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# Tumour-Natural Killer and CD8+ T Cells Interaction Model with Delay

by 1,* and
1
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru 81310, Malaysia
2
Department School of Health Science, Universiti Sains Malaysia, Kubang Kerian, Kota Bharu 16150, Malaysia
*
Author to whom correspondence should be addressed.
Academic Editors: Hamid Reza Karimi, Chang-Hua Lien and Sundarapandian Vaidyanathan
Mathematics 2022, 10(13), 2193; https://doi.org/10.3390/math10132193
Received: 18 May 2022 / Revised: 18 June 2022 / Accepted: 19 June 2022 / Published: 23 June 2022
The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD${8}^{+}$ T cells. NK cells are spontaneously active against infected target cells, whereas CD${8}^{+}$ T cells take some times to activate cell called as cell-specific targeting, to kill the virus. The interaction between NK cells and tumour cells has produced the other CD${8}^{+}$ T cell, called tumour-specific CD${8}^{+}$ T cells. We illustrate the tumour–immune interaction through mathematical modelling by considering the cell cycle. The interaction of the cells is described by a system of delay differential equations, and the delay, $\tau$ represent time taken for tumour cell reside interphase. The stability analysis and the bifurcation behaviour of the system are analysed. We established the stability of the model by analysing the characteristic equation to produce a stability region. The stability region is split into two regions, tumour decay and tumour growth. By applying the Routh–Hurwitz Criteria, the analysis of the trivial and interior equilibrium point of the model provides conditions for stability and is illustrated in the stability map. Numerical simulation is carried out to show oscillations through Hopf Bifurcation, and stability switching is found for the delay system. The result also showed that the interaction of NK cells with tumour cells could suppress tumour cells since it can increase the population of CD${8}^{+}$ T cells. This concluded that the inclusion of delay and immune responses (NK-CD${8}^{+}$ T cells) into consideration gives us a deep insight into the tumour growth and helps us understand how their interactions contribute to kill tumour cells. View Full-Text
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MDPI and ACS Style

Awang, N.A.; Maan, N.; Sulain, M.D. Tumour-Natural Killer and CD8+ T Cells Interaction Model with Delay. Mathematics 2022, 10, 2193. https://doi.org/10.3390/math10132193

AMA Style

Awang NA, Maan N, Sulain MD. Tumour-Natural Killer and CD8+ T Cells Interaction Model with Delay. Mathematics. 2022; 10(13):2193. https://doi.org/10.3390/math10132193

Chicago/Turabian Style

Awang, Nor A., Normah Maan, and Mohd D. Sulain. 2022. "Tumour-Natural Killer and CD8+ T Cells Interaction Model with Delay" Mathematics 10, no. 13: 2193. https://doi.org/10.3390/math10132193

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