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

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


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.
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][19][20][21][22][23][24][25][26][27][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][30][31][32][33][34][35][36][37][38][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 where B i : ðC½a, bÞ 3 ⟶ ℝ and i = 1, 2, 3 are continuous functions linearly independent and verifying the generalized impulsive conditions Δy 1 t j À Á = H 1j t j , y 1 t j À Á , y 2 t j À Á , y 3 t j À Á À Á , Δy 2 t j À Á = H 2j t j , y 1 t j À Á , y 2 t j À Á , y 3 t j À Á À Á , where H ij : ½a, b × ℝ 3 ⟶ ℝ is continuous functions for i = 1, 2, 3,j = 1, 2, ⋯, n, with Δy i ðt j Þ = y i ðt + j Þ − y i ðt − j Þ, and t j fixed points such that a ≔ t 0 < t 1 < t 2 < ⋯<t n < t n+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 functions Our 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: Section 2 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.
Next definition will be used in the main theorem:

Main Result
The main result will provide the existence of, at least, a solution for the problems (1)-(3).
As g i is a L 1 -Carathéodory function, H ij and the truncatures δ i , δ * i are continuous; therefore, T i are well defined and continuous. Therefore, T is well defined and continuous.
Consider a bounded set D ⊂ X 3 : So, there is k > 0 such that kðx, y, zÞk X 3 < k, for ðx, y, zÞ ∈ D: T i D is uniformly bounded, as, for i = 1, 2, 3, where ψ i,k is the positive function given by Definition 1.
T i D is equicontinuous because, for i = 1, 2, 3, and t 1 , t 2 ∈ a, b with t 1 < t 2 (without loss of generality), as t 1 ⟶ t 2 : T i D is equiconvergent on the impulsive moments, asjT i ðy 1 , y 2 , y 3 ÞðtÞ − lim t⟶t + j T i ðy 1 , y 2 , y 3 ÞðtÞj ≤ j∑ j:t j <t jH ij ðt j , when t ⟶ t + j : Therefore, T i and T are compact operators. Consider now the closed, bounded, and convex set Ω ⊂ X 3 , defined by with R > 0 such that Journal of Function Spaces By the above calculus, T Ω⊂Ω, and from Schauders' fixed point theorem, T has a fixed point y * = ðy * 1 , y * 2 , y * 3 Þ, which is solution of the problems (21), (24), and (26).

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.

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.
Motivated by the papers above, we apply our technique to a vital dynamic SIRS system composed by the differential equations for t in a normalized interval ½0, 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.
Applying an adequate mathematical software, these inequalities can be illustrated by the graph of the correspondent solution, given in Figure 3, considering the population in percentage in the first 100 days.

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 Theorem 3.
Moreover, in Section 4, 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.

Data Availability
No data were used to support the study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.