## Mathematical Models of Pesticide Effectiveness and Resistance

Modeling pesticide-induced mortality is an important problem usually ignored by treating the pesticide effects as instantaneous. Starting in the late 1970s, many mathematical models of pesticide effectiveness have been formulated, coupled with models of insect population dynamics, and evaluated using data from pesticide efficacy trials. Although the models were developed for different purposes and modeling situations, they are closely related in construction and purpose and contain many similarities. For example, they include the concept that the effective pesticide dose received by an individual insect is related in a simple way to the pesticide concentration in the environment. Most base the probability of dying from pesticide effects on a sigmoid relationship (probit or logistic) to the log of the amount of pesticide that the insects have received. In addition, they commonly assume that pesticide residues follow a first-order degradation curve in the environment. A 'unified' approach for modeling pesticide effectiveness was implemented (Figure 1), consisting of: (1) differential equation submodels for pesticide deposition-degradation; (2) intake-clearance submodels of the pesticide in insects; and (3) a hazard function submodel (i.e., the instantaneous per capita rate of mortality for survival of the insects) based

Pesticide application

Degradation (e.g., ■I microbial, chemical, and photodegradation)

Intake

on the curve of pesticide retained by the insects. Thus, the overall model consists of explicit (and easily substitutable) submodels for the processes considered (i.e., pesticide intake-clearance) and is also based on the hazard function from which any kind of desired mortality can be derived. Therefore, this simulation approach should be more flexible and adaptable to specific needs ofnew modeling situations, that is, model components or algorithms can be modified in accordance with specific physiological and behavioral properties of the organism, as well as the physical and toxic properties ofthe pesticide. Additional improvements could be incorporated to improve this type of mathematical model for pesticide effectiveness. For example, stochastic differential equations could be used to allow variability in the pesticide deposition, degradation, intake, or clearance processes. Also, an assumption is made that pesticide degradation is a time-homogeneous process, but time-varying phenomena on which the degradation process depends (e.g., temperature and rainfall) could be incorporated into the model.

Other simulation models are available that predict the effects of various pesticide uses and insect management regimes on the frequency of susceptible and resistant insects. One such model, DEMANIR (Development and Management of Insecticide Resistance), was developed using the VENSIM® simulation environment and enables simulation of various insecticide use patterns and management strategies. DEMANIR allows for the density-dependent effects on insect reproductive rate and migration, effects of farm hygiene, and variation in fitness levels between susceptible and resistant insects. Insecticide efficacy includes time since treatment, insecticide resistance level, and insecticide application efficiency. Simulation runs cover 2 years, with a harvest in each year and ensuing analysis of various cleanup (hygiene) and insecticide treatment strategies.

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