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Year 2021, Volume: 4 Issue: 2, 110 - 123, 01.06.2021

Ümit Çakan

https://doi.org/10.33401/fujma.863224

Abstract

References

  • [1] F. Evirgen, S. Uçar, N. Özdemir, System analysis of HIV infection model with CD4+T under non-singular kernel derivative, Appl. Math. Nonlinear Sci., 5(1) (2020), 139-146, https://doi.org/10.2478/amns.2020.1.00013.
  • [2] E. Uçar, N. Özdemir, A fractional model of cancer-immune system with Caputo and Caputo–Fabrizio derivatives, Eur. Phys. J. Plus, 136(43) (2021), 17 pages, https://doi.org/10.1140/epjp/s13360-020-00966-9.
  • [3] S. Uçar, N. Özdemir, İ. Koca, E. Altun, Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative, Eur. Phys. J. Plus, 135, (414) (2020), 18 pages, https://doi.org/10.1140/epjp/s13360-020-00420-w.
  • [4] P.A. Naik, K.M. Owolabi, M. Yavuz, J. Zu, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, Chaos Solitons & Fractals, 140 (2020) 110272, 13 pages, https://doi.org/10.1016/j.chaos.2020.110272.
  • [5] M. Yavuz, E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393, https://doi.org/10.1016/j.physa.2019.03.069.
  • [6] M. Yavuz, N. O¨ zdemir , Analysis of an epidemic spreading model with exponential decay law, Math. Sci. Appl. E-Notes, 8(1) (2020), 142-154, https://doi.org/10.36753/mathenot.691638.
  • [7] W.O. Kermack, A.G. Mc Kendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115(772) (1927), 700-721.
  • [8] T. Kesemen, M. Merdan, Z. Bekiryazıcı, Analysis of the dynamics of the classical epidemic model with beta distributed random components, Ig˘dır U¨ niv. Fen Bil Enst. Der., 10(3) (2020), 1956-1965, DOI: 10.21597/jist.658471.
  • [9] M. Merdan, Z. Bekiryazici, T. Kesemen, T. Khaniyev, Deterministic stability and random behavior of a Hepatitis C model, PLoS ONE, 12(7) (2017), e0181571, 17 pages, https://doi.org/10.1371/journal.pone.0181571.
  • [10] İ. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus, 133(100) (2018), 11 pages, https://doi.org/10.1140/epjp/i2018-11949-4.
  • [11] J. Jia, S. Han, On the analysis of a class of SIR model with impulsive effect and vertical infection, Math. Practice Theory, 37(24) (2007), 96-101.
  • [12] J. Jia, Q. Li, Qualitative analysis of an SIR epidemic model with stage structure, Appl. Math. Comput., 193 (2007), 106-115.
  • [13] 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.
  • [14] W. Zhao, T. Zhang, Z. Chang, X. Meng, Y. Liu, Dynamical analysis of SIR epidemic models with distributed delay, J. Appl. Math., 2013 (2013), 15 pages, https://doi.org/10.1155/2013/154387.
  • [15] A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control, 15(3) (2010), 299-306.
  • [16] S.A. Al-Sheikh, Modeling and analysis of an SEIR epidemic model with a limited resource for treatment, Glob. J. Sci. Front. Res. Math. Decis. Sci., 12(14) (2012), 57-66.
  • [17] N. Yi, Q. Zhang, K. Mao, D. Yang, Q. Li, Analysis and control of an SEIR epidemic system with nonlinear transmission rate, Math. Comput. Modelling, 50 (2009), 1498-513.
  • [18] J. Zhang, J. Li, Z. Ma, Global dynamics of an SEIR epidemic model with immigration of different compartment, Acta Math. Sci. Ser. B, 26(3) (2006), 551-567.
  • [19] K. Cooke, P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996) 240-260.
  • [20] M. De la. Sen, S. Alonso-Quesada, A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953-976.
  • [21] X. Zhou, J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4438-4450.
  • [22] H. Shu, D.Fan, J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real World Appl., 13(4) (2012), 1581-1592.
  • [23] M.A. Safi, A.B. Gumel, Global asymptotic dynamics of a model for quarantine and isolation, Discrete Contin. Dyn. Syst. Ser. B., 14(1) (2010), 209-231.
  • [24] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.
  • [25] J.M. Drazen, R. Kanapathipillai, E.W. Campion, E.J. Rubin, S.M. Hammer, S. Morrissey, L.R. Baden, Ebola and quarantine, N. Engl. J. Med., 371 (2014), 2029-2030.
  • [26] Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Modelling, 47(1-2) (2008), 235-245.
  • [27] A. D´enes, A.B. Gumel, Modeling the impact of quarantine during an outbreak of Ebola virus disease, Infect. Dis. Model., 4 (2019), 12-27.
  • [28] H.B. Fredj, F. Cherif, Novel corona virus disease infection in Tunisia: Mathematical model and the impact of the quarantine strategy, Chaos Solitons & Fractals, 138 (2020), 109969, 10 pages, https://doi.org/10.1016/j.chaos.2020.109969.
  • [29] C. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17(3) (2020), 2708-2724.
  • [30] A. Atangana, S. İ. Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: Theory, methods and applications, medRxiv DOI: 10.1101/2020.05.08.20095588.
  • [31] Md. S. Islam , J.I. Ira, K.M.A. Kabir, Md. Kamrujjaman, COVID-19 Epidemic compartments model and Bangladesh. Preprints (www.preprints.org), Posted: 12 April 2020 doi:10.20944/preprints202004.0193.v1, 2020.
  • [32] S. Djilali, B. Ghanbari, Coronavirus pandemic: A predictive analysis of the peak outbreak epidemic in South Africa, Turkey and Brazil, Chaos Solitons & Fractals, 138 (2020), 9 pages, 109971, https://doi.org/10.1016/j.chaos.2020.109971.
  • [33] P.A. Naik, M. Yavuz, S. Qureshi, J. Zu, S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, Eur. Phys. J. Plus, 135(795) (2020), 42 pages, https://doi.org/10.1140/epjp/s13360-020-00819-5.
  • [34] A. Raza, A. Ahmadian, M. Rafiq, S. Salahshour, M. Ferrara, An analysis of a nonlinear susceptible-exposed-infected-quarantine-recovered pandemic model of a novel coronavirus with delay effect, Results Phys., 21 (2021), 7 pages, 103771, https://doi.org/10.1016/j.rinp.2020.103771.
  • [35] N. Sene, Analysis of the stochastic model for predicting the novel coronavirus disease, Adv. Differ. Equ., 568 (2020), 19 pages, https://doi.org/10.1186/s13662-020-03025-w.
  • [36] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley and Sons 2000.
  • [37] J.P. LaSalle, Stability of non autonomous systems, Nonlinear Anal., 1(1) (1976), 83-91.
  • [38] https://www.tga.gov.tr/fight-against-covid-19-in-turkey/, Date of Available: 10.01.2021

Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy

Year 2021, Volume: 4 Issue: 2, 110 - 123, 01.06.2021

Ümit Çakan

https://doi.org/10.33401/fujma.863224

Abstract

In this study, using a system of delay nonlinear ordinary differential equations, we introduce a new compartmental epidemic model considered the effect of filiation (contamination) control strategy to the spread of Covid-19. Firstly, the formulation of this new $SI_{u}I_{a}QR$ epidemic model with delay process and the parameters arised from isolation and filiation is formed. Then the disease-free and endemic equilibrium points of the model is obtained. Also, the basic reproduction number $\mathcal{R}_{0}$ is found by using the next-generation matrix method, and the results on stabilities of the disease-free and endemic equilibrium points are investigated. Finally some examples are presented to show the effect of filiation control strategy.

Keywords

Basic reproduction number COVID-19 Filiation control strategy Lyapunov Function Mathematical epidemiology Quarantine Stability Analysis

References

  • [1] F. Evirgen, S. Uçar, N. Özdemir, System analysis of HIV infection model with CD4+T under non-singular kernel derivative, Appl. Math. Nonlinear Sci., 5(1) (2020), 139-146, https://doi.org/10.2478/amns.2020.1.00013.
  • [2] E. Uçar, N. Özdemir, A fractional model of cancer-immune system with Caputo and Caputo–Fabrizio derivatives, Eur. Phys. J. Plus, 136(43) (2021), 17 pages, https://doi.org/10.1140/epjp/s13360-020-00966-9.
  • [3] S. Uçar, N. Özdemir, İ. Koca, E. Altun, Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative, Eur. Phys. J. Plus, 135, (414) (2020), 18 pages, https://doi.org/10.1140/epjp/s13360-020-00420-w.
  • [4] P.A. Naik, K.M. Owolabi, M. Yavuz, J. Zu, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, Chaos Solitons & Fractals, 140 (2020) 110272, 13 pages, https://doi.org/10.1016/j.chaos.2020.110272.
  • [5] M. Yavuz, E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393, https://doi.org/10.1016/j.physa.2019.03.069.
  • [6] M. Yavuz, N. O¨ zdemir , Analysis of an epidemic spreading model with exponential decay law, Math. Sci. Appl. E-Notes, 8(1) (2020), 142-154, https://doi.org/10.36753/mathenot.691638.
  • [7] W.O. Kermack, A.G. Mc Kendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115(772) (1927), 700-721.
  • [8] T. Kesemen, M. Merdan, Z. Bekiryazıcı, Analysis of the dynamics of the classical epidemic model with beta distributed random components, Ig˘dır U¨ niv. Fen Bil Enst. Der., 10(3) (2020), 1956-1965, DOI: 10.21597/jist.658471.
  • [9] M. Merdan, Z. Bekiryazici, T. Kesemen, T. Khaniyev, Deterministic stability and random behavior of a Hepatitis C model, PLoS ONE, 12(7) (2017), e0181571, 17 pages, https://doi.org/10.1371/journal.pone.0181571.
  • [10] İ. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus, 133(100) (2018), 11 pages, https://doi.org/10.1140/epjp/i2018-11949-4.
  • [11] J. Jia, S. Han, On the analysis of a class of SIR model with impulsive effect and vertical infection, Math. Practice Theory, 37(24) (2007), 96-101.
  • [12] J. Jia, Q. Li, Qualitative analysis of an SIR epidemic model with stage structure, Appl. Math. Comput., 193 (2007), 106-115.
  • [13] 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.
  • [14] W. Zhao, T. Zhang, Z. Chang, X. Meng, Y. Liu, Dynamical analysis of SIR epidemic models with distributed delay, J. Appl. Math., 2013 (2013), 15 pages, https://doi.org/10.1155/2013/154387.
  • [15] A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control, 15(3) (2010), 299-306.
  • [16] S.A. Al-Sheikh, Modeling and analysis of an SEIR epidemic model with a limited resource for treatment, Glob. J. Sci. Front. Res. Math. Decis. Sci., 12(14) (2012), 57-66.
  • [17] N. Yi, Q. Zhang, K. Mao, D. Yang, Q. Li, Analysis and control of an SEIR epidemic system with nonlinear transmission rate, Math. Comput. Modelling, 50 (2009), 1498-513.
  • [18] J. Zhang, J. Li, Z. Ma, Global dynamics of an SEIR epidemic model with immigration of different compartment, Acta Math. Sci. Ser. B, 26(3) (2006), 551-567.
  • [19] K. Cooke, P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996) 240-260.
  • [20] M. De la. Sen, S. Alonso-Quesada, A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953-976.
  • [21] X. Zhou, J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4438-4450.
  • [22] H. Shu, D.Fan, J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real World Appl., 13(4) (2012), 1581-1592.
  • [23] M.A. Safi, A.B. Gumel, Global asymptotic dynamics of a model for quarantine and isolation, Discrete Contin. Dyn. Syst. Ser. B., 14(1) (2010), 209-231.
  • [24] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.
  • [25] J.M. Drazen, R. Kanapathipillai, E.W. Campion, E.J. Rubin, S.M. Hammer, S. Morrissey, L.R. Baden, Ebola and quarantine, N. Engl. J. Med., 371 (2014), 2029-2030.
  • [26] Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Modelling, 47(1-2) (2008), 235-245.
  • [27] A. D´enes, A.B. Gumel, Modeling the impact of quarantine during an outbreak of Ebola virus disease, Infect. Dis. Model., 4 (2019), 12-27.
  • [28] H.B. Fredj, F. Cherif, Novel corona virus disease infection in Tunisia: Mathematical model and the impact of the quarantine strategy, Chaos Solitons & Fractals, 138 (2020), 109969, 10 pages, https://doi.org/10.1016/j.chaos.2020.109969.
  • [29] C. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17(3) (2020), 2708-2724.
  • [30] A. Atangana, S. İ. Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: Theory, methods and applications, medRxiv DOI: 10.1101/2020.05.08.20095588.
  • [31] Md. S. Islam , J.I. Ira, K.M.A. Kabir, Md. Kamrujjaman, COVID-19 Epidemic compartments model and Bangladesh. Preprints (www.preprints.org), Posted: 12 April 2020 doi:10.20944/preprints202004.0193.v1, 2020.
  • [32] S. Djilali, B. Ghanbari, Coronavirus pandemic: A predictive analysis of the peak outbreak epidemic in South Africa, Turkey and Brazil, Chaos Solitons & Fractals, 138 (2020), 9 pages, 109971, https://doi.org/10.1016/j.chaos.2020.109971.
  • [33] P.A. Naik, M. Yavuz, S. Qureshi, J. Zu, S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, Eur. Phys. J. Plus, 135(795) (2020), 42 pages, https://doi.org/10.1140/epjp/s13360-020-00819-5.
  • [34] A. Raza, A. Ahmadian, M. Rafiq, S. Salahshour, M. Ferrara, An analysis of a nonlinear susceptible-exposed-infected-quarantine-recovered pandemic model of a novel coronavirus with delay effect, Results Phys., 21 (2021), 7 pages, 103771, https://doi.org/10.1016/j.rinp.2020.103771.
  • [35] N. Sene, Analysis of the stochastic model for predicting the novel coronavirus disease, Adv. Differ. Equ., 568 (2020), 19 pages, https://doi.org/10.1186/s13662-020-03025-w.
  • [36] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley and Sons 2000.
  • [37] J.P. LaSalle, Stability of non autonomous systems, Nonlinear Anal., 1(1) (1976), 83-91.
  • [38] https://www.tga.gov.tr/fight-against-covid-19-in-turkey/, Date of Available: 10.01.2021
There are 38 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ümit Çakan 0000-0002-9766-5863

Publication Date June 1, 2021
Submission Date January 17, 2021
Acceptance Date March 9, 2021
Published in Issue Year 2021 Volume: 4 Issue: 2

Cite

APA Çakan, Ü. (2021). Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy. Fundamental Journal of Mathematics and Applications, 4(2), 110-123. https://doi.org/10.33401/fujma.863224
AMA Çakan Ü. Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy. Fundam. J. Math. Appl. June 2021;4(2):110-123. doi:10.33401/fujma.863224
Chicago Çakan, Ümit. “Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 With the Effect of Contamination Control (Filiation) Strategy”. Fundamental Journal of Mathematics and Applications 4, no. 2 (June 2021): 110-23. https://doi.org/10.33401/fujma.863224.
EndNote Çakan Ü (June 1, 2021) Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy. Fundamental Journal of Mathematics and Applications 4 2 110–123.
IEEE Ü. Çakan, “Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy”, Fundam. J. Math. Appl., vol. 4, no. 2, pp. 110–123, 2021, doi: 10.33401/fujma.863224.
ISNAD Çakan, Ümit. “Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 With the Effect of Contamination Control (Filiation) Strategy”. Fundamental Journal of Mathematics and Applications 4/2 (June 2021), 110-123. https://doi.org/10.33401/fujma.863224.
JAMA Çakan Ü. Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy. Fundam. J. Math. Appl. 2021;4:110–123.
MLA Çakan, Ümit. “Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 With the Effect of Contamination Control (Filiation) Strategy”. Fundamental Journal of Mathematics and Applications, vol. 4, no. 2, 2021, pp. 110-23, doi:10.33401/fujma.863224.
Vancouver Çakan Ü. Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy. Fundam. J. Math. Appl. 2021;4(2):110-23.
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