Modelling And Dynamics Of The Distributed State-structure Models For Infectious Diseases On Complex Networks

In the thesis,a series of distributed state-structured epidemic models based on complex networks are proposed and investigated.These epidemic models include a discrete state-structured model,a distributed delay state-structured model and a state-structured model in which the disease can spread vertically as well as horizontally.Our models allow newly infected individuals to distribute into any infected state and in-corporate possible transfers(deterioration or amelioration)between any two different infected states.These models can simulate complex propagating process of epidemic diseases,e.g.human immunodeficiency virus(HIV)and Tuberculosis(TB)with an-tiretroviral therapy(ART)or antibiotic treatment.In the first chapter,we introduce the brief history of mathematical epidemiology and the stability theory of epidemic dynamical system,especially the graph-theoretic method for constructing the Lyapunov function is significant to the following global stability analysis.In the second chapter,we propose and investigate a distributed state-structured epidemic model in which the state structure is discrete.By the next generation method,the basic reproduction number is derived.We obtain some significant results on the basic reproduction number,e.g.its biological interpretation and upper-lower bound estimation.We also prove that the basic reproduction number is a sharp threshold parameter:if basic reproduction number is less than or equal to 1,the disease free equilibrium is globally asymptotically stable,the disease will eventually die out;if basic reproduction number is greater than 1,the disease free equilibrium is unstable,but there is a unique endemic equilibrium which is globally asymptotically stable,the disease will persist at a positive level.The impact of parameters to endemic equilibrium and the impact of state-transfer to basic reproduction number are discussed through numerical simulations.In the third chapter,we investigate a class of distributed state-structure model with both horizontal and vertical transmission.This model can be used to describe the complex progress of infectious diseases which proportionally passed on to the offspring of the infected parentage.We derive the basic reproduction number Ro =Rh + Rv,where Rh and Rv are the horizontal reproduction number and vertical reproduction number,respectively.The requirement of vertical reproduction number is Rv<1 which means that vertical transmission cannot sustain an epidemic by itself.The global dynamics are completely determined by basic reproduction number:if Rv<1 and R0 ? 1,the disease-free equilibrium is globally asymptotically stable and the disease always dies out;if Rv<1 and R0>1,the disease-free equilibrium is unstable,but there is a unique endemic equilibrium which is globally asymptotically stable,the disease persists a positive level among the population.The special case without horizontal transmission and the impact of vertical transmission to the global dynamics are discussed by numerical simulations.In the last chapter,we consider a class of distributed state-structure model with distributed delay which dramatically increases the complexity of the model and makes the analysis more challenging.This delay model can be used to simulate different mechanisms of epidemics,such as age structure,latent period,seasonal or diurnal variations,interactions across spatial distances or through complicated paths.We prove that this realistic model retain most of the key properties of the previous one in chapter 2,including the threshold behavior and particularly the global stabilities of the equilibria.