On The Study Of An SEIV Epidemic Model And A Turbidostat Model

Some variable quantities in the epidemic models or microorganism models will change as the change of time, which will have an effect on the whole models. In the epidemic models, for instance, we considers the vaccination, and after the susceptible person vaccinates, then a part of them can produce the immune body, that is immunity, which will diminish gradually even vanish as one moves backward in time. Also, in the cultivation of microorganism, duing to the change of time, the concentration of microorganisms will get higher and higher, which will cause a series of adverse effects, such as the substrate inhibition, a loss of the mass transfer efficiency caused by higher viscosity, and so on. What particularly serious is that many secondary metabolite of microbial are suppressed by highly concentrated glucose, the carbohydrate and the azotic compound degradation product, and then give an impact on the microbial yield. Therefore, it is realistic to take into account such variable quantities which affect the model, thus make our model more realistic and more meaningful. The primary coverage of this article may summarize as follows:In the second chapter, we consider an SEIV epidemic model with vaccination and vertical transmission. The natural birth rate and death rate are assumed to be identical and the immunity acquired by preventive vaccine or by infection will gradually wane after a period of time. We provide the basic reproduction number R0 of the system, the sufficient conditions for the stability of equilibrium is derived and the existence of bifurcation are analyzed.In the third chapter, we present and analyze a turbidostat model with variable dilution and impulsive feedback control. When the microorganism concentration achieves the threshold value,the impulsive control will occur. According to the types and the preset values of positive equilibrium for the system without impulsive effect and using the existence criteria of periodic solution of a general planar impulsive autonomous system, we get the conditions for the existence of periodic solution of order one. It is shown that the system either tends to a stable state or has a periodic solution. Finally, we obtain the conditions for stability of periodic solution of order one.