The Study On A Review Of The Schisotomiasis Japonicum Dynamical Models And The Schisotomiasis Models With Time Delay

Schistosomiasis is a widely spread infection disease that has been jeopardizing human healthfor a long time. Mathematical models have been increasingly applied in the description ofSchistosomiasis. Thus, we introduce some typical models to summarize their advancement in thesecond charpter. Five types of mathematical models for schistosomiasis transmission wereidentified, with an emphasis on the key parameters to establishment of the models. These modelsare analyzed to gain insights into the qualitative features of the equilibria which allow thedetermination of the basic reproduction rateR0. The role and application ofR0in the filed ofschistosomiasis transmission assessment were reviewed. New insights in the transmissiondynamics of Schistosoma japonicum assessed byR0provided the qualitative approach todetermine the progress of the national control programme.From the lifestyle of schistosoma, it is necessary to study the impact of the prepatent periodon schistosomiasis transmission. However, there is little concern on the bifurcation of delayedschistosomiasis models. Therefore, the remainder of this thesis we mainly deal with dynamicalproperties of two kinds of delay differential equations. The main work is summarized as follows:In the third charpter, a schistosomiasis japonicum model is proposed that incorporates timedelay which represents the developmental time from cercaria penetration through skins of humanhosts to egg laying. By linearizing the system at the positive equilibrium and analyzing theassociated characteristic equation, the asymptotic stability of the equilibria is investigated. And itis proved that Hopf bifurcations occur when the time delay pass through a sequence of criticalvalue. Furthermore, the explicit formulae for determining the stability and the direction of theHopf bifurcation periodic solutions are derived by using techniques from the normal form theoryand Center Manifold Theorem. Some numerical simulations for verifying the theoretical analysisare also carried out.In the forth charpter, a two-dimensional system is studied that incorporates two time delayswhich are the incubation period of human and snail, respectively. Our purpose is to demonstratethat the time delays are harmless for stability of equilibria of the system. Further, sufficientconditions of stability of equilibria are obtained. Finally, our theoretical results are confirmed bynumerical simulation.