Study On Traveling Wave Solution Of Water Borne Infectious Disease Model

Waterborne infectious diseases are caused by the contamination of pathogenic microorganisms in water sources and transmission through unclean water sources.Cholera is an acute intestinal infection caused by bacterium Vibrio cholera,which can lead to dehydration and even death if untreated,this outbreak is spreading rapidly and the initial case-fatality rate is very high.Cholera epidemic has multiple transmission pathways,including direct human-to-human transmission pathway and indirect human-to-environment transmission pathway.In this paper,we aim to study the existence of traveling wave solutions of the cholera model,in order to understand the dynamical behavior of cholera deeply.The traveling wave solution has translation invariance.It can effectively study the propagation velocity of the disease and predict whether the epidemic will spread in the future.It mainly includes the following two aspects:In Chapter 2,a partial differential equation model with diffusion term is constructed,and the unique positive disease-free equilibrium point and the unique positive local equilibrium point of the model are solved respectively.The basic regeneration number R₀ is obtained by using the method of regeneration matrix.First,we explore the existence of traveling wave solution when R₀>1 and c?c*,where c*is the minimum wave velocity,by using the Schauder's fixed point theorem associated with the upper-lower solutions.Moreover,the Lyapunov functional is used to show the boundary asymptotic behaviour of traveling wave solution;When 00>1,c?c*,it is linearized at the disease-free equilibrium point.If the linearized system meets the non negative nontrivial solution of the boundary condition,the upper and lower solutions of the two types of models can be constructed respectively,and then the auxiliary system is introduced,and the set of travel wave solutions is defined according to the constructed upper and lower solutions,A compact continuous operator suitable for Schauder fixed point theorem is defined on this set.After that,we prove the existence of the traveling wave solutions of this model,and also the boundary asymptotic behavior of traveling wave solution;When 0