Sanyi Tang

Models for integrated pest control and their biological implications

Successful integrated pest management (IPM) control programmes depend on many factors which include host-parasitoid ratios, starting densities, timings of parasitoid releases, dosages and timings of insecticide applications and levels of host-feeding and parasitism. Mathematical models can help us to clarify and predict the effects of such factors on the stability of host-parasitoid systems, which we illustrate here by extending the classical continuous and discrete host-parasitoid models to include an IPM control programme. The results indicate that one of three control methods can maintain the host level below the economic threshold (ET) in relation to different ET levels, initial densities of host and parasitoid populations and host-parasitoid ratios. The effects of host intrinsic growth rate and parasitoid searching efficiency on host mean outbreak period can be calculated numerically from the models presented. The instantaneous pest killing rate of an insecticide application is also estimated from the models. The results imply that the modelling methods described can help in the design of appropriate control strategies and assist management decision-making. The results also indicate that a high initial density of parasitoids (such as in inundative releases) and high parasitoid inter-generational survival rates will lead to more frequent host outbreaks and, therefore, greater economic damage. The biological implications of this counter intuitive result are discussed.

Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases.

Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest-natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided. Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications. Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics.

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 hosts 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.

Sliding Bifurcations of Filippov Two Stage Pest Control Models with Economic Thresholds

In order to control pests, a specific management strategy called the threshold policy is proposed, which can be described by Filippov systems (or piecewise smooth systems). The aim of this work is to investigate a variety of bifurcation phenomena of the equilibria and sliding cycles of Filippov two stage structured population models with density dependent per capita birth rates and transition rates from the juvenile class into the adult class. It is shown that interadult competition alone can give rise to multiple sliding segments and multiple pseudoequilibria, whilst interadult and interjuvenile competition together can result in rich sliding bifurcations. As the threshold value varies, local sliding bifurcations including boundary node (saddle), tangency, and pseudo--saddle-node bifurcations occur sequentially, and global sliding bifurcations including buckling bifurcations of the sliding cycles, sliding crossing bifurcations, and pseudohomoclinic bifurcations can be present. Threshold policy control ha...

Dynamics of an infectious diseases with media/psychology induced non-smooth incidence

This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).

Sliding Mode Control of Outbreaks of Emerging Infectious Diseases

This paper proposes and analyzes a mathematical model of an infectious disease system with a piecewise control function concerning threshold policy for disease management strategy. The proposed models extend the classic models by including a piecewise incidence rate to represent control or precautionary measures being triggered once the number of infected individuals exceeds a threshold level. The long-term behaviour of the proposed non-smooth system under this strategy consists of the so-called sliding motion—a very rapid switching between application and interruption of the control action. Model solutions ultimately approach either one of two endemic states for two structures or the sliding equilibrium on the switching surface, depending on the threshold level. Our findings suggest that proper combinations of threshold densities and control intensities based on threshold policy can either preclude outbreaks or lead the number of infecteds to a previously chosen level.

Community-based measures for mitigating the 2009 H1N1 pandemic in China.

Since the emergence of influenza A/H1N1 pandemic virus in March–April 2009, very stringent interventions including Fengxiao were implemented to prevent importation of infected cases and decelerate the disease spread in mainland China. The extent to which these measures have been effective remains elusive. We sought to investigate the effectiveness of Fengxiao that may inform policy decisions on improving community-based interventions for management of on-going outbreaks in China, in particular during the Spring Festival in mid-February 2010 when nationwide traveling will be substantially increased. We obtained data on initial laboratory-confirmed cases of H1N1 in the province of Shaanxi and used Markov-chain Monte-Carlo (MCMC) simulations to estimate the reproduction number. Given the estimates for the exposed and infectious periods of the novel H1N1 virus, we estimated a mean reproduction number of 1.68 (95% CI 1.45–1.92) and other A/H1N1 epidemiological parameters. Our results based on a spatially stratified population dynamical model show that the early implementation of Fengxiao can delay the epidemic peak significantly and prevent the disease spread to the general population but may also, if not implemented appropriately, cause more severe outbreak within universities/colleges, while late implementation of Fengxiao can achieve nothing more than no implementation. Strengthening local control strategies (quarantine and hygiene precaution) is much more effective in mitigating outbreaks and inhibiting the successive waves than implementing Fengxiao. Either strong mobility or high transport-related transmission rate during the Spring Festival holiday will not reverse the ongoing outbreak, but both will result in a large new wave. The findings suggest that Fengxiao and travel precautions should not be relaxed unless strict measures of quarantine, isolation, and hygiene precaution practices are put in place. Integration and prompt implementation of these interventions can significantly reduce the overall attack rate of pandemic outbreaks.

Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models

We review mathematical modeling and related statistical issues of HIV dynamics primarily in response to antiretroviral drug therapy in this article. We start from a basic model of virus infection and then review a number of more advanced models with consideration of pharmacokinetic factors, adherence and drug resistance. Specifically, we illustrate how mathematical models can be developed and parameterized to understand the effects of long-term treatment and different treatment strategies on disease progression. In addition, we discuss a variety of parameter estimation methods for differential equation models that are applicable to either within- or between-host viral dynamics.

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.

One-compartment model with Michaelis-Menten elimination kinetics and therapeutic window: an analytical approach

The purpose of this article is to provide the analytical solutions of one-compartment models with Michaelis-Menten elimination kinetics for three different inputs (single intravenous dose, multiple-dose bolus injection and constant). All analytical solutions obtained in present paper can be described by the well defined Lambert W function which can be easily implemented in most mathematical softwares such as Matlab and Maple. These results will play an important role in fitting the Michaelis-Menten parameters and in designing a dosing regimen to maintain steady-state plasma concentrations. In particular, the analytical periodic solution for multi-dose inputs is also given, and we note that the maximum and minimum values of the periodic solution depends on the Michaelis-Menten parameters, dose and time interval of drug administration. In practice, it is important to maintain a concentration above the minimum therapeutic level at all times without exceeding the minimum toxic concentration. Therefore, the one-compartment model with therapeutic window is proposed, and further the existence of periodic solution, analytical expression and its period are analyzed. The analytical formula of period plays a key role in designing a dose regimen to maintain the plasma concentration within a specified range over long periods of therapy. Finally, the completely analytical solution for the constant input rate is derived and discussed which depends on the relations between constant input rate and maximum rate of change of concentration.