Abstract
A hybrid SIR vector disease model with incubation is established, where susceptible host population satisfies the logistic equation and the recovered host individuals are commercially harvested. It is utilized to discuss the transmission mechanism of infectious disease and dynamical effect of commercial harvest on population dynamics. Positivity and permanence of solutions are analytically investigated. By choosing economic interest of commercial harvesting as a parameter, dynamical behavior and local stability of model system without time delay are studied. It reveals that there is a phenomenon of singularity induced bifurcation as well as local stability switch around interior equilibrium when economic interest increases through zero. State feedback controllers are designed to stabilize model system around the desired interior equilibria in the case of zero economic interest and positive economic interest, respectively. By analyzing corresponding characteristic equation of model system with time delay, local stability analysis around interior equilibrium is discussed due to variation of time delay. Hopf bifurcation occurs at the critical value of time delay and corresponding limit cycle is also observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied. Numerical simulations are carried out to show consistency with theoretical analysis.
Highlights
In recent decades, plenty of mathematical models describing the population dynamics of infectious disease have been extensively utilized to understand the transmission mechanism of infectious disease within population ecosystem
We will mainly concentrate on dynamical behavior and local stability analysis around interior equilibrium of model system (4) in this paper
The economic interest of commercial harvesting should be restricted within certain interval that guarantees the existence of interior equilibrium, which can be found in Remark 16
Summary
Plenty of mathematical models describing the population dynamics of infectious disease have been extensively utilized to understand the transmission mechanism of infectious disease within population ecosystem (see [1,2,3,4] and references therein). Along with the line of this research, Enatsu et al [12] analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease and the proposed model with a nonlinear incidence rate satisfying suitable monotonicity conditions. It is necessary to discuss the coexistence and interaction mechanism of population within harvested epidemiological ecosystem as well as dynamical effect of harvest effort due to variation of economic interest. The main objective of this paper is to investigate the transmission mechanism of infectious disease and dynamical effect of commercial harvest on population dynamics, especially the complex dynamical behavior and stability switch due to variation of incubation and commercial harvest economic interest.
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