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Math Biosci. 2019 Feb;308:8-19. doi: 10.1016/j.mbs.2018.12.009. Epub 2018 Dec 8.

Mathematical analysis of a tumour-immune interaction model: A moving boundary problem.

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Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X 20, Hatfield, Pretoria 0028, South Africa. Electronic address:
Department of Mathematical Sciences, Stellenbosch University, Private Bag X1 Matieland, 7602, South Africa. Electronic address:


A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level.


Cancer dormancy; Hyperbolic tangent method; Moving boundary problem; Travelling wave solutions; Tumour radius; Tumour-immune interactions


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