=1n‍vk{3Sk(2)*∑j=1n‍βkj(2)Ij*×∫0h‍fk(2)(τ)Y×([Ik*Sk(2)(t-τ)Ij(t-τ)Sk(2)IkIj*]1/3)dτ+2Sk(1)*∑j=1n‍βkj(1)Ij*×∫0h‍fk(1)(τ)Y×([Ik*Sk(1)(t-τ)Ij(t-τ)Sk(1)IkIj*]1/2)dτ-∑i=12‍∑j=1n‍βkj(i)×∫0h‍fk(i)(τ)Sk(i)*Ij*×lnIk*Sk(i)(t-τ)Ij(t-τ)IkIj*Sk(i)dτ-Δ}.
By Theorem 2 and the fact that Y(x)≤0, where equality holds if and only if x=1, we obtain
(61)B2≤∑k=1n‍vk{-∑i=12‍∑j=1n‍βkj(i)∫0h‍fk(i)(τ)Sk(i)*Ij*×lnIk*Sk(i)(t-τ)Ij(t-τ)IkIj*Sk(i)dτ-Δ}=-∑k=1n‍vk∑i=12‍∑j=1n‍ck(i)βkj(i)Sk(i)*Ij*lnIk*IjIkIj*=-∑k,j=1n‍vkβ¯kjlnIk*IjIkIj*=0.
From (58) and (59), we see that if V˙=0, then
(62)Sk(i)=Sk(i)*,i=1,2,k=1,2,…,n.
If (62) holds, it follows from (21) that
(63)0=φk(Sk(1)*)-∑j=1n‍βkj(1)Sk(1)*Ij-akSk(1)*,0=akSk(1)*-∑j=1n‍βkj(2)Sk(2)*Ij-dk(2)Sk(2)*,k=1,2,…,n.
Then, we obtain that
(64)∑j=1n‍βkj(1)Sk(1)*Ij=φk(Sk(1)*)-akSk(1)*,∑j=1n‍βkj(2)Sk(2)*Ij=akSk(1)*-dk(2)Sk(2)*,k=1,2,…,n.
Thus,
(65)I˙k=ck(1)(φk(Sk(1)*)-akSk(1)*)+ck(2)(akSk(1)*-dk(2)Sk(2)*)-mkIk,k=1,2,…,n.
This implies that
(66)limt→+∞Ik=ck(1)(φk(Sk(1)*)-akSk(1)*)+ck(2)(akSk(1)*-dk(2)Sk(2)*)mk=Ik*.
Hence, the largest invariant subset of the set where V˙=0 is the singleton {P*}. By LaSalle’s Invariance Principle, P* is globally attractive. By a similar argument to that in the proof of Theorem 7, P* is globally asymptotically stable in Γ for R0>1.Following [17], we set matrices
(67)F:=(∑i=12‍ck(i)βkj(i)Sk0(i))n×n,V:=diag(m1,m2,…,mn).
The next generation matrix for system (16) is
(68)Q:=FV-1=(∑i=12ck(i)βkj(i)Sk0(i)mk)n×n.
Let r0=ρ(Q), where ρ denotes the spectral radius.By Lemma6 and Theorems 2, 4, 7, and 8, we can obtain the following results.Corollary 9.
Assume thatB=[∑i=12‍βkj(i)] is irreducible and ck(1)≥ck(2), k=1,2,…,n. If r0≤1, then P0 of system (21) is globally asymptotically stable in Γ.Corollary 10.
Assume thatB=[∑i=12‍βkj(i)] is irreducible and ck(1)≥ck(2), k=1,2,…,n. If r0>1, then P* of system (21) is globally asymptotically stable in Γ.Remark 11.
Note that, for Theorems7 and 8, we do not assume that B=[∑i=12‍βkj(i)] is irreducible. If Theorems 7 and 8 hold, then Corollaries 9 and 10 hold too, respectively. However, it is easy to show some examples which conform to the conditions of Theorem 7 or Theorem 8 but are not in accord with the conditions of Corollary 9 or Corollary 10. It seems that all authors use the the graph theory under the assumption that the relative matrix is irreducible which is analogous with the conditions of Corollaries 9 and 10 (see, e.g., [1–13]).



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*Source: 276579-2013-12-08.xml*