Dynamics Of Nonlocal Epidemic Models With Free Boundary

Infectious diseases caused great threat to human survival.A large number of mathematicians predict the dynamic behavior of infectious diseases by establishing a mathematical model,and they provides a theoretical basis for preventing and controlling disease by analyzing mathematical models.In 1937,infectious diseases(such as,cholera et al.)is prevalent in European Mediterranean,the research found that once infected with the disease,human will make the number of agents in the environment increase due to human activities.In order to study its dynamics for such transmission mechanism,Capasso and Paveri-Fontana presented a cooperative ODE model to characterize the spread of the disease.However the ODE model ignores the influence of spatial diffusion.Assume that the agents diffuse randomly and the diffusion rate of infective human is relatively small so that it can be ignored,by introducing free boundary conditions,Ahn[1]considered the spatial spread of the disease over changing area.Due to the agents not only spread locally,but also spread to relatively farther location,and so the dispersal of agents may be better described by a nonlocal diffusion operator.At the same time,the growth of the agents at somewhere is not only caused by the infective human here,it may also be caused by the infected human from other places,and so it is more practical to consider the problem with nonlocal effect.This paper will mainly consider the influence of nonlocal diffusion and nonlocal effect on the spread of disease.Firstly,we consider the asymptotic spreading speed of the spreading front,which is described by Ahn et in[1].By using the corresponding semi-wave problem,we obtain the precise estimate of spreading speed.Secondly,we consider a partially degenerate epidemic model with free bound-ary and the nonlocal effect.We obtain the global existence and uniqueness of the solution to this problem,and then discuss the property of the principal eigenvalue of the corresponding eigenvalue problem,namely,the monotonicity of the principal eigenvalue about the size of the area.By this property,we deduce the criteria for the disease spreading and vanishing.Our results show that the size of the initial area of the problem with nonlocal effects making the disease break out is larger than that of problem without nonlocal effects.Thirdly,following the approach of the work by Cao et al.,we propose a partial degenerate epidemic model with free boundary and nonlocal diffusion,and study the dynamic behavior of the disease.We prove the global existence and uniqueness of the solution,and then obtain a spreading-vanishing dichotomy of disease and its criteria.Our results show that when the basic reproduction number of ODE model R0>1,if the nonlocal diffusion rate of agents d>0 is sufficiently small,then the disease will spread regardless of the initial data.This means that nonlocal diffusion of the infectious agents may increase the chance of epidemic disease spreading,compared with the work by Ahn et al.Finally,based on the former part,we analyze the partial degenerate epidemic model with nonlocal diffusion and nonlocal effect.Similarly,we get the global existence and uniqueness of the solution to this problem,and then give a spreading-vanishing dichotomy of disease and its criteria.But we found that after considering the influence of the nonlocal effects,when R0>1,the disease will not always spread if the nonlocal diffusion rate of agents d>0 is sufficiently small.We find that the nonlocal effect may decrease the chance of epidemic disease spreading,compared with the results in the former part.