Mathematical Modeling And Numerical Simulations Of Avian Influenza

Avian influenza is a zoonotic disease caused by the transmission of the avian influenza A virus. Avian influenza viruses normally infect only specific species, but have, on rare occasions, crossed the species barrier to infect humans, such as avian influenza A H5N1 and H7N9 viruses. After the outbreak of the first avian influenza A virus (H5N1) in Hong Kong in 1997, the avian influenza A H5N1 virus has caused more than 500 human infections worldwide with nearly a 60% death rate; And more than 400 human cases were caused by avian influenza A H7N9 with a death rate of nearly 40% since it was firstly reported in China in 2013. Avian influenza has caused the great attention of the world health organization and has been listed as one of the legal infectious diseases in China because of its high morbidity and mortality. Many countries have taken the appropriate measures to control the prevalence of avian influenza. In this thesis, we construct three avian influenza dynamics transmission models according to the modeling thought of epidemic dynamics method, combining the characteristic of the spread of avian influenza and three different factors.This thesis consists of five chapters. The first chapter introduces the background of avian influenza, the current research status at home and abroad of avian influenza models and the main research content.In Chapter 2, we consider the psychological effect in human population on the transmission rule of avian influenza. Cross-sectional surveys conducted in Thailand and China after the outbreaks of the avian influenza A H5N1 and H7N9 viruses show a high degree of awareness of human avian influenza in both urban and rural populations, a higher level of proper hygienic practice among urban residents, and in particular a dramatically reduced number of visits to live markets in urban popula-tion after the influenza A H7N9 outbreak in China in 2013. In this paper, taking into account the psychological effect toward avian influenza in the human population, a bird-to-human transmission model in which the avian population exhibits satura-tion effect is constructed. The dynamical behavior of the model is studied by using the basic reproduction number. The results demonstrate that the saturation effect within avian population and the psychological effect in human population cannot change the stability of equilibria but can affect the number of infected humans if the disease is prevalent.Our numerical simulations are given to support the theoretical results and sensitivity analyses of the basic reproduction number in terms of model parameters are performed to seek for effective control measures for avian influenza.It should be noted that incubation periods of avian influenza virus in avian and human population are different and the survival probability of infectives during incubation period is different too. In Chapter 3, we take account of the incubation periods of avian influenza A virus and the survival probability of infectives on the influence of endemic law of avian influenza. Due to lag effect of incubation peri-od, we construct a time delay differential equation with different time delays. By analyzing the dynamical behavior of the model, we obtain the threshold value for the prevalence of avian influenza and investigate the local and global asymptoti-cal stability of equilibria of the system. And numerical simulations confirmed our theoretical results.Related study has shown that the possibility that migrant birds are viewed as the original infection source is the largest. Migrant birds do not subject to constant growth by carrying capacity of environment. In Chapter 4, we consider the effect of the growth of avian population on the transmission rule of avian influenza. In terms of different growth laws of the avian population, that is the logistic growth and Allee effect, We construct two avian influenza bird-to-human transmission models and analyze their dynamical behavior completely. We obtain the threshold value for the prevalence of avian influenza and investigate the local or global asymptotical stability of each equilibrium of these systems by linear analysis or combining Lia-punov function method and LaSalle’s invariance principle, respectively. Moreover, we obtain necessary and sufficient conditions for the occurrence of periodic solutions in the avian influenza system with Allee effect of the avian population.In chapter 5, we summarize the current work and set the research targets in the future.