Delay differential system for interaction between tumour growth and immune responses

Abstract

This paper presents a model of tumour growth that describes the interactions between immune responses and tumour cells. The model is based on the model discussed in [1] with considering another immune response (NK cell) instead of merely an immune response (CD8+ T cell). The new proposed model becomes a four-population model, represented as delay differential system that includes population of tumour during interphase, population of tumour during mitosis and immune responses (NK and CD8+ T cells) . The cycling of population of tumour is subdivided into phases: interphase (G1, S and G2 phases) and mitosis phase (M-phase). The stability of the system is then analysed by Routh-Hurwitz criteria in order to determine the stability of the fixed points. Routh-Hurwitz criteria is used for non-delay case. The comparison between the proposed model and the existing one is also considered. For the case when there is no delay, a st ability map shows that the curve limits tumour growth region and the curve for the proposed model shown to lie consistently below the curve of the existing one. In delay case, we used Geometric Argument [2] to establish the stability of the system. By modifying the argument principle, we show that the stability switching continuously as delay increase.