Study On Dynamical Models Of Epidemic Diseases With Nonmonotone Incidence Rate

Over the past decades, infectious disease is the main disease of harming human. With the improvement of health facilities, and the progress of the level of medical treatment ,human has made a great breakthrough in the prevention and control of infectious disease. Nonetheless, infectious disease is still not disappear in some backward underdeveloped areas , even very rampant. So the research of the pathogenesis of infectious diseases,propagation and preventive measures has the very vital significances.Commonly the research method used by workers is converting the actual situation to the mathematical model, and theoretically analysis influence of epidemic diseases. Through data analysis, the conclusions will be more realistic. The method of theoretical qualitative research of infectious diseases is called infectious dynamics method. Over the past decades, the dynamics research of infectious diseases has developped rapidly, and theoretical analysis is more perfect. People used to have a lot of analysis method such as:Lyapunov function method, LaSalle invariability theory, Characteristic root method, etc.Dynamical models of epidemic diseases with nonmonotone incidence rate are the clas-sical models of infectious diseases. But, the models have many limitations. Researchers usually improve infectious disease model through the methods of improved infection rates, and study the dynamics properties of outbreaks of disease or destroy according to the dynam-ics method. We all know that the nonlinear infection rates have wider dynamic properties than linear infection rates in the model. In this paper, we hypothesis that the infection rate is nonlinear infection rates, and establish three kind of infectious disease model. Through comparing basic reproductive number (decide whether the disease developped endemic dis-eases),we have the conclusion:a variety of epidemiological models without vital dynamics (i.e., births and deaths) have similar repertoires of dynamical behaviors,In what follows, the fractions of the population that are suceptible, exposed (but not yet infectious), infectious, and recovered with immunity, are denoted by S, E, I, and R, respectively. First,in order to simplify research, we give some basic assumptions(1) The meaning of infection rates:Positive parameter "α" relys on patients, environ- mental conditions and the ability of infectious germs, etc. Positive parameters "β" denotes the probability of infected after susceptible persons contact with infected. "βf(S)" denotes an infected person has the non-linear infectious, and f(S) satisfies the following conditionsA1)When S∈[0,∞), there are f(S)≥0;f(0)= 0.A2)When f(S)∈C1 [0,∞), S∈(0,∞), there are f'(S)> 0.A3)Intheinterval (0,K), thefunctionof g(S)=βf(S)Iαhas a unique extreme point(2) Suppose:disease spread quickly in a closed environment, and pop in a relatively short time. Population has no birth and death in the short-term or birth and death is able to balance.Disease mortality can be neglected. And,suppose that the total population is a constant.Based on these assumption, we constructed the following epidemiological models:The first model:Infectious disease model of having no immunity note:the constant "p" is the recovery rate,-is average recovery,that is,the average infected period. Using the method of characteristic root and Routh-Huriwiz criteria,we can get the following results: The second model:The epidemic model with immunity note:The positive constant "p" is the proportion of the recovered in the patients,that is ,removed rate coefficient. The positive constant "p" is the probability of lossing immunity. Using the method of characteristic root and Routh-Huriwiz criteria,we can get the following results:The third model:The epidemic model with delay. note:r is fixed recovery. Using the method of characteristic root and Routh-Huriwiz criteria,we can get the following results:According to the table,we will find,With the variation of parameters,these models have similar dynamics nature:1)whenα=1,Model has a threshold condition Ro. R0<1:Diseases will gradually disappear; R0>1,infectious will remain popular,arrive the only stable equilibrium position,and become endemic.2)whena=1/2, Threshold conditions disappear. Infectious will remain popular,arrive the only stable equi-librium position,and become endemic.3)whenα= 2,R0< 1, Diseases will gradually disappear; R0> 1:Situation is more complicated.For some initial conditions,diseases will gradually disappear; But under certain conditions,Infectious will remain popular. And under some special conditions,Epidemics will appear stable periodic oscillation.