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

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 G₁ 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.

A two-sex model of human papillomavirus infection: Vaccination strategies and a case study

Vaccination is effective in preventing human papillomavirus (HPV) infection. It still remains debatable whether males should be included in a vaccination program and unclear how to allocate the vaccine in genders to achieve the maximum benefits. In this paper, we use a two-sex model to assess HPV vaccination strategies and use the data from Guangxi Province in China as a case study. Both mathematical analysis and numerical simulations show that the basic reproduction number, an important indicator of the transmission potential of the infection, achieves its minimum when the priority of vaccination is given to the gender with a smaller recruit rate. Given a fixed amount of vaccine, splitting the vaccine evenly usually leads to a larger basic reproduction number and a higher prevalence of infection. Vaccination becomes less effective in reducing the infection once the vaccine amount exceeds the smaller recruit rate of the two genders. In the case study, we estimate the basic reproduction number is 1.0333 for HPV 16/18 in people aged 15-55. The minimal bivalent HPV vaccine needed for the disease prevalence to be below 0.05% is 24050 per year, which should be given to females. However, with this vaccination strategy it would require a very long time and a large amount of vaccine to achieve the goal. In contrast with allocating the same vaccine amount every year, we find that a variable vaccination strategy with more vaccine given in the beginning followed by less vaccine in later years can save time and total vaccine amount. The variable vaccination strategy illustrated in this study can help to better distribute the vaccine to reduce the HPV prevalence. Although this work is for HPV infection and the case study is for a province in China, the model, analysis and conclusions may be applicable to other sexually transmitted diseases in other regions or countries.

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

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.

An age-structured model of the human papillomavirus dynamics and optimal vaccine control

We formulate an age-structured model based on a system of nonlinear partial differential equations to assist the early and catch up female vaccination programs for human papillomavirus (HPV) types 6 and 11. Since these HPV types do not induce permanent immunity, the model, which stratifies the population based on age and gender, has a susceptible-infectious-susceptible (SIS) structure. We calculate the effective reproduction number [Formula: see text] for the model and describe the local-asymptotic stability of the disease-free equilibrium using [Formula: see text]. We prove the existence of an endemic equilibrium for [Formula: see text] for the no vaccine case. However, analysis of the model for the vaccine case reveals that it undergoes the phenomenon of backward bifurcation. To support our theoretical results, we estimate the age and time solution with the given data for Toronto population, when an early and catch up female vaccine program is adopted, and when there is no vaccine. We show that early and catch up female vaccine program eliminates the infection in both male and female populations over a period of 30 years. Finally, we introduce the optimal control to an age-dependent model based on ordinary differential equations and solve it numerically to obtain the most cost-effective method for introducing the catch up vaccine into the population.