From economic threshold to economic injury level: Modeling the residual effect and delayed response of pesticide application.

Integrated Pest Management (IPM) poses a challenge in determining the optimal timing of pesticide sprays to ensure that pest populations remain below the Economic Injury Level (EIL), due to the long-term residual effects of many pesticides and the delayed responses of pest populations to pesticide sprays. To address this issue, a specific pesticide kill-rate function is incorporated into a deterministic exponential growth model and a subsequent stochastic model. The findings suggest the existence of an optimal pesticide spraying cycle that can periodically control pests below the EIL. The results regarding stochasticity indicate that random fluctuations promote pest extinction and ensure that the pest population, under the optimal cycle, does not exceed the EIL on average, even with a finite number of IPM strategies. All those confirm that the modeling approach can accurately reveal the intrinsic relationship between the two key indicators Economic Threshold and EIL in the IPM strategy, and further realize the precise characterization of the residual effect and delayed response of pesticide application.

Modeling and analysis of a multilayer solid tumour with cell physiological age and resource limitations.

We study an avascular spherical solid tumour model with cell physiological age and resource constraints in vivo. We divide the tumour cells into three components: proliferating cells, quiescent cells and dead cells in necrotic core. We assume that the division rate of proliferating cells is nonlinear due to the nutritional and spatial constraints. The proportion of newborn tumour cells entering directly into quiescent state is considered, since this proportion can respond to the therapeutic effect of drug. We establish a nonlinear age-structured tumour cell population model. We investigate the existence and uniqueness of the model solution and explore the local and global stabilities of the tumour-free steady state. The existence and local stability of the tumour steady state are studied. Finally, some numerical simulations are performed to verify the theoretical results and to investigate the effects of different parameters on the model.

Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration.

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and $ \widetilde{\mathcal{R}}_0 < 1 $, smokers will eventually become extinct. If $ \widetilde{\mathcal{R}}_0 > 1 $, smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as $ R_0/ > 1 $. Finally, conclusions are explained by numerical simulations.

Threshold dynamics of a stochastic mathematical model for Wolbachia infections.

A stochastic mathematical model is proposed to study how environmental heterogeneity and the augmentation of mosquitoes with Wolbachia bacteria affect the outcomes of dengue disease. The existence and uniqueness of the positive solutions of the system are studied. Then the V-geometrically ergodicity and stochastic ultimate boundedness are investigated. Further, threshold conditions for successful population replacement are derived and the existence of a unique ergodic steady-state distribution of the system is explored. The results show that the ratio of infected to uninfected mosquitoes has a great influence on population replacement. Moreover, environmental noise plays a significant role in control of dengue fever.

Dynamic analysis of an age structure model for oncolytic virus therapy.

Cancer is recognized as one of the serious diseases threatening human health. Oncolytic therapy is a safe and effective new cancer treatment method. Considering the limited ability of uninfected tumor cells to infect and the age of infected tumor cells have a significant effect on oncolytic therapy, an age-structured model of oncolytic therapy involving Holling-Ⅱ functional response is proposed to investigate the theoretical significance of oncolytic therapy. First, the existence and uniqueness of the solution is obtained. Furthermore, the stability of the system is confirmed. Then, the local stability and global stability of infection-free homeostasis are studied. The uniform persistence and local stability of the infected state are studied. The global stability of the infected state is proved by constructing the Lyapunov function. Finally, the theoretical results are verified by numerical simulation. The results show that when the tumor cells are at the appropriate age, injection of the right amount of oncolytic virus can achieve the purpose of tumor treatment.

Effects of pesticide dose on Holling II predator-prey model with feedback control.

We establish a Holling II predator-prey system with pesticide dose response non-linear pulses and then study the global dynamics of the model. First, we construct the Poincaré map in the phase set and discuss its main properties. Second, threshold conditions for the existence and stability of boundary periodic solution and order-[Formula: see text] periodic solutions have been provided. The results show that the pesticide dose increases when the period of control increases, while it will decrease as threshold increases. Sensitivity analyses reveal that critical condition for the stability of boundary periodic solution is very sensitive to control parameters and pesticide doses. The bifurcation analysis reveals that the proposed model exists complex dynamics. Compared to the model with fixed moments, it demonstrates that the density of pest population not only can be controlled below the threshold but also can avoid some negative effects due to pesticide application, confirming the importance of biological control.

Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties.

The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results.

Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties.

This paper addresses the problem of robust stability for quaternion-valued neural networks (QVNNs) with leakage delay, discrete delay and parameter uncertainties. Based on Homeomorphic mapping theorem and Lyapunov theorem, via modulus inequality technique of quaternions, some sufficient conditions on the existence, uniqueness, and global robust stability of the equilibrium point are derived for the delayed QVNNs with parameter uncertainties. Furthermore, as direct applications of these results, several sufficient conditions are obtained for checking the global robust stability of QVNNs without leakage delay as well as complex-valued neural networks (CVNNs) with both leakage and discrete delays. Finally, two numerical examples are provided to substantiate the effectiveness of the proposed results.

Threshold conditions for integrated pest management models with pesticides that have residual effects.

Impulsive differential equations (hybrid dynamical systems) can provide a natural description of pulse-like actions such as when a pesticide kills a pest instantly. However, pesticides may have long-term residual effects, with some remaining active against pests for several weeks, months or years. Therefore, a more realistic method for modelling chemical control in such cases is to use continuous or piecewise-continuous periodic functions which affect growth rates. How to evaluate the effects of the duration of the pesticide residual effectiveness on successful pest control is key to the implementation of integrated pest management (IPM) in practice. To address these questions in detail, we have modelled IPM including residual effects of pesticides in terms of fixed pulse-type actions. The stability threshold conditions for pest eradication are given. Moreover, effects of the killing efficiency rate and the decay rate of the pesticide on the pest and on its natural enemies, the duration of residual effectiveness, the number of pesticide applications and the number of natural enemy releases on the threshold conditions are investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications and predator releases. Latin Hypercube Sampling/Partial Rank Correlation uncertainty and sensitivity analysis techniques are employed to investigate the key control parameters which are most significantly related to threshold values. The findings combined with Volterra's principle confirm that when the pesticide has a strong effect on the natural enemies, repeated use of the same pesticide can result in target pest resurgence. The results also indicate that there exists an optimal number of pesticide applications which can suppress the pest most effectively, and this may help in the design of an optimal control strategy.