SIQRV Model and Numerical Application

Abstract

There is a pandemic situation caused by the COVID-19 epidemic almost all over the world. Despite the measures taken to prevent this epidemic and the vaccine studies developed, the course of the epidemic changes day by day as the virus mutates and changes its structure. The common opinion of the experts is that the most important weapon in the fight against the epidemic is the vaccine. In this paper, a new SIQRV model was obtained by adding a class of individuals who have been vaccinated to the SIQR (susceptible-infected-quarantine-recovered) model. In the newly created SIQRV model, the total population is divided into five sections. Numerical results were obtained through Euler method and graphs were drawn by mathematical analysis class of the susceptible individual (S) , the class of the infected individual (I), the class of the in quarantine (Q), the class of the recovered (R) and the class of the vaccinated individuals (V).

Keywords

• SIQRV Model
• Pandemic
• Euler Method
• Quarantine
• Vaccination.

SIQRV Modeli ve Nümerik Uygulaması

Abstract

Hemen hemen tüm dünya genelinde COVID-19 salgının neden olduğu bir pandemik durum söz konusudur. Bu salgının önlenmesi için alınan önlemler ve geliştirilen aşı çalışmalarına rağmen virüsün mutasyona uğrayarak yapısının değişmesiyle birlikte salgının seyri her geçen gün değişim göstermektedir. Uzmanların ortak kanaatleri ise salgınla mücadelenin en önemli silahın aşı olduğu yönündedir. Bu çalışmada SIQR (Hassas-Enfekte-Karantina-İyileşmiş) salgın hastalık modeline aşılanmış bireyler sınıfı eklenerek yeni bir SIQRV modeli elde edildi. Yeni oluşturulan SIQRV modelinde toplam nüfus beş bölüme ayrılmıştır. Duyarlı birey sınıfı (S), enfektif birey sınıfı (I), karantinada olan birey sınıfı (Q), iyileşmiş bireylerin sınıfı (R) ve aşılanmış bireylerin sınıfı (V) ile ilgili matematiksel analizler yapılarak nümerik sonuçlar Euler methodu yardımı ile elde edildi ve grafikler çizildi.

Keywords

• SIQRV Modeli
• Pandemi
• Euler Methodu
• Karantina
• Aşılama.

References

1. Köksal, S.S., 2008. "Epidemiyoloji", Halk Sağlığı Ders Kitabı, Baltaş Z, Ed., İstanbul Üniversitesi, İstanbul, 46-144.
2. Valleron, A. J., 2000. Roles of Mathematical Modeling in Epidemiology, Comptes Rendus de I’Academie des Sciences - Series III- Sciences de la Vie, 323 (5): 429-433.
3. Allen L. J. S., 2007, An Introduction to Mathematical Biology, Department of Mathematics and Statistics, Texas Tech University, Pearson Education., 348.
4. W.O. Kermack and A.G. McKendrick, 1927. A contribution to the mathematical theory of epidemics, in: Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 115 (772), 700-721,
5. S.T.A. Shah, M. Mansoor, A.F. Mirza, M. Dilshad, M.I. Khan, R. Farwa, M.A. Khan, M. Bilal and H.M.N. Iqbal, 2020. Predicting COVID-19 spread in Pakistan using the SIR Model, J. Pure Appl. Microbiol. 14 (2), 1423-1430.
6. R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, 1991).
7. N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Application (Hafner Press, New York, 1975)
8. H. Hethcote, M. Zhien, L. Shengbing, 2002. Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences 180, 141160.

9. Muroya, Y.; Enatsu, Y.; Nakata, Y., 2011. Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate. J. Math. Anal. Appl., 377, 1–14.
10. Mishra, B.K.; Jha, N. SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Model. 2010, 34, 710–715.
11. Liu, X.; Takeuchi, Y.; Iwami, S., Liu, X.; Takeuchi, Y.; Iwami, S., 2008. SVIR epidemic models with vaccination strategies. J. Theor. Biol. , 253, 1–11.
12. Trawicki, M.B., 2017. Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity. Mathematics, 5, 7.
13. Sun, C.; Yang, W., 2010. Global results for an SIRS model with vaccination and isolation. Nonlinear Anal. Real World Appl., 11, 4223–4237.
14. Eckalbar, J.C.; Eckalbar, W.L. Dynamics of an SIR model with vaccination dependent on past prevalence with high-order distributed delay. Biosystems 2015, 129, 50–65.
15. Odagaki, T., 2021. Exact properties of SIQR model for COVID-19. Physica A: Statiscal Mechanics and its Applications, 564, 125564.
16. Mascagni, M. , 1990. The backward euler method for numerical solution of the Hodgkin–Huxley equations of nerve conduction. SIAM journal on numerical analysis, 27(4), 941-962.