Stability analysis of an age-structured model of cervical cancer cells and HPV dynamics

Vitalii V. Akimenko; Fajar Adi-Kusumo; Vitalii V. Akimenko; Fajar Adi-Kusumo

doi:10.3934/mbe.2021308

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 6155-6177. doi: 10.3934/mbe.2021308

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Stability analysis of an age-structured model of cervical cancer cells and HPV dynamics

Vitalii V. Akimenko ^{1
,
,},
Fajar Adi-Kusumo ²

1.
Faculty of Computer Sciences and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, Kyiv 01030, Ukraine
2.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta 55281, Indonesia

Academic Editor: Sophia Jang

Received: 12 May 2021 Accepted: 07 July 2021 Published: 14 July 2021

Stability analysis of an autonomous epidemic model of an age-structured sub-populations of susceptible, infected, precancerous and cancer cells and unstructured sub-population of human papilloma virus (HPV) (SIPCV epidemic model) aims to gain an insight into the features of cervical cancer disease. The model considers the immune functional response of organism to the virus population growing by the HPV-density dependent death rate, while the death rates of infected, precancerous and cancerous cells do not depend on the HPV quantity because the immune system of organism does not respond to its own cells. Interaction between susceptible cells and HPV is described by the Lotka-Voltera incidence rate and leads to the growth of infected cells. Some of infected cells become precancerous cells, and the other apoptosis when viruses leave infected cells and are ready to infect new susceptible cells. Precancerous cells partially become cancer cells with the density-dependent saturated rate. Conditions of existence of the endemic equilibrium of system were obtained. It was proved that this equilibrium is always locally asymptotically stable whenever it exists. We obtained: (i) the conditions of cancer tumor localization (asymptotically stable dynamical regimes), (ii) outbreak of cancer cell population (that may correspond to metastasis), (iii) outbreak of dysplasia (precancerous cells) which induces the outbreak of cancer cells (that may correspond to metastasis). In cases (ii), (iii) the conditions of existence of endemic equilibrium do not hold. All cases are illustrated by numerical experiments.
- SIPCV epidemic model,
- age-structured model,
- HPV,
- stability analysis
Citation: Vitalii V. Akimenko, Fajar Adi-Kusumo. Stability analysis of an age-structured model of cervical cancer cells and HPV dynamics[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 6155-6177. doi: 10.3934/mbe.2021308

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Abstract

Stability analysis of an autonomous epidemic model of an age-structured sub-populations of susceptible, infected, precancerous and cancer cells and unstructured sub-population of human papilloma virus (HPV) (SIPCV epidemic model) aims to gain an insight into the features of cervical cancer disease. The model considers the immune functional response of organism to the virus population growing by the HPV-density dependent death rate, while the death rates of infected, precancerous and cancerous cells do not depend on the HPV quantity because the immune system of organism does not respond to its own cells. Interaction between susceptible cells and HPV is described by the Lotka-Voltera incidence rate and leads to the growth of infected cells. Some of infected cells become precancerous cells, and the other apoptosis when viruses leave infected cells and are ready to infect new susceptible cells. Precancerous cells partially become cancer cells with the density-dependent saturated rate. Conditions of existence of the endemic equilibrium of system were obtained. It was proved that this equilibrium is always locally asymptotically stable whenever it exists. We obtained: (i) the conditions of cancer tumor localization (asymptotically stable dynamical regimes), (ii) outbreak of cancer cell population (that may correspond to metastasis), (iii) outbreak of dysplasia (precancerous cells) which induces the outbreak of cancer cells (that may correspond to metastasis). In cases (ii), (iii) the conditions of existence of endemic equilibrium do not hold. All cases are illustrated by numerical experiments.

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