Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data

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DOI: https://doi.org/10.28924/2291-8639-22-2024-32
Published: Feb 13, 2024
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I. K. Youssef, M. Khalifa Saad, Ali Dumlu, Somia A. Asklany, Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data, Int. J. Anal. Appl., 22 (2024), 32.

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I. K. Youssef, M. Khalifa Saad, Ali Dumlu, Somia A. Asklany

Abstract

The updated version of the Picard method for solving systems of differential equations is employed to solve the SIR system. A local performance of the Picard iteration algorithm combined with the Gauss-Seidel approach is applied to the SIR model. The integral form of the SIR model, in addition to the use of Gauss-Seidel philosophy (using the most recent calculated values), achieved more accuracy in the computational work than those obtained using the differential forms. Documented data regarding the spread of corona virus 19 in the Kingdom of Saudi Arabia region from April to the end of December 2020 were used to calculate the corresponding actual values for the parameters and the initial conditions. Due to efficient management and to obtain representable behaviours, we restricted the size of the study to only 1% of the population. The global characteristics of the integral formulation have affected the calculations accurately. The initial conditions and the model’s parameters are established depending on the documented data. The results illustrate the superiority of the updated Picard formulation over the classical Picard within their domain of convergence. The results of this study illuminate and validate the importance of mathematical modeling. These findings can provide valuable insights into mathematical modeling for those involved in environmental health research, especially those responsible for devising strategic plans.

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References

  1. H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev. 42 (2000), 599–653. https://doi.org/10.1137/s0036144500371907.
  2. F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, NY, 2012. https://doi.org/10.1007/978-1-4614-1686-9.
  3. I.K. Youssef, M.H.M. Hassan, A Comparative Study for Some Mathematical Models of Epidemic Diseases with Application to Strategic Management, Appl. Sci. 12 (2022), 12639. https://doi.org/10.3390/app122412639.
  4. Y. Okabe, A. Shudo, A Mathematical Model of Epidemics–A Tutorial for Students, Mathematics. 8 (2020), 1174. https://doi.org/10.3390/math8071174.
  5. J. Enrique Amaro, J. Dudouet, J. Nicolás Orce, Global Analysis of the COVID-19 Pandemic Using Simple Epidemiological Models, Appl. Math. Model. 90 (2021), 995–1008. https://doi.org/10.1016/j.apm.2020.10.019.
  6. I.K. Youssef, H.A. El-Arabawy, Picard Iteration Algorithm Combined With Gauss–seidel Technique for Initial Value Problems, Appl. Math. Comp. 190 (2007), 345–355. https://doi.org/10.1016/j.amc.2007.01.058.
  7. O.D. Makinde, Adomian Decomposition Approach to a Sir Epidemic Model With Constant Vaccination Strategy, Appl. Math. Comp. 184 (2007), 842–848. https://doi.org/10.1016/j.amc.2006.06.074.
  8. M. Rafei, H. Daniali, D.D. Ganji, Variational Iteration Method for Solving the Epidemic Model and the Prey and Predator Problem, Appl. Math. Comput. 186 (2007), 1701–1709. https://doi.org/10.1016/j.amc.2006.08.077.
  9. H. Khan, R.N. Mohapatra, K. Vajravelu, S.J. Liao, The Explicit Series Solution of SIR and SIS Epidemic Models, Appl. Math. Comp. 215 (2009), 653–669. https://doi.org/10.1016/j.amc.2009.05.051.
  10. W. Liu, S.A. Levin, Y. Iwasa, Influence of Nonlinear Incidence Rates Upon the Behavior of SIRS Epidemiological Models, J. Math. Biology. 23 (1986), 187–204. https://doi.org/10.1007/bf00276956.
  11. S. Bushnaq, Shafiullah, M. Sarwar, H. Alrabaiah, Existence Theory and Numerical Simulations of Variable Order Model of Infectious Disease, Results Appl. Math. 19 (2023), 100395. https://doi.org/10.1016/j.rinam.2023.100395.
  12. S.M. O’Regan, T.C. Kelly, A. Korobeinikov, M.J.A. O’Callaghan, A.V. Pokrovskii, Lyapunov Functions for SIR and SIRS Epidemic Models, Appl. Math. Lett. 23 (2010), 446–448. https://doi.org/10.1016/j.aml.2009.11.014.
  13. G. Zaman, I.H. Jung, Stability Techniques in SIR Epidemic Models, Proc. Appl. Math. Mech. 7 (2007), 2030063– 2030064. https://doi.org/10.1002/pamm.200701147.
  14. WHO, WHO COVID-19 Dashboard. https://covid19.who.int.
  15. I.K. Youssef, R.A. Ibrahim, Boundary Value Problems, Fredholm Integral Equations, SOR and KSOR Methods, Life Sci. 10 (2013), 304–312.
  16. G. Bärwolff, Mathematical Modeling and Simulation of the COVID-19 Pandemic, Systems. 8 (2020), 24. https://doi.org/10.3390/systems8030024.
  17. H.M. Ahmed, R.A. Elbarkouky, O.A.M. Omar, M.A. Ragusa, Models for COVID-19 Daily Confirmed Cases in Different Countries, Mathematics. 9 (2021), 659. https://doi.org/10.3390/math9060659.
  18. P. Shahrear, S.M.S. Rahman, M.M.H. Nahid, Prediction and Mathematical Analysis of the Outbreak of Coronavirus (COVID-19) in Bangladesh, Results Appl. Math. 10 (2021), 100145. https://doi.org/10.1016/j.rinam.2021.100145.
  19. A.M. Almeshal, A.I. Almazrouee, M.R. Alenizi, S.N. Alhajeri, Forecasting the Spread of COVID-19 in Kuwait Using Compartmental and Logistic Regression Models, Appl. Sci. 10 (2020), 3402. https://doi.org/10.3390/app10103402.