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Received: 6 January 2019 / Revised: 13 February 2019 / Accepted: 13 February 2019 / Published: 16 February 2019

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder's fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.

1. Introduction

In this paper two different problems are analyzed.

Part one is concerned with the study of a fully nonlinear coupled system of equations

f i: [a,b]× R 3→R and i=1,2,3 are L 1-Carathéodory functions, subject to the nonlinear functional boundary conditions

where L i: C[a,b]3× R 5→R, i=1,2,3, are continuous functions with properties later to be defined. The technique used for the functional problem is based on the Arzèla-Ascoli theorem and Schauder's fixed point theorem.

Part two, more precisely, Section 4, deals with the fully nonlinear couple system of Equation ( 1) coupled with the periodic boundary conditions

Given that the conditions on L i, do not allow the problem ( 1)-( 2) to cover the periodic case, a different approach for the problem ( 1)-( 3) is required. In this case, in order to obtain the existence and location of periodic solutions, the upper and lower solutions method, along with some adequate local monotone assumptions on the nonlinearities, is used.

Mathematical modelling and applications are becoming increasingly popular nowadays. With the sudden outburst of keywords such as big data, data analytics and modelling, the quest for mathematical models is on high demand. In the area of mathematical modelling, systems of differential equations are a must, due to their high applicability in areas such as population dynamics [ 1, 2, 3], finance [ 4], medicine [ 5], biotechnology [ 6] and physics [ 7, 8], and also examples treated in [ 9, 10].

Nevertheless, in the literature available, the cases dealing with coupled systems of equations are not abundant. Such systems can be found in [ 11, 12, 13], however, in this paper the authors present a problem where both equations are coupled. In addition to that, to the best of our knowledge, it is the first time where coupled systems are considered with coupled functional boundary conditions.

This feature allows to generalize the classical boundary data in the literature, such as two-point or multi-point, nonlinear, nonlocal, integro-differential conditions, among others. Indeed, the functional part can deal with global boundary assumptions, such as minimum or maximum arguments, infinite multi-point data, and integral conditions on the several unknown functions. Functional problems, along with their features, can be seen in [ 14, 15, 16, 17, 18, 19] and the references therein.

The methods and techniques applied in this paper can be easily adapted to coupled systems with n equations and variables. However, as the notation and writing appear to be heavy, and may avoid the clarity of the results, we prefer to prove our theoretical part for which is adequate for our application.

This paper has the following structure: Section 2 contains some definitions and generic assumptions on the nonlinearities. Section 3 shows the main result for problems with functional boundary conditions. In Section 4 it is studied the periodic problem via lower and upper solutions technique together with some local growth conditions. The final section presents an application of ( 1)-( 2) to an epidemic SIRS model to illustrate the applicability of the problem discussed and to show the potentialities of the functional boundary conditions, exploring global initial boundary conditions on the system.

2. Definitions and Assumptions

Throughout this work we consider the space of continuous functions in [a,b], on the Banach space E:= C[a,b]3, equipped with the norm

where u i= max t∈ u i t .

The functional boundary functions verify the assumption:

Hypothesis 1.

The admissible nonlinearities will be L 1-Carathéodory functions, according the following definition:

Definition 1.

To demonstrate the final result, Schauder's fixed point theorem will be an important tool to guarantee the existence of fixed points for the operator to be defined:

Theorem 1.

3. Main Result for Functional Problems

In this section, we present and prove the main existence result for ( 1)-( 2), given by the following theorem:

Theorem 2.

Hypothesis 2.

with

Proof.

and the operator

defined by

Consider

with k>0 such that

Claim 2.

Remark that (Hypothesis 2) implies that the periodic case is not covered by ( 2).

4. Existence and Localization Result for the Periodic Case

Consider now the system ( 1) with the periodic boundary conditions ( 3). As it is well known, in nonlinear differential equations, the periodic case is more delicate and requires a different approach for general nonlinearities.

The method to be used will apply lower and upper solutions technique, based on the definition:

Definition 2.

This method allows to obtain an existence and localization theorem:

Theorem 3.

Proof.

for

Then the operator

given by

is completely continuous in .

□

5. An Epidemic Model of an SIRS System With Nonlinear Incidence Rate and Interaction from Infectious to Susceptible Subjects

The existent literature has innumerous examples of applications of SIR models, namely in [ 21, 22, 23] where the population is divided into Susceptible S , Infectious I and Recovery R . However, in SIR models, recovered individuals are assumed to develop lifelong immunity, which for some diseases such as seasonal flu, influenza or venereal diseases is not necessarily true. A recovered individual becomes susceptible and possibly infected after some time. In this case, SIRS models, where recovered individuals lose immunity and become susceptible again, are far more adequate. Examples can be found in [ 24, 25, 26, 27, 28, 29]. In [ 30], the authors develop and explore a mathematical model of an SIRS epidemic model where a transfer between infectious and susceptible rate is included, as shown in Figure 1.

This approach was designed to model the cases where recovery cannot generate immunity for a long time. Infected individuals may recover after some treatments and go back directly to the susceptible class. In addition, a nonlinear incidence with the average number of new cases of a disease per unit time, S g(I), is included, as suggested in [ 31]. This nonlinear incidence replicates a more complex dynamic than the ones presented on bilinear or standard incidence models, and it seems to perform better when modelling more complex cholera cases, as shown in [ 31].

The model presented is then

where:

Λ represents the recruitment rate of susceptible individuals;

μ is the natural death rate;

γ 1 is the transfer rate from the infected class to the susceptible class;

γ 2 is the transfer rate from the infected class to the recovered class;

α is the disease-induced death rate;

δ the immunity loss rate.

In this model Λ and μ are assumed to be positive and δ, γ 1, and α are assumed to be nonnegative. As per the nonlinear incidence, S g(I), g is a real locally Lipschitz function on R+ with the following conditions, as presented in [ 32]:

g 0 =0 and g I >0 for I>0,

g I /I is continuous and monotonously increasing for I>0 and lim I→ 0+ g(I)I exists as β>0.

Unlike the previous models presented, in this paper we couple the system ( 12) with the boundary conditions, for t∈ 0,T and

These functional boundary conditions fully reflect the considerations and conclusions presented in [ 30]. These types of boundary conditions have the following practical meaning:

S(0), the initial number of susceptible subjects, is equal to the maximum of the infected subjects;

I 0 , the initial number of infected subjects, is a weighted average of the susceptible individuals, weighted by the final value of the susceptible S, at time T;

R 0 , the initial number of individuals who recovered, is equal to a weighted average of the infected individuals, weighted by the final value of the susceptible, S, at time T.

Functional boundary conditions allow these assumptions to be examined in full, as operators can be considered as boundary conditions. These types of global conditions, that can include somewhat more abstract conditions, can only be contemplated via functional boundary conditions.

This model can therefore be presented in the form of ( 1)-( 2), where u 1=S, u 2=I, u 3=R, and b=T,

It is clear that the boundary conditions L 1, and satisfy (H1).

Moreover, , and are L 1-Carathéodory functions such that, for max u 1, u 2, u 3<k,

To satisfy (Hypothesis 2), one must verify that there exists , such that,

For , condition (Hypothesis 2) is verified and therefore, by Theorem 2, the system ( 12)-( 13) has at least one solution S,I,R∈ C 0,1 3, for the values considered.

6. Conclusions

In this paper, the authors show the existence of solution for a first order fully coupled system of three equations, involving two different cases. The first case, with coupled functional boundary conditions, is an existence result. The second case, with periodic boundary conditions, which is not covered by the first result, is an existence and location result. The extra information obtained in this second case is associated with the technique used, as it relies on the upper and lower solution method.

The application to an SIRS model, with global boundary conditions, underlines the key advantage and flexibility of the functional boundary conditions. The example shown illustrates not only the theorem proved in this paper, but it also provides guidance on how to arrange global conditions, in order to conform with the layout in ( 2).

As a matter of fact, a similar approach can be taken in several other models, allowing global conditions to be considered as boundary conditions, highly increasing the level of applicability of these models.

Author Contributions

The authors equally contributed to this work.

Funding

This project was supported by Fundação para a Ciência e a Tecnologia (FCT) via project UID/MAT/04674/2019.

Conflicts of Interest

The authors declare no conflict of interest.

References

1. Deuflhard, P. Differential equations in technology and medicine: Computational concepts, adaptive algorithms, and virtual labs. In Computational Mathematics Driven by Industrial Problems; Springer: Berlin/Heidelberg, Germany, 2000; pp. 69-125. [Google Scholar]

2. Fialho, J.; Minhós, F. Multiplicity and location results for second order functional boundary value problems. Dyn. Syst. Appl. 2014, 23, 453-464. [Google Scholar]

3. Graef, J.; Kong, L.; Minhós, F. Higher order boundary value problems with ϕ -Laplacian and functional boundary conditions. Comput. Math. Appl. 2011, 61, 236-249. [Google Scholar] [CrossRef]

4. Zeidler, E. Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems; Springer: New York, NY, USA, 1986. [Google Scholar]

Figure 1. SIRS model diagram.

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