Nonlinear pulse vaccination in an SIR epidemic model with resource limitation

Mathematical models can assist in the design and understanding of vaccination strategies when resources are limited. Here we propose and analyse an SIR epidemic modelwith a nonlinear pulse vaccination to examine how a limited vaccine resource affects the transmission and control of infectious diseases, in particular emerging infectious diseases. The threshold condition for the stability of the disease free steady state is given. Latin Hypercube Sampling/Partial Rank Correlation Coefficient uncertainty and sensitivity analysis techniques were employed to determine the key factors which are most significantly related to the threshold value. Comparing this threshold value with that without resource limitation, our results indicate that if resources become limited pulse vaccination should be carried out more frequently than when sufficient resources are available to eradicate an infectious disease. Once the threshold value exceeds a critical level, both susceptible and infected populations can oscillate periodically. Furthermore, when the pulse vaccination period is chosen as a bifurcation parameter, the SIR model with nonlinear pulse vaccination reveals complex dynamics including period doubling, chaotic solutions, and coexistence of multiple attractors. The implications of our findings with respect to disease control are discussed

Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak

Host-parasitoid models including integrated pest management (IPM) interventions with impulsive effects at both fixed and unfixed times were analyzed with regard to host-eradication, host-parasitoid persistence and host-outbreak solutions. The host-eradication periodic solution with fixed moments is globally stable if the host's intrinsic growth rate is less than the summation of the mean host-killing rate and the mean parasitization rate during the impulsive period. Solutions for all three categories can coexist, with switch-like transitions among their attractors showing that varying dosages and frequencies of insecticide applications and the numbers of parasitoids released are crucial. Periodic solutions also exist for models with unfixed moments for which the maximum amplitude of the host is less than the economic threshold. The dosages and frequencies of IPM interventions for these solutions are much reduced in comparison with the pest-eradication periodic solution. Our results, which are robust to inclusion of stochastic effects and with a wide range of parameter values, confirm that IPM is more effective than any single control tactic

The effects of resource limitation on a predator-prey model with control measures as nonlinear pulses

The dynamical behavior of a Holling II predator-prey model with control measures as nonlinear pulses is proposed and analyzed theoretically and numerically to understand how resource limitation affects pest population outbreaks. The threshold conditions for the stability of the pest-free periodic solution are given. Latin hypercube sampling/partial rank correlation coefficients are used to perform sensitivity analysis for the threshold concerning pest extinction to determine the significance of each parameter. Comparing this threshold value with that without resource limitation, our results indicate that it is essential to increase the pesticide’s efficacy against the pest and reduce its effectiveness against the natural enemy, while enhancing the efficiency of the natural enemies. Once the threshold value exceeds a critical level, both pest and its natural enemies populations can oscillate periodically. Furthermore,when the pulse period and constant stocking number as a bifurcation parameter, the predator-prey model reveals complex dynamics. In addition, numerical results are presented to illustrate the feasibility of our main results

A locust phase change model with multiple switching states and random perturbation

Insects such as locusts and some moths can transform from a solitarious phase when they remain in loose populations and a gregarious phase, when they may swarm. Therefore, the key to effective management of outbreaks of species such as the desert locust Schistocercagregaria is early detection of when they are in the threshold state between the two phases, followed by timely control of their hopper stages before they fledge because the control of flying adult swarms is costly and often ineffective. Definitions of gregarization thresholds should assist preventive control measures and avoid treatment of areas that might not lead to gregarization. In order to better understand the effects of the threshold density which represents the gregarization threshold on the outbreak of a locust population, we developed a model of a discrete switching system. The proposed model allows us to address: (1) How frequently switching occurs from solitarious to gregarious phases and vice versa; (2) When do stable switching transients occur, the existence of which indicate that solutions with larger amplitudes can switch to a stable attractor with a value less than the switching threshold density?; and (3) How does random perturbation influence the switching pattern? Our results show that both subsystems have refuge equilibrium points, outbreak equilibrium points and bistable equilibria. Further, the outbreak equilibrium points and bistable equilibria can coexist for a wide range of parameters and can switch from one to another. This type of switching is sensitive to the intrinsic growth rate and the initial values of the locust population, and may result in locust population outbreaks and phase switching once a small perturbation occurs. Moreover, the simulation results indicate that the switching transient patterns become identical after some generations, suggesting that the evolving process of the perturbation system is not related to the initial value after some fixed number of generations for the same stochastic processes. However, the switching frequency and outbreak patterns can be significantly affected by the intensity of noise and the intrinsic growth rate of the locust population

Global dynamics of a state-dependent feedback control system

The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order-k (k ≥ 3) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control

Modeling the effects of augmentation strategies on the control of Dengue fever with an impulsive differential equation

Dengue fever has rapidly become the world’s most common vector-borne viral disease. Use of endosymbiotic Wolbachia is an innovative technology to prevent vector mosquitoes from reproducing and so break the cycle of dengue transmission. However, strategies such as population eradication and replacement will only succeed if appropriate augmentations with Wolbachia-infected mosquitoes that take account of a variety of factors are carried out. Here, we describe the spread of Wolbachia in mosquito populations using an impulsive differential system with four state variables, incorporating the effects of cytoplasmic incompatibility and the augmentation of Wolbachia-infected mosquitoes with different sex ratios.We then evaluated (a) how each parameter value contributes to the success of population replacement; (b) how different release quantities of infected mosquitoes with different sex ratios affect the success of population suppression or replacement; and (c) how the success of these two strategies can be realized to block the transmission of dengue fever. Analysis of the system’s stability, bifurcations and sensitivity reveals the existence of forward and backward bifurcations, multiple attractors and the contribution of each parameter to the success of the strategies. The results indicate that the initial density of mosquitoes, the quantities of mosquitoes released in augmentations and their sex ratios have impacts on whether or not the strategies of population suppression or replacement can be achieved. Therefore, successful strategies rely on selecting suitable strains of Wolbachia and carefully designing the mosquito augmentation program

Cumulative effects of incorrect use of pesticides can lead to catastrophic outbreaks of pests

Modeling external perturbations such as chemical control within each generation of discrete populations is challenging. Based on a method proposed in the literature, we have extended a discrete single species model with multiple instantaneous pesticide applications within each generation, and then discuss the existence and stability of the unique positive equilibrium. Further, the effects of the timing of pesticide applications and the instantaneous killing rate on the equilibrium were investigated in more detail and we obtained some interesting results, including a paradox and the cumulative effects of the incorrect use of pesticides on pest outbreaks. In order to show the occurrences of the paradox and of hormesis, several special models have been extended and studied. Further, the biological implications of the main results regarding successful pest control are discussed. All of the results obtained confirm that the cumulative effects of incorrect use of pesticides may result in more severe pest outbreaks and thus, in order to avoid a paradox in pest control, control strategies need to be designed with care, including decisions on the timing and number of pesticide applications in relation to the effectiveness of the pesticide being used

Linking key intervention timing to rapid decline of the COVID-19 effective reproductive number to quantify lessons from mainland China

Effective reproductive numbers (Rt) were calculated from data on the COVID-19 outbreak in China and linked to dates in 2020 when different interventions were enacted. From a maximum of 3.98 before the lockdown in Wuhan City, the values of Rt declined to below 1 by the second week of February, after the construction of hospitals dedicated to COVID-19 patients. The Rt continued to decline following additional measures in line with the policy of “early detection, early report, early quarantine, and early treatment.” The results provide quantitative evaluations of how intervention measures and their timings succeeded, from which lessons can be learned by other countries dealing with future outbreaks

Analytical methods for detecting pesticide switches with evolution of pesticide resistance

After a pest develops resistance to a pesticide, switching between different unrelated pesticides is a common management option, but this raises the following questions: (1) What is the optimal frequency of pesticide use? (2) How do the frequencies of pesticide applications affect the evolution of pesticide resistance? (3) How can the time when the pest population reaches the economic injury level (EIL) be estimated and (4) how can the most efficient frequency of pesticide applications be determined? To address these questions, we have developed a novel pest population growth model incorporating the evolution of pesticide resistance and pulse spraying of pesticides. Moreover, three pesticide switching methods, threshold condition-guided, density-guided and EIL-guided, are modelled, to determine the best choice under different conditions with the overall aim of eradicating the pest or maintaining its population density below the EIL. Furthermore, the pest control outcomes based on those three pesticide switching methods are discussed. Our results suggest that either the density-guided or EIL-guided method is the optimal pesticide switching strategy, depending on the frequency (or period) of pesticide applications