Existence Of Traveling Wave Solutions For A Class Of Waterborne Infectious Disease Model

Waterborne infectious diseases are infectious diseases that seriously endanger human life and health.They are mainly spread through unclean water and food,usually with acute onset and rapid spread,and may be life-threatening if not treated in time.According to the statistics of the World Health Organization,more than 50 million people worldwide die from infectious diseases caused by water pollution every year.Cholera is a typical water-borne infectious disease,which has many modes of transmission,including direct transmission between people and indirect transmission between people and unclean water sources.Since 1817,cholera has caused seven large-scale epidemics around the world,resulting in a large number of deaths,which directly threatened human health and survival and brought great harm to human society.Considering the influence of vaccination and time-delay factors on cholera epidemic,a delayed reaction-diffusion model with vaccination is established.When R₀>1,c>c^*,the upper and lower solution functions are skillfully constructed and the model is proved to have traveling wave solutions by using Schauder’s immovable point theorem;in addition,the asymptotic behavior of traveling wave solutions at positive infinity is discussed by constructing a suitable Lyapunov function.Finally,the model is verified to have no traveling wave solution by bilateral Laplace transform and Fatou’s theorem when the wave speed is less than the minimum wave speed.A class of water-borne infectious disease models with generalized non-local diffusion terms are investigated,and different functions are used to represent the incidence between people and between people and the environment,as well as the growth function of cholera germs.When R₀>1,c>c^*,the existence of traveling wave solutions for this model is discussed by constructing the upper and lower solution functions,combined with Schauder’s immobility point theorem,and then a suitable Lyapunov function is constructed to discuss the asymptotics of traveling wave solutions;when 0<c<c^*,the idea of the converse method is used to prove that there is no traveling wave solution for this model by bilateral Laplace transform and Fatou’s theorem.