# First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Assumptions

**Hypothesis**

**1.**

**Definition**

**1.**

- (i)
- for each$\left({y}_{1},{y}_{2},{y}_{3}\right)\in {\mathbb{R}}^{3}$,$t\mapsto {f}_{i}(t,{y}_{1},{y}_{2},{y}_{3})$are measurable on$[a,b],$for$i=1,2,3;$
- (ii)
- for almost every$t\in [a,b],$$\left({y}_{1},{y}_{2},{y}_{3}\right)\mapsto {f}_{i}(t,{y}_{1},{y}_{2},{y}_{3})$are continuous on${\mathbb{R}}^{3},$for$i=1,2,3;$
- (iii)
- for each$L>0$, there exists a positive function${\psi}_{iL}\in {L}^{1}\left[a,b\right],$$i=1,2,3,$such that, for$max\left\{\u2225{y}_{i}\u2225,i=1,2,3\right\}<L$,$$\left|{f}_{i}(t,{y}_{1}\left(t\right),{y}_{2}\left(t\right),{y}_{3}\left(t\right))\right|\le {\psi}_{iL}\left(t\right),\phantom{\rule{4.pt}{0ex}}a.e.\phantom{\rule{4.pt}{0ex}}t\in [a,b],\phantom{\rule{4.pt}{0ex}}i=1,2,3.$$

**Theorem**

**1.**

## 3. Main Result for Functional Problems

**Theorem**

**2.**

**Hypothesis**

**2.**

**Proof.**

**Claim**

**1.**

**Claim**

**2.**

## 4. Existence and Localization Result for the Periodic Case

**Definition**

**2.**

**Theorem**

**3.**

**Proof.**

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

- $\mathsf{\Lambda}$ represents the recruitment rate of susceptible individuals;
- $\mu $ is the natural death rate;
- ${\gamma}_{1}$ is the transfer rate from the infected class to the susceptible class;
- ${\gamma}_{2}$ is the transfer rate from the infected class to the recovered class;
- $\alpha $ is the disease-induced death rate;
- $\delta $ the immunity loss rate.

- $g\left(0\right)=0$ and $g\left(I\right)>0$ for $I>0,$
- $g\left(I\right)/I$ is continuous and monotonously increasing for $I>0$ and ${lim}_{I\to {0}^{+}}\frac{g\left(I\right)}{I}$ exists as $\beta >0.$

- $S\left(0\right)$, the initial number of susceptible subjects, is equal to the maximum of the infected subjects;
- $I\left(0\right),$ 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 $T;$
- $R\left(0\right),$ 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.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Al-Moqbali, M.K.A.; Al-Salti, N.S.; Elmojtaba, I.M. Prey–Predator Models with Variable Carrying Capacity. Mathematics
**2018**, 6, 102. [Google Scholar] [CrossRef] - Gumus, O.A.; Kose, H. On the Stability of Delay Population Dynamics Related with Allee Effects. Math. Comput. Appl.
**2012**, 17, 56–67. [Google Scholar] [CrossRef][Green Version] - Song, H.S.; Cannon, W.R.; Beliaev, A.S.; Konopka, A. Mathematical Modeling of Microbial Community Dynamics: A Methodological Review. Processes
**2014**, 2, 711–752. [Google Scholar] [CrossRef][Green Version] - Osakwe, C.J.U. Incentive Compatible Decision Making: Real Options with Adverse Incentives. Axioms
**2018**, 7, 9. [Google Scholar] [CrossRef] - 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]
- Jang, S.S.; de la Hoz, H.; Ben-zvi, A.; McCaffrey, W.C.; Gopaluni, R.B. Parameter estimation in models with hidden variables: An application to a biotech process. Can. J. Chem. Eng.
**2012**, 90, 690–702. [Google Scholar] [CrossRef] - Akarsu, M.; Özbaş, Ö. Monte Carlo Simulation for Electron Dynamics in Semiconductor Devices. Math. Comput. Appl.
**2005**, 10, 19–26. [Google Scholar] [CrossRef][Green Version] - Malinzi, J.; Quaye, P.A. Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method. Math. Comput. Appl.
**2018**, 23, 35. [Google Scholar] [CrossRef] - Nieto, J.J. Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal.
**2002**, 51, 1223–1232. [Google Scholar] [CrossRef] - Zhang, W.; Fan, M. Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Math. Comput. Model.
**2004**, 39, 479–493. [Google Scholar] [CrossRef] - Agarwal, R.P.; O’Reagan, D. A coupled system of boundary value problems. Appl. Anal.
**1998**, 69, 381–385. [Google Scholar] [CrossRef] - Asif, N.A.; Talib, I.; Tunc, C. Existence of solutions for first-order coupled system with nonlinear coupled boundary conditions. Bound. Val. Prob.
**2015**, 2015, 134. [Google Scholar] [CrossRef] - Asif, N.A.; Khan, R.A. Positive solutions to singular system with four-point coupled boundary conditions. J. Math. Anal. Appl.
**2012**, 386, 848–861. [Google Scholar] [CrossRef][Green Version] - Cabada, A.; Fialho, J.; Minhós, F. Extremal solutions to fourth order discontinuous functional boundary value problems. Math. Nachr.
**2013**, 286, 1744–1751. [Google Scholar] [CrossRef][Green Version] - Cabada, A.; Pouso, R.; Minhós, F. Extremal solutions to fourth-order functional boundary value problems including multipoint condition. Nonlinear Anal. Real World Appl.
**2009**, 10, 2157–2170. [Google Scholar] [CrossRef] - Fialho, J.; Minhós, F. Higher order functional boundary value problems without monotone assumptions. Bound. Val. Prob.
**2013**, 2013, 81. [Google Scholar] [CrossRef][Green Version] - Fialho, J.; Minhós, F. Multiplicity and location results for second order functional boundary value problems. Dyn. Syst. Appl.
**2014**, 23, 453–464. [Google Scholar] - 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] - Graef, J.; Kong, L.; Minhós, F.; Fialho, J. On the lower and upper solution method for higher order functional boundary value problems. Appl. Anal. Discret. Math.
**2011**, 5, 133–146. [Google Scholar] [CrossRef] - Zeidler, E. Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems; Springer: New York, NY, USA, 1986. [Google Scholar]
- Angstmann, C.N.; Henry, B.I.; McGann, A.V. A Fractional-Order Infectivity and Recovery SIR Model. Fract. Fract.
**2017**, 1, 11. [Google Scholar] [CrossRef] - Cui, Q.; Qiu, Z.; Liu, W.; Hu, Z. Complex Dynamics of an SIR Epidemic Model with Nonlinear Saturate Incidence and Recovery Rate. Entropy
**2017**, 19, 305. [Google Scholar] [CrossRef] - Secer, A.; Ozdemir, N.; Bayram, M. A Hermite Polynomial Approach for Solving the SIR Model of Epidemics. Mathematics
**2018**, 6, 305. [Google Scholar] [CrossRef] - Alexander, M.E.; Moghadas, S.M. Bifurcation analysis of an SIRS epidemic model with generalized incidence. SIAM J. Appl. Math.
**2005**, 65, 1794–1816. [Google Scholar] [CrossRef] - Chen, J. An SIRS epidemic model. Appl. Math. J. Chin. Univ.
**2004**, 19, 101–108. [Google Scholar] [CrossRef] - Hu, Z.; Bi, P.; Ma, W.; Ruan, S. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discret. Contin. Dyn. Syst. Ser. B
**2011**, 15, 93–112. [Google Scholar] [CrossRef] - Liu, J.; Zhou, Y. Global stability of an SIRS epidemic model with transport-related infection. Chaos Solitons Fract.
**2009**, 40, 145–158. [Google Scholar] [CrossRef] - Teng, Z.; Liu, Y.; Zhang, L. Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality. Nonlinear Anal. Theory Methods Appl.
**2008**, 69, 2599–2614. [Google Scholar] [CrossRef] - Jin, Y.; Wang, W.; Xiao, S. An SIRS model with a nonlinear incidence rate. Chaos Solitons Fract.
**2007**, 34, 1482–1497. [Google Scholar] [CrossRef] - Li, T.; Zhang, F.; Liu, H.; Chen, Y. Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible. Appl. Math. Lett.
**2017**, 70, 52–57. [Google Scholar] [CrossRef] - Capasso, V.; Serio, G. A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci.
**1978**, 42, 43–61. [Google Scholar] [CrossRef] - Li, J.; Yang, Y.; Xiao, Y.; Liu, S. A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence. J. Appl. Anal. Comput.
**2016**, 6, 38–46. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Fialho, J.; Minhós, F. First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model. *Axioms* **2019**, *8*, 23.
https://doi.org/10.3390/axioms8010023

**AMA Style**

Fialho J, Minhós F. First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model. *Axioms*. 2019; 8(1):23.
https://doi.org/10.3390/axioms8010023

**Chicago/Turabian Style**

Fialho, João, and Feliz Minhós. 2019. "First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model" *Axioms* 8, no. 1: 23.
https://doi.org/10.3390/axioms8010023