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.

Two-sex logistic model for human papillomavirus and optimal vaccine

We formulate a two-gender susceptible–infectious–susceptible (SIS) model to search for optimal childhood and catch-up vaccines over a 20-year period. The optimal vaccines should minimize the cost of Human Papillomavirus (HPV) disease in random logistically growing population. We find the basic reproduction number [Formula: see text] for the model and use it to describe the local-asymptotic stability of the disease-free equilibrium (DFE). We estimate the solution of the model to show the role of vaccine in reducing [Formula: see text] and controlling the disease. We formulate some optimal control problems to find the optimal vaccines needed to control HPV under limited resources. The optimal vaccines needed to keep [Formula: see text] are the catch-up vaccine rates of 0.004 and 0.005 for females and males, respectively; 100[Formula: see text] is needed to reduce [Formula: see text] to its minimum value. To reduce the expenses for HPV disease and its vaccines, we need 100[Formula: see text] childhood vaccines (both genders) for the first 13–14 years and then gradually reduce the vaccine to reach [Formula: see text] at year 20. For adults (both genders), we need maximum rates (one) for the first 9 years, then [Formula: see text] for the next 3–4 years before reducing gradually to zero rate at year 20. Although the childhood vaccines provide very early protection strategy against HPV, its time to control HPV is longer than that for adult vaccines. Thus, full adults’ only vaccines for enough period is a viable choice to control HPV at minimal cost and short time.

Computational modeling of human papillomavirus with impulsive vaccination

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.