Research Of Dynamics And Profiles Of Reaction-Diffusion Cholera Model With Bacterial Hyperinfectivity

Cholera is defined as an acute intestinal infectious disease.It not only threatens human health and life,but also leads to severe social decline and even national extinction.With the development of science,to some extent,though,the cholera epidemic has been effectively contained.However,in general,the epidemic and prevention of cholera epidemic is still serious.It is known that the research on the mathematical models of the cholera epidem-ic is helpful to deepen people’s understanding of the transmission mechanism of cholera.Therefore,the main contents are as follows:1.Based on a reaction-diffusion cholera model with distinct dispersal rates,the model to be studied in the second chapter of this paper is established by considering the bacterial hyperinfectivity.Firstly,we establish the well-posedness of the model.To cope with the lack of compactness of solution semiflow,we verify the asymptotic smoothness of semi-flow implied with κ-contraction condition,and prove the model possesses a global attractor.Then,we identify the basic reproduction number,R0,for the model and it is identified as a threshold,predicting whether or not the disease extinction and persistence will occur.R0 is also equivalently characterized by some principle spectral conditions of an associated elliptic eigenvalue problem and the concrete expression of R0 is also given.Next,the asymptotic profiles of the positive steady state are investigated for the cases when the dispersal rate of the susceptible humans is small and large.Finally,in a homogeneous case,we also confirm the global attractivity of a unique positive equilibrium by utilizing the technique of Lyapunov function when R0>1.2.The third chapter of this article gives a supplement to the recent work of Wang and Wang(2019)in the sense that:(1)for the critical case where R0=1,cholera-free steady state is globally asymptotically stable by proving the local asymptotic stability and global attractivity;(2)in a homogeneous case,the positive constant steady-state is globally asymp-totically stable with additional condition when R0>1 by Lyapunov function.