Qualitative Analysis Of Some Classes Of Infectious Disease Models With Reaction Or Diffusion Effects

To better understand and control the spread of infectious diseases,in this thesis,we present three types of infectious disease models with different characteristics:two network-based models coupling epidemic spread and information diffusion,a diffusive influenza model with multiple strains,and a class of predator-prey type ecoepidemiological models;and conduct a relatively comprehensive study on their dynamical behavior.We mainly focus on the qualitative structure,uniform persistence,stability and bifurcation phenomena of these nonlinear systems,such as the existence and stability of equilibria and periodic orbits,the existence and robustness of heteroclinic connections,and various types of bifurcation(namely,Hopf bifurcation,bifurcation of heteroclinic orbits and saddle-node bifurcation).The main content of the thesis consists of three parts,which is stated as follows.Firstly,we give a qualitative analysis of two network-based models coupling epidemic spread and information diffusion: the interplay model and the epidemic control model.More specifically,we obtain the existence of the disease-free equilibrium,endemic equilibrium and synchronization manifold,and their global asymptotic stability.Furthermore,we perform some numerical simulations to complement the theoretical analysis above.Secondly,we are concerned with the existence of traveling wave solutions of a complex reaction-diffusion system describing the spatiotemporal spread of influenza with multiple strains.By introducing an auxiliary system and using Schauder's fixed-point theorem,we carry out a limiting argument,and establish the existence of semi-traveling waves starting from the disease-free equilibrium.Conditions for the existence of two other types of traveling waves,i.e.,strong and weak(persistent)traveling waves,are obtained by constructing an appropriate Lyapunov functional and applying persistence theory of dynamical systems.We further discuss several situations in which semi-traveling waves starting from the disease-free equilibrium do not exist,and give an estimation of minimal wave speed for influenza transmission.Finally,we analyze a class of eco-epidemiological models where prey is subject to Allee effect and infection.We establish the existence,uniqueness,positivity and uniformultimate boundedness of the solutions for the proposed system in the positive octant.For some subsystems,we use and develop the Conley index and restricted Conley index to determine the existence of the bifurcation points(Hopf bifurcation point and bifurcation point of heteroclinic orbits)and heteroclinic orbits(cycles)and to show the robustness of heteroclinic orbits.We show that the strong Allee effect can create a separatrix curve(or surface),leading to multi-stability.We find that the heteroclinic cycles form a heteroclinic network and identify an interior periodic orbit by applying Poincar?e map and bifurcation theory.