# Global Stability of Two-Group Epidemic Models with Distributed Delays and Random Perturbation

Authors: Xiaoming Fan; Zhigang Wang; Xuelian Xu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We discuss a two-group SEIR epidemic model with distributed delays, incorporating random fluctuation around the endemic equilibrium. Our research shows that the endemic equilibrium of the model with distributed delays and random perturbation is stochastically asymptotically stable in the large. In addition, a sufficient stability condition is obtained by constructing suitable Lyapunov function.


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## Body

## 1.  Introduction

For the research of control of disease in populations, significant progress has been made in the theory and application of epidemiology modeling by mathematical research [1–8]. One of the main problems for the theory of differential equations and their applications is connected with stability. Most traditional compartmental models in mathematical epidemiology descend from the classical SIR model of Kermack and McKendrick [9], where the population is divided into the classes of susceptible, infected, and recovered individuals. For some diseases, such as influenza and tuberculosis, on adequate contact with an infectious individual, a susceptible becomes exposed for a while, that is, infected but not yet infectious. Thus it is realistic to introduce a latent compartment; the total population can be partitioned into four compartments: susceptible, latent or exposed, infectious, and recovered, with sizes denoted by S, E, I, and R, respectively. The resulting models are of SEI, SEIR, or SEIRS type, respectively. SEIR model has been widely discussed in the literature. Local and global stability analysis of the disease-free and endemic equilibria has been carried out using different assumptions and contact rates in [6, 7]. Greenhalgh [5] considered SEIR models that incorporate density dependence in the death rate. Korobeinikov [6] considers the global properties for SEIR and SEIS by means of the Lyapunov functions. In fact, there are real benefits to be gained in using stochastic models because real life is full of randomness and stochasticity. Recently, some stochastic epidemic models have been studied by many authors, see [3, 10]. Dalal et al. [4] showed that stochastic models had nonnegative solutions and carried out analysis on the asymptotic stability of models. Tornatore et al. [10] studied the stability of disease-free equilibrium of a stochastic SIR model with or without distributed time delay. On the other hand, taking into account environmental variability, white noise stochastic perturbations around the positive endemic equilibrium of epidemic models was considered in [3, 11]. Beretta et al. proved the stability of epidemic model with stochastic time delays influenced by probability under certain conditions [3]. Such type of stochastic perturbations firstly was proposed in [3, 12] and later was successfully used in many other papers for many other different systems (see, for instance, [13–20]). A more general multigroup epidemic model is proposed to describe the disease spread in a heterogeneous host population with general age structure and varying infectivity by  Li et al. [1]. They investigated a class of multigroup epidemic models with distributed delays and established the global dynamics determined by the basic reproduction number R0. More specifically, they proved that, if R0≤1, then the disease-free equilibrium is globally asymptotically stable; if R0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. However, to the best of the authors’ knowledge, no literature exists regarding SEIR model with random perturbation. Thus, the current study hopes to serve such a need and is inspired by the report of [1]. In this paper, based on the SEIR model of [1], we consider the white noise stochastic perturbations around its endemic equilibrium and use the methods, which is similar to [3]. We construct a class of the Lyapunov functions, as it is useful to study the global properties of stochastic models. By means of it, we prove the SEIR model is stochastically asymptotically stable in the large under certain condition.The paper is organized as follows. In Section2 we recall the deterministic SEIR model and its main results by Li et al. [1]. We introduce the model with stochastic perturbations around the endemic equilibrium in Section 3. In Section 4 the global stability of the endemic equilibrium is proved by the method of the Lyapunov functions.


## 2. Preliminaries

We briefly review the following results obtained by Li et al. [1]. Let Sk,Ek,Ik, and Rk denote the susceptible, infected but noninfectious, infectious, and recovered populations in the kth group, respectively. Let ik(t,r) denote the population of infectious individuals in the kth group with respect to the age of infection r at time t, and Ik(t)=∫r=0∞ik(t,r)dr. Let hk(r)⩾0 be a continuous kernel function that represents the infectivity at the age of infection r. The disease incidence in the k-th group, assuming a bilinear incidence form, can be calculated as ∑j=1nβkjSk(t)∫r=0∞hj(r)ij(t,r)dr, where the sum takes into account cross-infections from all groups and βkj represents the transmission coefficient between compartments Sk and Ij. In the special case hk(r)≡1, the incidence becomes ∑j=1nβkjSk(t)Ij(t) as in [2]. Therefore, the model in [2] can be generalized to the following system of differential equations(2)Sk′=Λk-∑j=1nβkjSk(t)∫r=0∞hj(r)ij(t,r)dr-dkSSk,Ek′=∑j=1nβkjSk(t)∫r=0∞hj(r)ij(t,r)dr-(dkE+ϵk)Ek,Ik′=ϵkEk-(dkI+γk)Ik,Rk′=γkIk-dkRRk,k=1,2,…,n.
Here Λk represents influx of individuals into the kth group, dkS,dkE,dkI, and dkR represent death rates of S,E,I, and R populations in the kth group, respectively, ϵk represents the rate of becoming infectious after a latent period in the k-th group, and γk represents the recovery rate of infectious individuals in the k-th group. All parameter values are assumed to be nonnegative and Λk,dkS,dkE>0 for all k. Note that(2.2)(∂∂t+∂∂r)ik(t,r)=-(dkI+γk)ik(t,r),ik(t,0)=ϵkEk(t),
whose solution is(2.3)ik(t,r)=ik(t-r,0)e-(dkI+γk)r=ϵkEk(t-r)e-(dkI+γk)r.
Substituting (2.3) into (2), we obtain(2.4)Sk′=Λk-∑j=1nβkjSk(t)∫r=0∞hj(r)ϵjEj(t-r)e-(djI+γj)rdr-dkSSk,Ek′=∑j=1nβkjSk(t)∫r=0∞hj(r)ϵjEj(t-r)e-(djI+γj)rdr-(dkE+ϵk)Ek,Ik′=ϵkEk-(dkI+γk)Ik,Rk′=γkIk-dkRRk,k=1,2,…,n.Since the variablesIk and Rk do not appear in the first two equations of (2.4), Li et al. consider the following reduced system with distributed time delays and general kernel functions [1]:(2.5)Sk′=Λk-∑j=1nβkjSk(t)∫r=0∞fj(r)Ej(t-r)dr-dkSSk,Ek′=∑j=1nβkjSk(t)∫r=0∞fj(r)Ej(t-r)dr-(dkE+ϵk)Ek.Here the kernel functionfk(r)⩾0 is continuous and 1⩾∫r=0∞fk(r)dr=hk>0. System (2.5) can be interpreted as a multigroup model for an infectious disease whose latent period r in hosts has a general probability density function (1/hk)fk(r)dr, for the k-th group. Let Sk0=Λk/dkS,hk=∫r=0∞fk(r)dr. The next-generation matrix for system (2.5) is(2.6)M0=(βkjSk0hkdkE+ϵk)n×n.
Define the basic reproduction number as the spectral radius of M0,(2.7)R0=ρ(M0).
In the special case when fk(r) is an exponential function, R0 reduces to that for the resulting ODE models. Make the following assumption on the kernel function fk(r) in (2.5):(2.8)∫r=0∞fk(r)e-λkrdr<∞,
where λk is a positive number, k=1,2,…,n. Define the following Banach space of fading memory type:(2.9)Ck={ϕ∈((-∞,0],R)ϕ(s)eλksisuniformlycontinuouson⁡(-∞,0]andsup⁡s⩽0|ϕ(s)|eλks<∞}
with norm ∥ϕ∥k=sups⩽0|ϕ(s)|eλks. For ϕ∈Ck, let ϕt∈Ck be such that ϕt(s)=ϕ(t+s),s∈(-∞,0]. Let Sk,0∈ℝ+ and ϕk∈Ck such that ϕk(s)⩾0,s∈(-∞,0]. We consider solutions (S1(t),E1t,S2(t),E2t,…,Sn(t),Ent) of system (2.5) with initial conditions(2.10)Sk(0)=Sk,0,Ek,0=ϕk,k=1,2,…,n.Standard theory of functional differential equations impliesEkt∈Ck for t>0. We consider system (2.5) in the phase space(2.11)X=∏k=1n(R×Ck).
It can be verified that solutions of (2.5) in X initial conditions (2.10) remain nonnegative. In particular, Sk(t)>0 for t>0. The following set is positively invariant for system (2.5):(2.12)Θ={(S1,E2(⋅),S2,E2(⋅),…,Sn,En(⋅))∈X∣0⩽Sk⩽ΛkdkS,0⩽Sk+Ek(0)⩽Λkdk*,Ek(s)⩾0,s∈(-∞,0],k=1,2,…,nΛkdkS}.
All positive semiorbits in Θ are precompact in X and thus have nonempty ω-limit sets. We have the following results [1].Lemma 2.1.
All positive semi-orbits inΘ have non-empty ω-limit sets. Let
(2.13)Θ°={(S1,E2(⋅),S2,E2(⋅),…,Sn,En(⋅))∈X∣0<Sk<ΛkdkS,0<Sk+Ek(0)<Λkdk*,Ek(s)>0,s∈(-∞,0],k=1,2,…,nΛkdkS}.
It can be shown that Θ° is the interior of Θ.Lemma 2.2.
Assume thatB=(βkj) is irreducible.(1)
IfR0⩽1, then P0 is the only equilibrium for system (2.5) in Θ.(2)
IfR0>1, then there exist two equilibria for system (2.5) in Θ: the disease-free equilibrium P0 and a unique endemic equilibrium P*.Lemma 2.3.
Assume thatB=(βkj) is irreducible.(1)
IfR0⩽1, then the disease-free equilibrium P0 of system (2.5) is globally asymptotically stable in Θ. If R0>1, then P0 is unstable.(2)
IfR0>1, then the endemic equilibrium P* of system (2.5) is globally asymptotically stable in Θ°.Biologically, Lemma2.3 implies that if the basic reproduction number R0⩽1, then the disease always dies out from all groups; if R0>1, then the disease always persists in all groups at the unique endemic equilibrium level, irrespective of the initial conditions.


## 3. Stochastic Model Derivation

In this paper, based on system (2.5), we consider the case of k=1,2 in the following system(3.1):(3.1)Ṡk=Λk-∑j=12βkjSk∫r=0∞fj(r)Ej(t-r)dr-dkSSk,Ėk=∑j=12βkjSk∫r=0∞fj(r)Ej(t-r)dr-(dkE+ϵk)Ek,İk=ϵkEk-(dkI+γk)Ik,Ṙk=γkIk-dkRRk.
It is easy to see that equilibrium for system (3.1) is given by P*=(S*;E*;I*;R*),(3.2)Sk*=dkE+ϵk∑j=12βkjhj,Ek*=Λk-dkS∑j=12βkjSk*hj,Ik*=ϵkEk*dkI+γk,Rk*=γkIk*dkR.We assume stochastic perturbations are of white noise type, which are directly proportional to distancesSk(t),Ek(t),Ik(t),Rk(t) from values of Sk*,Ek*,Ik*,Rk*, influence the Ṡk(t),Ėk(t),İk(t),Ṙk(t) respectively. So system (2.4) results in(3.3)Ṡk=Λk-∑j=12βkjSk∫r=0∞fj(r)Ej(t-r)dr-dkSSk+σ1k(Sk-Sk*)Ḃ1k,Ėk=∑j=12βkjSk∫r=0∞fj(r)jEj(t-r)dr-(dkE+ϵk)Ek+σ2k(Ek-Ek*)Ḃ2k,İk=ϵkEk-(dkI+γk)Ik+σ3k(Ik-Ik*)Ḃ3k,Ṙk=γkIk-dkRRk+σ4k(Rk-Rk*)Ḃ4k,k=1,2,
where B1k(t),B2k(t),B3k(t),B4k(t) are independent standard Brownian motions and σik2>0 represent the intensities of Bik(t)(i=1,2,3,4), respectively. Obviously, stochastic system (3.3) has the same equilibrium points as system (3.1). In the next section, we will investigate the stability of the equilibrium P* of system (3.3). Below we will construct a class of different Lyapunov functions to achieve our proof under certain conditions.


## 4. Stochastic Stability of the Endemic Equilibrium

In this paper, unless otherwise specified, let(Ω,ℱ,{ℱt}t⩾t0,P) be a complete probability space with a filtration {ℱt}t⩾t0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all P-null sets). Let βi(t) be the Brownian motions defined on this probability space. If R0>1, then the stochastic system (3.3) can be centered at its endemic equilibrium P*=(Sk*,Ek*,Ik*,Rk*), by the change of variables(4.1)uk=Sk-Sk*,vk=Ek-Ek*,wk=Ik-Ik*,zk=Rk-Rk*.
we obtain
(4.2)u̇k=-∑j=12βkjuk∫r=0∞fj(r)vj(t-r)dr-∑j=12βkjSk*∫r=0∞fj(r)vj(t-r)dr-∑j=12βkjukEj*hj-dkSuk+σ1kukḂ1k,v̇k=∑j=12βkjuk∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjSk*∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjukEj*hj-(dkE+ϵk)vk+σ2kvkḂ2k,ẇk=ϵkvk-(dkI+γk)wk+σ3kwkḂ3k,żk=γkwk-dkRzk+σ4kzkḂ4k,k=1,2.It is easy to see that the stability of the equilibrium of the system (3.3) is equivalent to the stability of zero solution of system (4.2). Before proving the main theorem we put forward a lemma in [21]. Consider the d-dimensional stochastic differential equation(4.3)dx(t)=f(x(t),t)dt+g(x(t),t)dB(t),t⩾t0.
Assume that the assumptions of the existence-and-uniqueness theorem are fulfilled. Hence, for any given initial value x(t0)=x0∈Rd, (4.3) has a unique global solution that is denoted by x(t;t0,x0). Assume furthermore that f(0,t)=0 and g(0,t)=0 for all t⩾t0. So (4.3) has the solution x(t)≡0 corresponding to the initial value x(t0)=0. This solution is called the trivial solution or equilibrium position. Denote by C2,1(Rd×[t0,∞);R+) the family of all nonnegative functions V(x,t) defined on Rd×[t0,∞) such that they are continuously twice differentiable in x and once in t. Define the differential operator L associated with (4.3) by(4.4)L=∂∂t+∑i=1dfi(x,t)∂∂xi+12∑i,j=1d[gT(x,t)g(x,t)]ij∂2∂xixj.
If L acts on a function V∈C2,1(Rd×[t0,∞);R+), then(4.5)LV(x,t)=Vt(x,t)+Vt(x,t)f(x,t)+12Tr⁡[gT(x,t)Vxx(x,t)g(x,t)].Definition 4.1.
(1) The trivial solution of (4.3) is said to be stochastically stable or stable in probability if for every pair of ɛ∈(0,1) and r>0, there exists a δ=δ(ɛ,r,t0)>0 such that
(4.6)P{|x(t;t0,x0)|<r∀t⩾t0}⩾1-ɛ
whenever |x0|<δ. Otherwise, it is said to be stochastically unstable.
(2) The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable, and, moreover, for everyɛ∈(0,1), there exists a δ0=δ0(ɛ,t0)>0 such that
(4.7)P{lim⁡t→∞x(t;t0,x0)=0}⩾1-ɛ
whenever |x0|<δ0.
(3)The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically asymptotically stable and, moreover, for allx0∈Rd,
(4.8)P{lim⁡t→∞x(t;t0,x0)=0}=1.Lemma 4.2 (see [21]).
If there exists a positive-definite decrescent radially unbounded functionV(x,t)∈C2,1(Rd×[t0,∞);R+) such that LV(x,t) is negative definite, then the trivial solution of (4.3) is stochastically asymptotically stable in the large.From the above lemma, we can obtain the stochastically asymptotically stability of equilibrium as follows.Theorem 4.3.
Assume thatB=(βkj) is irreducible and R0>1; then, if the following condition is satisfied
(4.9)σ1k2<2dkS,σ2k2<2(dkE+ϵk),σ3k2<2(dkI+γk),σ4k2<2dkR,
the endemic equilibrium P* of system (3.3) is stochastically asymptotically stable in the large.Proof.
It is easy to see that we only need to prove the zero solution of (4.2) is stochastically asymptotically stable in the large. Let x(t)=(u(t),v(t),w(t),z(t))T. We define the Lyapunov function V(x(t)) as follows:
(4.10)V(x)=∑k=12[akuk2+vk2+bk(uk+vk)2+ak2wk2+ak4zk2+∑j=12βkjSk*(1+ak)∫r=0∞fj(r)∫t-rtvk2(τ)dτdr],
where ak>0,bk>0 are real positive constants to be chosen later. So it is obvious that V(x) is positive definite and decrescent.
Using Itô’s formula, we compute(4.11)LV=∑k=12{∑j=12βkj2(akuk+bk(uk+vk))×[-∑j=12βkjuk∫r=0∞fj(r)vj(t-r)dr-∑j=12βkjSk*∫r=0∞fj(r)vj(t-r)dr-∑j=12βkjukEj*hj-dkSuk]+2(vk+bk(uk+vk))[∑j=12βkjuk∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjSk*∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjukEj*hj-(dkE+ϵk)vk]+2ak2wk(ϵkvk-(dkI+γk)wk)+2ak4zk(γkwk-dkRzk)+(ak+bk)σ1k2uk2+(1+bk)σ2k2vk2+ak2σ3k2wk2+ak4σ4k2zk2+∑j=12βkjSk*(1+ak)hjvj2-∑j=12βkjSk*(1+ak)∫r=0∞fj(r)vj2(t-r)dr}.
Using (3.2), we obtain
(4.12)LV=∑k=12{-uk2[(ak+bk)(2dkS-σ1k2)+2∑j=12akβkjEj*hj]-vk2[(1+bk)(2∑j=12βkjSk*hj-σ2k2)]-ak2wk2(2(dkI+γk)-σ3k2)-ak4zk2(2dkR-σ4k2)+2(∑j=12βkjEj*hj-bk(dkS+∑j=12βkjSk*hj))×ukvk+2(ak2ϵkvkwk+γkwkzk)+2∑j=12βkjSk*(vk-akuk)∫r=0∞fj(r)vj(t-r)dr+2∑j=12βkjuk(vk-akuk)∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjSk*(1+ak)hjvj2-∑j=12βkjSk*(1+ak)∫r=0∞fj(r)vj2(t-r)dr}.
In (4.12), we choose
(4.13)bk=∑j=12βkjSk*hj(dkS+∑j=12βkjSk*hj).
Then
(4.14)LV=∑k=12{-uk2[(ak+bk)(2dkS-σ1k2)+2∑j=12akβkjEj*hj]-vk2[(1+bk)(2∑j=12βkjSk*hj-σ2k2)]-ak2wk2(2(dkI+γk)-σ3k2)-ak4zk2(2dkR-σ4k2)+2ak2(ϵkvkwk+ak2γkwkzk)+2∑j=12βkjSk*(vk-akuk)∫r=0∞fj(r)vj(t-r)dr+2∑j=12βkjuk(vk-akuk)∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjSk*(1+ak)hjvj2-∑j=12βkjSk*(1+ak)∫r=0∞fj(r)vj2(t-r)dr}.
Moreover, using the Cauchy inequality to 2ak2ϵkvkwk and 2ak4γkwkzk, we can obtain
(4.15)2ak2ϵkvkwk⩽ak2ϵk(vk2ak+akwk2),2ak4γkwkzk⩽ak4γk(wk2ak+akzk2),2∑j=12βkjSk*(vk-akuk)∫r=0∞fj(r)vj(t-r)dr⩽ak∑j=12βkjSk*(uk2+∫r=0∞fj(r)vj2(t-r)dr)+∑j=12βkjSk*(vk2+∫r=0∞fj(r)vj2(t-r)dr).
Substituting (4.15) into (4.14) as well as using 1⩾∫r=0∞fk(r)dr=hk>0, yields
(4.16)LV⩽∑k=12{-uk2[bk(2dkS-σ1k2)-∑j=12akβkjSk*]-vk2[(1+bk)(2∑j=12βkjSk*-σ2k2)-ak(ϵk+∑j=12βkjSk*)]-ak2wk2(2(dkI+γk)-σ3k2-ak(ϵk+γk))-ak4zk2(2dkR-σ4k2-akγk)+2∑j=12βkjuk(vk-akuk)∫r=0∞fj(r)vj(t-r)dr+∑j=12βkjSk*(1+ak)vj2}=L0V+2∑k=12∑j=12βkjuk(vk-akuk)∫r=0∞fj(r)vj(t-r)dr,
where
(4.17)L0V=∑k=12{-uk2[bk(2dkS-σ1k2)-∑j=12akβkjSk*]-vk2[2bk∑j=12βkjSk*-(1+bk)σ2k2-ak(ϵk+∑j=12βkjSk*)]-ak2wk2(2(dkI+γk)-σ3k2-ak(ϵk+γk))-ak4zk2(2dkR-σ4k2-akγk)-uk2[bk(2dkS-σ1k2)-∑j=12akβkjSk*]-vk2[2bk∑j=12βkjSk*-(1+bk)σ2k2-ak(ϵk+∑j=12βkjSk*)]}.
From (4.9) it follows that there exists ak>0 such that
(4.18)ak<min⁡{bk(2dkS-σ1k2)∑j=12βkjSk*,2bk∑j=12βkjSk*-(1+bk)σ2k2ϵk+∑j=12βkjSk*,2(dkI+γk)-σ3k2ϵk+γk,2dkR-σ4k2γk}.
Therefore, there exists 𝒞>0 such that LV0⩽-𝒞|(uk,vk,wk,zk)|2.
Let us suppose thatP{|vk(t)|<δk}=1. Then
(4.19)2∑j=12βkjuk(vk-akuk)∫r=0∞fj(r)vj(t-r)dr⩽∑j=12βkjδk(uk2(1+2ak)+vk2).
Therefore
(4.20)LV⩽∑k=12{-uk2[bk(2dkS-σ1k2)-∑j=12akβkjSk*-∑j=12βkjδk(1+2ak)]-vk2[(1+bk)(2∑j=12akβkjSk*hj-σ2k2)-ak(ϵk+∑j=12βkjSk*)-∑j=12βkjδk]-ak2wk2(2(dkI+γk)-σ3k2-ak(ϵk+γk))-ak4zk2(2dkR-σ4k2-akγk)-uk2[bk(2dkS-σ1k2)-∑j=12akβkjSk*-∑j=12βkjδk(1+2ak)]}.
Hence for sufficiently small δk>0, LV(x,t) is negative definite in a sufficiently small neighborhood of x=0 for t⩾0. According to Lemma 4.2, we therefore conclude that the zero solution of (4.2) is stochastically asymptotically stable in the large. The proof is complete.



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