A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.
Roguing and replanting are the most common strategies to control plant diseases and pests. How to build the mathematical models of plant virus transmission and consider the impact of roguing and replanting strategies on plant virus eradication is of great practical significance. In the present paper, we propose the mathematical models for plant virus transmission with continuous and impulsive roguing control. For the model with continuous control strategies, the threshold values for the existences and stabilities of multiple equilibria have been given, and the effect of roguing strategies on the threshold values is also addressed. Furthermore, the model with impulsive roguing control tactics is proposed, and the existence and stability of the plant-only and disease-free periodic solutions of the model are investigated by calculating several threshold values. Moreover, when selecting the design control strategy to minimize the threshold, we systematically analyze the existence of the optimal times of roguing infected plants within a replanting cycle, which is of great significance to the design and optimization of the prevention and control strategy of plant virus transmission. Finally, numerical investigations are given to reveal the main conclusions, and the biological implications of the main results are briefly discussed in the last section.
<p style='text-indent:20px;'>How to develop the mathematical model of plant disease transmission and consider the effect of saturated farming awareness strategies on plant disease control is of great practical significance. To do this, the plant disease transmission models with continuous and pulse farming awareness control strategy have been proposed in the present paper. For the model with continuous control strategies, the threshold values for the existence and stability of multiple equilibria have been given, and the effect of farming awareness on disease persistence is discussed; For the model with nonlinear impulsive control tactics, the existence and stability of the plant disease-free periodic solutions are investigated and the threshold condition is given, further we get the conditions for the permanence of the system and obtain the sufficient conditions under which the positive periodic solution exists by bifurcation theory. Numerical simulations reveal that people's farming awareness, especially global awareness, plays an important role in controlling and eradicating the plant disease. Moreover, the nonlinear impulsive control could result in the complex dynamic behavior, including period doubling, chaos and multiple attractors, which makes it difficult to design a successful plant disease control strategy.</p>
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