An SIR Model of Influenza with the Effects of Treatment and Vaccination

Year 2024, , 51 - 59, 31.08.2024

Elif Demir , Canan Vural

https://doi.org/10.33187/jmsm.1472066

Abstract

We produced an SIR model of influenza which is a global infectious disease, by using Caputo fractional derivative. In this model, we separated S and I into different groups. Separation is made according to the group of people in S who get vaccinated and are protected from influenza, also people in S who get vaccinated but are not protected besides people in S who do not get vaccinated. Furthermore, infected people are separated as treated and untreated people in I. We did stability analysis of the model and produced the basic reproduction number. We emphasized the importance of influenza vaccine and treatment for infected people by varying the values of the parameters and was shown with graphics.

Keywords

Basic Reproduction Number, Fractional SIR Model, Influenza, Rates Of Treatment And Vaccination, Stability Analysis

Project Number

FYL-2023-5925

References

• [1] C. Nypaver, C. Dehlinger, C. Carter, Influenza and influenza vaccine: a review, Journal of Midwifery & Women’s Health, 66(1) (2021), 45-53.
• [2] A. D. Iuliano, et al., Estimates of global seasonal influenza-associated respiratory mortality: A modelling study, The Lancet, 391 (10127) (2018), 1285-1300.
• [3] Y. Wang, et al., Vaccination coverage with the pneumococcal and influenza vaccine among persons with chronic diseases in Shanghai, China, 2017, BMC Public Health, 20 (2020), 1-9.
• [4] R. Allard, et al, Diabetes and the severity of pandemic influenza A (H1N1) infection, Diabetes care, 33(7) (2010), 1491-1493.
• [5] https://www.who.int/news-room/spotlight/history-of-vaccination/history-of-influenza-vaccination?topicsurvey=ht7j2q)&gclid=Cj0KCQiAwbitBhDIARIsABfFYIJGDMPmzAm9bfYs7KULeumVIdTyBz8jYArZ40HX6oRQbYoQzhpXm1YaAqUqEALw wcB
• [6] https://grip.saglik.gov.tr/tr/tedavi.html
• [7] R. Kumar, S. Kumar, A new fractional modelling on susceptible-infected-recovered equations with constant vaccination rate, Nonlinear Engineering, 3(1) (2014), 11-19.
• [8] Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186(1) (2007), 286-293.
• [9] W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332(1) (2007), 709-726.
• [10] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(1-2) (2002), 29-48.
• [11] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application, In: Multiconference, IMACS, IEEE-SMC, Lille, France, 2 (1996), 963-968.
• [12] E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka, On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler Chua and Chen systems, Phys. Lett. A, 358(1) (2006), 1-4.
• [13] P. Tomasek, On Euler methods for Caputo fractional differential equations, Arch. Math., 59 (2023), 287-294.