# Stability Analysis of a Multigroup SEIR Epidemic Model with General Latency Distributions

Authors: Nan Wang; Jingmei Pang; Jinliang Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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 numberℜ0 is defined and proved as the role of a threshold; that is, the disease-free equilibrium P0 is globally asymptotically stable if ℜ0≤1, while an endemic equilibrium P* exists uniquely and is globally asymptotically stable if ℜ0>1. For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach.


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## Body

## 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 [1–6]. 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. [5] 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 r≥0 is the rate at which infective individuals recover. N is constant total populations. It is assumed in [5] 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, lim⁡t→∞P(t)=0 with ∫0∞‍P(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 [8–17] 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 inton distinct groups (n≥2). For 1≤k≤n, 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 [18], 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 [19], 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 [19] 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=1n‍∫0t‍fkj(Sk(u),Ij(u))Ij(u)e-δk(t-u)gk(t-u)du-δkEk(t),Ik′(t)=∑j=1n∫0tfkj(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 Section3, we prove the global asymptotic stability of the disease-free equilibrium P0 for ℜ0≤1 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 ℜ0>1 using the graph-theoretic approach.


## 2. Preliminaries

Since the variablesEk 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=1n∫0tfkj(Sk(u),Ij(u))Ij(u)e-δk(t-u)gk(t-u)du‍-(δk+γk)Ik(t).The incidence functionfkj(Sk,Ij) in (7) is assumed to be continuously differentiable in the interior of ℝ+2 and to satisfy the following hypotheses:(S1)
fkj(0,Ij)=0, for all Ij≥0;(S2)
∂fkj(Sk,Ij)/∂Sk>0, for all Sk>0 and Ij≥0;(S3)
∂fkj(Sk,Ij)/∂Ij≤0, for all Sk≥0 and Ij≥0;assume that the functionsφk satisfy the following conditions:(S4)
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 0≤Sk<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[21],fkj(Sk,Ij)=ηSkIj1+θSk[1].
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 [22]. 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)lim⁡sup⁡t→∞⁡Sk(t)≤Sk0,for1≤k≤n.
From the biological significance, we only need to consider (7) in the following region:
(10)Γ≔{ℝ+2n:Sk,Ik≥0,Sk+Ik≤Sk0(S1,I1,S2,I2,…,Sn,In)mnm∈ℝ+2n:Sk,Ik≥0,Sk+Ik≤Sk0,1≤k≤n}.
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)=∫ξ∞gk(u)e-δkudu,Qk=J(0)=∫0∞gk(u)e-δkudu.
It can be verified that Qk∈(0,1).For finite timet, 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=1n‍∫0∞‍fkj(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={sup⁡s≤0|ϕk(s)|ϕk∈C((-∞,0],ℝ):ϕk(s)eσksmisuniformlycontinuouson⁡(-∞,0],msup⁡s≤0|ϕk(s)|eσks<∞},YΔ={ϕk∈Ck:ϕk(s)≥0∀s≤0}
with norm ∥ϕ∥k=sup⁡s≤0|ϕ(s)|eσks. Let ψt∈Ci and t>0 be such that ψt(s)=ψ(t+s), s∈(-∞,0].Letϕk,ψk∈Ck 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 [24] implies (S1t,I1t,…,Snt,Int)∈Ck for all t>0. We study system (12) in the following phase space:
(15)𝕏𝕘=∏k=1n‍(ℝ×Ck).
It can be verified that solutions of (12) in 𝕏𝕘 with initial conditions (14) remain nonnegative.An equilibriumP*=(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 [26], we have the following lemma.Lemma 1.
For system (7), the disease-free equilibrium P0 is locally asymptotically stable if ℜ0<1 while it is unstable if ℜ0>1.


## 3. Global Stability of the Disease-Free Equilibrium

Theorem 2.
Assume that the functionsφk and fkj satisfy (S1)–(S4), and M0 is irreducible. (i)
Ifℜ0≤1, then P0 is the unique equilibrium of system (7), and P0 is globally asymptotically stable in Γ.(ii)
Ifℜ0>1, then P0 is unstable and system (7) is uniformly persistent.Proof.
It follows from the Perron-Frobenius theorem (see Theorem2.1.4 in [27]) 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=1n‍∫0t‍fkj(Sk(u),Ij(u))Ij(u)mmmmmmmmmm×e-δk(t-u)gk(t-u)dummmm-Ik(t)1δk+γk]≤∑k=1n‍ωk[1δk+γk∑j=1n‍fkj(Sk,0)Ij(t)Qk-Ik(t)]≤∑k=1n‍ωk[1δk+γk∑j=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+γk∑j=1n‍fkj(Sk,0)Ij(t)Qk]=∑k=1n‍ωkIk(t).
If Sk≠Sk0, then
(22)∑k=1n‍ωk[1δk+γk∑j=1n‍fkj(Sk,0)Ij(t)Qk]≤∑k=1n‍ωk[1δk+γk∑j=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.
Ifℜ0>1 and I≠0, 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 [28] and similar arguments as in [4, 10, 13, 16, 17], we know that, if ℜ0>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)lim⁡inf⁡t→∞⁡Sk(t)≥ϵ,lim⁡inf⁡t→∞⁡Ik(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 [29] or Theorem D.3 of [30]). Summarizing the statements above, if ℜ0>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-ln⁡⁡u,∀u>0.
Obviously, H:ℝ+→ℝ+ attains its strict global minimum at u=1 and H(1)=0.To get the global stability ofP*, we make the following assumptions:(S5)
(φk(Sk)-φk(Sk*))(Sk-Sk*)≤0 for Sk≥0;(S6)
(φk(Sk)-φk(Sk*))[fkk(Sk,Ik*)-fkk(Sk*,Ik*)]<0 for Sk≠Sk*;(S7)
(((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 functionsφk and fkj satisfy (S1)–(S7), and the matrix M0 is irreducible. If ℜ0>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*=Qk∫Sk*Sk(t)fkk(η,Ik*)-fkk(Sk*,Ik*)fkk(η,Ik*)dη+Ik*H(Ik(t)Ik*)+V+,
where
(27)V+=∑j=1n∫0∞fkj(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=1n∫0∞fkj(Sk*,Ij*)Ij*J(u)ddt×H(fkj(Sk(t-u),Ij(t-u))Ij(t-u)fkj(Sk*,Ij*)Ij*)du=-∑j=1n∫0∞fkj(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=1n‍∫0∞‍fkj(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=1n‍∫0∞‍fkj(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*×ln⁡⁡fkj(Sk(t),Ij(t))Ij(t)fkj(Sk*,Ij*)Ij*)-∑j=1n‍∫0∞‍gk(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*.×ln⁡⁡fkj(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=1n‍∫0∞‍fkj(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=1n‍∫0∞‍fkj(Sk(t-u),Ij(t-u))mmmmn×Ij(t-u)e-δkugk(u)du-Ik(t)Ik*∑j=1n‍∫0∞‍fkj(Sk*,Ij*)Ij*e-δkugk(u)du-Ik*Ik(t)∑j=1n‍∫0∞‍fkj(Sk(t-u),Ij(t-u))mmmmmmn×Ij(t-u)e-δkugk(u)du+∑j=1n‍∫0∞‍fkj(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=1n‍∫0∞‍fkj(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-ln⁡⁡fkj(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=1n‍∫0∞‍fkj(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*-ln⁡Ij(t)Ij*+ln⁡Ik(t)Ik*].
Furthermore, under (S5)–(S7), we have
(33)VP*′≤∑j=1n‍Qkfkj(Sk*,Ij*)×Ij*[Ij(t)Ij*-Ik(t)Ik*-ln⁡Ij(t)Ij*+ln⁡Ik(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 [12] is a Lyapunov function for system (12). Using similar arguments as in [4, 8–13, 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.



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