# Global Dynamics of an SVEIR Model with Age-Dependent Vaccination, Infection, and Latency

Authors: Rodrigue Yves M’pika Massoukou; Suares Clovis Oukouomi Noutchie; Richard Guiem
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

Vaccine-induced protection is substantial to control, prevent, and reduce the spread of infectious diseases and to get rid of infectious diseases. In this paper, we propose an SVEIR epidemic model with age-dependent vaccination, latency, and infection. The model also considers that the waning vaccine-induced immunity depends on vaccination age and the vaccinated individuals fall back to the susceptible class after losing immunity. The model is a coupled system of (hyperbolic) partial differential equations with ordinary differential equations. The global dynamics of the model is established through construction of appropriate Lyapunov functionals and application of Lasalle’s invariance principle. As a result, the global stability of the infection-free equilibrium and endemic equilibrium is obtained and is fully determined by the basic reproduction numberR0.


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## Body

## 1. Introduction

Protection induced by vaccines plays a significant role in preventing and reducing the transmission of infectious diseases. One of the greatest successful events of vaccination is illustrated through the eradication of small-pox. It is reported in [1] that the case of small-pox was last recorded in 1977. Immunity conveyed by vaccination depends on different vaccines and vaccination policies—lifelong immunity occurs for certain vaccines while immunization period is induced by some vaccines. However, waning vaccine-induced immunity takes place (naturally) after immunization process. It is reported in [2] that a significant decay in the proportion of chicken pox took place in US in 1995 after conducting a universal vaccination campaign. But, surprisingly new cases of chicken pox appeared mainly in highly vaccinated school communities in US. Some studies were conducted and revealed waning vaccine-induced immunity in children under protection induced by vaccine. Moreover, this was also investigated in [3–5] and it was proved that such waning of immunity is attached to the time since vaccination and the age at vaccination. In this regard, it was published in [6, 7] that the time of vaccine-induced immunity depends on individuals features and vaccine age.From the abovementioned statements and citations, we see that it is necessary to associate waning vaccine-induced immunity with vaccination in infectious disease models and it is interesting to investigate the impact of waning vaccine-induced immunity on the dynamics of infectious diseases. Many mathematical models on vaccination have been already investigated and, to cite a few of them, see [2, 8–16]. Some of the above-cited model considered either age-dependent vaccination, while some did not.Despite vaccination age-structure being the main and appropriate feature required in the dynamics of infectious diseases with waning induced-vaccine immunity, most of epidemiological models with vaccination including waning induced-vaccine immunity were studied after assuming a constant rate of immunity loss (to name a few, see [9, 10, 15]). Age-dependent vaccination was considered in some epidemiological models studied recently in [1, 2, 17–20]. However, some of these works considered either waning vaccine-induced immunity or not, either vaccine-age-dependent waning vaccine-induced immunity or not, either age-dependent latency or not, either age-dependent relapse or not, and either age-dependent infection or not. In [17] an SVIR epidemic model with continuous age-dependent vaccination was formulated to establish the global stability of equilibria. In [18] an SVIJS epidemic model with age-dependent vaccination was considered to study the asymptotical behavior of the equilibria, after assuming that age-dependent vaccine-induced immunity decays with time after vaccination. In [19] an SVIS epidemic model with age-dependent vaccination and vaccine-age-dependent waning vaccine-induced immunity and treatment was formulated to investigate backward bifurcations. In [2] an SVIR epidemic model with vaccination age was considered to establish global stability of equilibria, after assuming that vaccine-induced immunity decays with time after vaccination. In [20] an SVEIR epidemic model with age-dependence vaccination, latency, and relapse was formulated to establish the global stability of the equilibria. In [1] a multigroup SVEIR epidemic model with latent class and vaccination age was formulated to study global stability of equilibria, after assuming that vaccine-induced immunity decays with time after vaccination and likewise in [17–19].Recently, in [14] an SVEIR epidemic model with continuous age-structure in the latent and infectious classes and without continuous age-structure in the vaccinated class was formulated to prove the global stability of equilibria, while in [20] an SVEIR epidemic model with continuous age-structure in the latent, infectious, and recovered classes and with vaccine-age-dependent waning vaccine-induced immunity was formulated. Moreover, in [14] the latency and infection ages are denoted by the same variable a. Similarly, in [20] the latency, relapse, and vaccination ages are denoted by the same variable a. In spite of this, to the best of our knowledge, the global dynamics of an SVEIR epidemic model with continuous age-structure in latency, infection, vaccination, and vaccine-age-dependence waning vaccine-induced immunity has not yet been neither considered nor investigated using the approach of Lyapunov functionals. The aim of this work is to fill this gap by investigating the global dynamics of an SVEIR epidemic model as above defined. Motivated by [14, 20], we propose a new SVEIR epidemic model originated from an existing SVEIR formulated in [1], by considering continuous age-structure in latency and infection in addition to age-dependence vaccination and vaccine-age-dependence waning vaccine-induced immunity (which the authors took into account in [1]). However, the latency, infection, and vaccination ages are all denoted by a, as in [14, 20]. Moreover, in this paper, we also consider a more significant incidence rate (taking into account transmission by both age-mates infective individuals and infective individuals of any age) of the form (1)St∫0∞K0aia,t+∫0∞Ka,a′ia′,tda′da,where K0(a) and K(a,a′) are defined below, instead of the classical incidence rate of the form (2)St∫0∞βaia,tda,where β(a) denotes the coefficient of disease transmission from infective individual, with infection age a, to susceptible individual. The latter is considered in the references therein where continuous age-structure in infection is taken into account.The model splits the total population into five epidemiological compartments, namely, the susceptible compartment, the vaccinated compartment, the latent compartment, infected compartment, and the removed compartment. LetS(t) and R(t) be the number of individuals in the susceptible and removed compartments at time t, respectively. Let v(a,t), e(a,t), and i(a,t) be the density of vaccinated, (latently) infected, and (actively) infected individuals with vaccination, latency, and infection age a at time t, respectively. It follows that V(t), E(t), and I(t) defined by (3)Vt=∫0∞va,tda,Et=∫0∞ea,tda,It=∫0∞ia,tda,are the number of individuals in the vaccinated, latent, and infected compartments, respectively.The model to be investigated consists of a hybrid system of nonlinear coupled ordinary differential equations and partial differential equations of the form(4)ddtSt=Λ-ν+μ0St+∫0∞αava,tda-St∫0∞K0aia,t+∫0∞Ka,a′ia′,tda′da∂∂tva,t=-∂∂ava,t-ηava,t∂∂tea,t=-∂∂aea,t-ϱaea,t∂∂tia,t=-∂∂aia,t-σaia,tddtRt=∫0∞γaia,tda-μ0Rt,where η(a)=α(a)+μ(a), ϱ(a)=ε(a)+μ(a), and σ(a)=γ(a)+μ(a), with boundary conditions(5)v0,t=νSte0,t=St∫0∞K0aia,t+∫0∞Ka,a′ia′,tda′dai0,t=∫0∞εaea,tda.And initial conditions(6)S0=S0>0,va,0=v0a,ea,0=e0a>0,ia,0=i0a>0,R0=R0>0,where S0 and R0 are initial size of susceptible and removed individuals, respectively, and v0(a), e0(a), and i0(a) are initial age-density of vaccinated, latent, and infective individuals, respectively. Moreover, v0, e0, and i0 are Lebesgue integrable functions, and it is assumed that the recruitment of newly vaccinated individuals in the vaccinated compartment is done at age zero.The meaning of parameters in (4)-(5) is given below:Λ: constant recruitment rate of susceptible individualsν: rate of vaccination of susceptible individualsμ0: natural mortality rate of individualsα(a): age-specific rate of waning vaccine-induced immunityμ(a): age-specific natural mortality rateγ(a): age-specific removal rateε(a): age-specific rate moving from latent to infectiveK0(a): age-specific infection rate of susceptible individuals by infective individuals (of the same age–intracohort contagion)K(a,a′): probability that an infective individual of age a′ will successfully infect a susceptible individual of age a, after contactIn the sequel, we make following assumptions on parameters in (4)–(5):A1 (i)
Λ , ν , μ 0 > 0 with ν<μ0. (ii)
α , η , ϱ , σ , γ , ε , K 0 ∈ L + ∞ ( 0 , ∞ ) and K∈L1(0,∞),L+∞(0,∞) with essential upper bounds α¯,η¯,ϱ¯,σ¯,γ¯,ε¯,K¯0 and K¯(a,·), respectively. (iii)
K 0 ( a ) , K ( · , a ′ ) , α ( a ) , γ ( a ) , ε ( a ) are Lipschitz continuous on R+ with coefficients MK0,MK,Mα,Mγ, and Mε, respectively. (iv)
There existsμ~∈(0,μ0] such that η(a)-α(a),ϱ(a)-ε(a),σ(a)-γ(a)>μ~.A2. R(·) is a decreasing function of t for any constant removal rate γ0 such that γ0≥γ¯.A3. η(a)Λ<(η(a)-α(a))1-∫0∞α(a)e-∫0aη(s)dsda, for every a.This paper is composed sections, in addition to the introduction, which are structured as follows. In Section2, we present some preliminary result on compactness property of the semiflow generated by (4)-(5) and we discuss its asymptotic smoothness property. The uniform persistence property of (4)-(5) is addressed in Section 4. Section 3 deals with the existence of equilibria and the formulation of the threshold parameter R0 (the basic reproduction number). Local stability of equilibria for (4) is established in Section 5, while global stability of equilibria for (4) is examined in Section 6.


## 2. Preliminaries

We consider the Banach space(7)X=R×L10,∞×L10,∞×L10,∞×Rendowed with the norm (8)x,φ,ψ,ϕ,yX=x+φ1+ψ1+ϕ1+y,where ·1=·L1, for any (x,φ,ψ,ϕ,y)∈X. Let us denote by X+ the positive cone of the Banach space X such that (9)X+=R+×L+10,∞×L+10,∞×L+10,∞×R+.For any initial value X0=(S0,v0(·),e0(·),i0(·),R0)∈X+ satisfying the conditions (10)v00=νS0e00=S0∫0∞K0ai0a+∫0∞Ka,a′i0a′da′dai00=∫0∞εae0adasystem (4) is well-posed, under assumption A1, due to [21]. Thus, a continuous semiflow Φ:R+×X+→X+ is obtained and it is defined by (4) such that(11)Φt,X0=ΦtX0=St,v·,t,e·,t,i·,t,Rt,t∈R+,X0∈X+.Now, we introduce the functions (12)χa=e-∫0aηsds,ϑa=e-∫0aϱsds,ζa=e-∫0aσsds.By integrating the second, third, and fourth equations of (4) along the characteristic line t-a=constant, respectively, we get(13)va,t=v0,t-aχa,0≤a≤t,v0a-tχaχa-t,0≤t≤a,ea,t=e0,t-aϑa,0≤a≤t,e0a-tϑaϑa-t0≤t≤a,ia,t=i0,t-aζa,0≤a≤t,i0a-tζaζa-t0≤t≤a,where(14)v0,t-a=νSt-ae0,t-a=St-a∫0∞K0aia,t-a+∫0∞Ka,a′ia′,t-ada′dai0,t-a=∫0∞εaea,t-ada.Taking the norm of Φt(X0) and using the positiveness of the components of Φt(X0), we get(15)ΦtX0X=St+v·,t1+e·,t1+i·,t1+Rt.Differentiating (15) with respect to t leads to(16)ddtΦtX0X=dStdt+ddtv·,t1+ddte·,t1+ddti·,t1+dRtdt.Next, we seek for the estimates of each time-derivative on the right hand side (16). First, we have(17)ddtv·,t1=ddt∫0tv0,t-aχada+∫t∞v0a-tχaχa-tda=ddt∫0tv0,sχt-sds+∫0∞v0ςχςddtχt+ςdς.Applying the Leibniz Integral Rule to the first integral in (17) yields(18)ddtv·,t1=χ0v0,t+∫0tv0,sddtχt-sds+∫0∞v0ςχςddtχt+ςdς=v0,t-∫0tv0,sηt-sχt-sds-∫0∞v0ςηt+ςχt+ςχςdς=νSt-∫0∞ηava,tda.Likewise, we also have(19)ddte·,t1=e0,t-∫0∞ϱaea,tdaand(20)ddti·,t1=∫0∞εaea,tda-∫0∞σaia,tda.Therefore, we get(21)ddtSt+v·,t1+e·,t1+i·,t1+Rt=Λ-μ0St+Rt-∫0∞ηa-αava,tda-∫0∞ϱa-εava,tda-∫0∞σa-γaia,tda.Using (iv) of A1, (21) yields(22)ddtSt+v·,t1+e·,t1+i·,t1+Rt≤Λ-μ~St+v·,t1+e·,t1+i·,t1+Rt,that is, (23)ddtΦtX0X≤Λ-μ~ΦtX0X.Thus, we obtain (24)ΦtX0X≤Λμ~-e-μ~tΛμ~-X0X,where Φ0(X0)X=X0X.If we consider the state spaceΓ of (4), defined by(25)Γ=x,φ,ψ,ϕ,y∈X+:x,φ,ψ,ϕ,yX≤Λμ~,we get (26)ΦtX0X≤Λμ~,for any t≥0, whenever X0∈Γ. Moreover (27)limsupt→∞ΦtX0X≤Λμ~,for any X0∈X+.Then, we state the following result.Lemma 1.
The setΓ is positively invariant for Φ; that is, (28)ΦtX0⊂Γ,∀t≥0,X0∈Γ.Moreover, Φt(X0) is point dissipative and the set Γ attracts all points in X+.As we aim to make use of the Lasalle’s Invariance Principle, we are required to establish the relative compactness of the orbitΦt(X0):t≥0 in X+ due to the infinite dimensional Banach space X. For this, we consider two operators Θ and Ψ, (Θ,Ψ:R+×X+→X+), such that(29)Θt,X0=ΘtX0=0,θ~v·,a,θ~e·,a,θ~i·,a,0Ψt,X0=ΨtX0=St,v~·,a,e~·,a,i~·,a,Rt,where(30)θ~va,t=0,0≤a≤t,va,t,0≤t≤a,v~a,t=va,t,0≤a≤t,00≤t≤a,θ~ea,t=0,0≤a≤t,ea,t0≤t≤a,e~a,t=ea,t,0≤a≤t,00≤t≤a,θ~ia,t=0,0≤a≤t,ia,t0≤t≤a,i~a,t=ia,t,0≤a≤t,00≤t≤a.Thus, Φt(X0)=Θt(X0)+Ψt(X0), for any t≥0; and from the proof of [22, Proposition 3.13] and Lemma 1, we get to the following result.Theorem 2.
ForX0∈Γ, the orbit Φt(X0):t≥0 has a compact closure in X+ if the following conditions are satisfied:(i)
There exists a functionΔ:R+×R+→R+ such that, for any r>0, limt→∞Δ(t,r)=0, and if X0∈Γ with X0X≤r, then Θt(X0)X≤Δ(t,r) for any t≥0.(ii)
For anyt≥0, Ψt(·) maps any bounded sets of Γ into a set with compact closure in X+.For verifying conditions (i) and (ii) of Theorem2, we need lemmas.Lemma 3.
Forr>0, let Δ(t,r)=e-μ~tr. Then limt→∞Δ(t,r)=0. Then (i) of Theorem 2 holds.Proof.
Clearly, we see thatlimt→∞Δ(t,r)=0. Making use of some equations in (13), we get(31)θ~va,t=0,0≤a≤t,v0a-tχaχa-t,0≤t≤a,θ~ea,t=0,0≤a≤t,e0a-tϑaϑa-t0≤t≤a,θ~ia,t=0,0≤a≤t,i0a-tζaζa-t0≤t≤a.Taking the initial condition X0∈Γ such that X0X≤r and t≥0, we have(32)ΘtX0X=0+θ~v·,t1+θ~e·,t1+θ~i·,t1+0=∫t∞v0a-tχaχa-tda+∫t∞e0a-tϑaϑa-tda+∫t∞i0a-tζaζa-tda=∫0∞v0sχs+tχsds+∫0∞e0sϑs+tϑsds+∫0∞i0sζs+tζsds=∫0∞v0se-∫ss+tητdτds+∫0∞e0se-∫ss+tϱτdτds+∫0∞i0se-∫ss+tστdτds≤e-μ~tv01+e01+i01≤e-μ~tX0X≤e-μ~tr=Λt,r.Lemma 4.
Fort≥0, Ψt(·) maps any bounded set of Γ into a set with a compact closure in X+.Proof.
SinceS(t) and R(t) remain in the compact set [0,Λ/μ~] by Lemma 1, it is sufficient to show that v~(a,t), e~(a,t), and i~(a,t) remain in a precompact subset of L+1(0,∞), which does not depend on the initial data X0∈Γ. To achieve this, the following conditions (see [23, Theorem B.2]) must be satisfied for v~(a,t), e~(a,t), and i~(a,t):(i)
The supremum ofz~(·,t)1 with respect to X0∈Γ is finite;(ii)
limh→∞∫h∞z~(a,t)da=0 uniformly with respect to X0∈Γ;(iii)
limh→0+∫h∞z~(a+h,t)-z~(a,t)da=0 uniformly with respect to X0∈Γ;(iv)
limh→0+∫0hz~(a,t)da=0 uniformly with respect to X0∈Γ,
wherez~∈v~,e~,i~. It follows from (13) and (30) that(33)e~a,t=e0t-aϑa,0≤a≤t,0,0≤t≤a,and hence, using Lemma 1, we get(34)0≤v~a,t≤νΛμ~e-μ~a,and hence, (i), (ii), and (iv) follow. To establish (iii), we take a sufficiently small h such that h∈(0,t) and we show that(35)∫0∞v~a+h,t-v~a,tda=∫t-ht0-νSt-aχada+ν∫0t-hSt-a-hχa+h-Sa,tχada≤νΛμ~h+ν∫0t-hSt-a-hχa-χa+hda+ν∫0t-hχaSt-a-h-Sa,tda≤νΛμ~h+νΛμ~∫0t-hχada-∫0t-hχa+hda+ν∫0t-hχaSt-a-h-Sa,tda≤νΛμ~h+νΛμ~∫0tχada+∫thχada+ν∫0t-hχaSt-a-h-Sa,tda≤ν2Λ+lShμ~.Indeed, χ(a) is a nondecreasing function of a such that 0≤χ(a)≤1 satisfying (36)∫0t-hχa+h-χada=∫0t-hχa-χa+hda=∫0t-hχada-∫0t-hχa+hda=∫0t-hχada-∫htχada=∫0t-hχada-∫ht-hχada-∫t-htχada=∫0hχada-∫t-htχada≤∫0hχada≤h.On the other hand, the Lipschitz continuity of S(·) is obtained from the first equation of (4), using the boundedness of the solution of (4) (see Lemma 1); that is, there exists lS>0 such that |S(t1)-S(t2)|≤lS|t1-t2| for any t1,t2≥0.
Sinceν(2Λ+lS)h/μ~ does not depend on the initial data X0∈Γ and ν(2Λ+lS)h/μ~→0 as h→0+, it follows from (35) that (iii) is satisfied.Therefore, From Lemma1 and Theorem 2, the existence result of global attractors (see [24]) follows.Theorem 5.
The semiflowΦt(X0):t≥0 has a global attractor in X+, which attracts any bounded subset of X+.


## 3. Equilibria and Basic Reproduction Number

System (4) has a unique disease-free equilibrium E0=(S0,v0(a),e0(a),i0,R0), where(37)S0=Λμ0+ν1-∫0∞αae-∫0aηsdsda,v0a=νS0e-∫0aηsds,e0a=i0a=0,R0=0.Apart fromE0, system (4) could also have an endemic equilibrium. We suppose that there exists an endemic equilibrium for system (4) denoted by E∗=(S∗,v∗,e∗,i∗,R∗). Therefore, (38)0=Λ-ν+μ0S∗+∫0∞αav∗ada-S∗∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′da0=-ddav∗a-ηav∗a0=-ddae∗a-ϱae∗a0=-ddai∗a-σai∗a0=∫0∞γai∗ada-μ0R∗.are satisfied. In addition, E∗ also satisfies (5), i.e.,(39)v∗0=νS∗e∗0=S∗∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′dai∗0=∫0∞εae∗ada.The second equation of (38) and the first equation of (39) give(40)v∗a=νS∗e-∫0aηsds.It follows from the third equation of (38) that(41)e∗a=e∗0e-∫0aϱsds.Equations (5) and (41) together with the fourth equation of (38) yield(42)i∗a=e∗0e-∫0aσsds∫0∞εae-∫0aϱsdsda.We introduce parameters R0, L, J, and P such that(43)R0=S0L∫0∞K0ae-∫0aσsds+∫0∞Ka,a′e-∫0a′σsdsda′daL=∫0∞εae-∫0aϱsdsdaJ=∫0∞γae-∫0aσsdsdaP=∫0∞αae-∫0aηsdsda.By substituting (42) into the second equation of (39), we have(44)S∗=S0R0And hence,(45)v∗a=v0aR0.Then, substituting (37), (40), (42), and (44) into the first equation of (38) yields(46)e∗0=Λ1-1R0.Therefore, it easily follows that(47)e∗a=Λ1-1R0ϑai∗a=Λ1-1R0LζaR∗=Λμ01-1R0LJ.A threshold condition is derived from the existence condition for the endemic equilibriumE∗ such that R0>1. Thus, the parameter R0, given by the first equation of (43), can be called the basic reproduction number of system (4). Moreover, R0 can also be expressed as(48)R0=Rintra+Rinter,where(49)Rintra=ΛLμ0+ν1-P∫0∞K0aζada,Rinter=ΛLμ0+ν1-P∫0∞∫0∞Ka,a′ζa′da′da.R i n t r a and Rinter can be understood, respectively, as the basic reproduction numbers for the corresponding model with purely intracohort infection mechanism (i.e., situation in which individuals can only be infected by their age-mates) and for the corresponding model with purely intercohort infection mechanism (i.e., situation in which individuals can be infected by those of any age).Therefore, we state the following.Theorem 6.
IfR0≤1, then system (4) has only a disease-free equilibrium E0, while if R0>1, then system (4) also has an endemic equilibrium E∗ in addition to the disease-free equilibrium E0.


## 4. Uniform Persistence

This section is devoted to the uniform persistence of system (4) under the condition R0>1. For this, we introduce a function ρ:X+→R+ defined by (50)ρx,ς,ϖ,υ,y=x∫0∞K0aυa+∫0∞Ka,a′υa′da′da,where (x,ς,ϖ,υ,y)∈X+. Furthermore, we consider the set X0 defined by (51)X0=X0∈X+:ρΦt0X0>0forsomet0∈R+such that ΦtX0→E0 as t→∞ whenever X0∈X+∖X0.Definition 7 (see [23, p. 61]).
System (4) is uniformly weakly ρ-persistent (respectively, uniformly strongly ρ-persistent) if there exists a positive ϵ∗, independent of initial conditions, such that (52)limsupt→∞ρΦtX0>ϵ∗respectively,liminft→∞ρΦtX0>ϵ∗for X0∈X+.Theorem 8.
IfR0>1, then system (4) is uniformly weakly ρ-persistent.Proof.
We assume that, for anyϵ∗>0, we can find X0ϵ∗∈X+ such that (53)limsupt→∞ρΦtX0ϵ∗≤ϵ∗.Since R0>1, then we can find a small enough ϵ0∗>0 such that(54)1<Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞εtϑtdt∫0∞K0aζa+∫0∞Ka,a′ζa′da′da.In particular, we can find X0ϵ0∗/2∈X+ such that (55)limsupt→∞ρΦtX0ϵ0∗/2≤ϵ0∗2.We can assume that, for any t≥0, ρΦt(X0ϵ0∗/2)≤ϵ0∗. It follows from the first equation of (4) that (56)ddtSt≥Λ-ϵ0∗-ν+μ0St+∫0tαava,tda=Λ-ϵ0∗-ν+μ0St+∫0tαaζav0,t-ada≥Λ-ϵ0∗-ν+μ0St+ν∫0tαaχaSt-ada.We apply the Laplace transform L to the above inequality and we obtain (57)λLSt-S0≥Λ-ϵ0∗λ-ν+μ0LSt+νLαtχtLSt.It follows that (58)LSt≥S0λ+Λ-ϵ0∗λλ+μ0+ν1-Lαtχt≥S0λ+Λ-ϵ0∗λλ+μ0+ν=Λ-ϵ0∗ν+μ0·1λ+S0-Λ-ϵ0∗ν+μ01λ+ν+μ0And hence, (59)St≥Λ-ϵ0∗ν+μ0L-11λ+S0-Λ-ϵ0∗ν+μ0L-11λ+ν+μ0=Λ-ϵ0∗ν+μ0+e-ν+μ0tS0-Λ-ϵ0∗ν+μ0,for  anyt≥0.This yields limsupt→∞⁡St≥Λ-ϵ0/ν+μ0. We can assume that, for any t≥0, (60)St≥Λ-ϵ0∗ν+μ0-ϵ0∗.Now, we consider the boundary condition defined by the second equation of (5) and we get (61)e0,t≥St∫0tK0aia,t+∫0tKa,a′ia′,tda′da≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0tK0aia,tda+∫0t∫0tKa,a′ia′,tda′da≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0tK0aζai0,t-ada+∫0t∫0tKa,a′ζa′i0,t-a′da′da≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0tK0aζa∫0t-aεbϑbe0,t-a-bdbda+∫0t∫0tKa,a′ζa′∫0t-a′εb′ϑb′e0,t-a′-b′db′da′da.We apply the Laplace Transform L to the above inequality so that (62)Le0,t≥Λ-ϵ0∗ν+μ0-ϵ0∗LK0tζt+1λLK·,tζtLεtϑtLe0,t.Dividing the above inequality by Le(0,t) yields (63)1≥Λ-ϵ0∗ν+μ0-ϵ0∗LK0tζt+1λLK·,tζtLεtϑt≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞e-λtεtϑtdt×∫0∞e-λtK0tζtdt+∫0∞e-λt∫0tKt,sζsdsdt=Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞e-λtεtϑtdt×∫0∞e-λtK0tζtdt+∫0∞∫0te-λtKt,sζsdsdt=Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞e-λtεtϑtdt×∫0∞e-λtK0tζt+∫0te-λtKt,sζsdsdt.First, taking the limit inferior as t→∞ on both sides of the above inequality, we obtain (64)1≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞e-λtεtϑtdt×∫0∞e-λtK0tζt+liminft→∞∫0te-λtKt,sζsdsdt=Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞e-λtεtϑtdt×∫0∞e-λtK0tζt+∫0∞e-λtKt,sζsdsdt.Next, take the limit as λ→0 on both sides of the above inequality, we get(65)1≥Λ-ϵ0∗ν+μ0-ϵ0∗∫0∞εtϑtdt×∫0∞K0tζt+∫0∞Kt,sζsdsdt,which contradicts the inequality given in (54).Combining the results from Theorems5 and 8 with [25, Theorem 3.2] leads to the uniform (strong) ρ-persistence as follows.Theorem 9.
IfR0>1, then the semiflow Φ is uniformly (strongly) ρ-persistent.Definition 10.
A total trajectory of a continuous semiflowΦ, defined by (11), is a function X:R→X+ such that Φt(X(r))=X(t+r) for any (t,r)∈R+×R.Note that a global attractor will only contain points with total trajectories through them as it needs to be invariant. So, theα-limit point of a total trajectory X, passing through X(0)=X0, is given by (66)αX0=⋂t≥0⋃s≥tXs¯.A total trajectoryX(t)=S(t),v(·,t),e(·,t),i(·,t),R(t) satisfies (67)va,r=νSr-aχa,a,r∈R+×R,ea,r=e0,r-aϑa,a,r∈R+×R,ia,r=i0,r-aζa,a,r∈R+×R.Corollary 11.
LetA and X(t) be, respectively, a global attractor of Φ in X+ and a total trajectory of Φ in A∩X+. If R0>1, then there exists ϵ>0 such that(68)St,v0,t,e0,t,i0,t,Rt≥ϵ,for  anyt≥0.Proof.
We consider the boundary condition given by the second equation of (5). Using (25) and (ii) of A1, we get(69)e0,t≤4it1Stmax⁡K¯0,K¯1≤4Λ2μ~2max⁡K¯0,K¯1≕K¯.From the first equation of (4), we have (70)ddtSt≥Λ-ν-μ0St-St∫0∞K0aζai0,t-a+∫0∞Ka,a′ζai0,t-a′da′da≥Λ-ν-μ0St-LK¯St∫0∞K0aζa+∫0∞Ka,a′ζada′da≥Λ-ν+μ0+K¯S0R0St;that is, (71)ddtSt≥Λ-ν+μ0+K¯S0R0St.This yields(72)St≥ΛS0ν+μ0S0+K¯R0+e-ν+μ0+K¯/S0R0tS0-ΛS0ν+μ0S0+K¯R0.Taking the limit inferior as t→∞ in (72) leads to (73)liminft→∞St≥ΛS0ν+μ0S0+K¯R0≕ϵ1.Therefore, S(t)≥ϵ1. It follows that v(0,t)≥ϵ1ν≕ϵ2.
Now, we consider again the boundary condition given by the second equation of (5). It is easy to see that (74)e0,t=ρΦtX0=ρXtAnd hence, (75)e0,t≥liminft→∞ρXt.It follows from Theorem 9 and Definition 7 that e(0,t)≥ϵ∗≕ϵ3.
Further, considering the boundary condition given by the third equation of (5), we obtain (76)i0,t=∫0∞εaϑae0,t-ada≥ϵ3∫0∞εaϑada≕ϵ4.
Finally, the fifth equation of (4) leads to (77)ddtRt≥ϵ4∫0∞γaζada-μ0RtAnd hence,(78)Rt≥ϵ4μ0∫0∞γaζada+e-μ0tR0-ϵ4μ0∫0∞γaζada.Taking the limit inferior as t→∞ in (78), we obtain (79)liminft→∞Rt≥ϵ4μ0∫0∞γaζada≕ϵ5.And therefore R(t)≥ϵ5. By choosing ϵ such that ϵ=miniϵi, for i∈1,2,3,4,5, we get (80)St,v0,t,e0,t,i0,t,R≥ϵ.


## 5. Local Stability of Equilibria

The conditions of stability for each equilibrium will be derived through linearization technique around the equilibrium.The conditions of stability for the disease-free equilibriumE0 can be investigated through the following result.Theorem 12.
IfR0<1 then E0 is locally asymptotically stable; if R0>1 then E0 is unstable.Proof.
To investigate the stability of the disease-free equilibriumE0, we denote by S~(t), v~(a,t), e~(a,t), i~(a,t), and R~(t) the perturbations of S(t), v(a,t), e(a,t), i(a,t), and R(t), respectively, such that(81)S~t=St-S0,v~a,t=va,t-v0ae~a,t=ea,t,i~a,t=ia,t,R~t=Rt.The perturbations satisfy the following:(82)ddtS~t=-ν+μ0S~t+∫0∞αav~a,tda-S0∫0∞K0ai~a,t+∫0∞Ka,a′i~a′,tda′da∂∂tv~a,t=-∂∂av~a,t-ηav~a,t∂∂te~a,t=-∂∂ae~a,t-ϱae~a,t∂∂ti~a,t=-∂∂ai~a,t-σai~a,tddtR~=∫0∞γai~a,tda-μ0R~,after substituting (81) into (4) and neglecting the terms of order higher or equal to two, with boundary conditions(83)v~0,t=νS~te~0,t=S0∫0∞K0ai~a,t+∫0∞Ka,a′i~a′,tda′dai~0,t=∫0∞εae~a,tda,after substituting (81) into (5) and neglecting the terms of order higher or equal to two.
Now, we consider the exponential solutions of systems (82)-(83) of the form(84)S~t=S¯eλt,v~a,t=v¯aeλt,e~a,t=e¯aeλt,i~a,t=i¯aeλt,R~t=R¯eλt,where S¯, v¯(a), e¯(a), i¯(a), and λ (real or complex number) satisfy the following system:(85)λS¯=-ν+μ0S¯+∫0∞αav¯ada-S0∫0∞K0ai¯a+∫0∞Ka,a′i¯a′da′daλv¯a=-ddav¯a-ηav¯aλe¯a=-ddae¯a-ϱae¯aλi¯a=-ddai¯a-σai¯aλR¯=∫0∞γai¯ada-μ0R¯,with boundary conditions(86)v¯0=νS¯e¯0=S0∫0∞K0ai¯a+∫0∞Ka,a′i¯a′da′dai¯0=∫0∞εae¯ada.
From the second, third, and fourth equations of (85), we get(87)v¯a=v¯0e-λa-∫0aηsds,e¯a=e¯0e-λa-∫0aϱsds,i¯a=i¯0e-λa-∫0aσsds,respectively, where v¯(0), e¯(0), and i¯(0) are given by (86).
Substituting the last equation of (87) into the boundary condition given by the second equation of (86) yields the characteristic equation(88)C0λ=1,where (89)C0λ=S0∫0∞εae-λa-∫0aϱsdsda×∫0∞K0ae-λa-∫0aσsds+∫0∞Ka,a′e-λa′-∫0a′σsdsda′da,such that C0(0)=R0. It is not difficult to see that d/dλC0(λ)=-C0(λ)<0. Thus, C0(λ) is a decreasing continuous function of λ which approaches ∞ as λ→-∞ and 0 as λ→∞. Hence, the characteristic equation (88) admits a real solution λ∗ such that λ∗<0 whenever C0(0)<1 and λ∗>0 whenever C0(0)>1.
On the other hand, by assuming a complex solutionλ=α+iβ of the characteristic equation C0(λ)=1, we can notice that Reλ≤eR(λ) is always true. Thus, we clearly get RC0(λ)≤C0(Rλ). It follows from the characteristic equation C0(λ)=1 that RC0(λ)=1 and IC0(λ)=0. Therefore, we obtain 1≤C0(Rλ), i.e., C0(λ∗)≤C0(Rλ). Hence, Rλ≤λ∗, since C0(λ) is a decreasing function.
It results from the above statements that all eigenvalues of the characteristic equationC0(λ)=1 have negative real part whenever C0(0)<1, i.e., R0<1. Thus, the disease-free equilibrium E0 is locally asymptotically stable if R0<1. Otherwise, C0(0)≥1, i.e., the unique real solution of the characteristic equation C0(λ)=1 is positive, and hence the disease-free equilibrium E0 is unstable.Theorem 13.
IfR0>1 then E∗ is locally asymptotically stable.Proof.
Likewise for the disease-free equilibrium, we perturb the disease-endemic equilibrium by letting(90)S~t=St-S∗,v~a,t=va,t-v∗a,e~a,t=ea,t-e∗a,i~a,t=ia,t-i∗a,R~t=Rt-R∗.Substituting S(t)=S~(t)+S∗, v(a,t)=v~(a,t)+v∗(a), e(a,t)=e~(a,t)+e∗(a), and i(a,t)=i~(a,t)+i∗(a), R(t)=R~(t)+R∗ into (4) and neglecting the terms of second order and above, the perturbations satisfy the linear system(91)ddtS~t=-ν+μ0S~t+∫0∞αav~a,tda-S∗∫0∞K0ai~a,t+∫0∞Ka,a′i~a′,tda′da-S~t∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′da∂∂tv~a,t=-∂∂av~a,t-ηav~a,t∂∂te~a,t=-∂∂ae~a,t-ϱae~a,t∂∂ti~a,t=-∂∂ai~a,t-σai~a,tddtR~=∫0∞γai~a,tda-μ0R~,with boundary conditions(92)v~0,t=νS~te~0,t=S∗∫0∞K0ai~a,t+∫0∞Ka,a′i~a′,tda′da+S~t∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′dai~0,t=∫0∞εae~a,tda,after substituting S(t)=S~(t)+S∗, v(a,t)=v~(a,t)+v∗(a), e(a,t)=e~(a,t)+e∗(a), i(a,t)=i~(a,t)+i∗(a), and R(t)=R~(t)+R∗ into (5) and neglecting the terms of second order and above.
Now, we consider the exponential solutions of systems (91)-(92) of the form(93)S~t=S¯eλt,v~a,t=v¯aeλt,e~a,t=e¯aeλt,i~a,t=i¯aeλt,R~t=R¯eλt,where S¯, v¯(a), e¯(a), i¯(a), and λ (real or complex number) satisfy the system(94)λS¯=-ν+μ0S¯+∫0∞αav¯ada-S∗∫0∞K0ai¯a+∫0∞Ka,a′i¯a′da′da-S¯∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′daλv¯a=-ddav¯a-ηav¯aλe¯a=-ddae¯a-ϱae¯aλi¯a=-ddai¯a-σai¯aλR¯=∫0∞γai¯ada-μ0R¯,with boundary conditions(95)v¯0=νS¯e¯0=S∗∫0∞K0ai¯a+∫0∞Ka,a′i¯a′da′da+S¯∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′dai¯0=∫0∞εae¯ada.Similarly to the process leading to the characteristic equation (88), the characteristic equation at the disease-endemic equilibrium E∗ is given by(96)λ+μ0+νAλC∗λ-B-λ+μ0+νAλ=0,where (97)C∗λ=C0λR0>0Aλ=1-∫0∞αae-λa-∫0aηsdsda>0B=∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′da>0.
It is sufficient to prove that (96) has no root with nonnegative real part. So, we suppose that (96) has a complex root with nonnegative real part denoted by (98)λ=α+iβ,
whereα≥0 and β≠0. It follows from (96) that (99)α+iβ+μ0+νAα+iβC∗α+iβ-B-α+iβ+μ0+νAα+iβ=0,where (100)C∗α+iβ=C0α+iβR0>0Aα+iβ=1-∫0∞αae-αa-∫0aηsdse-iβada>0B=∫0∞K0ai∗a+∫0∞Ka,a′i∗a′da′da>0,and hence (101)C∗α+iβ=RC∗α+iβ+IC∗α+iβ,where(102)RC∗α+iβ=1+BKαβα+μ0+ν-ν∫0∞αae-αa-∫0aηsdscos⁡βada>1and (103)IC∗α+iβ=-BKαββ-ν∫0∞αae-αa-∫0aηsdssin⁡βada<0,with (104)Kαβ=α+μ0+ν-ν∫0∞αae-αa-∫0aηsdscos⁡βada2+β-ν∫0∞αae-αa-∫0aηsdssin⁡βada2>0.Since RC∗(α+iβ)≤C∗(α) is always true and C∗(α)≤C∗(0)=1 (C∗ is a decreasing function of λ), thus RC∗(α+iβ)≤1.
Therefore, the latter statement contradicts (102).


## 6. Global Stability of Equilibria

To investigate the global asymptotic stability of equilibria of system (4) we shall use suitable Volterra-type Lyapunov functions of the form(105)GX=X-1-ln⁡X,X>0.We state the following results.Theorem 14.
The disease-free equilibriumE0 of (4) is globally asymptotically stable if R0≤1.Proof.
A Lyapunov functionL0 of the form (106)L0t=L10t+L20t+L30t+L40t+L50tis considered, where (107)L10t=S0GStS0,L20t=∫0∞v0aGva,tv0adaL30t=∫0∞φaea,tda,L40t=∫0∞ωaia,tda,L50t=C0Rt.The functions φ and ω are nonnegative functions to be chosen suitably and carefully. L˙i0, i=1,…,5, denote the derivatives of Li0 with respect to t along the solution to (4) and are given by(108)L˙10·=ν+μ0S02-S0S-SS0+1-S0S∫0∞αava,·1-v0ava,·da+S0∫0∞K0aia,·+∫0∞Ka,a′ia′,·da′da-e0,·L˙20·=-lima→∞va,·+νS-S0ln⁡S+νS01-ln⁡ν-∫0∞ηava,·-v0aln⁡va,·da,L˙30·=-lima→∞φaea,·+φ0e0,·+∫0∞φ˙a-ϱaφaea,·da,L˙40·=-lima→∞ωaia,·+ω0i0,·+∫0∞ω˙a-σaωaia,·da,L˙50·=C0∫0∞γaia,·da-C0μ0R.Therefore, (109)
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,i.e.,(110)L˙0·=ν+μ0S02-S0S-SS0+φ0-1e0,·+νS-S0ln⁡S+νS01-ln⁡ν-1-S0S∫0∞αav0ada-C0μ0R-∫0∞γaia,·da-S0S∫0∞αava,·da+∫0∞αava,·da-∫0∞ηava,·-v0aln⁡va,·da+∫0∞ω′a-σaωa+S0K0a+S0∫0∞Ka,a′e-∫aa′σsdsda′ia,·da+∫0∞φ′a-ϱaφa+ω0εaea,·da,after choosing a and a′ such that (111)ia′,tζa′=ia,tζa.
Note that(112)νS-S0ln⁡S+νS01-ln⁡ν-1-S0S∫0∞αav0ada=-νS01+ln⁡νS+1SνS2+2νS0+Λ-ν+μ0S0S-Λ-ν+μ0S0S0,after using ∫0∞α(a)v0(a)da=-Λ+(ν+μ0)S0. Thus, (110) becomes(113)L˙0·=ν+μ0S02-S0S-SS0+φ0-1e0,·-C0μ0R-∫0∞γaia,·da-νS01+ln⁡νS+1SνS2+2νS0+Λ-ν+μ0S0S-Λ-ν+μ0S0S0-S0S∫0∞αava,·da-∫0∞ηa-αava,·-ηav0aln⁡va,·da+∫0∞ω′a-σaωa+S0K0a+S0∫0∞Ka,a′e-∫aa′σsdsda′ia,·da+∫0∞φ′a-ϱaφa+ω0εaea,·da.Using assumption A2, we show that(114)μ0Rt-∫0∞γaia,tda≥μ0Rt-γ¯∫0∞ia,tda≥μ0Rt-γ0It>0,where (115)It=∫0∞ia,·da.By assumption A3, obtain(116)∫0∞ηa-αava,·-ηav0aln⁡va,·da>∫0∞ηa1-v0a-αava,·da>∫0∞ηa1-νS0-αava,·da>∫0∞ηa1-Λ1-∫0∞αae-∫0aηsdsda-1-αava,·da>0.Next, we choose functions ω and φ such that(117)ωa=S0∫0∞K0u+∫0∞Ku,a′e-∫ua′σsdsda′e-∫auσsdsduand(118)φa=ω0∫0∞εue-∫auϱsdsdu.It follows from (117) and (118) that φ(0)=R0 and (119)ω0=S0∫0∞K0ue-∫0uσsds+∫0∞Ku,a′e-∫0a′σsdsda′du.Moreover, by differentiation of (117) and (118) with respect to age a, we obtain (120)ω′a-σaωa+S0K0a+S0∫0∞Ka,a′e-∫aa′σsdsda′=0and (121)φ′a-ϱaφa+ω0εa,respectively. Therefore, (113) is reduced to(122)L˙0·=-ν+μ0SS-S02+R0-1e0,·-C0μ0R-∫0∞γaia,·da-νS01+ln⁡νS+1SνS2+2νS0+Λ-ν+μ0S0S-Λ-ν+μ0S0S0-S0S∫0∞αava,·da-∫0∞ηa-αava,·-ηav0aln⁡va,·da.We denote(123)FS=νS2+2νS0+Λ-ν+μ0S0S-Λ-ν+μ0S0S0.Since (124)Λμ0+ν<S0<Λμ0-ν,it is easy to check that F(S) has two negative (real) roots. Moreover, F(S)>0 for every S≥0. Therefore, the sign of L˙0(·) will be determined by the sign of (125)F0-ν+μ0St-S02=-ΛS0-ν+μ0StSt-2S0.Thus, three cases occur:
Case 1 (S(t)>2S0). It is easy to see that F(0)-(ν+μ0)(S(t)-S0)2<0 and therefore L˙0(t)≤0 if R0≤1.
Case 2 (S(t)=2S0). For this value of S(t), we have F(0)-(ν+μ0)(S(t)-S0)2=-ΛS0<0 and therefore L˙0(t)≤0 if R0≤1.
Case 3 (0<S(t)<2S0). For such values of S(t) we have F(0)-(ν+μ0)(S(t)-S0)2=-(Λ-2(ν+μ0)S(t))S0-(ν+μ0)S2. Since 2-1(ν+μ0)-1Λ<2S0, it follows that F(0)-(ν+μ0)(S(t)-S0)2<0 for any 0<S(t)<2-1(ν+μ0)-1Λ. Therefore L˙0(t)≤0 if R0≤1.
It results from the above that the derivative ofL0(t) along the solutions of (4) is L˙0(t)≤0. If S(t)=S0, v(a,t)=v0(a), and e(a,t)=i(a,t)=R(t)=0 are simultaneously satisfied with R0=1, then L˙0(t)=0 holds. Moreover, it can be verified that (S,v,e,i,R):L˙0(t)=0=E0. Therefore, it results from Lasalle’s Invariance Theorem [26, p. 200] that E0 is globally asymptotically stable, if R0≤1.Theorem 15.
The endemic equilibriumE∗ of (4) is globally asymptotically stable on the set, if R0>1.Proof.
To prove the above result we consider a Lyapunov function of the form given below(126)L∗t=L1∗t+L2∗t+L3∗t+L4∗t+L5∗t,where (127)L1∗t=S∗GStS∗,L2∗t=∫0∞v∗aGva,tv∗ada,L3∗t=∫0∞e∗aGea,te∗adaL4∗t=∫0∞i∗aGia,ti∗ada,L5∗t=R∗GRtR∗,with G(X)=X-1-ln⁡X, for any X>0.
Thus, we have(128)L˙1∗·=1-S∗SΛ-ν+μ0S+∫0∞αava,·da-S∫0∞K0aia,·+∫0∞Ka′,aia′,·da′daUsing (37), (44), and (45) together with the first equation of (38), we obtain (129)L˙1∗·=ΛR02-S∗S-SS∗+2-S∗S-SS∗∫0∞αav∗ada+1-S∗S-∫0∞αav∗ada+∫0∞αava,·da-S∫0∞K0aia,·+∫0∞Ka′,aia′,·da′da-S∗∫0∞K0ai∗a+∫0∞Ka′,ai∗a′da′da.We add and subtract, carefully, some terms to the above expression, and we also identify group of terms in the form given by G(X)=X-1-ln⁡X. We obtain(130)L˙1∗·=ΛR02-S∗S-SS∗-∫0∞αav∗aGS∗va,·Sv∗a-GS∗S-Gva,·v∗ada-2GS∗S∫0∞αav∗ada-S∗∫0∞K0ai∗aGSia,·S∗i∗a-Gia,·i∗ada-S∗∫0∞∫0∞Ka′,ai∗a′GSia′,·S∗i∗a′-Gia′,·i∗(a′)da′da.By differentiating L2∗ with respect to t, we obtain(131)L˙2∗·=-∫0∞1-v∗ava,·∂∂tva,·+ηava,·da.Since (132)v∗a∂∂tGva,·v∗a=1-v∗ava,·∂∂tva,·+ηava,·thus (131) yields(133)L˙2∗·=-∫0∞v∗a∂∂tGva,·v∗ada.Using the second equation of (38) after integrating by part, we get(134)L˙2∗·=v∗0GSS∗-∫0∞ηav∗aGva,·v∗ada.Moreover, (135)v∗0GSS∗=v∗0∫0∞ηae-∫0∞ηsdsGSS∗da=∫0∞ηav∗aGSS∗da,yields(136)L˙2∗·=-∫0∞ηav∗aGva,·v∗a-GSS∗da.Similarly to L2∗, from L3∗ and L4∗ we get, respectively,(137)L˙3∗·=-∫0∞ϱae∗aGea,·e∗a-Ge0,·e∗0daand(138)L˙4∗·=-∫0∞σai∗aGia,·i∗a-Gi0,·i∗0da,while using the fifth equation of (38) L5∗ leads to(139)L˙5∗·=μ0R∗2-RR∗-R∗R-∫0∞γai∗aGR∗ia,·Ri∗a-GR∗R-Gia,·i∗ada.By combining (130), (136), (137), (138), and (139), we obtain (140)L˙∗·=ΛR02-S∗S-SS∗-∫0∞αav∗aGS∗va,·Sv∗a-GS∗S-Gva,·v∗ada-2GS∗S∫0∞αav∗ada-S∗∫0∞K0ai∗aGSia,·S∗i∗a-Gia,·i∗ada-S∗∫0∞∫0∞Ka′,ai∗a′GSia′,·S∗i∗a′-Gia′,·i∗(a′)da′da+GSS∗∫0∞ηav∗ada-∫0∞ηav∗aGva,·v∗ada-∫0∞ϱae∗aGea,·e∗a-Ge0,·e∗0da-∫0∞σai∗aGia,·i∗a-Gi0,·i∗0da+μ0R∗2-RR∗-R∗R-∫0∞γai∗aGR∗ia,·Ri∗a-GR∗R-Gia,·i∗ada.This yields (141)L˙∗·=ΛR02-S∗S-SS∗-∫0∞αav∗aGS∗va,·Sv∗a-GS∗S-Gva,·v∗ada-S∗∫0∞K0ai∗aGSia,·S∗i∗a-GSS∗-Gia,·i∗ada-S∗∫0∞∫0∞Ka′,ai∗a′GSia′,·S∗i∗a′-GSS∗-Gia′,·i∗a′da′da-2GS∗S∫0∞αav∗ada-∫0∞ηav∗aGva,·v∗ada-S∗∫0∞K0ai∗a+∫0∞Ka′,ai∗a′da′da-∫0∞ηav∗adaGSS∗-∫0∞ϱae∗aGe∗0ea,·e0,te∗ada-∫0∞ϱae∗a1-e∗0e0,·ea,·e∗a-e0,·e∗0da-∫0∞σai∗aGi∗0ia,·i0,ti∗ada-∫0∞σai∗a1-i∗0i0,·ia,·i∗a-i0,·i∗0da+μ0R∗2-RR∗-R∗R-∫0∞γai∗aGR∗ia,·Ri∗a-GR∗R-Gia,·i∗ada.Since (142)S∗∫0∞K0ai∗a+∫0∞Ka′,ai∗a′da′da=1,∫0∞ηae-∫0∞ηsdsda=1,thus(143)L˙∗·=ΛR02-S∗S-SS∗-∫0∞αav∗aGS∗va,·Sv∗a-GS∗S-Gva,·v∗ada-S∗∫0∞K0ai∗aGSia,·S∗i∗a-GSS∗-Gia,·i∗ada-S∗∫0∞∫0∞Ka′,ai∗a′GSia′,·S∗i∗a′-GSS∗-Gia′,·i∗a′da′da-2GS∗S∫0∞αav∗ada-∫0∞ηav∗aGva,·v∗ada-1-νS∗GSS∗-∫0∞ϱae∗aGe∗0ea,·e0,te∗ada-∫0∞ϱae∗a1-e∗0e0,·ea,·e∗a-e0,·e∗0da-∫0∞σai∗aGi∗0ia,·i0,ti∗ada-∫0∞σai∗a1-i∗0i0,·ia,·i∗a-i0,·i∗0da+μ0R∗2-RR∗-R∗R-∫0∞γai∗aGR∗ia,·Ri∗a-GR∗R-Gia,·i∗ada.Using assumption A3, it is easy to see that Λ<1-∫0∞α(a)e-∫0aη(s)dsda. Hence, S0/R0<1/ν for R0>1; i.e., 1-νS∗>0 for R0>1. Moreover, (i)
GS∗v(a,·)/Sv∗(a)-GS∗/S-Gv(a,·)/v∗(a) = 1-S∗/S1-v(a,·)/v∗(a)>0 ⇔ S/S∗<1andv(a,·)/v∗(a)>1 or S/S∗>1andv(a,·)/v∗(a)<1;(ii)
GSi(a,·)/S∗i∗(a)-GS/S∗-Gi(a,·)/i∗(a) = 1-S/S∗1-i(a,·)/i∗(a)>0 ⇔ S/S∗>1andi(a,·)/i∗(a)>1 or S/S∗<1andi(a,·)/i∗(a)<1;(iii)
1-e∗(a)/e(0,·)e(a,·)/e∗(a)-e(0,·)/e∗(0)>0 ⇔ e(a,·)/e∗(a)<e(0,·)/e∗(0)<1 or e(a,·)/e∗(a)>e(0,·)/e∗(0)>1;(iv)
1-i∗(a)/i(0,·)i(a,·)/i∗(a)-i(0,·)/i∗(0)>0 ⇔ i(a,·)/i∗(a)<i(0,·)/i∗(0)<1 or i(a,·)/i∗(a)>i(0,·)/i∗(0)>1;(v)
GR∗i(a,·)/Ri∗(a)-GR∗/R-Gi(a,·)/i∗(a) = 1-R∗/R1-i(a,·)/i∗(a)>0 ⇔ R/R∗<1andi(a,·)/i∗(a)>1 or R/R∗>1andi(a,·)/i∗(a)<1.
Therefore, we obtain(144)L˙∗t<0,in the following cases: (i)
S/S∗<1, v(a,·)/v∗(a)>1, e(a,·)/e∗(a)>e(0,·)/e∗(0)>1, i(a,·)/i∗(a)<i(0,·)/i∗(0)<1, and R/R∗>1;(ii)
S/S∗<1, v(a,·)/v∗(a)>1, e(a,·)/e∗(a)<e(0,·)/e∗(0)<1, i(a,·)/i∗(a)<i(0,·)/i∗(0)<1, and R/R∗>1;(iii)
S/S∗>1, v(a,·)/v∗(a)<1, e(a,·)/e∗(a)>e(0,·)/e∗(0)>1, i(a,·)/i∗(a)>i(0,·)/i∗(0)>1, and R/R∗<1;(iv)
S/S∗>1, v(a,·)/v∗(a)<1, e(a,·)/e∗(a)<e(0,·)/e∗(0)<1, i(a,·)/i∗(a)>i(0,·)/i∗(0)>1, and R/R∗<1.
From (144), we say that the derivative of L∗(t) along the solutions of (4) is L˙∗(t)<0. If S(t)=S∗, v(a,t)=v∗(a), e(a,t)=e∗(a), i(a,t)=i∗(a), and R(t)=R∗ are simultaneously satisfied, then we obtain L˙∗(t)=0 from (76). Moreover, it can be verified that (S,v,e,i,R):L˙∗(t)=0=E∗. Therefore, it results from Lasalle’s Invariance Theorem [26, p. 200] that E∗ is globally asymptotically stable, if R0>1.The results from Theorems14 and 15 show that there exists a threshold parameter, called the basic reproduction number and denoted by R0, that is essential in the stability analysis of the global dynamics of the model defined by system (4). Moreover, such a parameter can play a crucial role in the implementation of human vaccination policies. As we can see in(145)∂R0∂ν=-Λ1-Pμ0+ν1-P2∫0∞K0aζa+∫0∞Ka,a′ζa′da′da<0,the rise in rate of vaccination of (infant) susceptible individuals against an SEIR infection can reduce the spread of the infection and help to elaborate control measures to prevent and reduce the spread of the infection.



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*Source: 8479638-2018-08-14.xml*