Study On Some SIS Type Epidemic Models In The Heterogeneous Environment

The development of human society has been accompanying epidemic.Histor-ically,the outbreak and the spread of infectious diseases have made people suffer from big disasters.Although science and technology have been advanced substan-tially and medical facilities have been greatly improved,infectious diseases remain a great threat to human's health according to WHO recent claimant.Hence,there is an essential need for more information on the spatial and temporal spread of disease,as well as its distribution and control strategies.The most influential and theoretical model,SIR model,was given by Kermack and McKendrick in 1927.Since then,mathematical models have been becoming an important tool in iden-tifying disease transmission processes,assessing infection risk and prevalence,and optimizing control strategies.In the early stage,researchers mainly concentrate on the spatially-independent systems which,could only reflect the epidemiological and demographic process as the time goes on.To closely match the reality,researchers realize that the spatial transmission is important in the spreading of disease.And a great number of dif-ferential equations with spatially-dependent epidemic models were formulated and investigated.Recently,it has been commonly recognized that spatial diffusion and environmental heterogeneity are significant factors that should be considered in the spread of many diseases,e.g.,influenza,malaria,WNv.Besides,temporal periodic-ity,advection,media coverage and the limited hospital resources have also attracted much more attention in the field of epidemiology.The present dissertation focuses on the impact of spatial heterogeneity,peri-odicity,advection,nonlinear contact transmission rate and nonlinear recovery rate on the persistence and extinction of the SIS type epidemic models.The main research work in this dissertation is organized as follows.We begin with a brief introduction of the background and the development about the research topic in Chapter 1.Chapter 2 deals with an SIS epidemic reaction-diffusion-advection model with free boundary in heterogenous environment.Firstly,the global existence,unique-ness and regularity of the solution to free boundary problem are presented by using the contraction mapping principle.We present the basic reproduction numbers and their properties,and implications for the corresponding reaction-diffusion system-s.Subsequently,we introduce the risk index for the SIS epidemic model,which is changing with timet,and the analytical properties are exhibited.By means of the properties of the basic reproduction numbers and risk index R0F(t),sufficient conditions for the disease to be eradicated or to spread are given.When spreading happens,we derive the different asymptotic spreading speeds which are influenced by the advection intensity using the semi-wave method.Numerical simulations il-lustrate the impacts of the advection intensity and the expanding capability on the spreading fronts.This result is quite different from that have been obtained in spatially-independent systems or spatially-dependent systems in fixed domains.Chapter 3 is devoted to the SIS epidemic reaction-diffusion-advection model with free boundary in time-periodic heterogeneous environment.We present the basic reproduction number and the explicit formula for special cases.Hereby,we define the risk index R0F(?)for the free boundary problem by the spectral radius of the next infection operator,which is related to the eigenvalues of correspond-ing periodic-parabolic eigenvalue problems.By employing the maximal principle,upper-lower solutions methods,the spectral theory and some techniques for partial differential equations,the sufficient conditions for the spreading and vanishing of the disease are given.When spreading happens,the asymptotic spreading speeds induced by advection for the left and right fronts are also obtained.Simulation-s given in last section imply that the advection intensity,the diffusion rate and the expanding capability also affect the transmission machanism of the infectious disease.Chapter 4 discusses a new diffusive SIS epidemic model incorporating the medi-a coverage impact in the heterogeneous environment.The media coverage impact is described by a nonlinear function which represents the disease contact transmission rate.Firstly,we introduce the basic reproduction number R0D that is dependent on media coverage index and diffusion rate of infected individuals by the variation-al method.Subsequently,we obtained the existence and asymptotic stability of disease-free equilibrium and endemic equilibrium.More specifically,by using the method of upper and lower solutions and its associated monotone iterations,togeth-er with the classical semigroup theory and the strong maximal principle,we prove that the disease-free equilibrium is globally asymptotically stable when RoD<1;for R0D>1,we verify the existence of endemic equilibrium,and that the endemic equilibrium is globally asymptotically stable in the special case ds = dI.Finally,simulations also show that if we strengthen the intensity of the mass media report-s,then the infection risk will become smaller,and the epidemic will be controlled rapidly and effectively.Chapter 5 is contributed to a novel SIS epidemic model with nonlinear recovery rate which captures the impact of spatial heterogeneity of environment and avail-able resource of the public health system on the persistence and extinction of the infectious disease.Firstly,we introduce the thresholds R0*and R0*,which are de-pendent on the minimal and maximal recovery rates,respectively,by the variational method.In addition,we establish the dynamical behaviors of the deterministic epi-demic model in term of the thresholds R0*and R0*.Namely,the existence and the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are obtained by the upper and lower solutions method and its associated itera-tion sequences,together with the multiply-multiply-subtract-integrate technique.Theoretical results and numerical simulations show that the preparedness of the minimum number of hospital beds in case of an emerging infectious disease is a crit-ical issue.Our results can help the public health agencies or administration arrange the appropriate number of hospital beds so as to optimize the allocation of public health resources.At last,we summarize the results in this thesis,and also present some problems for future research in Chapter 6.