Establishment And Study Of Infectious Diseases Model Considering Medical Resources

In this paper, we establish three mathematical models, using the theory of diferentialequation qualitative theory and branch theory to study the dynamic properties of infectiousdiseases, reveal the epidemic regularity and reasons, and fnd the optimal strategy to controlthe disease.Firstly, we discuss the SIVS epidemic model with vaccination and recovery rate func-tion, prove that when vaccine partial efective, the disease-free equilibrium always exist.Ifthe basic reproduction number Rσc<1, disease-free equilibrium is locally asymptoticallystable, under some conditions, the system from only has the disease-free equilibrium toappear an endemic equilibrium,then appear two endemic equilibrium, occur the backwardbifurcation.When the system have two endemic equilibria, one for the saddle point, thestability of another endemic equilibrium changes over parameters;if the basic reproductionnumber Rσc>1, the disease-free equilibrium is unstable, there exists unique endemic equi-librium.When the vaccine is completely efective,the disease-free equilibrium always exist. Ifthe basic reproduction number R0c<1,the disease-free equilibrium is locally asymptoticallystable, under some conditions, there are two endemic equilibria, the smaller of them is a sad-dle point, the larger endemic equilibrium changes its stability over parameters;if the basicreproduction number R0c>1, the disease-free equilibrium is unstable, there exists uniqueendemic equilibrium.When the vaccine is useless,the disease-free equilibrium always exist.Ifthe basic reproduction number R1c<1, the disease-free equilibrium is locally asymptoti-cally stable, under some conditions there are two endemic equilibria, the smaller of them is asaddle point,the larger endemic equilibrium is locally asymptotically stable, namely appearbistability phenomenon;if the basic reproduction number R1c>1, disease-free equilibrium isunstable, there exists unique endemic equilibrium and it is locally asymptotically stable.Bythe numerical simulation, when vaccine partial efective,the existence and stability of theequilibrium have been proved,and the result showed that increase the size of the medicalresources inputs can reduce the disease population, it may contribute to control the dis- ease.The presence of backward bifurcation,show that the basic reproduction number doesnot determine the disease disappear or not, it depends on the initial value of the infectedpopulation.Secondly, we consider the SIS mathematical model with nonlinear incidence rate andrecovery rate function,defne the basic reproduction number, discuss the existence and stabil-ity of the equilibrium, the disease-free equilibrium is always exists and locally asymptoticallystable. There may be two, one or zero endemic equilibrium of the system, under some con-ditions, the system has two endemic equilibria, the smaller of them is a saddle point, thestability of the larger endemic equilibrium changes over parameters.Furthermore, we get theexistence conditions of parameters for the existence condition of Hopf bifurcation and itsdirection at the larger endemic equilibrium.Finally, we study the SEIS dynamic model with the incubation period and recoveryrate function,obtain the basic reproduction number,the disease-free equilibrium is alwaysexists.When the basic reproduction number is less than1, the disease-free equilibrium islocally asymptotically stable; when the basic reproduction number is greater than1, thedisease-free equilibrium is unstable.When the basic reproductive number is less than1, thesystem may be have two, one or zero endemic equilibrium.Under some conditions, the en-demic equilibrium is locally asymptotically stable.And the system at most have two endemicequilibrium, and they can exist at the same time.According to the numerical simulation,weverify that the local stability of the disease-free equilibrium.By analysis the images,we canobtain that sufcient medical resources have a signifcant efect in the defence and controlof the disease.