AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 740256 10.1155/2014/740256 740256 Research Article Stability Analysis of a Multigroup SEIR Epidemic Model with General Latency Distributions http://orcid.org/0000-0003-4275-8541 Wang Nan Pang Jingmei http://orcid.org/0000-0002-9646-2765 Wang Jinliang Wang Kaifa 1 School of Mathematical Science Heilongjiang University Harbin 150080 China hlju.edu.cn 2014 1742014 2014 11 02 2014 27 03 2014 17 4 2014 2014 Copyright © 2014 Nan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction number 0 is defined and proved as the role of a threshold; that is, the disease-free equilibrium P0 is globally asymptotically stable if 01, while an endemic equilibrium P* exists uniquely and is globally asymptotically stable if 0>1. For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach.

1. Introduction

Mathematical models have become important tools in analyzing the spread and control of infectious diseases. The SIR model is one of the most popular ones in this field, for which the total population is subdivided into three compartments: susceptible, infectious, and removed. For some diseases, it is reasonable to include a latent (or exposed) class for those susceptible individuals who are infected with the disease but are not yet infectious, which leads to SEIR model . Let S(t), E(t), I(t), and R(t) be the numbers of individuals in the susceptible, exposed, infectious, and removed compartments, respectively, with the total population N(t)=S(t)+E(t)+I(t)+R(t). Suppose that d>0 represents the constant recruitment rate and the natural mortality rate. Assuming mass action for the disease transmission and letting β>0 denote the effective contact rate, the rate of change of S(t) is (1)S(t)=d-βS(t)I(t)-dS(t). Taking into consideration a general exposed distribution, van den Driessche et al.  formulated and studied the following model: (2)S(t)=d-βS(t)I(t)-dS(t),E(t)=0tβS(u)I(u)e-d(t-u)P(t-u)du,R(t)=  rI(t)-dR(t),I(t)=N-S(t)-E(t)-R(t), where r0 is the rate at which infective individuals recover. N is constant total populations. It is assumed in  that individuals rarely die of the disease and the disease-induced death is negligible, which ensures a constant population; that is, N(t)=N·P(t) denotes the probability (without taking death into account) that an exposed individual still remains in the exposed class t time units after entering the exposed class and it satisfies the following.

(A1) P:[0,)[0,1] is nonincreasing, piecewise continuous with possibly finitely many jumps and satisfies P(0+)=1, limtP(t)=0 with 0P(u)du being positive and finite.

In fact, the integral term in model (2) is in the sense of Riemann-Stieltjes integrals; the second equation of (2) takes the following form: (3)E(t)=βS(t)I(t)-dE(t)+0tβS(u)I(u)e-d(t-u)dtP(t-u)du, where dtP(t-u)=dP(t-u)/dt. It follows from total population size N which is constant that the rate of change of I is governed by (4)I(t)=-0tβS(u)I(u)e-d(t-u)dtP(t-u)du-(d+r)I(t). Thus, model (2) can be written as the system (5)S(t)=d-βS(t)I(t)-dS(t),E(t)=βS(t)I(t)-dE(t)+0tβS(u)I(u)e-d(t-u)dtP(t-u)du,I(t)=-0tβS(u)I(u)e-d(t-u)dtP(t-u)du-(d+r)I(t),R(t)=  rI(t)-dR(t). Recently, a model of this type including the possibility of disease relapse has been proposed in [5, 6] to study the transmission and spread of some infectious diseases such as herpes, and its global dynamics have been completely investigated in [5, 7].

Heterogeneity in the host population can result from different contact modes such as those among children and adults for childhood diseases (e.g., measles and mumps) or different behaviors such as the numbers of sexual partners for some sexually transmitted infections (e.g., herpes and condyloma acuminatum). Taking into consideration different contact patterns, distinct number of sexual partners, or different geography and so forth, it is more proper to divide individual hosts into groups. Therefore, lots of multigroup models have been proposed in the literature to describe the transmission of infectious disease in heterogeneity environment (see  and references cited therein).

In multigroup epidemic models, a heterogeneous host population is divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions, so that within-group and intergroup interactions can be modeled separately. In this paper, we formulate a multigroup SEIR epidemic model with general exposed distribution and general incidence rates. The population is divided into n distinct groups (n2). For 1kn, the kth group is further partitioned into four compartments: susceptible, exposed, infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), Ik(t), and Rk(t), respectively. Within the kth group, φk(Sk) represents the growth rate of Sk, which includes both the production and the natural death of susceptible individuals.

In , Zhang et al. studied a multigroup SEIR epidemic model with general exposed distribution and general incidence rates. By using the well-known “linear chain trick,” the authors reformulate the model into an equivalent ordinary differential equations system. The global stability results of equilibria are obtained by constructing suitable Lyapunov functionals for general incidence rate function fkj(Sk(t),Ij(t)). In , Hattaf et al. introduced a general incidence rate f(S,I)I in a delayed SIR epidemic model.

Motivated by these facts, in this paper, we incorporate the general incidence rate presented in  to the following system of differential and integral equations: (6)Sk(t)=φk(Sk(t))-j=1nfkj(Sk(t),Ij(t))Ij(t),Ek(t)=j=1nfkj(Sk(t),Ij(t))Ij(t)-j=1n0tfkj(Sk(u),Ij(u))Ij(u)e-δk(t-u)gk(t-u)du-δkEk(t),Ik(t)=j=1n0tfkj(Sk(u),Ij(u))Ij(u)e-δk(t-u)gk(t-u)du-(δk+γk)Ik(t),Rk(t)=γkIk(t)-δkRk(t), where gj(t)=-Pj(t), the nonlinear term fkj(Sk(t),Ij(t))Ij(t) represents the cross-infection from group j to group k, δk denotes the natural death rates of exposed and infectious classes in the kth group, and γk denotes the production of the recovered individuals from infectious ones in the kth group. All constants δk, γk,  k=1,2,,n, are assumed to be positive.

The organization of this paper is as follows; in the next section, we give some preliminaries of our main model. In Section 3, we prove the global asymptotic stability of the disease-free equilibrium P0 for 01 using the classical method of Lyapunov. The existence of endemic equilibrium is also proved. In Section 4, we prove global asymptotic stability of an endemic equilibrium P* for 0>1 using the graph-theoretic approach.

2. Preliminaries

Since the variables Ek and Rk do not appear in the first and third equations of (6), we can only consider the reduced system as follows: (7)Sk(t)=φk(Sk(t))-j=1nfkj(Sk(t),Ij(t))Ij(t),Ik(t)=j=1n0tfkj(Sk(u),Ij(u))Ij(u)e-δk(t-u)gk(t-u)du-(δk+γk)Ik(t).

The incidence function fkj(Sk,Ij) in (7) is assumed to be continuously differentiable in the interior of +2 and to satisfy the following hypotheses:

fkj(0,Ij)=0, for all Ij0;

fkj(Sk,Ij)/Sk>0, for all Sk>0 and Ij0;

fkj(Sk,Ij)/Ij0, for all Sk0 and Ij0;

assume that the functions φk satisfy the following conditions:

φk are local Lipschitz on [0,) with φk(0)>0, and there is a unique positive solution ξ=Sk0 for the equation φk(ξ)=0; φk(Sk)>0 for 0Sk<Sk0, and φk(Sk)<0 for Sk>Sk0.

Typical examples of fkj(Sk,Ij) satisfying (S1)(S3) include common incidence functions such as (8)fkj(Sk,Ij)=SkIj  [20,2,3],fkj(Sk,Ij)=SkqIj  ,fkj(Sk,Ij)=ηSkIj1+θSk  . The class of φk(Sk) that satisfies (S4) includes both λk-dkSSk and λk-dkSSk+rkSk(1-Sk/Nk), which have been widely used in the literature of population dynamics [1, 8].

For model (7), the existence, uniqueness, and continuity of solutions follow from the theory for integrodifferential equations in . It can be easily verified that every solution of (7) with nonnegative initial conditions remains nonnegative. It follows from (S4) and the first equation in (7) that Sk(t)φk(Sk(t)), and thus (9)limsuptSk(t)Sk0,for  1kn. From the biological significance, we only need to consider (7) in the following region: (10)Γ{+2n:Sk,Ik0,Sk+IkSk0(S1,I1,S2,I2,,Sn,In)mnm+2n:Sk,Ik0,Sk+IkSk0,1kn}. Indeed, one can easily verify that the set Γ is positively invariant with respect to (7).

It is clear that system (7) has a disease-free equilibrium P0=(S10,0,S10,0,,Sn0,0) in Γ. Next, we will give some notations which will be useful for our main results.

Let (11)J(ξ)=ξgk(u)e-δkudu,Qk=J(0)=0gk(u)e-δkudu. It can be verified that Qk(0,1).

For finite time t, system (7) may not have an endemic equilibrium. If system (7) has an endemic equilibrium, the endemic equilibrium must satisfy the limiting system (12)Sk(t)=φk(Sk(t))-j=1nfkj(Sk(t),Ij(t))Ij(t),Ik(t)=j=1n0fkj(Sk(t-u),Ij(t-u))mmm×Ij(t-u)e-δkugk(u)du-(δk+γk)Ik(t).

Since the limiting system (12) contains an infinite delay, its associated initial condition needs to be restricted in an appropriate fading memory space. For any σk(0,δk), define the following Banach space of fading memory type (see [23, 24] and references therein): (13)Ck={sups0|ϕk(s)|ϕkC((-,0],):ϕk(s)eσksmisuniformlycontinuouson  (-,0],msups0|ϕk(s)|eσks<},YΔ={ϕkCk:ϕk(s)0  s0} with norm ϕk=sups0|ϕ(s)|eσks. Let ψtCi and t>0 be such that ψt(s)=ψ(t+s), s(-,0].

Let ϕk,ψkCk such that ϕk(s),ψk(s)0 for all s(-,0]. We consider solutions of system (12), (S1t,I1t,,Snt,Int), with initial conditions (14)(ϕ1,ψ1,ϕ2,ψ2,,ϕn,ψn). The standard theory of functional differential equations  implies (S1t,I1t,,Snt,Int)Ck for all t>0. We study system (12) in the following phase space: (15)𝕏𝕘=k=1n(×Ck). It can be verified that solutions of (12) in 𝕏𝕘 with initial conditions (14) remain nonnegative.

An equilibrium P*=(S1*,I1*,S2*,I2*,,Sn*,In*) in the interior of Γ is called an endemic equilibrium of system (12), where Sk*,Ik*>0 satisfy the equilibrium equations (16)φk(Sk*)=j=1nfkj(Sk*,Ij*)Ij*,j=1nfkj(Sk*,Ij*)Ij*Qk=(δk+γk)Ik*. Set R0=ρ(M0) to denote the special radius of the matrix M0, where (17)M0=(fkj(Sk0,0)Qkδk+γk)n×n. The parameter R0 is defined as the basic reproduction number [25, 26]. Since it can be verified that system (7) satisfies conditions (A1)(A5) of Theorem 2 of , we have the following lemma.

Lemma 1.

For system (7), the disease-free equilibrium P0 is locally asymptotically stable if 0<1 while it is unstable if 0>1.

3. Global Stability of the Disease-Free Equilibrium Theorem 2.

Assume that the functions φk and fkj satisfy (S1)(S4), and M0 is irreducible.

If 01, then P0 is the unique equilibrium of system (7), and P0 is globally asymptotically stable in Γ.

If 0>1, then P0 is unstable and system (7) is uniformly persistent.

Proof.

It follows from the Perron-Frobenius theorem (see Theorem  2.1.4 in ) that the nonnegative irreducible matrix M0 has a positive eigenvector (ω1,ω2,,ωn) such that (18)(ω1,ω2,,ωn)ρ(M0)=(ω1,ω2,,ωn)M0. Now, we construct a Lyapunov functional (19)VP0=k=1nωkδk+γkIk. Differentiating VP0 along the solution of system (7) and under (S2) and (S3), we obtain (20)VP0=k=1nωk[1δk+γkmmmm×j=1n0tfkj(Sk(u),Ij(u))Ij(u)mmmmmmmmmm×e-δk(t-u)gk(t-u)dummmm-Ik(t)1δk+γk]k=1nωk[1δk+γkj=1nfkj(Sk,0)Ij(t)Qk-Ik(t)]k=1nωk[1δk+γkj=1nfkj(Sk0,0)Ij(t)Qk-Ik(t)]=(ω1,ω2,,ωn)[M0I-I]=[ρ(M0)-1](ω1,ω2,,ωn)I, where I=(I1,I2,,In)T. Suppose that ρ(M0)<1. Then, VP0=0 if and only if I=0. Suppose that ρ(M0)=1. Then, it follows from (20) that VP0=0 implies (21)k=1nωk[1δk+γkj=1nfkj(Sk,0)Ij(t)Qk]=k=1nωkIk(t). If SkSk0, then (22)k=1nωk[1δk+γkj=1nfkj(Sk,0)Ij(t)Qk]k=1nωk[1δk+γkj=1nfkj(Sk0,0)Ij(t)Qk](ω1,ω2,,ωn)M0I=(ω1,ω2,,ωn)ρ(M0)I=(ω1,ω2,,ωn)I, which implies that (21) has only the trivial solution I=0. Therefore, VP0=0 if and only if Ik=0 or Sk=Sk0 provided ρ(M0)=1. It can be verified that the only compact invariant subset of the set where VP0=0 is the singleton {P0}. By LaSalle’s Invariance Principle, P0 is globally asymptotically stable in Γ if ρ(M0)1.

If 0>1 and I0, it is easy to see that (23)[ρ(M0)-1](ω1,ω2,,ωn)I>0. It follows from the continuity that VP0>0 holds in a small neighborhood of P0. This implies that P0 is unstable. Using a uniform persistence result from  and similar arguments as in [4, 10, 13, 16, 17], we know that, if 0>1, the instability of P0 implies the uniform persistence of (7) in Γ; that is, there exists a positive constant ϵ>0 such that (24)liminftSk(t)ϵ,liminftIk(t)ϵ,k=1,2,,n. The uniform persistence of system (7) together with the uniform boundedness of solutions in Γ, which follows from the positive invariance of Γ, implies the existence of an endemic equilibrium P* in Γ (see Theorem 2.8.6 of  or Theorem D.3 of ). Summarizing the statements above, if 0>1, system (7) is uniformly persistent and there exists at least one endemic equilibrium P* in Γ. This completes the proof.

4. Global Stability of an Endemic Equilibrium

Denote (25)H(u)=u-1-lnu,u>0. Obviously, H:++ attains its strict global minimum at u=1 and H(1)=0.

To get the global stability of P*, we make the following assumptions:

(φk(Sk)-φk(Sk*))(Sk-Sk*)0 for Sk0;

(φk(Sk)-φk(Sk*))[fkk(Sk,Ik*)-fkk(Sk*,Ik*)]<0 for SkSk*;

(((fkk(Sk*,Ik*)fkj(Sk,Ij)Ij)/(fkk(Sk,Ik*)fkj(Sk*,Ij*)Ij*)) −  1)(1-((fkk(Sk,Ik*)fkj(Sk*,Ij*))/(fkk(Sk*,Ik*)fkj(Sk, Ij))))0 for Sk,Ij>0.

Theorem 3.

Assume that the functions φk and fkj satisfy (S1)(S7), and the matrix M0 is irreducible. If 0>1, then there is a unique endemic equilibrium P* for system (12), and P* is globally asymptotically stable in the interior of Γ.

Proof.

Define a Lyapunov functional as (26)VP*=QkSk*Sk(t)fkk(η,Ik*)-fkk(Sk*,Ik*)fkk(η,Ik*)dη+Ik*H(Ik(t)Ik*)+V+, where (27)V+=j=1n0fkj(Sk*,Ij*)Ij*J(u)×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)du. First, we calculate the derivative of V+; then, we have (28)V+=j=1n0fkj(Sk*,Ij*)Ij*J(u)ddt×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)du=-j=1n0fkj(Sk*,Ij*)Ij*J(u)ddu×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)du=-j=1nfkj(Sk*,Ij*)Ij*J(u)×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)|u=0+j=1n0fkj(Sk*,Ij*)Ij*×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)dJ(u)=j=1nQkfkj(Sk*,Ij*)Ij*H(fkj(Sk(t),Ij(t))Ij(t)fkj(Sk*,Ij*)Ij*)-j=1n0fkj(Sk*,Ij*)Ij*gk(u)e-δku×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)du=j=1nQk(fkj(Sk(t),Ij(t))Ij(t)fkj(Sk*,Ij*)Ij*fkj(Sk(t),Ij(t))Ij(t)-fkj(Sk*,Ij*)Ij*×lnfkj(Sk(t),Ij(t))Ij(t)fkj(Sk*,Ij*)Ij*)-j=1n0gk(u)e-δku×[fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*fkj(Sk(t-u),Ij(t-u))Ij(t-u)-fkj(Sk*,Ij*)Ij*.×lnfkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*]du. Calculating the time derivative of VP* along the solution of system (12), we have (29)VP*=Qk(1-fkk(Sk*,Ik*)fkk(Sk(t),Ik*))×[φk(Sk(t))-j=1nfkj(Sk(t),Ij(t))Ij(t)]+(1-Ik*Ik(t))×[j=1n0fkj(Sk(u),mmmmmmmmmIj(u))Ij(u)e-δk(t-u)gk(t-u)dumm-(δk+γk)Ik(t)j=1n]+V+. Using equilibrium equations (16), we have (30)VP*=Qk(φk(Sk(t))-φk(Sk*))(1-fkk(Sk*,Ik*)fkk(Sk(t),Ik*))+j=1nQkfkj(Sk*,Ij*)Ij*-j=1nQkfkj(Sk(t),Ij(t))Ij(t)-j=1nQkfkj(Sk*,Ij*)Ij*fkk(Sk*,Ik*)fkk(Sk(t),Ik*)+j=1nQkfkj(Sk(t),Ij(t))Ij(t)fkk(Sk*,Ik*)fkk(Sk(t),Ik*)+j=1n0fkj(Sk(t-u),Ij(t-u))mmmmn×Ij(t-u)e-δkugk(u)du-Ik(t)Ik*j=1n0fkj(Sk*,Ij*)Ij*e-δkugk(u)du-Ik*Ik(t)j=1n0fkj(Sk(t-u),Ij(t-u))mmmmmmn×Ij(t-u)e-δkugk(u)du+j=1n0fkj(Sk*,Ij*)Ij*e-δkugk(u)du+V+. Using V+, we rewrite (30) as (31)VP*=Qk(φk(Sk(t))-φk(Sk*))(1-fkk(Sk*,Ik*)fkk(Sk(t),Ik*))+j=1nQkfkj(Sk*,Ij*)Ij*mmm×[2-fkk(Sk*,Ik*)fkk(Sk(t),Ik*)mmmmm+fkk(Sk*,Ik*)fkj(Sk(t),Ij(t))Ij(t)fkk(Sk(t),Ik*)fkj(Sk*,Ij*)Ij*mmmmm-Ik(t)Ik*]-j=1n0fkj(Sk*,Ij*)Ij*gk(u)e-δkummnn·[Ik*fkj(Sk(t-u),Ij(t-u))Ij(t-u)Ik(t)fkj(Sk*,Ij*)Ij*mmmnn-lnfkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk(t),Ij(t))Ij(t)Ik*fkj(Sk(t-u),Ij(t-u))Ij(t-u)Ik(t)fkj(Sk*,Ij*)Ij*]du. Therefore, (32)VP*=Qk(φk(Sk(t))-φk(Sk*))(1-fkk(Sk*,Ik*)fkk(Sk(t),Ik*))-j=1nQkfkj(Sk*,Ij*)Ij*×[(fkk(Sk(t),Ik*)fkj(Sk*,Ij*)fkk(Sk*,Ik*)fkj(Sk(t),Ij(t)))H(fkk(Sk*,Ik*)fkk(Sk(t),Ik*))mmm+H(fkk(Sk(t),Ik*)fkj(Sk*,Ij*)fkk(Sk*,Ik*)fkj(Sk(t),Ij(t)))]+j=1nQkfkj(Sk*,Ij*)m×(fkk(Sk*,Ik*)fkj(Sk(t),Ij(t))Ij(t)fkk(Sk(t),Ik*)fkj(Sk*,Ij*)Ij*-1)m×(1-fkk(Sk(t),Ik*)fkj(Sk*,Ij*)fkk(Sk*,Ik*)fkj(Sk(t),Ij(t)))-j=1n0fkj(Sk*,Ij*)Ij*gk(u)e-δkummmmn×H(Ik*fkj(Sk(t-u),Ij(t-u))Ij(t-u)Ik(t)fkj(Sk*,Ij*)Ij*)du+j=1nQkfkj(Sk*,Ij*)mmmm×Ij*[Ij(t)Ij*-Ik(t)Ik*  -lnIj(t)Ij*+lnIk(t)Ik*]. Furthermore, under (S5)–(S7), we have (33)VP*j=1nQkfkj(Sk*,Ij*)×Ij*[Ij(t)Ij*-Ik(t)Ik*-lnIj(t)Ij*+lnIk(t)Ik*]. Obviously, the equalities in (33) hold if and only if Sk=Sk* and Ik=Ik*, k=1,2,,n. Therefore, the functional V=k=1nvkVP* as defined in Theorem 3.1 of  is a Lyapunov function for system (12). Using similar arguments as in [4, 813, 16, 17], one can show that the largest invariant subset where Vp*=0 is the singleton {P*}. By LaSalle’s Invariance Principle, P* is globally asymptotically stable in the interior of Γ. This completes the proof of Theorem 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of our original paper. J. Wang is supported by National Natural Science Foundation of China (no. 11201128), the Science and Technology Research Project of the Department of Education of Heilongjiang Province (no. 12531495), the Natural Science Foundation of Heilongjiang Province (no. A201211), and the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province.

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