On the stability analysis of solutions of an integral equation with an application in epidemiology

Volume 7, Issue 1, pp 16--25 http://dx.doi.org/10.22436/mns.07.01.02

Publication Date: June 27, 2021 Submission Date: February 03, 2021 Revision Date: May 25, 2021 Accteptance Date: May 27, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 904 Downloads
• 2226 Views

Authors

Ümit Çakan - Department of Mathematics, İnönü University, Malatya, Turkey.

Abstract

This paper concerns a nonlinear integral equation modeling the spread of epidemics in which immunity does not occur after recovery. The model is mainly based on the return of some of the individuals who have been exposed to the pathogen and who have completed the incubation period, into the susceptible class. We first prove the uniqueness of the global solution of the model with the given initial conditions. After determining the positively invariant region for the model, using LaSalle invariance principle [J. P. LaSalle, IRE Trans. CT, \({\bf 7} (1960)\), 520--527] and the concept of persistence we present some results about the stability analysis of the solutions according to the case of the reproduction number \(\mathcal{R}_{0}\) which is a vital threshold in the spread of diseases.

Share and Cite

Keywords

• Global stability analysis
• Lyapunov function
• LaSalle invariance principle
• mathematical epidemiology
• persistence

MSC

•  34D05
•  34D08
•  34D20
•  92B05
•  92D25
•  92D30

References

• [1] O. Adebimpe, K. A. Bashiru, T. A. Ojurongbe, Stability analysis of an SIR epidemic with non-linear incidence rate and treatment, Open J. Model. Simul., 3 (2015), 104--110
  • View Article


• [2] E. Balcı, İ. Öztürk, S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos Solitons Fractals, 123 (2019), 43--51
  • View Article


• [3] M. V. Barbarossa, M. Polner, G. Röst, Stability switches induced by immune system boosting in an SIRS model with discrete and distributed delays, SIAM J. Appl. Math., 77 (2017), 905--923
  • View Article


• [4] E. Beretta , Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995), 250--260
  • View Article


• [5] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107--4115
  • View Article


• [6] F. Brauer, C. Castillo-Chavez, Mathematical models in population biology and epidemiology: second edition, Springer, New York (2012)
  • View Article


• [7] H. Cao, Y. Zhou, B. Song, https://www.hindawi.com/journals/ddns/2011/653937/, Discrete Dyn. Nat. Soc., 2011 (2011), 21 pages
  • View Article


• [8] E. B. Cox, M. A. Woodbury, L. E. Myers, A new model for tumor growth analysis based on a postulated inhibitory substance, Comp. Biomed. Res., 13 (1980), 437--445
  • View Article


• [9] W. E. Diewert, K. Spremann, F. Stehling, Mathematical modelling in economics: Essays in Honor of Wolfgang Eichhorn, Springer, Berlin (1993)
  • View Article


• [10] P. V. D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29--48
  • View Article


• [11] N. Farajzadeh Tehrani, M. R. Razvan, S. Yasaman, Global analysis of a delay SVEIR epidemiological model, Iran. J. Sci. Technol. Trans. A Sci., 37 (2013), 483--489
  • View Article


• [12] A. D. Freed, K. Diethelm, Y. Luchko, Fractional-order viscoelasticity (FOV): Constitutive developments using the fractional calculus: First annual report, Technical Memorandum, TM-2002-211914, NASA Glenn Research Center, Cleveland (2002)
  • View Article


• [13] M. Golgeli, A mathematical model of Hepatitis B transmission in Turkey, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (2019), 1586--1595
  • View Article


• [14] G. Gripenberg, On some epidemic models, Quart. Appl. Math., 39 (1981), 317--327
  • View Article


• [15] H. W. Hethcote, D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37--47
  • View Article


• [16] S. Hu, M. Khavanin, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal., 34 (1989), 261--266
  • View Article


• [17] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Part I Proc. Roy. Soc. A, 115 (1927), 700--721
  • View Article


• [18] I. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus, 133 (2018), 11 pages
  • View Article


• [19] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75--83
  • View Article


• [20] Y. Kuang, Delay differential equations: with applications in population dynamics, Academic Press, Boston (1993)
  • View Article


• [21] J. P. LaSalle, Some extensions of Liapunov’s second method, IRE Trans. CT, 7 (1960), 520--527
  • View Article


• [22] Z. Lu, X. Chi, L. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math. Comput. Modelling, 36 (2002), 1039--1057
  • View Article


• [23] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141--1145
  • View Article


• [24] W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581--591
  • View Article


• [25] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55--59
  • View Article


• [26] M. M. Ojo, F. O. Akinpelu, Lyapunov functions and global properties of SEIR epidemic model, IJCMP, 1 (2017), 11--16
  • View Article


• [27] S. Side, Y. M. Rangkuti, D. G. Pane, M. S. Sinaga, Stability analysis susceptible, exposed, infected, recovered (SEIR) model for spread model for spread of dengue fever in medan, J. Phys.: Conf. Ser., 954 (2018), 11 pages
  • View Article


• [28] H. L. Smith, H. R. Thieme, Dynamical systems and population persistence, American Mathematical Society,, 118 (2011), 10 pages
  • View Article


• [29] P. J. Torvik, R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294--298
  • View Article


• [30] B. Yang, Stochastic dynamics of an SEIS epidemic model, Adv. Difference Equ., 226 (2016), 11 pages
  • View Article