Mathematical Models Of Disease Transmission In The Predator-Prey System And Its Kinetics

In recent years, there have a great development after the pioneering work of Lotka and Volterra focusing on population dynamics, and Kermack and McKendrick majoring in epidemic dynamics, those help us to understand the importance in exploring the resource and reducing disease transmission. In real word, species does not exist alone, and they spread the disease, so we should merge these two areas of research, but little attention has been paid so far. In this paper, we studied the disease transmission in the predator-prey system, the mathematical models were established, through mathematical analysis and numerical simulation, we obtain the following main results:1 .When the disease is spread among the prey population, a predator-prey model with disease in the prey is established, the boundness of solution and the sufficient condition of locally asymptotically stable of the equilibria are studied. Furthermore, the global stability of the equilibria, and the sufficient condition of boundary equilibrium and positiveequilibrium global stability are also obtained. We find the basic reproduce number R₀,provide the basic theory of controlling disease.2.When the predator spread disease, a predator-prey model with disease in the predatoris established. Boundedness of solutions and stability of the equilibriua of the model are discussed. Morever, we assume that the environment perturbations are of white noise type, then the stochastic differential model is established, and we find a Lyapunov function,suggesting that the stability of the positive equilibrium is robust with respect to stochastic perturbations.3.For a simple Eco-Epidemiology modelcan have complex dynamic behavior, such as chaos.4.To control the chaotic behavior of the above system, we provide three possible methods: incorporate the refuge effect; the harvest effect; the delay effect, mathematical models were established, through mathematical analysis and numerical simulation, we find the three effects can all prevent oscillations and improve stability in the above system. Finally, we discussed the effects of the incidence rate, response function and spatial factor that affect the dynamics of the systems in detail. Based on the works of early researchers, we suggest our effort in the future.