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Year 2025, Volume: 8 Issue: 3, 129 - 143, 11.09.2025

Emrah Yılmaz , Gülcan Tokay

https://doi.org/10.33187/jmsm.1569583

Abstract

References

  • [1] X. Xang, Review of Classical Epidemic Models, SAMSI Undergraduate Workshop Spring, 2010.
  • [2] W.H. Hamer, The Milroy lectures on epidemic disease in England—The evidence of variability and persistence of type, The Lancet, 1 (1906), 733-739.
  • [3] W. O. Kermack, A. G. McKendrick, A contribution to mathematical theory of epidemics, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 115(772) (1927), 700-721. https://doi.org/10.1098/rspa.1927.0118
  • [4] B. Dubey, P. Dubey, U. S. Dubey, Dynamics of an SIR model with nonlinear incidence and treatment rate, Appl. Appl. Math., 10(2) (2015), 718-737.
  • [5] P. Bhattacharya, S. Paul, P. Biswas, Mathematical modeling of treatment SIR model with respect to variable contact rate, International Proceedings of Economics Development and Research, 83 (2015), 34-41.
  • [6] R. Uddin, E. A. Algehyne, Mathematical analysis of COVID-19 by using SIR model with convex incidence rate, Results Phys., 23 (2021), Article ID 103970. https://doi.org/10.1016/j.rinp.2021.103970
  • [7] W. J. Zhu, S. F. Shen, An improved SIR model describing the epidemic dynamics of the COVID-19 in China, Results Phys., 25 (2021), Article ID 104289. https://doi.org/10.1016/j.rinp.2021.104289
  • [8] S. Ghersheen, V. Kozlov, V. Tkachev, et al., Mathematical analysis of complex SIR model with coinfection and density dependence, Comput. Math. Methods, 1 (2019), Article ID e1042. https://doi.org/10.1002/cmm4.1042
  • [9] A. Atangana, Modelling the spread of COVID-19 with new fractional-fractal operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136(2020), Article ID 109860. https://doi.org/10.1016/j.chaos.2020.109860
  • [10] B. Wacker, J. Schlüter, Time-continuous and time-discrete SIR models revisited: theory and applications, Adv. Differ. Equations, 556 (2020), 1-44. https://doi.org/10.1186/s13662-020-02995-1
  • [11] P. Van Den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2(3) (2017), 288-303. https://doi.org/10.1016/j.idm.2017.06.002
  • [12] H. Talukder, K. Debnath, A. Raquib, et al., Estimation of basic reproduction number (Ro) of novel coronavirus (COVID-19) from SEIR Model in perspective of Bangladesh, J. Infect. Dis. Epidemiol., 6(4) (2020), 1-6. https://doi.org/10.23937/2474-3658/1510144
  • [13] O. D. Makinde, Adomian Decomposition Approach to a SIR epidemic model with constant vaccination strategy, Appl. Math. Comput., 184(2) (2007), 842-848. https://doi.org/10.1016/j.amc.2006.06.074
  • [14] S. T. Akinyemi, M. O. Ibrahim, I. G. Usman, et al., Global stability analysis of SIR epidemic model with relapse and immunity loss, International Journal of Applied Science and Mathematical Theory, 2(1) (2016), 1-14.
  • [15] D. R. Anderson, S. G. Georgiev, Conformable Dynamic Equations on Time Scales, Chapman and Hall/CRC, 2020. https://doi.org/10.1201/9781003057406
  • [16] K. Ergen, A. Cilli, N. Yahnioğlu, Predicting epidemic diseases using mathematical modelling of SIR, Acta Phys. Pol. A, 128 (2015), 273-275. https://doi.org/10.12693/APhysPolA.128.B-273
  • [17] B. Uzunoğlu, SIS Salgın hastalıkların matematiksel modeli ve kararlılık analizi, Master’s thesis, Kayseri Erciyes University, 2013.
  • [18] L. A. Urena-Lopez, A. X. Gonzalez-Morales, Time-dependent SI model for epidemiology and applications to Covid-19, Rev. Mex. Fis., 67(5) (2021), 1-15. https://doi.org/10.31349/revmexfis.67.050706
  • [19] L. J. Allen, F. Brauer, P. Van den Driessche, et al., Mathematical Epidemiology (Vol. 1945, pp. 3-17). J. Wu (Ed.). Berlin, Springer, 2008.
  • [20] M. Bohner, S. Streipert, D.F. Torres, Exact solution to a dynamic SIR model, Nonlinear Anal. Hybrid Syst, 32 (2019), 228-238. https://doi.org/10.1016/j.nahs.2018.12.005
  • [21] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2nd ediition, 1975, 430 pages.
  • [22] M. Y. Li, J. R. Graef, L. Wang, et al., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160(2) (1999), 191–213. https://doi.org/10.1016/S0025-5564(99)00030-9
  • [23] W. R. Derrick, P. Van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31(5) (1993), 495-512. https://doi.org/10.1007/BF00173889
  • [24] A. Rachah, D. F. M. Torres, Analysis, simulation and optimal control of a SEIR model for Ebola virus with demographic effects, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 67(1) (2018), 179-197. https://doi.org/10.1501/Commua1 0000000841
  • [25] N. T. J. Bailey, Spatial models in the epidemiology of infectious diseases, Biological Growth and Spread, Lecture Notes in Biomathematics, 38 (1980), 233-261.
  • [26] J. Arino, R. Jordan, P. Van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206(1) (2007), 46-60. https://doi.org/10.1016/j.mbs.2005.09.002
  • [27] R. Rifhat, L. Wang, Z. Teng, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Phys. A: Stat. Mech. Appl., 481 (2017), 176-190. https://doi.org/10.1016/j.physa.2017.04.016
  • [28] C. Yuan, D. Jiang, D. O’Regan, et al., Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Commun. Nonlinear Sci. Numer. Simul., 17(6) (2012), 2501-2516. https://doi.org/10.1016/j.cnsns.2011.07.025
  • [29] K. Fan, Y. Zhang, S. Gao, et al., A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A, 481 (2017), 198-208. https://doi.org/10.1016/j.physa.2017.04.055
  • [30] J. Djordjevic, C. J. Silva, D. F. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168-175. https://doi.org/10.1016/j.aml.2018.05.005
  • [31] S. Çakan, Dağılımlı zaman gecikmeli bir SIS salgın modelinin kararlılığına ilişkin bazı sonuçlar, Türk Doğa ve Fen Dergisi, 10(2) (2021), 18-28. https://doi.org/10.46810/tdfd.814302
  • [32] H. Khan, R. N. Mohapatra, K. Vajravelu, et al., The explicit series solution of SIR and SIS epidemic models, Appl. Math. Comput., 215(2) (2009), 653–669. https://doi.org/10.1016/j.amc.2009.05.051
  • [33] R. Pettersson, T. Lindström, The Role of Explicit Solutions in the Analysis of Epidemic Models, Bachelor’s Degree Project Thesis in Applied Mathematics, Linnaeus University, Sweden, 2021.
  • [34] M. Bohner, S. H. Streipert, The SIS-model on time scales, Pliska Stud. Math., 26 (2016), 11-28.

Proportional Epidemic Models on Time Scales

Year 2025, Volume: 8 Issue: 3, 129 - 143, 11.09.2025

Emrah Yılmaz , Gülcan Tokay

https://doi.org/10.33187/jmsm.1569583

Abstract

In this study, some epidemic models such as SI (Susceptible-Infectious), SIR (Susceptible-Infectious-Recovered), and SIS (Susceptible-Infectious-Susceptible) have been formulated using proportional derivative on time scales, and solutions have been obtained through the application of time scale analysis techniques and proportional derivative rules.

Keywords

Epidemic models Proportional derivative Time scales

References

  • [1] X. Xang, Review of Classical Epidemic Models, SAMSI Undergraduate Workshop Spring, 2010.
  • [2] W.H. Hamer, The Milroy lectures on epidemic disease in England—The evidence of variability and persistence of type, The Lancet, 1 (1906), 733-739.
  • [3] W. O. Kermack, A. G. McKendrick, A contribution to mathematical theory of epidemics, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 115(772) (1927), 700-721. https://doi.org/10.1098/rspa.1927.0118
  • [4] B. Dubey, P. Dubey, U. S. Dubey, Dynamics of an SIR model with nonlinear incidence and treatment rate, Appl. Appl. Math., 10(2) (2015), 718-737.
  • [5] P. Bhattacharya, S. Paul, P. Biswas, Mathematical modeling of treatment SIR model with respect to variable contact rate, International Proceedings of Economics Development and Research, 83 (2015), 34-41.
  • [6] R. Uddin, E. A. Algehyne, Mathematical analysis of COVID-19 by using SIR model with convex incidence rate, Results Phys., 23 (2021), Article ID 103970. https://doi.org/10.1016/j.rinp.2021.103970
  • [7] W. J. Zhu, S. F. Shen, An improved SIR model describing the epidemic dynamics of the COVID-19 in China, Results Phys., 25 (2021), Article ID 104289. https://doi.org/10.1016/j.rinp.2021.104289
  • [8] S. Ghersheen, V. Kozlov, V. Tkachev, et al., Mathematical analysis of complex SIR model with coinfection and density dependence, Comput. Math. Methods, 1 (2019), Article ID e1042. https://doi.org/10.1002/cmm4.1042
  • [9] A. Atangana, Modelling the spread of COVID-19 with new fractional-fractal operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136(2020), Article ID 109860. https://doi.org/10.1016/j.chaos.2020.109860
  • [10] B. Wacker, J. Schlüter, Time-continuous and time-discrete SIR models revisited: theory and applications, Adv. Differ. Equations, 556 (2020), 1-44. https://doi.org/10.1186/s13662-020-02995-1
  • [11] P. Van Den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2(3) (2017), 288-303. https://doi.org/10.1016/j.idm.2017.06.002
  • [12] H. Talukder, K. Debnath, A. Raquib, et al., Estimation of basic reproduction number (Ro) of novel coronavirus (COVID-19) from SEIR Model in perspective of Bangladesh, J. Infect. Dis. Epidemiol., 6(4) (2020), 1-6. https://doi.org/10.23937/2474-3658/1510144
  • [13] O. D. Makinde, Adomian Decomposition Approach to a SIR epidemic model with constant vaccination strategy, Appl. Math. Comput., 184(2) (2007), 842-848. https://doi.org/10.1016/j.amc.2006.06.074
  • [14] S. T. Akinyemi, M. O. Ibrahim, I. G. Usman, et al., Global stability analysis of SIR epidemic model with relapse and immunity loss, International Journal of Applied Science and Mathematical Theory, 2(1) (2016), 1-14.
  • [15] D. R. Anderson, S. G. Georgiev, Conformable Dynamic Equations on Time Scales, Chapman and Hall/CRC, 2020. https://doi.org/10.1201/9781003057406
  • [16] K. Ergen, A. Cilli, N. Yahnioğlu, Predicting epidemic diseases using mathematical modelling of SIR, Acta Phys. Pol. A, 128 (2015), 273-275. https://doi.org/10.12693/APhysPolA.128.B-273
  • [17] B. Uzunoğlu, SIS Salgın hastalıkların matematiksel modeli ve kararlılık analizi, Master’s thesis, Kayseri Erciyes University, 2013.
  • [18] L. A. Urena-Lopez, A. X. Gonzalez-Morales, Time-dependent SI model for epidemiology and applications to Covid-19, Rev. Mex. Fis., 67(5) (2021), 1-15. https://doi.org/10.31349/revmexfis.67.050706
  • [19] L. J. Allen, F. Brauer, P. Van den Driessche, et al., Mathematical Epidemiology (Vol. 1945, pp. 3-17). J. Wu (Ed.). Berlin, Springer, 2008.
  • [20] M. Bohner, S. Streipert, D.F. Torres, Exact solution to a dynamic SIR model, Nonlinear Anal. Hybrid Syst, 32 (2019), 228-238. https://doi.org/10.1016/j.nahs.2018.12.005
  • [21] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2nd ediition, 1975, 430 pages.
  • [22] M. Y. Li, J. R. Graef, L. Wang, et al., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160(2) (1999), 191–213. https://doi.org/10.1016/S0025-5564(99)00030-9
  • [23] W. R. Derrick, P. Van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31(5) (1993), 495-512. https://doi.org/10.1007/BF00173889
  • [24] A. Rachah, D. F. M. Torres, Analysis, simulation and optimal control of a SEIR model for Ebola virus with demographic effects, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 67(1) (2018), 179-197. https://doi.org/10.1501/Commua1 0000000841
  • [25] N. T. J. Bailey, Spatial models in the epidemiology of infectious diseases, Biological Growth and Spread, Lecture Notes in Biomathematics, 38 (1980), 233-261.
  • [26] J. Arino, R. Jordan, P. Van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206(1) (2007), 46-60. https://doi.org/10.1016/j.mbs.2005.09.002
  • [27] R. Rifhat, L. Wang, Z. Teng, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Phys. A: Stat. Mech. Appl., 481 (2017), 176-190. https://doi.org/10.1016/j.physa.2017.04.016
  • [28] C. Yuan, D. Jiang, D. O’Regan, et al., Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Commun. Nonlinear Sci. Numer. Simul., 17(6) (2012), 2501-2516. https://doi.org/10.1016/j.cnsns.2011.07.025
  • [29] K. Fan, Y. Zhang, S. Gao, et al., A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A, 481 (2017), 198-208. https://doi.org/10.1016/j.physa.2017.04.055
  • [30] J. Djordjevic, C. J. Silva, D. F. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168-175. https://doi.org/10.1016/j.aml.2018.05.005
  • [31] S. Çakan, Dağılımlı zaman gecikmeli bir SIS salgın modelinin kararlılığına ilişkin bazı sonuçlar, Türk Doğa ve Fen Dergisi, 10(2) (2021), 18-28. https://doi.org/10.46810/tdfd.814302
  • [32] H. Khan, R. N. Mohapatra, K. Vajravelu, et al., The explicit series solution of SIR and SIS epidemic models, Appl. Math. Comput., 215(2) (2009), 653–669. https://doi.org/10.1016/j.amc.2009.05.051
  • [33] R. Pettersson, T. Lindström, The Role of Explicit Solutions in the Analysis of Epidemic Models, Bachelor’s Degree Project Thesis in Applied Mathematics, Linnaeus University, Sweden, 2021.
  • [34] M. Bohner, S. H. Streipert, The SIS-model on time scales, Pliska Stud. Math., 26 (2016), 11-28.
There are 34 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Emrah Yılmaz 0000-0002-7822-9193

Gülcan Tokay 0000-0002-6317-6164

Early Pub Date August 18, 2025
Publication Date September 11, 2025
Submission Date October 18, 2024
Acceptance Date July 26, 2025
Published in Issue Year 2025 Volume: 8 Issue: 3

Cite

APA Yılmaz, E., & Tokay, G. (2025). Proportional Epidemic Models on Time Scales. Journal of Mathematical Sciences and Modelling, 8(3), 129-143. https://doi.org/10.33187/jmsm.1569583
AMA Yılmaz E, Tokay G. Proportional Epidemic Models on Time Scales. Journal of Mathematical Sciences and Modelling. September 2025;8(3):129-143. doi:10.33187/jmsm.1569583
Chicago Yılmaz, Emrah, and Gülcan Tokay. “Proportional Epidemic Models on Time Scales”. Journal of Mathematical Sciences and Modelling 8, no. 3 (September 2025): 129-43. https://doi.org/10.33187/jmsm.1569583.
EndNote Yılmaz E, Tokay G (September 1, 2025) Proportional Epidemic Models on Time Scales. Journal of Mathematical Sciences and Modelling 8 3 129–143.
IEEE E. Yılmaz and G. Tokay, “Proportional Epidemic Models on Time Scales”, Journal of Mathematical Sciences and Modelling, vol. 8, no. 3, pp. 129–143, 2025, doi: 10.33187/jmsm.1569583.
ISNAD Yılmaz, Emrah - Tokay, Gülcan. “Proportional Epidemic Models on Time Scales”. Journal of Mathematical Sciences and Modelling 8/3 (September2025), 129-143. https://doi.org/10.33187/jmsm.1569583.
JAMA Yılmaz E, Tokay G. Proportional Epidemic Models on Time Scales. Journal of Mathematical Sciences and Modelling. 2025;8:129–143.
MLA Yılmaz, Emrah and Gülcan Tokay. “Proportional Epidemic Models on Time Scales”. Journal of Mathematical Sciences and Modelling, vol. 8, no. 3, 2025, pp. 129-43, doi:10.33187/jmsm.1569583.
Vancouver Yılmaz E, Tokay G. Proportional Epidemic Models on Time Scales. Journal of Mathematical Sciences and Modelling. 2025;8(3):129-43.
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