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

Abstract

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.

Keywords

• 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

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