Abstract
ABSTRACTA previously published mathematical model, governing tumour growth with mixed immunotherapy and chemotherapy treatments, is modified and studied. The search time, which is assumed to be neglectable in the previously published model, is incorporated into the functional response for tumour-cell lysis by effector cells. The model exhibits bistability where a tumour-cell population threshold exists. A tumour with an initial cell population below the threshold can be controlled by the immune system and remains microscopic and asymptomatic called cancer without disease while that above the threshold grows to lethal size. Bifurcation analysis shows that (a) the chemotherapy-induced damage may cause a microscopic tumour, which would never grow to become lethal if untreated, to grow to lethal size, (b) applying chemotherapy alone requires a large dosage to be successful, (c) immunotherapy is ineffective, and (d) a combination of chemotherapy and immunotherapy can produce a synergistic effect on the outcome of a treatment.
Highlights
Cancer has become a leading cause of death worldwide
A tumour with an initial cell population below the threshold can be controlled by the immune system and remains microscopic and asymptomatic called cancer without disease while that above the threshold grows to lethal size
Chemotherapy is the use of drugs to destroy cancer cells, and immunotherapy is a treatment that enhances the immune system in fighting cancer
Summary
Cancer has become a leading cause of death worldwide. Cancer cells grow rapidly and have the ability to metastasize to other organs. Due to the intrinsic property of the ratio-dependent functional response, the mathematical model [18] is not differentiable at the tumour-free equilibrium. Wei [42] has proven that the tumour-free equilibrium is stable under a range of parameter values without the treatment terms in the mathematical model. Due to the lack of differentiability at the tumour-free equilibrium, the numerical simulation conducted by Wei [42] did not provide the result of the stability property of the tumour-free fixed point. The model with Beddington–DeAngelis-like functional response is expected to have a stable equilibrium of a microscopic tumour size.
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