Some Results Related to Stabilities of an SIS Epidemic Model with Distributed Time Delay

Yıl 2021, Cilt: 10 Sayı: 2, 18 - 28, 31.12.2021

Sümeyye Çakan

https://doi.org/10.46810/tdfd.814302

Öz

In this study, as differently from other SIS mathematical epidemic models which there exist in the literature on the spread of epidemic diseases, a mathematical epidemic model has been presented by means of the system of nonlinear distributed delay integro-differential equations, taking into account that the effect of latent period which varies according to individuals on the spread of the disease. The disease-free and endemic equilibrium points of this nonlinear system have been obtained, also the number of secondary infections (basic reproduction number) related to the model has been found. Then some results about the stability of the equilibrium points and so the stability of the system have been obtained according to whether the number of secondary infections, which is a crucial parameter on the spread of diseases, is less than 1 or not.

Anahtar Kelimeler

Mathematical Model, Asymptotic Stability, Disease-free Equilibrium Point, Endemic Equilibrium Point, Basic Reproduction Number (The Number of Secondary Infections)

Dağılımlı Zaman Gecikmeli Bir SIS Salgın Modelinin Kararlılığına İlişkin Bazı Sonuçlar

Öz

Bu çalışmada salgın hastalıkların yayılması konusunda literatürde mevcut olan diğer SIS matematiksel salgın modellerinden farklı olarak, bireylere göre değişen latent periyodunun hastalığın yayılmasına ilişkin süreçteki etkisi dikkate alınarak lineer olmayan dağılımlı gecikmeli bir integro-diferensiyel denklem sistemi yardımıyla matematiksel bir model sunulmuştur. Lineer olmayan bu sistemin hastalıktan bağımsız ve hastalıkla ilişkili denge noktaları elde edilerek, modele ilişkin ikincil enfeksiyon sayısı (temel çoğalma sayısı) bulunmuştur. Ardından salgının seyrinde kritik bir parametre olan ikincil enfeksiyon sayısının 1‘den küçük olup olmayışına göre denge noktalarının ve dolayısıyla sistemin kararlılığına dair bazı sonuçlar elde edilmiştir.

Anahtar Kelimeler

Matematiksel Model, Asimptotik Kararlılık, Hastalıktan Bağımsız Denge Noktası, Hastalıkla İlişkili Denge Noktası, İkincil enfeksiyon sayısı (Temel çoğalma sayısı)

Kaynakça

• [1] Kermack WO, McKendrick AG. A contributions to the mathematical theory of epidemics. Proc. Roy. Soc. A. 1927;115:700-721.
• [2] Vargas-De-León C. Stability analysis of a SIS epidemic model with standart incidence. Foro-Red-Mat: Revista Electronica de Contenido Matematico. 2011;28(4):1-11.
• [3] Uzunoğlu B. SIS salgın hastalıkların matematiksel modeli ve kararlılık analizi [Yüksek Lisans Tezi]. Kayseri: Erciyes Üniversitesi Fen Bilimleri Enstitüsü; 2013.
• [4] Agaba GO, Kyrychko YN, Blyuss KB. Time-delayed SIS epidemic model with population awareness. Ecological Complexity. 2017;31:50-56.
• [5] Cui J, Tao X, Zhu H. An SIS infection model incorporating media coverage. Rocky Mountain Journal of Mathematics. 2008;38:1323-1334.
• [6] Hethcote HW, Driessche P. van den. Two SIS epidemiologic models with delays. J. Math. Bio. 2000;40:3-26.
• [7] Hethcote HW, Driessche P. van den. An SIS epidemic model with variable population size and a delay. J. Math. Biol. 1995;34:177-194.
• [8] Li Y, Cui J. The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage. Commun. Nonlinear Sci. Numer. Simul. 2009;14:2353-2365.
• [9] Liu J, Zhang T. Bifurcation analysis of an SIS epidemic model with nonlinear birth rate. Chaos, Solitons & Fractals. 2009;40:1091-1099.
• [10] Nakata Y, Röst G. Global stability of an SIS epidemic model with a finite infectious period. Differential and Integral Equations. 2018;31(3-4):161-172.
• [11] Thirthar AA, Naji RK. The dynamics of an SIS epidemic model with two delays. Jour of Adv Research in Dynamical & Control Systems. 2018;10:1917-1933.
• [12] Ucakan Y. SIRS matematik modelinin incelenmesi ve kararlılık analizi [Yüksek Lisans Tezi]. Kayseri: Erciyes Üniversitesi Fen Bilimleri Enstitüsü; 2014.
• [13] Weia J, Zou X. Bifurcation analysis of a population model and the resulting SIS epidemic model with delay. J. Comput. Appl. Math. 2006;197(1):169-187.
• [14] Allen LJS. An Introduction to mathematical biology. Upper Saddle River, New Jersey; 2007.
• [15] LaSalle JP. Some extensions of Liapunovs second method. IRE Trans. Circs. Th. 1960;7:520-527.
• [16] Lakshmikantham V, Leela S, Martynyuk AA. Stability analysis of nonlinear systems, Marcel Dekker, Inc. New York; 1989.
• [17] Diekmann O, Heesterbeek JAP, Metz JAJ. On the definition and the computation of the basic reproduction ratio _0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 1990;28(4):365-382.
• [18] Driessche PVD, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002;180:29-48.