Traveling Waves And Entire Solutions In A Delayed Diffusive Epidemic Model

Since 1937, the theory of traveling wave solutions of the reaction diffusion equa-tions has been widely used to describe and explain different problems in physics, chemistry and biology. One of the most typical application is that the spatial spread of epidemic. In 1979, Capasso and Paveri-Fontana proposed an mathematical model for the 1973 cholera epidemic in the European Mediterranean region. Later, based on this, Capasso and Maddalena proposed and analyzed a reaction diffusion system with delay modeling the spatial spread of a class of bacterial and viral diseases: (?)There are many results about the system. For example, the existence of traveling waves fronts, the existence of the asymptotic speeds of spread, and the global asymp-totic stability of the traveling waves have been widely studied. These results are all based on d2= 0, namely, the small mobility of the infective human population is neglected. In the real life, however, the mobility of the infective human population is inevitable. So, in this paper we consider the case of d2> 0, that is to say, we put the impaction of the mobility of the infective human population into consideration. Firstly, we use the way of sub-sup solutions and Schauder's fixed point theorem to study the existence of the traveling waves of the system, when g is monotone and g is not monotone. Secondly, the asymptotic stability of traveling waves is derived by virtue of squeezing technique based on the comparison principle. At last, by combining sub-sup solutions, some comparison arguments and using the traveling-solution and spatially independent heteroclinic orbits of the system, we establish the existence of entire solution when g is monotone and g is not monotone.