Study On Dynamical Behavior Of Biological Model

Biological mathematics is a subject that applies mathematical tools to the study of real life in different areas of biology. With the development of the times, biological mathematics not only covers a wide range and is a combination of other disciplines, but also itself has many branches. As one of the important branches, biodynamics has been an indication that human life is dependent on the research of biological mathematics. In the research process, we often need relevant knowledge of dynamics, and some methods that are applied to create the exact mathematical model on the basis of practical problems. Then through calculation, analysis and numerical simulation, we can get corresponding theories. These theories can be further applied to the society and human life, to better promote the progress of the world.The first part of this paper focuses on the research background and research status of the epidemic model, the branch of dynamics of infectious disease. In order to research epidemics much deeper, then this paper provides the knowledge of dynamical system.The second part of this paper introduces the thoughts of epidemics firstly, then expounds how to build a desirable epidemic model, and mainly introduces SIR epidemic model in which Kermack-Mckendrick is most classic.The third part of this paper mainly studies bifurcation analysis of a SIR epidemic model with nonmonotoic infection rate under linear treatment. Using qualitative theory of planar system and the specification theory, we find that the model has disease-free equilibrium and endemic equilibrium, which can be nodes, focuses, centers and saddles under different parameters. Through specification theory of Hopf bifurcation, exact variable substitutions and calculating the first Lyapunov coefficient, the system will occur supercritical Hopf bifurcation near the weak center or subcritical Hopf bifurcation.Moreover, this paper makes the other variable substitution into the normal form of BT bifurcation, further we can get bifurcation curves. Concretely, the behavior of the SIR epidemic models is: if and 0?d?1, the first Liapunov number ??0, then system has a supercritical Hopf bifurcation; if the first Liapunov number ??0, then system has a subcritical Hopf bifurcation. When are satisfied, the system is a cusp with codimension two, that is a BT singularity. At last, through relevant numerical simulations, they can prove theories.