Modeling a Tumor Growth under Immunological Activity

Abstract

In this paper, a mathematical model which consists of system of differential equations with piecewise constant argument is constructed to describe tumor-immune system interaction. The system is based on the tumor growth model constructed by Kuznetsov et all. A solution of the system with piecewise constant arguments leads to a system of difference equations. Using Schur-Cohn criterion and a Lyapunov function, sufficient conditions are obtained for the local and global asymptotic stability of a positive equilibrium point of the system of difference equations. Neimark-Sacker bifurcations analysis shows that stable limit cycle occurs at the bifurcation point, thus resulting oscillations for tumor and immune system.

Keywords

• Tumor –immune system interaction, stability, system of difference equation, bifurcation.

Bağışıklık Sistemi Etkisi Altında Tümör Büyümesinin Modellenmesi

Abstract

Bu çalışmada, tümör-bağışıklık sistemi etkileşimini tanımlamak için tam değer fonksiyonlu diferansiyel denklem sisteminden oluşan bir matematiksel model kurulmuştur. Sistem, Kuznetsov ve arkadaşlarının tümör büyümesi için önermiş olduğu matematiksel modele dayanmaktadır. Oluşturulan tam değer fonksiyonlu diferansiyel denklem sisteminin çözümünden fark denklem sistemi elde edilmiştir. Schur-Cohn kriteri ve Lyapunov fonksiyonun kullanılmasıyla fark denklem sisteminin pozitif denge noktasının yerel ve global kararlı olmasını sağlayan yeterli koşullar belirlenmiştir. Neimark-Sacker çatallanma analizi, çatallanma noktasında kararlı limit döngüsünün oluştuğu ve bunun sonucunda tümör ve bağışıklık sisteminin salınıma gittiğini göstermektedir.

Keywords

• Tümör –bağışıklık sistemi etkileşimi, kararlılık, fark denklem sistemi, çatallanma.

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