A mathematical model for estimating the LC50 (or LD50) among an insect life cycle

In this study, a mathematical model is made to estimate the median lethal concentration or dose (LC50 or LD50). The model is based on the data of different insecticide groups, where each one is represented by the effect of three insecticides over different orders of insects by using different application technique. The trend of change of the LC50 or LD50 is observed among the insect life cycle for each group of insecticides. It is shown that for an insecticide group, there is a clear trend for the change of the LC50 (or LD50) when going from an age stage to another. That trend is simulated for each group to predict the LC50 or LD50 at an age stage by knowing it at another stage and method of treatment used.


INTRODUCTION
Resistance of insect pests to different insecticide groups is the most serious problem in insect pest control. Resistance can develop to virtually any human, animal and crop protection product that is designed to kill pests. The likelihood of resistance occurring and the speed with which it develops depends on a combination of factors that make up the "selection pressure" (Georghiou and Taylor 1977a, b). These factors include (a) the biology and ecology of the pest, (b) how toxic and persistent a pesticide is and (c) the frequency of product use. Once a pest has developed resistance to one pesticide it may also be "cross-resistant" to other pesticides that have the same mode of action. In rare cases, a pest can develop "multiple resistances" to more than one class of pesticide with different modes of action (Lo et al. 2000).
The development of organochlorine, organophosphate, carbamate and pyrethroid resistance in different insect groups (Sparks 1981, Wolfenbarger et al. 1981, and reports of increased IGR and plant extracts tolerance reported. In addition to resistance problem presence of cross-resistance between the insecticides from different groups such as pyrethroids and DDT (Ahmad and McCaffery 1988). Also a resistance of lepidopteron insects to teflubenzuron, tebufenozide, bifenthrin, and lambda-cyalothtin reveals a cross-resistance to these different insecticides (Sauphanor et al. 1998). Resistance in B. tabaci is known to be multi-factorial, based on both enhanced detoxification of insecticides and modifications to three of their major target proteins: (1) Acetyl-cholinesterase (AChE), targeted by organophosphates (OPs) and carbamates.
Estimation of median lethal concentration or dosage (LC 50 and LD 50 respectively) is very valuable. LC 50 or LD 50 is indicator to the level of resistance of population response to pesticides. So in this study we focused on estimation of this term by using mathematical models.
In this study a mathematical simulation has been presented for different insecticides groups, which were represented by three insecticides from each group. Then study of their effect on the different insect stages of various insect orders by using the most common methods of exposure at different unites of insecticide concentrations or doses. The importance of the model is that it allows us of predicting the variation of response of different stages (egg, immature stages and mature stage) of various orders to insecticides by using different methods of exposure.
The data were fitted to continuous curves to enable the process of predicting LC 50 or LD 50 of certain stage by knowing LC 50 or LD 50 and of others at the same technique of exposure.

MATERIAL AND METHODS
Here we describe our mathematical model and the approach taken in its analysis. First, we provide a brief perspective on the insecticides resistance problem. Second, we illustrate in details the general notes about the behavior of changing of the LC 50 from age stage to another for each group of insecticides. Finally, the programming, computation and analysis of the model are described.
For estimation of LC 50 or LD 50 we use in this paper one method of exposure as example, it is topical method for bio-insecticides. Topical technique: Test-material solutions were applied by topical application to test insects. The test insects were anesthetized by using suitable method to insects used. Then, insecticide dilutions were applied to the ventral abdomen, thorax, between mesothoracic coxa or just behind the head on the ventral side of insect. One micro liter of test-material solution containing the appropriate concentration of insecticides was applied to stage of insect used by a standard digital micrometer syringe and the 24 h mortality was subjected to determine LC 50 and LC 90 values (Gouamene-Lamine et al. 2003, Lorini and Galley 1998, Meink et al. 1998, Nathan et al. 2008, Ugurlu and Gurkan 2007, Wing et al. 2000and Wright et al. 2000.

General notes about the data
We stress on the following important notes that will constitute a guideline for choosing the assumptions of the mathematical model:  Despite the differences between the values of the LC 50 for two kinds of insecticides or insects, they have a similar trend for the variation of the LC 50 when going from a stage to another.
 We can't conclude that the LC 50 is different in a group of insects than another one. The values are varying with no observed trend along a group of insects.  The only observed trend is in the change of the value of the LC 50 when moving from a stage to another one along the life cycle of the insect. All the relatively small or large values of m are within less than 3 standard deviations about the mean, i.e., they can't be considered extreme values.  All the relatively small or large values of m are within less than 3 standard deviations about the mean, i.e., they can't be considered extreme values.  The differences between the relatively small or large values of m and the mean (measured in standard deviations) is always less than the differences between the corresponding values of the LC 50.  A parameter is suggested to describe the change of the LC 50 when going on the life cycle from a stage to another stage. We call it "m". It simply represents the ratio of the difference between the value of the LC 50 in the second stage and the first stage to the LC 50 of the first stage.

Assumptions of the mathematical Model
To get a mathematical model that is consistent with the data collected above, we follow the following assumptions, 1-The model categorized the data according to insecticide groups.

2-No distinguish is made among different groups of insects.
3-There is a clear trend of change of the LC 50 along a life cycle.

4-
The variable m is assumed to be normally distributed for each insecticide between two age stages. 5-For each group of values of m corresponding to transformation between two stages for an insecticide, the mean and standard deviation are used to calculate confidence intervals to estimate the LC 50 at an age stage given the LC 50 of the previous stage along the life cycle of the insect.

Calculations of the Model
In the following table, we illustrate the details of the calculations required for the model. Consider the following table of the LC 50 values of an insecticide for two different stages a and b.  For a sample of k (<30) elements, whose standard deviation is  , the value of "e" for a 1- confidence interval is given by

Application to the case of bio-insecticides, topical exposure
In what follows, the simulation process is illustrated through an example showing the application of the above technique to table 1 of LD 50 values of response of Coleoptera, and Lepidoptera when exposed to bio-insecticides by topical technique.
In table (2), the values of m between the different age stages are calculated as well as their means and standard deviations as shown below.
Of course the negative value is rejected, so all we can say about this case is that the Larva LD 50 is expected to be less than 0.121x with 95% confident.  For example if the LD 50 at the egg stage is 15, it will be estimated with 95% confident to be less than 0.121(15) = 1.815 in the Larva state.  As estimated in this example we can predict any LC 50 or LD 50 to any stage at by knowing LC 50 or LD 50 to other stage and the method of exposure used. It is to be noted that, if we were seeking a point estimate of the LC 50 rather than an interval estimate (as in our case) the formula of the LC 50 at the b-stage would be just ) 1 ( m x  , but it would be of course less accurate. This work is a first trial for predicting the LC 50 of insecticides along an insect life cycle. We hope that future work be carried for each insecticide group seeking a more accurate and closer values for the mean and the standard deviation.
In the case that a study collects a sample of more than 30 results of the same unit, the t-distribution will be replaced by the z-distribution.

ACKNOWLEDGEMENT
I would like to express my great gratitude to Prof. Dr. Reda Fadeel bakr, Prof. of Entomology, Fac. of Science, Ain Shams University, for the suggestion of the point, and his valuable advices and discussions during the work. Also, I would like to deep thank Dr. Ahmad Mostafa Kamel for his help and support.