nonlinear and is solved numerically. A systemic characterization of the bean weevil treatment process is used to

describe the model, which uses three functions of time: the number of beans, the pest population, and the

amount of diatomaceous earth. The three functions offer users four applications: forecasting, control,

formulation of treatment efficacy estimators, and simulation of different types of pest control. The model is

built for closed (isolated) experiments typical of laboratories, but this feature makes it extensible to other

treatments to combat bean weevils in closed spaces characteristic of the storage of beans in silos.

Key-Words: -mathematical model, bean weevil, treatments, experiences, efficacy, control

Received: April 13, 2022. Revised: December 12, 2022. Accepted: January 7, 2023. Published: February 14, 2023.

(mathematical) modelling of biological systems [8],

[9], [10], [11], [12], and [38]. These are just a few

titles from a large number of papers on the subject.

The computer simulation of the integrated

management of spruce worms was already used in

1982 [26].

Many concerns similar to the ones described in this

paper are also found in the literature. In [13], the

authors mathematically model the control of

harmful insects by interrupting mating and capturing

equations), which, as in our case, uses the

calibration operation, is dedicated to a deadly

tropical disease for humans[17]. Differential models

for mosquitoes carrying Wolbachia bacteria and

models is part of a class of hybrid models, which are

often described by continuous dynamics but

repeatedly experience discrete changes at certain

times [16].

Concerns at a higher-level aim at an optimal design

of the dynamic models of the pest populations [27].

Works dedicated to determining the efficacy of

alternative treatments, even for stored beans and,

among others, diatomaceous earth, using statistical

the body of the bruchids Callosobruchus maculatus

(F.) or Acanthoscelidesobtectus (Coleoptera:

Bruchidae) is proposed by the authors [29], [31]. In

[37], the authors also consider the influence of

ambient temperature and humidity on the bean

weevil population.At the level of descriptive

statistics, the results can be used to extend

mathematical models to the two variables, which in

the present research are considered constants.

Mathematical models are constructed to explain the

existence of optimal values for the parameters of the

phenomenon and, in addition, calculate these values.

Based on mathematical models, the authors [40]

optimize the control of mosquito populations.

Several mathematical models with which heuristic

optimizations are performed are reviewed by the

authors [41]. Mathematical modelling with control

and optimization also exposes the authors [42] to

sugar beet pests. The articles of the authors [43],

[44], [45], [46], [47], and many others are inscribed

system. The purpose of the model is to estimate the

effectiveness of the treatment.

2.1 Material of the study

The physical support of the study consists of the

closed system made in a cylindrical container with

transparent walls, in which there are three main

The behaviour of the components of the closed or

isolated system described above allowed us to

formulate the following hypotheses, which were

used in the mathematical modelling of the system:

i1) The prey (beans) interacts only with the primary

predator (bean weevil);

i2) The prey does not record its growth or

reproduction;

2.4 Estimation of the constant parameters

involved in the model

The mass of the beans with which the experiments

described in [1] were carried out was calculated by

averaging 1000 grains, which together weighed 145-

450 g. Therefore, we work with the average mass of

a bean having the value  = 0.0002975 kg. For

average of 0.1176 g, therefore, the mass of a bean

weevil is = 0.000002352 kg. As for each

experiment, 50 weevils were introduced into

containers; the result is the initial mass of bean

weevils, which is  = 0.0001176 kg. The initial

dose of diatomaceous earth was limited between

100 and 900 ppm (by mass ratio to the initial mass

of beans). As a result, the initial mass of diatomite,

, is between 0.0000001 and 0.0000009 kg,

calculated according to the formula:.

was taken, in the conditions in which the

temperature in the experiment container is kept

constant. For the first tests, it will be assumed that

the pest control operation is done only by the initial

dose of diatomite and that no additional amounts are

added during the 14 days of the experiment.

Therefore, the function 󰇛󰇜 is identical to zero. The

parameter , which describes the consumption of

diatomaceous earth, is used only to control the

amount of diatomite consumed in the control

used using the data from the experiments described

in [1]. The experimental data we rely on are the

mortality rates achieved by the treatments with

doses of 100, 300, 500, and 900 ppm at observation

times of 3 and 7 days. Let the experimental

mortality  and , = 1, ..., 4, correspond to

the four doses of diatomite applied, at times of 3 and

Repellence index

The repellence index function is defined by [6]

according to the formula (11).

The numerical analysis used the following data:

 ppm,  kg,

which corresponds to a uniform load on the third

day of the experiment with a loading speed of 70

ppm/day.

The function of loading the pest with the attack

of the system described above is graphically

presented in Fig. 10.

4 Comments

The mathematical model proposed in this article

manages to simulate the main aspects of the process

of experimental estimation of the efficacy of

treatments against pests.

The shape of the mortality curves is in agreement

with the experimental data, at least in the area of

intermediate control times. Differences occur at low

case of loading in terms of pest mortality is given in

Fig. 11, and the evolution in time of the additional

load and the loading speed is given in Fig. 10. It is

observed that starting treatment at a lower dose and

supplementing with the active substance along the

way may increase the efficacy of treatment at a

higher initial dose. In [29], interpolation curves

similar to those of Figs. 7, 9, and 11 appear in

connection with the accumulation of diatomic soils

more deeply embedded in the mathematical model,

describing the phenomenon and interpreting the

results.

The mathematical model of experiments to estimate

the efficacy of substances for controlling bean

weevils can be extended to common storage spaces,

such as silos. In this case, it must be taken into

account that during the storage time there may be

fluctuations in the loading with stored material, and

the loading with substances to control pests varies

term 󰇛󰇜 in the third equation of the same system.

The mathematical model proposed in this paper

goes through the three stages indicated in a

reference paper [8]: "conceptualization of the

biological system into a model mathematical

formalization of the previous conceptual model and

optimization and system management derived from

the analysis of the mathematical model." Because

the answer of the model proposed in this paper has

main elements of similarity with the real isolated

system of experiments to estimate the efficacy of

A specification of principle for the possible

character. Man tends to exterminate any species that