Mathematical analysis of a SIPC age-structured model of cervical cancer

Eminugroho Ratna Sari; Fajar Adi-Kusumo; Lina Aryati; Eminugroho Ratna Sari; Fajar Adi-Kusumo; Lina Aryati

doi:10.3934/mbe.2022281

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 6013-6039. doi: 10.3934/mbe.2022281

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Mathematical analysis of a SIPC age-structured model of cervical cancer

1.
Department of Mathematics, Universitas Gadjah Mada, Yogyakarta 55281, Indonesia
2.
Department of Mathematics Education, Universitas Negeri Yogyakarta, Yogyakarta 55281, Indonesia

Academic Editor: Sandra Saliani

Received: 22 February 2022 Revised: 26 March 2022 Accepted: 31 March 2022 Published: 12 April 2022

Human Papillomavirus (HPV), which is the main causal factor of cervical cancer, infects normal cervical cells on the specific cell's age interval, i.e., between the $ G_1 $ to $ S $ phase of cell cycle. Hence, the spread of the viruses in cervical tissue not only depends on the time, but also the cell age. By this fact, we introduce a new model that shows the spread of HPV infections on the cervical tissue by considering the age of cells and the time. The model is a four dimensional system of the first order partial differential equations with time and age independent variables, where the cells population is divided into four sub-populations, i.e., susceptible cells, infected cells by HPV, precancerous cells, and cancer cells. There are two types of the steady state solution of the system, i.e., disease-free and cancerous steady state solutions, where the stability is determined by using Fatou's lemma and solving some integral equations. In this case, we use a non-standard method to calculate the basic reproduction number of the system. Lastly, we use numerical simulations to show the dynamics of the age-structured system.
- cervical cancer,
- cells dynamics,
- age-structured model,
- force of infection,
- basic reproduction number,
- global stability
Citation: Eminugroho Ratna Sari, Fajar Adi-Kusumo, Lina Aryati. Mathematical analysis of a SIPC age-structured model of cervical cancer[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6013-6039. doi: 10.3934/mbe.2022281

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Abstract

Human Papillomavirus (HPV), which is the main causal factor of cervical cancer, infects normal cervical cells on the specific cell's age interval, i.e., between the $ G_1 $ to $ S $ phase of cell cycle. Hence, the spread of the viruses in cervical tissue not only depends on the time, but also the cell age. By this fact, we introduce a new model that shows the spread of HPV infections on the cervical tissue by considering the age of cells and the time. The model is a four dimensional system of the first order partial differential equations with time and age independent variables, where the cells population is divided into four sub-populations, i.e., susceptible cells, infected cells by HPV, precancerous cells, and cancer cells. There are two types of the steady state solution of the system, i.e., disease-free and cancerous steady state solutions, where the stability is determined by using Fatou's lemma and solving some integral equations. In this case, we use a non-standard method to calculate the basic reproduction number of the system. Lastly, we use numerical simulations to show the dynamics of the age-structured system.

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