As commonly known, cancer is one of the fatal diseases to which considerable attention needs to be paid. The purpose of the research concerned here was to form a mathematical model of the spread of cancer with chemotherapy and to know the dynamics of its solution. As for the stages in achieving the purpose, they were forming a mathematical model, determining the point of equilibrium, determining the basic reproduction number, analyzing the stability around the equilibrium point, and conducting numerical simulation with the parameters given. The pattern of how cancer cells spread could be modeled in the form of a mathematical equation according to the system of differential equation. From the system formed, an equilibrium solution and an analysis of the behavioral dynamics of the cell spread with treatment in the form of chemotherapy were attained. Simulation with graphs indicates that the growth rate of cancer cells influences the population of the said cells. The greater the growth rate of cancer cells, the greater the population of those cells. Besides, it is also obtained that the increasing dosage of the drug given with the limits allowed, the lower of those cancer cells.
<abstract><p><italic>Human Papillomavirus</italic> (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.</p></abstract>
The bacterium Vibrio cholerae is the cause of cholera. Cholera is spread through the feces of an infected individual in a population. From a mathematical point of view, this problem can be brought into a mathematical model in the form of Susceptible-Infected-Recovered (SIR), which considers the birth rate. Because outbreaks that occur easily spread if not treated immediately, it is necessary to control the susceptible individual population by vaccination. The vaccine used is Oral Vibrio cholera. For this reason, the purposes of this study were to establish a model for the spread of cholera without vaccination, analyze the stability of the model around the equilibrium point, form a model for the spread of cholera with vaccination control, and describe the simulation results of numerical model completion. Based on the analysis of the stability of the equilibrium point of the model, it indicates that if the contact rate is smaller than the sum of the birth rate and the recovery rate, cholera will disappear over time. If the contact rate is greater than the sum of the birth rate and the recovery rate, then cholera is still present, or in other words, the disease can still spread. Because the spread is endemic, optimal control of the population of susceptible individuals is needed, in this case, control by vaccination, so that the population of susceptible individuals becomes minimum and the population of recovered individuals increases.
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