Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis

Zafer Öztürk; Halis Bilgil; Sezer Sorgun

doi:10.33187/jmsm.1196961

Araştırma Makalesi

Yıl 2023, Cilt: 6 Sayı: 2, 49 - 55, 07.08.2023

Zafer Öztürk Halis Bilgil Sezer Sorgun

https://doi.org/10.33187/jmsm.1196961

Öz

Kaynakça

[1] J. T. Townsend, D. E. Landon, Mathematical models of recognition and confusion in psychology, Math. Soc. Sci., 4(1) (1983), 25-71.
[2] W. K. Estes, Mathematical models in psychology, A Handbook for Data Analysis in the Behaviorial Sciences: Volume 1: Methodological Issues, Volume 2: Statistical Issues, 3 (2014).
[3] D. Wodarz, M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24(12) (2002), 1178-1187.
[4] I. Podlubny, Fractional Differential Equations, Academy Press, San Diego CA, 1999.
[5] F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical models in epidemiology, Springer, New York, 2019.
[6] H. Bilgil, A. Yousef, A. Erciyes, U¨ . Erdinc¸, Z. O¨ ztu¨rk, A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event, J. Comput. Appl. Math., (2022), 115015.
[7] N. T. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE, 1975.
[8] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180(1-2) (2002), 141-160.
[9] S. Wang, Y. Ding, H. Lu, S. Gong, Stability and bifurcation analysis of SIQR for the COVID-19 epidemic model with time delay, Math. Biosci. Eng., 18(5) (2021), 5505-5524.
[10] B. K. Mishra, N. Jha, SEIQRS model for the transmission of malicious objects in computer network, Appl. Math. Model., 34(3) (2010), 710-715.
[11] X. Liu, T. Yasuhiro, I. Shingo, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253(1) (2008), 1-11.
[12] M. B. Trawicki, Deterministic seirs epidemic model for modeling vital dynamics, vaccinations, and temporary immunity, Mathematics, 5(1) (2017), 7.
[13] Z. Öztürk, S. Sorgun, H. Bilgil, SIQRV Modeli ve Nu¨merik Uygulaması, Avrupa Bilim ve Teknoloji Dergisi, 28 (2021), 573-578.
[14] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, In: Proc. R. Soc. Lond., Series A, Containing Papers of a Mathematical and Physical Character, 115(772) (1927), 700-721.
[15] D. Yaro, S. K. Omari-Sasu, P. Harvim, A. W. Saviour, B. A. Obeng, Generalized Euler method for modeling measles with fractional differ ential equations, Int. J. Innov. Res. Dev., 4 (2015).
[16] Z. O¨ ztu¨rk, S. Sorgun, H. Bilgil, U¨ . Erdinc¸, New exact solutions of conformable time-fractional bad and good modified Boussinesq equations, J. New Theory, 37 (2021), 8-25.
[17] M. Braun, M. Golubitsky, Differential Equations and their Applications, Springer-Verlag, New York, 1983.
[18] https://tuikweb.tuik.gov.tr/UstMenu.do.
[19] P. Kumar, V. S. Erturk, M. Vellappandi, H. Trinh, V. Govindaraj, A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in Caputo sense, Chaos Solit. Fractals, 158 (2022), 112067.
[20] P. Kumar, V. S. Erturk, H. Abboubakar, K. S. Nisar, Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives, Alex. Eng. J., 60(3) (2021), 3189-3204.
[21] P. Kumar, V. Govindaraj, V. S. Erturk, A novel mathematical model to describe the transmission dynamics of tooth cavity in the human population, Chaos Solit. Fractals, 161 (2022), 112370.
[22] M. Vellappandi, P. Kumar, V. Govindaraj, Role of fractional derivatives in the mathematical modeling of the transmission of Chlamydia in the United States from 1989 to 2019, Nonlinear Dyn., (2022), 1-15.
[23] S. Abbas, S. Tyagi, P. Kumar, V. S. Erturk, S. Momani, Stability and bifurcation analysis of a fractional-order model of cell-to-cell spread of HIV-1 with a discrete time delay, Math. Meth. App. Sci., 45(11) (2022), 7081-7095.
[24] V. S. Erturk, E. Godwe, D. Baleanu, P. Kumar, J. Asad, A. Jajarmi, Novel fractional-order Lagrangian to describe motion of beam on nanowire, Acta Phys. Pol. A, 140(3) (2021), 265-272.
[25] P. Kumar, V. S. Erturk, A. Yusuf, S. Kumar, Fractional time-delay mathematical modeling of Oncolytic Virotherapy, Chaos Solit. Fractals, 150 (2021), 111123.
[26] A. Din, F. M. Khan, Z. U. Khan, A. Yusuf, T. Munir, The mathematical study of climate change model under nonlocal fractional derivative, Partial Differential Equations in Applied Mathematics, 5 (2022), 100204.
[27] E. Viera-Martin, J. F. Gomez-Aguilar, J. E. Solis-Perez, J. A. Hernandez-Perez, R. F. Escobar-Jimenez, Artificial neural networks: a practical review of applications involving fractional calculus, Eur. Phys. J. Spec. Top ., (2022), 1-37.
[28] V. S. Erturk, A. Ahmadkhanlu, P. Kumar, V. Govindaraj, Some novel mathematical analysis on a corneal shape model by using Caputo fractional derivative, Optik, 261 (2022), 169086.
[29] V. S. Erturk, A. K. Alomari, P. Kumar, M. Murillo-Arcila, Analytic solution for the strongly nonlinear multi-order fractional version of a BVP occurring in chemical reactor theory, Discrete Dyn. Nat. Soc., 2022 (2022), 8655340.
[30] Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: a review, Fract. Calc. Appl. Anal., 19(5) (2016), 1222-1249.
[31] P. Kumar, V. Govindaraj, V. S. Erturk, M. H. Abdellattif, A study on the dynamics of alkali-silica chemical reaction by using Caputo fractional derivative, Pramana, 96(3) (2022), 1-19.
[32] Z. O¨ ztu¨rk, H. Bilgil, U¨ . Erdinc¸, An optimized continuous fractional grey model for forecasting of the time dependent real world cases, Hacettepe J. Math. Stat., 51(1) (2022), 308-326.
[33] U¨ . Erdinc, H. Bilgil, Z. O¨ ztu¨rk, A novel fractional forecasting model for time dependent real world cases, Accepted: Revstat Stat. J., (2022).

Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis

Yıl 2023, Cilt: 6 Sayı: 2, 49 - 55, 07.08.2023

Zafer Öztürk Halis Bilgil Sezer Sorgun

https://doi.org/10.33187/jmsm.1196961

Öz

Psychological diseases and their treatment are problems related to public health. According to data from the World Health Organization, about a billion people have either mental illness or substance use disorder problems in 2017. Mental, neurological diseases and substance use disorders account for 30 percent of the global non-fatal disease burden and 10 percent of the global disease burden. It is noted that in the world Dec 2005 and 2015, the incidence of mental health diseases increased by about 16 percent. In this study, we have created a fractional-order mathematical modeling for the population of individuals suering from psychological diseases in a society. In this model, the total population was divided into three compartments: individuals who did not receive psychological treatment (S), individuals who received psychological support (P) and individuals who recovered after completing psychological treatment (R). As a fractional derivative, we used the Caputo derivative definitions. Numerical solutions were obtained with the help of Euler method by performing stability analysis related to the fractional SPR model created for the mathematical model of psychological patients. Thus, it was interpreted by creating dynamics for the number of individuals with psychological problems
in a population.

Anahtar Kelimeler

Fractional Psychological Disease Model, Mathematical Modeling, Euler Method, Caputo Derivative, Stability analysis.

Kaynakça

[1] J. T. Townsend, D. E. Landon, Mathematical models of recognition and confusion in psychology, Math. Soc. Sci., 4(1) (1983), 25-71.
[2] W. K. Estes, Mathematical models in psychology, A Handbook for Data Analysis in the Behaviorial Sciences: Volume 1: Methodological Issues, Volume 2: Statistical Issues, 3 (2014).
[3] D. Wodarz, M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24(12) (2002), 1178-1187.
[4] I. Podlubny, Fractional Differential Equations, Academy Press, San Diego CA, 1999.
[5] F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical models in epidemiology, Springer, New York, 2019.
[6] H. Bilgil, A. Yousef, A. Erciyes, U¨ . Erdinc¸, Z. O¨ ztu¨rk, A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event, J. Comput. Appl. Math., (2022), 115015.
[7] N. T. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE, 1975.
[8] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180(1-2) (2002), 141-160.
[9] S. Wang, Y. Ding, H. Lu, S. Gong, Stability and bifurcation analysis of SIQR for the COVID-19 epidemic model with time delay, Math. Biosci. Eng., 18(5) (2021), 5505-5524.
[10] B. K. Mishra, N. Jha, SEIQRS model for the transmission of malicious objects in computer network, Appl. Math. Model., 34(3) (2010), 710-715.
[11] X. Liu, T. Yasuhiro, I. Shingo, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253(1) (2008), 1-11.
[12] M. B. Trawicki, Deterministic seirs epidemic model for modeling vital dynamics, vaccinations, and temporary immunity, Mathematics, 5(1) (2017), 7.
[13] Z. Öztürk, S. Sorgun, H. Bilgil, SIQRV Modeli ve Nu¨merik Uygulaması, Avrupa Bilim ve Teknoloji Dergisi, 28 (2021), 573-578.
[14] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, In: Proc. R. Soc. Lond., Series A, Containing Papers of a Mathematical and Physical Character, 115(772) (1927), 700-721.
[15] D. Yaro, S. K. Omari-Sasu, P. Harvim, A. W. Saviour, B. A. Obeng, Generalized Euler method for modeling measles with fractional differ ential equations, Int. J. Innov. Res. Dev., 4 (2015).
[16] Z. O¨ ztu¨rk, S. Sorgun, H. Bilgil, U¨ . Erdinc¸, New exact solutions of conformable time-fractional bad and good modified Boussinesq equations, J. New Theory, 37 (2021), 8-25.
[17] M. Braun, M. Golubitsky, Differential Equations and their Applications, Springer-Verlag, New York, 1983.
[18] https://tuikweb.tuik.gov.tr/UstMenu.do.
[19] P. Kumar, V. S. Erturk, M. Vellappandi, H. Trinh, V. Govindaraj, A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in Caputo sense, Chaos Solit. Fractals, 158 (2022), 112067.
[20] P. Kumar, V. S. Erturk, H. Abboubakar, K. S. Nisar, Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives, Alex. Eng. J., 60(3) (2021), 3189-3204.
[21] P. Kumar, V. Govindaraj, V. S. Erturk, A novel mathematical model to describe the transmission dynamics of tooth cavity in the human population, Chaos Solit. Fractals, 161 (2022), 112370.
[22] M. Vellappandi, P. Kumar, V. Govindaraj, Role of fractional derivatives in the mathematical modeling of the transmission of Chlamydia in the United States from 1989 to 2019, Nonlinear Dyn., (2022), 1-15.
[23] S. Abbas, S. Tyagi, P. Kumar, V. S. Erturk, S. Momani, Stability and bifurcation analysis of a fractional-order model of cell-to-cell spread of HIV-1 with a discrete time delay, Math. Meth. App. Sci., 45(11) (2022), 7081-7095.
[24] V. S. Erturk, E. Godwe, D. Baleanu, P. Kumar, J. Asad, A. Jajarmi, Novel fractional-order Lagrangian to describe motion of beam on nanowire, Acta Phys. Pol. A, 140(3) (2021), 265-272.
[25] P. Kumar, V. S. Erturk, A. Yusuf, S. Kumar, Fractional time-delay mathematical modeling of Oncolytic Virotherapy, Chaos Solit. Fractals, 150 (2021), 111123.
[26] A. Din, F. M. Khan, Z. U. Khan, A. Yusuf, T. Munir, The mathematical study of climate change model under nonlocal fractional derivative, Partial Differential Equations in Applied Mathematics, 5 (2022), 100204.
[27] E. Viera-Martin, J. F. Gomez-Aguilar, J. E. Solis-Perez, J. A. Hernandez-Perez, R. F. Escobar-Jimenez, Artificial neural networks: a practical review of applications involving fractional calculus, Eur. Phys. J. Spec. Top ., (2022), 1-37.
[28] V. S. Erturk, A. Ahmadkhanlu, P. Kumar, V. Govindaraj, Some novel mathematical analysis on a corneal shape model by using Caputo fractional derivative, Optik, 261 (2022), 169086.
[29] V. S. Erturk, A. K. Alomari, P. Kumar, M. Murillo-Arcila, Analytic solution for the strongly nonlinear multi-order fractional version of a BVP occurring in chemical reactor theory, Discrete Dyn. Nat. Soc., 2022 (2022), 8655340.
[30] Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: a review, Fract. Calc. Appl. Anal., 19(5) (2016), 1222-1249.
[31] P. Kumar, V. Govindaraj, V. S. Erturk, M. H. Abdellattif, A study on the dynamics of alkali-silica chemical reaction by using Caputo fractional derivative, Pramana, 96(3) (2022), 1-19.
[32] Z. O¨ ztu¨rk, H. Bilgil, U¨ . Erdinc¸, An optimized continuous fractional grey model for forecasting of the time dependent real world cases, Hacettepe J. Math. Stat., 51(1) (2022), 308-326.
[33] U¨ . Erdinc, H. Bilgil, Z. O¨ ztu¨rk, A novel fractional forecasting model for time dependent real world cases, Accepted: Revstat Stat. J., (2022).

Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	Zafer Öztürk 0000-0001-5662-4670 Halis Bilgil 0000-0002-8329-5806 Sezer Sorgun 0000-0001-8708-1226
Yayımlanma Tarihi	7 Ağustos 2023
Gönderilme Tarihi	31 Ekim 2022
Kabul Tarihi	31 Aralık 2022
Yayımlandığı Sayı	Yıl 2023 Cilt: 6 Sayı: 2

Kaynak Göster

APA	Öztürk, Z., Bilgil, H., & Sorgun, S. (2023). Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis. Journal of Mathematical Sciences and Modelling, 6(2), 49-55. https://doi.org/10.33187/jmsm.1196961
AMA	Öztürk Z, Bilgil H, Sorgun S. Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis. Journal of Mathematical Sciences and Modelling. Ağustos 2023;6(2):49-55. doi:10.33187/jmsm.1196961
Chicago	Öztürk, Zafer, Halis Bilgil, ve Sezer Sorgun. “Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis”. Journal of Mathematical Sciences and Modelling 6, sy. 2 (Ağustos 2023): 49-55. https://doi.org/10.33187/jmsm.1196961.
EndNote	Öztürk Z, Bilgil H, Sorgun S (01 Ağustos 2023) Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis. Journal of Mathematical Sciences and Modelling 6 2 49–55.
IEEE	Z. Öztürk, H. Bilgil, ve S. Sorgun, “Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis”, Journal of Mathematical Sciences and Modelling, c. 6, sy. 2, ss. 49–55, 2023, doi: 10.33187/jmsm.1196961.
ISNAD	Öztürk, Zafer vd. “Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis”. Journal of Mathematical Sciences and Modelling 6/2 (Ağustos 2023), 49-55. https://doi.org/10.33187/jmsm.1196961.
JAMA	Öztürk Z, Bilgil H, Sorgun S. Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis. Journal of Mathematical Sciences and Modelling. 2023;6:49–55.
MLA	Öztürk, Zafer vd. “Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis”. Journal of Mathematical Sciences and Modelling, c. 6, sy. 2, 2023, ss. 49-55, doi:10.33187/jmsm.1196961.
Vancouver	Öztürk Z, Bilgil H, Sorgun S. Application of Fractional $SPR$ Psychological Disease Model in Turkey and Stability Analysis. Journal of Mathematical Sciences and Modelling. 2023;6(2):49-55.