The Predator-prey Models With Epidemics

Mathematical methods have been playing a prominent role in studying the spread of infectious diseases for a long time. Firstly, people use discrete time model to discuss the measles, then use differential equation model to discuss the spread of malaria, up to recently, bifurcation theory, chaos and some nonlinear methods have been used to study infectious diseases. So many mathematical methods make the study of infectious disease closer and closer to the reality, and more and more infectious diseases can be discussed by the mathematical models.Nowadays, people pay more attention to the emergence of a variety of zoonotic diseases. How to control the transmission of the diseases is becoming a very important research subject. These infectious diseases have some new characteristics different from others before, like being infected by preying, spreading within and across species, etc. How to involve these new features into the corresponding mathematical models is being considered by academia.Generally, ordinary differential equations can be used to present the predator-prey model with epidemics, including the new features. By researching the stability of equations, some conclusions about the spread of infectious disease and the control methods can be got.The thesis shows three kinds of predator-prey model with epidemics based on different backgrounds, and then discusses the dynamic of the system and the control method of the infectious diseases.In Chapter1, Introduction to epidemic model and predator-prey model, and the review of recent advances on these models are given. Our contributions to the models are introduced in the chapters that follow.In Chapter2, we use an infectious disease between shrimp and algae as a background to present a predator-prey model. This disease can spread in each one species, but can’t spread across species. Firstly, linear approximation and the Lyapunov Second Method are used to prove the stability of the nonnegative equilibrium. Secondly, thresholds in different steady state are calculated. Thirdly, the feasibility to control the spread of disease is analyzed.A predator-prey model based on the impact to human health caused by the addition of antibiotics in the feed of food animals is presented in Chapter3.Resistant bacteria not only spread between food animals, but also through the way of contact or prey on cross-species infection to human beings, and continue spread between humans. Aimed at the transmission characteristics, the process of resistant bacteria spread between animals and humans is analogy into infectious disease spread in the predator-prey model. In this model, there is only one infection disease, and it can spread from animals to human being. On the other hand, the necessary of controlling the addition of antibiotics in the feed, and the safe line of the addition are analyzed.In Chapter4, a predator-prey model with the incubation period of infectious diseases has been given, which is based on the transmission of Brucellosis between animals and human beings. The epidemics can be transmitted between preys, and also can be transmitted to predator from preys, but cannot be transmitted between predators. Furthermore, the effect of incubation period in the prey is considered. This transmission mode is the same with most kinds of zoonotic infectious diseases, and the threshold to control the transmission of such diseases is analyzed, finally, by discussion the key parameters the evidence and method are found to control further spread of infectious diseases.The thesis is then concluded in Chapter5.