# Functional Coupled Systems with Generalized Impulsive Conditions and Application to a SIRS-Type Model

Authors: Feliz Minhós; Rui Carapinha
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.


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## Body

## 1. Introduction

The study of impulsive boundary value problems is richer than the related differential equation theory without impulses and has strategic importance in multiple current scientific fields, from sociology and medical sciences to generalized industry production or in any other real-world phenomena where sudden variations occur.The classic impulsive theory can be seen in [1, 2]. In the last two decades, a vast literature on impulsive differential problems has been produced, such as [3–17], only to mention a few.Functional problems composed by differential equations and conditions with global dependence on the unknown variable generalize the usual boundary value problems and can include equations and/or conditions with deviating arguments, delays or advances, nonlinear, or nonlocal, increasing in this way the range of applications. The readers interested in results in this direction, on bounded or unbounded domains, may look for in [18–28] and the references therein.Recently, coupled systems have been studied by many authors, not only from a theoretical point of view but also due to the huge applications in many sciences and fields, with several methods and approaches. We recommend to the interested readers, for instance, [29–39].Motivated by the results contained in some of the above references, in this paper, we consider the first-order coupled impulsive system of equations(1)y1′t=g1t,y1t,y2t,y3t,y2′t=g2t,y1t,y2t,y3t,y3′t=g3t,y1t,y2t,y3t,a.e.t∈a,b\tj, where tj is fixed points, and j=1,2,⋯,n and gi:a,b×ℝ3⟶ℝ are L1-Carathéodory functions, for i=1,2,3, with the functional boundary conditions
(2)B1y1,y2,y3=0,B2y1,y2,y3=0,B3y1,y2,y3=0,where Bi:Ca,b3⟶ℝ and i=1,2,3 are continuous functions linearly independent and verifying the generalized impulsive conditions
(3)Δy1tj=H1jtj,y1tj,y2tj,y3tj,Δy2tj=H2jtj,y1tj,y2tj,y3tj,Δy3tj=H3jtj,y1tj,y2tj,y3tj,where Hij:a,b×ℝ3⟶ℝ is continuous functions for i=1,2,3,j=1,2,⋯,n, with Δyitj=yitj+−yitj−, and tj fixed points such that a≔t0<t1<t2<⋯<tn<tn+1≔b.As far as we know, it is the first time where those three features are taken together to have a coupled impulsive system with functional boundary conditions and generalized impulsive effects, which one including, eventually, impulses on the three unknown functions. We underline two novelties of this paper:(1)
Condition (2) generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and/or integrodifferential ones, or global arguments, as maxima or minima, among others. In this way, new types of problems and applications could be considered, enabling greater and wider information on the problems studied(2)
The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and asymptotic behavior of the unknown functionsOur method is based on lower and upper solution technique together with the fixed point theory. In short, the main result is obtained studying a perturbed and truncated system, with modified boundary and impulsive conditions and applying Schauder’s fixed point theorem to a completely continuous vectorial operator. Moreover, the paper contains a method to overcome the nonlinearities monotony through a combination with adequate changes in the definition of lower and upper solutions.The paper is structured in the following way: Section2 contains the functional framework, definitions, and other known properties. The main result is in Section 3, where the proof is divided into steps, for the reader’s convenience. In Section 4, it is shown a method where the definition of coupled lower and upper functions can be used to obtain different versions of the main theorem, with different monotone characteristics on the nonlinearities. The last section contains an application to a vital dynamic SIRS-type model, representing the dynamic epidemiological evolution of susceptible (S), infected (I), Recovered (R), and newly infected individuals in a population on a normalized period, subject to impulsive effects and global restrictions.


## 2. Definitions and Auxiliary Results

Defineyitk±≔limt⟶tk±yit, for i=1,2,3, and consider the sets
(4)PCka,b=y:y∈Cka,b,ℝ3continuousfort≠tj,yktj=yktj−,yktj+existsforj=1,2,⋯,n,for k=0,1 and the space X3≔PCa,b3 equipped with the norm
(5)y1,y2,y3X3=maxy1,y2,y3,where
(6)y≔supt∈a,byt.It is clear thatX3,.X3 is a Banach space.The tripley1,y2,y3 is a solution of problem (1)-(3) if y1,y2,y3∈X3 and verifies conditions (1), (2), and (3).Definition 1.
A functionw:a,b×ℝ3⟶ℝ for i=1,2,3 is L1−Carathéodory if
(i)
for eachx,y,z∈ℝ3, t↦wt,x,y,z is measurable on a,b(ii)
fora.e.t∈a,b,x,y,z↦wt,x,y,z is continuous on ℝ3(iii)
for eachρ>0, there exists a positive function ψρ∈L1a,b and for x,y,z∈ℝ3 such that(7)maxx,y,z<ρ,
one has(8)wt,x,y,z≤ψρt,a.e.t∈a,b.In this paper, the definition of lower and upper solutions plays a key role in our method.Next definition will be used in the main theorem:Definition 2.
Consider thePC1-functions αi,βi:a,b⟶ℝ,i=1,2,3.
The tripleα1,α2,α3∈X3 is a lower solution of the problem (1)-(3) if
(9)αi′t≤git,α1t,α2t,α3t,fori=1,2,3,B1α1,α2,α3≥0,B2α1,α2,α3≥0,B3α1,α2,α3≥0,and, for j=1,2,⋯,n,(10)Δα1tj≤H1jtj,α1tj,α2tj,α3tj,Δα2tj≤H2jtj,α1tj,α2tj,α3tj,Δα3tj≤H3jtj,α1tj,α2tj,α3tj.The tripleβ1,β2,β3∈X3 is an upper solution of the problems (1)-(3) if the reversed inequalities hold.


## 3. Main Result

The main result will provide the existence of, at least, a solution for the problems (1)-(3).Theorem 3.
Assume that there areα and β lower and upper solutions of problem (1)-(3), according Definition 2, such that
(11)αit≤βit,∀t∈a,b,fori=1,2,3.
Letgi:a,b×ℝ3⟶ℝ,i=1,2,3, be L1−Carathéodory functions, not identically null, on the set
(12)t,yi∈a,b×ℝ3:αit≤yi≤βit,i=1,2,3,(13)g1t,y1,α2t,α3t≤g1t,y1,y2,y3≤g1t,y1,β2t,β3t,for t∈a,b\tj,j∈1,2,⋯,n,y1∈ℝ,αit≤yi≤βit, and i=2,3.(14)g2t,α1t,y2,α3t≤g2t,y1,y2,y3≤g2t,β1t,y2,β3t,for t∈a,b\tj,j∈1,2,⋯,n,y2∈ℝ,αit≤yi≤βit, and i=1,3.(15)g3t,α1t,α2t,y3≤g3t,y1,y2,y3≤g3t,β1t,β2t,y3,for t∈a,b\tj,j∈1,2,⋯,n,y3∈ℝ,αit≤yi≤βit, and i=1,2, with
(16)Biα1,α2,α3≤Biy1,y2,y3≤Biβ1,β2,β3,for αi≤yi≤βi,i=1,2,3.
Assume thatHij:a,b×ℝ3⟶ℝ satisfies
(17)Hijtj,α1tj,α2tj,α3tj≤Hijtj,w1,w2,w3≤Hijtj,β1tj,β2tj,β3tj,
forαitj≤wi≤βitj,i=1,2,3, and j∈1,2,⋯,n.
If there isρ>0 such that
(18)maxi=1,2,3maxαi,βi+∑j=1nHijtj,w1,w2,w3+∫abψi,ρs+1ds<ρ,then there exists at least a triple y1,y2,y3∈X3, solution of (1)-(3), such that
(19)αit≤yit≤βit,∀t∈a,b,fori=1,2,3.Proof.
Letα1,α2,α3,β1,β2,β3∈X3 be, respectively, lower and upper solutions of (1)-(3), as in Definition 2, verifying (11).
Consider the continuous truncaturesδi:a,b×ℝ⟶ℝ,i=1,2,3, denoted, for short, as δit, defined by
(20)δit≔δit,yit=βit,yit>βit,yit,αit≤yit≤βit,αit,yit<αit,and consider the modified and perturbed problem composed by the differential system
(21)y1′t=g1t,δ1t,δ2t,δ3t+y1t−δ1t1+y1t−δ1t,(22)y2′t=g2t,δ1t,δ2t,δ3t+y2t−δ2t1+y2t−δ2t,(23)y3′t=g3t,δ1t,δ2t,δ3t+y3t−δ3t1+y3t−δ3t,for t∈a,b\tj,j∈1,2,⋯,n, together with the truncated boundary conditions
(24)yia=δia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3,for i=1,2,3, with the functional truncations
(25)δi∗w=βi,w>βi,w,αi≤w≤βi,αi,w<αi,and the truncated impulsive conditions
(26)Δyitj=Hijtj,δ1tj,δ2tj,δ3tj,
for j∈1,2,⋯,n.☐Claim 4.
The problems (21), (24), and (26) have at least a solution.
This claim will be proved by the fixed point theory, applied to the vectorial operator(27)T:X3⟶X3,given by
(28)Ty1,y2,y3=T1y1,y2,y3,T2y1,y2,y3,T3y1,y2,y3,where for i=1,2,3,(29)Ti:X3⟶Xdefined as
(30)Tiy1,y2,y3t≔δia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3+∑j:tj<tHijtj,δ1tj,δ2tj,δ3tj+∫atgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds.
Integrating (21) for t∈a,t1, by (24), we have
(31)yit=δia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3++∫atgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds.By integration of (21) for t∈t1,t2, by (26) and (31), we get
(32)yit=yit1++∫atgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds==yit1−+Hi1t1,δ1t1,δ2t1,δ3t1++∫t1tgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds=δia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3+Hi1t1,δ1t1,δ2t1,δ3t1+∫atgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds.Iterating these arguments, it is clear that the fixed points ofT, that is, the set of the fixed points of Ti, for i=1,2,3, are solutions of the problems (21), (24), and (26).Asgi is a L1-Carathéodory function, Hij and the truncatures δi,δi∗ are continuous; therefore, Ti are well defined and continuous. Therefore, T is well defined and continuous.Consider a bounded setD⊂X3. So, there is k>0 such that x,y,zX3<k, for x,y,z∈D.TiD is uniformly bounded, as, for i=1,2,3,(33)Tiy1,y2,y3=supt∈a,bTiy1,y2,y3t≤supt∈a,bδia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3+∑j:tj<tHijtj,δ1tj,δ2tj,δ3tj+∫atgis,δ1s,δ2s,δ3s+yis−δis1+yis−δisds≤maxi=1,2,3maxαi,βi+∑j=1nHijtj,δ1tj,δ2tj,δ3tj+maxi=1,2,3∫abψi,ks+1ds<∞,where ψi,k is the positive function given by Definition 1.TiD is equicontinuous because, for i=1,2,3, and t1,t2∈a,b] with t1<t2 (without loss of generality),
(34)Tiy1,y2,y3t1−Tiy1,y2,y3t2≤∑j:t1<tj<t2Hijtj,δ1tj,δ2tj,δ3tj+∫t1t2ψi,ks+1ds⟶0,ast1⟶t2.TiD is equiconvergent on the impulsive moments, asTiy1,y2,y3t−limt⟶tj+Tiy1,y2,y3t≤∑j:tj<tHijtj,δ1tj,δ2tj,δ3t−∑j:tj<tj+Hijtj,δ1tj,δ2tj,δ3t+∫atψi,ks+1ds−∫atj+ψi,ks+1ds⟶0,(42)when t⟶tj+. Therefore, Ti and T are compact operators.Consider now the closed, bounded, and convex setΩ⊂X3, defined by
(35)Ω=w∈X3:TwX3≤Rwith R>0 such that
(36)R>maxi=1,2,3maxαi,βi+∑j=1nHijtj,δ1tj,δ2tj,δ3tj+maxi=1,2,3∫abψi,Rs+1ds.By the above calculus,TΩ⊂Ω, and from Schauders’ fixed point theorem, T has a fixed point y∗=y1∗,y2∗,y3∗, which is solution of the problems (21), (24), and (26).Claim 5.
This functiony∗=y1∗,y2∗,y3∗ is a solution of problems (1)-(3), too.
To prove this claim is enough to show that, for every solutiony1,y2,y3∈X3 of problems (21), (24), and (26), the following inequalities hold:
(37)αit≤yit≤βit,fori=1,2,3,andt∈a,b,(38)αia≤yia+Biδ1a,y1∗a,δ2a,y2∗a,δ3a,y3∗a≤βia.
Lety≔y1,y2,y3∈X3 be a solution of the problems (21), (24), and (26).
To prove the first inequality of (37), for i=1, assume that there is t∈a,b such that α1t−y1t>0 and define
(39)supt∈a,bα1t−y1t≔α1t∗−y1t∗>0.
Remark thatt∗≠a, as, by (24) and (20),
(40)α1a−y1a=α1a−δ1a,yia+B1δ1∗y1,δ2∗y2,δ3∗y3≤0.Ift∗ is between two consecutive impulses, that is, t∈tp,tp+1, for fixed p=0,1,⋯,n, then α1′t∗−y1′t∗=0, by (24), (13), and Definition 2, this contradiction is achieved
(41)0≤α1′t∗−y1′t∗=α1′t∗−g1t∗,α1t∗,δ2t∗,δ3t∗−y1t∗−α1t∗1+y1t∗−α1t∗<α1′t∗−g1t∗,α1t∗,δ2t∗,δ3t∗≤α1′t∗−g1t∗,α1t∗,α2t∗,α3t∗≤0.Ift∗ is an impulsive moment, that is, there is j∈1,2,⋯,n such that t∗=tj+; then, by (26), (17), and Definition 2, we have
(42)0≤Δα1tj−Δy1tj=Δα1tj−H1jtj,δ1tj,δ2tj,δ3tj≤Δα1tj−H1jtj,α1tj,α2tj,α3tj≤0.Therefore,(43)Δy1tj−Δα1tj=0,that is, there are no jumps at any point tj. Then, by (39),
(44)0≤α1′tj−−y1′tj−,and the contradiction is obtained as in the previous case.Therefore,α1t≤y1t, for t∈a,b. With the same arguments, it can be proved that y1t≤β1t, for t∈a,b.A similar technique can be applied for functionsg2 and g3, applying conditions (14) and/or (15), respectively.Suppose now, by contradiction, that(45)αia>yia+Biδ1∗y1,δ2∗y2,δ3∗y3.Then, by (24),
(46)yia=δia,yia+Biδ1∗y1,δ2∗y2,δ3∗y3=αia,which is in contradiction with (45), by (16) and Definition 2,
(47)0=yia−αia>Biδ1∗y1,δ2∗y2,δ3∗y3−Biα1,α2,α3≥0,fori=1,2,3.The remaining inequalities can be proved with similar arguments.


## 4. Relation between Monotonies and Lower and Upper Definitions

The monotone assumptions required on the nonlinearities and on the impulsive functions, by conditions (13)-(15) and (17), although local, can seem too restrictive. Indeed, these monotonies can be modified since they are combined with different definitions of coupled lower and upper solutions, following the method described in this section.Definition 6.
Consider thePC1-functions αi,βi:a,b⟶ℝ,i=1,2,3.
The triplesα1,α2,α3,β1,β2,β3∈X3 are coupled lower and upper solutions of the problems (1)-(3) if
(48)α1′t≤g1t,α1t,β2t,α3t,αi′t≤git,α1t,α2t,α3t,fori=2,3,β1′t≥g1t,β1t,α2t,β3t,βi′t≥g1t,β1t,β2t,β3t,fori=2,3,and, for j=1,2,⋯,n,(49)Δα1tj≤H1jtj,β1tj,α2tj,α3tj,(50)Δα2tj≤H2jtj,α1tj,α2tj,α3tj,(51)Δα3tj≤H3jtj,α1tj,α2tj,β3tj,(52)Δβ1tj≥H1jtj,α1tj,β2tj,β3tj,(53)Δβ2tj≥H2jtj,β1tj,β2tj,β3tj,(54)Δβ3tj≥H3jtj,β1tj,β2tj,α3tj.The inequalities for boundary conditions are similar to Definition2.With this definition, the assumption on the local monotony of functiong1 and on the impulsive functions H1j and H3j can be replaced, as in the following version of Theorem 3:Theorem 7.
Assuming that all the assumptions of Theorem3, with coupled lower and upper solutions defined as in Definition 6, (13) is replaced by
(55)g1t,β1t,β2t,α3t≤g1t,y1,y2,y3≤g1t,α1,α2t,β3t,for t∈a,b\tj,j∈1,2,⋯,n,y1∈ℝ,αit≤yi≤βit,i=2,3, and (12) by
(56)H1jtj,β1tj,α2tj,α3tj≤H1jtj,w1,w2,w3≤H1jtj,α1tj,β2tj,β3tj,H2jtj,α1tj,α2tj,α3tj≤H2jtj,w1,w2,w3≤H2jtj,β1tj,β2tj,β3tj,H3jtj,α1tj,α2tj,β3tj≤H3jtj,w1,w2,w3≤H3jtj,β1tj,β2tj,α3tj,for αitj≤wi≤βitj,i=1,2,3, and j∈1,2,⋯,n. Then, there is at least y1,y2,y3∈X3 solution of (1)-(3) such that
(57)αit≤yit≤βit,∀t∈a,b,fori=1,2,3.Proof.
The proof of Theorem3 holds, and it remains to prove that every solution y1,y2,y3∈X3 of problems (21), (24), and (26) verifies
(58)α1t≤y1t≤β1t,fort∈a,b.
Assume that there ist∈a,b such that αit−yi∗t>0 and define
(59)supt∈a,bαit−yi∗t≔αit∗−yi∗t∗>0.
Considert∗ between two consecutive impulses, that is, t∈tp,tp+1. Then, by (24), (13), and Definition 2, this contradiction is achieved
(60)0≤α1′t−y1′∗t=α1′t−g1t,α1t,δ2t,δ3t−y1∗t−α1t1+y1∗t−α1t<α1′t−g1t,α1t,δ2t,δ3t≤α1′t−g1t,α1t,β2t,α3t≤0.
In the impulsive points, case wheret∗=tj+, by (49),
(61)0≤Δα1tj−Δy1tj=Δα1tj−H1jtj,δ1tj,δ2tj,δ3tj≤Δα1tj−H1jtj,β1tj,α2tj,α3tj≤0,
analogously, forΔα3−y3tj.☐Following these arguments, we can obtain different versions of Theorem3, combining adequate definitions of coupled lower and upper solutions, as in Definition 6, and alternative monotone assumptions on g2 and g3 and on the impulsive functions Hij.


## 5. Application to a Vital Dynamic SIRS Model

The study of epidemiological phenomena via compartmental models is currently a special concern as it simplifies the mathematical modeling of infectious diseases. These types of models try to predict, for instance, how a disease spreads, the duration of an epidemic, the variation of the number of infected people, and other epidemiological parameters. So, they are important tools to help the definition of rules for public health interventions and how they may affect the outcome of the epidemic.The classic SIR model is a basic compartmental model where the population is divided into three groups: susceptible (S), infected (I), and recovered (R). People may change groups, but the SIR model assumes that the population gains lifelong immunity to some disease upon recovery. This is true for some infectious diseases, such as measles, mumps, or rubella, but it is not the case for some airborne diseases, such as seasonal influenza, where the individual’s immunity may wane over time. In this situation, the SIRS model is more adequate as it allows that the recovered individuals can return to a susceptible state and be infected again.These compartmental models were introduced in the early 20th century, by Kermack and McKendrick in 1927, ([40]), but since then, many authors study these topics, under different and varied features, objectives, and techniques. As examples, we mention only some recent works on the field: [41–48].Motivated by the papers above, we apply our technique to a vital dynamic SIRS system composed by the differential equations(62)S′t=−γStIt+λRt,I′t=γStIt−μ+dIt,R′t=μIt−λRt,for t in a normalized interval 0,1,γ,μ, representing the infection and recover rates; λ is the rate of recovered individuals becoming susceptible again, and d is the death number by infection.Our method allows to consider global and asymptotic data as a particular case of functional boundary conditions:(63)inft∈0,1St=limt⟶14+St,maxt∈0,14It=limt⟶14+It,supt∈0,1Rt=R1,and generalized impulsive functions, with only one impulsive moment, for the sake of clarity,
(64)ΔS14=−5γS14I14+λR14I14,(65)ΔI14=5γS14I14,(66)ΔR14=−λR14I14.It is clear that problems (62)-(64) are a particular case of problem (1)-(3), with y1=S,y2=I,y3=R,a=0,b=1,n=j=1,t1=1/4,(67)g1t,St,It,Rt=−γStIt+λRt,g2t,St,It,Rt=γStIt−μ+dIt,g3t,St,It,Rt=μIt−λRt,and the functional boundary conditions
(68)B1S,I,R=inft∈0,1St−limt⟶14+St=0,B2S,I,R=maxt∈0,14It−limt⟶14+It=0,B3S,I,R=supt∈0,1Rt−R1=0,and the impulsive effects
(69)ΔS14=H11S14,I14,R14=−5γS14I14+λR14I14,ΔI14=H21S14,I14,R14=5γS14I14−kI14,ΔR14=H31S14,I14,R14=−λR14I14+kI14.As a numeric exampl.e we consider the ratesγ1=0.017,λ1=0.02, μ1=0.93, and d1=0.2 before the impulsive moment, that is, for 0≤t≤1/4 and γ2=0.001,λ2=0.1, μ2=0.162, and d2=0, after the impulsive effect, i.e., for 1/4<t≤1.For these values, the triple null functionsα1,α2,α3=0,0,0 and the piecewise one β1,β2,β3 given by
(70)β1t=112t6+17.12t5+4.6t4−0.4t3−0.02t2+0.02t+1.1,0≤t≤14,−14.5t6+20t5+10t4+14.6t3−0.02t2+0.006t+2,14<t≤1,β2t=112t6+17.12t5+4.6t4−0.4t3−0.02t2+0.02t+0.01,0≤t≤14,−14.5t6+20t5+10t4+14.6t3−0.02t2+0.006t+1.5,14<t≤1,β3t=112t6+17.12t5+4.6t4−0.4t3−0.02t2+0.02t+1,0≤t≤14,−14.5t6+20t5+10t4+14.6t3−0.02t2+0.006t+2,14<t≤1,are, respectively, lower and upper solutions of problems (62)-(64), according to Definition 6, as it can be seen in Figures 1 and 2.Figure 1
Graphs in0,1/4.Figure 2
Graphs in [0,1].All the assumptions of Theorem7 are fulfilled for 56.22≤ρ<137.23, and therefore, there is a solution of problems (62)-(64), such that
(71)0≤St≤β1t,0≤It≤β2t,0≤Rt≤β3t,fort∈0,1.Applying an adequate mathematical software, these inequalities can be illustrated by the graph of the correspondent solution, given in Figure3, considering the population in percentage in the first 100 days.Figure 3
Evolution of functionsS, I, and R in the first 100 days.


## 6. Conclusion

The paper’s main goal is to present sufficient conditions for the solvability of impulsive coupled systems with functional boundary conditions generalizing the classical boundary ones. In this way, problems may consider restrictions related to the global variation of solutions or their asymptotic behavior near the impulsive moments, as can be seen in Theorem3.Moreover, in Section4, it is shown how we can use the definition of lower and upper solutions to overcome restrictions on the monotone conditions on the nonlinearities.The application’s aim is to illustrate how the theoretical results could be applied to real phenomena.



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*Source: 3758274-2021-08-19.xml*