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Recent research indicates that quiescent tumor cells are significantly less radiosensitive with a greater repair capacity than proliferative cells. In order to better predict patient-specific responses to radiotherapy, we develop a mathematical model with treatment terms to describe dynamical behaviors of tumor growth. The global stabilities of the tumor-free equilibrium, the tumor-present equilibrium and the corresponding sufficient criteria are obtained. In addition, we simulate volumetric imaging data from 12 head-and-neck cancer patients and estimate the patient-specific responses to radiotherapy. Results indicate that radiosensitivity of proliferative cells is a critical factor that determines a successful radiotherapy. By comparison with previous simulation results, we find that the model presented in this paper is more suitable to describe the radiotherapy procedure of head-and-neck cancer. Finally, we discuss the influences of different radiotherapy strategies on therapeutic effect. The results show that treatment strategies with large dose or short treatment cycle can obtain better treatment effect.

Cancer incidence rates are significantly different all over the world. Breast cancer is affected by many factors, the most important being genetics and lifestyle. The aim of this paper is to study the mutation mechanisms of breast cancer for Japanese women by fitting the incidence data of three high-quality population areas in Japan from 1985 to 2010. To achieve this goal, we have set up multi-stage models within the mathematical model of Moolgavkar, Venzon, and Knudson. Such models take both mutation rates and clonal expansion rates as parameters in each compartment into consideration. Based on our simulation outcomes, two to twelve driver mutations are sufficient in the pathway to female breast cancer in Japan. On the other hand, a previous study demonstrated that breast cancer in American women requires two to fourteen gene mutations to get a cancerous cell. Moreover, the 3-stage mathematical model is the optimal model as it fits clinical data very nicely for all affected cases of females in Japan and the US. The genetic instability has a prominent effect on the tumorigenesis of Japanese females caused by the first four driver mutations. The calculated results for Japanese women are compared with previous works for American women.

In this work, in order to identify the most effective measure and combinations of several measures to control influenza spread, we propose an SVEIAR influenza model with imperfect vaccination, media coverage and antiviral treatment. The global dynamics of the model is explored. Sensitivity analysis of the basic reproduction number and the endemic equilibrium is conducted to evaluate the effectiveness of influenza control measures. Furthermore, an optimal control problem incorporating the three measures is formulated to design optimal control strategies for influenza. The cost-effectiveness analysis reveals that combining the three measures is the most cost-effective among the strategies considered. Numerical simulations show that media propaganda can play a dominant role in curbing influenza transmission.

A stochastic predator–prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large environmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.