Studies On Epidemic Models With Acute And Chronic Stages For Which Diseases May Recrudesce At Chronic Stages

Many infectious diseases such as Hepatitis C and HCV have acute and chronic stages. This paper studies in detail several epidemic models with acute and chronic stages for which the diseases may recrudesce at chronic stages. We establish an MSIS model with acute and chronic stages for which the diseases may recrudesce at chronic stages, an SIVS model with vertical transmission for which the diseases can recrudesce at chronic stages, and an SIVS model with nonlinear incidence rate for which the diseases may recrudesce at chronic stages. Conditions for the existence and the stability of disease-free equilibria and endemic equilibria of each model are obtained by means of the stability theory of ordinary differential equations.The introduction describes the importance of the research on epidemic diseases by means of mathematical modeling, introduces briefly the elementary notions of epidemic dynamics, known models and frequently used approaches, and gives a short survey of recent work on epidemic models with acute and chronic stages. Chapter One lists some important theorems in the qualitative theory and the stability theory of ordinary differential equations.Chapter Two establishes an MSIS model with acute and chronic stages for which the diseases may recrudesce at chronic stages, discusses conditions for the existence and the stability of disease-free equilibria and endemic equilibria, and simulates numerically the effect of the recrudesce of the disease at chronic stages to the transmission of the diseases.Since some infectious diseases, such as hepatitis B, hepatitis C and AIDS, with acute and chronic stages for which the effectives can transfer from chronic stages to acute stages and for which the newly born children may partly carry viruses without being protected by their mothers, Chapter Three is devoted to the study of an SIVS model with this kind of vertical transmission and the diseases can recrudesce at chronic stages, and discusses the affection of the vertical transmission on the horizontal transmission. The results of numerical simulations show that the population of the endemic equilibria will increase when the proportion of the vertical transmission rises. Therefore, it is also an indispensable aspect to decrease the proportions of the vertical transmission to effective control of the infectious diseases.Chapter Four of this thesis studies an SIVS model with nonlinear incidence rate for which the diseases may recrudesce at chronic stages, which better fits the real world problems. The existence of endemic equilibria is proved by the intermediate theorem. Numerical simulations show that the endemic equilibria really exist and are asymptotically stable globally as the threshold R₀ >1. In addition, the influence of the protection of thesusceptible on disease transmission and the strategy of controlling the infective are discussed according to the results of numerical simulations.The main purpose of the establishment of epidemic models for which the diseases may recrudesce at chronic stages is to study the effect on the epidemic transmission of disease recrudesce at chronic stages. Theoretical and numerical results conclude that the population of the endemic equilibria will be greater when the diseases may recrudesce at chronic stages. Therefore active treatment of the infectives at chronic stages so as to prevent the disease recrudesce is an effective measure for the control of the epidemic transmission. In case that there are vertical transmissions, reducing the proportion of the vertical transmission may make a positive contribution.