Establishment And Analysis Of An Epidemic Model With Treatment

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 dynamic model with nonlinear incidence rate and maximumtreatment capacity, prove that if the infective level at which the healthcare system before thecapacity reaches, the disease-free equilibrium is stable. Under some conditions, it is shownthat there exists an unstable saddle point and an asymptotically stable endemic equilibrium,that there will be bistability phenomena. If the infective level at which the health care systemafter the capacity reaches, under some conditions, there exists an unstable saddle point whilethe stability of another endemic equilibrium changes with the changing of parameters when itexist. Furthermore, we get the conditions of parameters for the existence of Bogdanov-Takensbifurcations. Secondly, we consider the mathematical model with saturated incidence rateand sectional treatment function, prove that when the infective level at which the healthcaresystem before the capacity reaches, if the basic reproduction number is less than1, thedisease-free equilibrium is stable, if the basic reproduction number is greater than1, thereexists an asymptotically stable endemic equilibrium, when the infective level at which thehealth care system after the capacity reaches, under some conditions, there exist two endemicequilibria. Finally, we study the dynamic behaviors for a class of vector-borne diseases model,obtain that if the basic reproduction number is less than1, there exists a unique disease-freeequilibrium, which means the disease will die out fnally. If the basic reproduction numberis greater than1, then there exists a unique endemic equilibrium which means the diseasewould become epidemic. And also prove the global stability of the disease’s disease-freeequilibrium and endemic equilibrium, calculate the critical value of each way, thus it playsa positive role to control disease.