Analysis of The Dynamics of The Classical Epidemic Model with Beta Distributed Random Components

Öz

In this study, the classical epidemic model of Kermack and McKendrick is analyzed with beta distributed random components. A random analysis is done for the deterministic epidemic model by transforming the parameters and initial values of the system to random variables with beta distribution. The approximations for the expectations of the model variables are compared with the deterministic results to comment on the randomness of the cases with random parameters and random initial values. Results for some numerical characteristics of these two cases are also given to investigate the accuracy of the approximations for the expected values.

Anahtar Kelimeler

• SIR Model
• Random Effect
• Beta Distribution
• Moment
• Random Differential Equation

Klasik Salgın Hastalik Modeli Dinamiklerinin Beta Dağılımına Sahip Rastgele Bileşenlerle İncelenmesi

Öz

Bu çalışmada Kermack ve McKendrick’in klasik salgın hastalık modeli beta dağılımına sahip rastgele bileşenlerle incelenmektedir. Deterministik model için sistemin parametreleri ve başlangıç koşulları beta dağılımına sahip rastgele değişkenlere dönüştürülerek bir rastgele inceleme yapılmaktadır. Model değişkenlerinin beklenen değerleri için elde edilen yaklaşımlar deterministik sonuçlarla karşılaştırılarak rastgele başlangıç koşulları ve rastgele parametre içeren durumların rastgele yapıları hakkında yorum yapılmaktadır. Beklenen değerlerin yaklaşımlarının doğruluğunun incelenmesi için iki durumun bazı sayısal karakteristiklerinin sonuçları da verilmektedir.

Anahtar Kelimeler

• Rastgele Etki
• SIR Modeli
• Beta Dağılımı
• Moment
• Rastgele Diferansiyel Denklem

Kaynakça

1. Araz SI, Durur H, 2018. Galerkin Method for Numerical Solution of Two Dimensional Hyperbolic Boundary Value Problem with Dirichlet Conditions. Kırklareli Üniversitesi Mühendislik ve Fen Bilimleri Dergisi 4(1):1-11.
2. Bekiryazici Z, Merdan M, Kesemen T, Khaniyev T, 2016. Mathematical Modeling of Biochemical Reactions Under Random Effects. Turkish Journal of Mathematics and Computer Science 5:8-18.
3. Dokuyucu MA, 2019. A fractional order alcoholism model via Caputo-Fabrizio derivative. AIMS Mathematics 5(2):781–797.
4. Dokuyucu MA, Celik E, Bulut H, Baskonus HM, 2018. Cancer treatment model with the Caputo-Fabrizio fractional derivative. The European Physical Journal Plus 133(3):92.
5. Durur H, 2020. Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method. Modern Physics Letters B 34(03):2050036.
6. Durur H, Şenol M, Kurt A, Taşbozan O, 2019. Zaman-Kesirli Kadomtsev-Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 12(2):796-806.
7. Feller W, 1968. An Introduction to Probability Theory and Its Applications (volume 1, 3rd edition). John Wiley & Sons. New York.
8. Hethcote HW, 2000. The mathematics of infectious diseases. SIAM review 42(4):599-653.

9. Kermack WO, McKendrick AG, 1927. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences 115(772):700-721.
10. Khan MA, Badshah Q, Islam S, Khan I, Shafie S, Khan SA, 2015. Global dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive vaccination. Advances in Difference Equations 2015(1):88.
11. Khudair AR, Haddad SAM, Khalaf SL, 2016. Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method. Open Journal of Applied Sciences 6:287–297.
12. Merdan M, Bekiryazici Z, Kesemen T, Khaniyev T, 2017. Comparison of stochastic and random models for bacterial resistance. Advances in Difference Equations 2017(1):133.
13. Merdan M, Khaniyev T, 2008. On the behavior of solutions under the influence of stochastic effect of avian-human influenza epidemic model. International Journal of Biotechnology & Biochemistry 4(1):75-100.
14. Prakasha DG, Veeresha P, Baskonus HM, 2019. Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative. The European Physical Journal Plus 134(5):241.
15. Pukhov G, 1982. Differential Transforms and Circuit theory. Circuit Theory and Applications 10:265–276.
16. Rasool G, Zhang T, Shafiq A, Durur H, 2019. Influence of chemical reaction on Marangoni convective flow of nanoliquid in the presence of Lorentz forces and thermal radiation: A numerical investigation. Journal of Advances in Nanotechnology 1(1):32.
17. Singh J, Kumar D, Hammouch Z, Atangana A, 2018. A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Applied Mathematics and Computation 316:504-515.
18. Suleiman MY, Hlaing Oo WM, Wahab MA, Zakaria A, 1999. Application of beta distribution to Malaysian sunshine data. Renewable Energy 18:573–579.
19. Villafuerte L, Cortés JC, 2013. Solving Random Differential Equations by Means of Differential Transform Methods. Advances in Dynamical Systems and Applications 8(2):413-423.
20. Wiley JA, Herschkorn SJ, Padian NS, 1989. Heterogeneity in the probability of HIV transmission per sexual contact: The case of male-to-female transmission in penile–vaginal intercourse. Statistics in Medicine 8:93–102.
21. Yokus A, 2019. Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68(1):353-361.
22. Yokus, A, 2020. Truncation and Convergence Dynamics: KdV Burgers Model in the Sense of Caputo Derivative, Boletim da Sociedade Paranaense de Matematica, doi: 10.5269/bspm.47472.
23. Yokus A, Yavuz M, 2020. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020258.
24. Zhou JK, 1986. Differential Transformation and its application for Electrical Circuits. Huarjung University Press. Wuuhahn. China.