Complex Dynamic Behavior Of Several Infectious Disease Models

In this paper,by analysizing the dynamic nature of two epidemic models, we getwhen the infectious disease will ultimately be cured, permanence , when the number ofinfectious will ultimately be stable. This study is divided into two parts.In the first part, we develop a three dimensional compartmental model to investigatethe impact of media coverage to the spread and control of infectious diseases. Stabilityanalysis of the model shows that the disease-free equilibrium is globally-asymptoticallystable if a certain threshold quantity, the basic reproduction number ((?)₀), is less thanunity. On the other hand, if (?)₀ > 1, it is shown that a unique endemic equilibrium appearsand it is locally-asymptotically stable. Numerical simulations suggest that when (?)₀ > 1,the effective media coverage stabilizes the oscillation, and less number of the individualsbecome infected in the course of transmission, which contributes to the prediction andcontrol of the outbreaks of infectious diseases.A mathematical model for a disease with a general exposed distribution, the possibil-ity of relapse and nonlinear incidence rate is proposed in the second part. By the methodof Lyapunov functionals, it is shown that the model's global dynamics are completelydetermined by the basic reproduction number (?)₀. More specifically, it is shown that thedisease dies out if (?)₀≤1 and that the disease becomes endemic if (?)₀ > 1. Applicationsare also made to the special case with a discrete delay and the result confirms that theendemic equilibrium is globally asymptotically stable as suggested.