Abstract
In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.
Highlights
It is well known that Poyang Lake located in the middle and lower reaches of the Yangtze River is the current largest freshwater lake in China
There are first-level and second-level national protected precious rare aquatic animals such as white-flag dolphin, cowfish, chinese sturgeon, hilsa herring and so on in Poyang Lake, making it known as the treasury of fishery resources and the fish species genetic base with a significant position in the ecology system of the fish industry of Yangtze River reaches [1]
In this paper, according to the above biological background, we investigate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and study dynamical behaviors of the model
Summary
Wang and Ma [18] investigated a nonautonomous SIS epidemic model with toxicant influence They showed the existence and global attractiveness of periodic solutions and obtained the threshold between extinction and weak persistence of the infected class. Liu and Duan [19] considering the biological population infected with some kinds of diseases and hunted by human beings, and they formulate two SI pollution-epidemic models with continuous and impulsive external effects, respectively, and investigate the dynamics of such systems. These previous models have invariably assumed that the exogenous input of toxicant is continuous or emitted in regular pulses. The system has a unique positive ω-periodic solution x* (t ) which is globally asymptotically stable
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