Traveling Wave Solution For A Diffusion SEIR Epidemic Model With Self-protection And Treatment

Outbreak of epidemics is always one of the most important factors which threaten hu-man life and health.In order to better research the nature of the spread of epidemics and take more effective measures to prevent from epidemics,a serious of models have been es-tablished and researched for different epidemics.Since an epidemic usually outbreaks in a small area and spreads to a larger area by diffusion of human,reaction-diffusion models can better fit in with this.Especially,traveling wave solution enables to show the transfer from one state to another(such as from disease-free state to endemic state).As a result,the study about traveling wave solution is meaningful for reaction-diffusion epidemic models.How-ever,most of epidemic models are non-monotone,so comparison principle does not hold,which makes the study about traveling wave solution more difficult for these models.With the advancement of information technologies,the access to information about epidemics has been becoming more abundant such that more and more people have been strengthening their self-protection against epidemics.The perfection of treatments has been effectively preventing from the spread of epidemics.Above all,for many epidemics with latent period,a diffusion SEIR epidemic model with self-protection and treatment has been established in Section 1.Moreover,we has worked on the existence and non-existence of traveling wave solution which tends to disease-free equilibrium state and endemic equilibri-um state at?,respectively.In Section 2,for proving the existence of traveling wave solution,the basic reproduction number R0 has been firstly obtained for corresponding reaction system.And the stability of disease-free equilibrium state and endemic equilibrium state has been also gained for corre-sponding reaction system.According to the principle eigenvalue of subsystem of infected compartments(the following is abbreviated as subsystem)and the relationship between R0with it,the minimal spread speed c~*has been found as R0>1.In the rest of this section,we have mainly proved the existence of traveling wave solution in case(1)R0>1,c>c~*and case(2)R0>1,c=c~*,respectively.Firstly,in case(1),sub and super solutions have been constructed by the principle eigenvalue of subsystem,and a closed convex set can be found.Moreover,a completely continue operator has been defined on this set via the cor-responding ellipse boundary-value problem.Secondly,it has followed from Schauder fixed point theorem that there exists traveling wave solution.Thirdly,sub and super solutions have been used to prove that traveling wave solution tends to disease-free equilibrium state at-?.And it has been implied traveling wave solution tends to endemic equilibrium state by constructing a suitable Lyapunov functional at+?.Finally,in case(2),we have used case(1)to approximate and tend to case(2),then the existence and asymptotic behavior can be similarly proved at+?.And asymptotic behavior can be shown via contradiction at-?.In Section 3,we have mainly researched the non-existence of traveling wave solution in case(1)R0<1,case(2)R0=1,and case(3)R0>1,c?(0,c~*),respectively.In case(1)and(2),the non-existence of traveling wave solution has also proved by contradiction.In case(3),we have firstly indicated traveling wave solution is exponential bounded.Then,it has followed from two-side Laplace transform and the principle eigenvalue of subsystem that there exists no traveling wave solution tending to disease-free and endemic equilibrium states at?.In Section 4,when the conditions in Section 2 have been satisfied,numerical analysis has displayed that there indeed exists a traveling wave solution tending to disease-free and endemic equilibrium states at?,and traveling wave solution is non-monotone.Finally,numerical analysis has also implied that it is a fact that both self-protection and treatment can reduce the spread of epidemic.