Mathematical Modeling And Research Of The Pest Control And Treatment Of Infectious Diseases

Two kinds of mathematical model is presented in this paper,the one is the dynamicsmodel of rodent population with barren control,the second is the SEIS and SEIRS modelwith saturated treatment. the concrete research content is as follows:Firstly, under the barren control, density-dependent and nonlinear infection-rate rodentpopulation models are established. Although some scholars research the nonlinear infection-rate rodent population models, the considered infection rate is oversimplifed. Now wediscuss the existence of equilibrium to the rodent population dynamics model under thebarren control. At the frst, we make use of Routh-Hurwitz criterion to demonstrate localstability of system’s equilibrium points. Then, we utilize Dulac function to prove the globalstability of positive equilibrium and analyze analyzes the infuence of parameters on thedynamic change of rodent population, carried on the numerical simulation.Secondly, considering the diference of epidemic diseases’ initial outbreak and mid andlate outbreaks, the SEISmodel with bilinear infection rate as well as saturated treatment.For the model, the paper obtained the existence condition of basic reproductive rate, itsbifurcation and endemic equilibriums. Then we utilize Routh-Hurwitz criterion to get thegradual stability of disease-free equilibrium and the epidemic equilibrium. At last, we uselyapunov function to prove the global stability of each equilibrium point and carry on thenumerical simulation.Thirdly, considering that vaccination is a common way to control the epidemic. There-fore, we build up the SEIRS model with continuous vaccination and saturated treatmentrate. As same as SEIRS model, we got the existing conditions of basic reproductive rate, itsbifurcation and endemic equilibriums.And the gradually local stability of the disease-freeequilibrium and the epidemic equilibrium are proved. The global stability of disease-freeequilibrium and endemic equilibriums are proved By lyapunov function and geometric ap-proach method.