A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results

Attractor of a nonlinear hybrid reaction–diffusion model of neuroendocrine transdifferentiation of human prostate cancer cells with time-lags

Prostate cancer is a serious disease that endangers men's health. The genetic mechanism and treatment of prostate cancer have attracted the attention of scientists. In this paper, we focus on the nonlinear mixed reaction diffusion dynamics model of neuroendocrine transdifferentiation of prostate cancer cells with time delays, and reveal the evolutionary mechanism of cancer cells mathematically. By applying operator semigroup theory and the comparison principle of parabolic equation, we study the global existence, uniqueness and boundedness of the positive solution for the model. Additionally, the global invariant set and compact attractor of the positive solution are obtained by Kuratowski's measure of noncompactness. Finally, we use the Pdepe toolbox of MATLAB to carry out numerical calculations and simulations on an example to check the correctness and effectiveness of our main results. Our results show that the delay has no effect on the existence, uniqueness, boundedness and invariant set of the solution, but will affect the attractor