Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму
Diabetes is a chronic disease affecting millions of people worldwide. Several studies have been carried out to control the diabetes problem, involving both linear and non-linear models. However, the complexity of linear models makes it impossible to describe the diabetic population dynamic in depth....
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2024
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System research and information technologies| _version_ | 1866302970504675328 |
|---|---|
| author | El Ouissari, Abdellatif El Moutaouakil, Karim |
| author_facet | El Ouissari, Abdellatif El Moutaouakil, Karim |
| author_sort | El Ouissari, Abdellatif |
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| datestamp_date | 2024-05-23T07:09:36Z |
| description | Diabetes is a chronic disease affecting millions of people worldwide. Several studies have been carried out to control the diabetes problem, involving both linear and non-linear models. However, the complexity of linear models makes it impossible to describe the diabetic population dynamic in depth. To capture more detail about this dynamic, non-linear terms were introduced into the mathematical models, resulting in more complicated models strongly consistent with reality (capable of re-producing observable data). The most commonly used methods for control estimation are Pantryagain’s maximum principle and Gumel’s numerical method. However, these methods lead to a costly strategy regarding material and human resources; in addition, diabetologists cannot use the formulas implemented by the proposed controls. In this paper, the authors propose a straightforward and well-performing strategy based on non-linear models and genetic algorithms (GA) that consists of three steps: 1) discretization of the considered non-linear model using classical numerical methods (trapezoidal rule and Euler–Cauchy algorithm); 2) estimation of the optimal control, in several points, based on GA with appropriate fitness function and suitable genetic operators (mutation, crossover, and selection); 3) construction of the optimal control using an interpolation model (splines). The results show that the use of the GA for non-linear models was successfully solved, resulting in a control approach that shows a significant decrease in the number of diabetes cases and diabetics with complications. Remarkably, this result is achieved using less than 70% of available resources. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.1.10 |
| first_indexed | 2025-07-17T10:28:32Z |
| format | Article |
| fulltext |
El Ouissari Abdellatif, El Moutaouakil Karim, 2024
134 ISSN 1681–6048 System Research & Information Technologies, 2024, № 1
UDC 62-50
DOI: 10.20535/SRIT.2308-8893.2024.1.10
INTELLIGENT OPTIMAL CONTROL OF NONLINEAR
DIABETIC POPULATION DYNAMICS SYSTEM USING
A GENETIC ALGORITHM
EL OUISSARI ABDELLATIF, EL MOUTAOUAKIL KARIM
Abstract. Diabetes is a chronic disease affecting millions of people worldwide.
Several studies have been carried out to control the diabetes problem, involving both
linear and non-linear models. However, the complexity of linear models makes it
impossible to describe the diabetic population dynamic in depth. To capture more
detail about this dynamic, non-linear terms were introduced into the mathematical
models, resulting in more complicated models strongly consistent with reality
(capable of re-producing observable data). The most commonly used methods for
control estimation are Pantryagain’s maximum principle and Gumel’s numerical
method. However, these methods lead to a costly strategy regarding material and
human resources; in addition, diabetologists cannot use the formulas implemented
by the proposed controls. In this paper, the authors propose a straightforward and
well-performing strategy based on non-linear models and genetic algorithms (GA)
that consists of three steps: 1) discretization of the considered non-linear model
using classical numerical methods (trapezoidal rule and Euler–Cauchy algorithm);
2) estimation of the optimal control, in several points, based on GA with appropriate
fitness function and suitable genetic operators (mutation, crossover, and selection);
3) construction of the optimal control using an interpolation model (splines). The
results show that the use of the GA for non-linear models was successfully solved,
resulting in a control approach that shows a significant decrease in the number of
diabetes cases and diabetics with complications. Remarkably, this result is achieved
using less than 70% of available resources.
Keywords: optimal control, differential equation, diabetes, genetic algorithms,
artificial intelligence, intelligent local search.
INTRODUCTION
Diabetes is a major public health problem. Diabetes is a major public health
problem and one of the most dangerous common diseases and is characterized by
high blood sugar [2]. Diabetes is the root of many diseases and costs many lives.
It is a serious chronic disease that occurs when the body does not properly use the
insulin it produces or when the pancreas does not produce enough insulin. There
are three types of diabetes: Type 1 diabetes, Type 2 diabetes and Gestational
Diabetes Mellitus (GDM) [3].
The number of people with diabetes has increased exponentially in recent
times. According to the World Health Organization (WHO) and the Interna- tional
Diabetes Federation (IDF) [1; 4], 463 million people had diabetes in 2019 and this
number is expected to reach 578 million in 2030 and 700 million in 2045 [11; 9].
The mathematical modeling of the phenomenon of diabetes is the subject of
several researchers in different mathematical fields. These include ordinary differ-
ential equations (ODE) that study, for example, the diabetic population as found
in [5–8], there are also research works that study the phenomenon of diabetes by
modeling has partial differential equations (PDE). Likewise, studies on this
Intelligent optimal control of nonlinear diabetic population dynamics system using …
Системні дослідження та інформаційні технології, 2024, № 1 135
phenomenon are carried out using delay differential equations (DDE), and oth-
ers, for example stochastic differential equations (SDE) and integro-differential
equations (IDE), Fredholm integral equations (FIE) [15–18].
In [18] the authors have given a thorough review of the delay differential
equation models and they are presented with some computational results and brief
summaries of the theoretical results for the cases of the insulin ultradian
oscillation models and the models for the diagnostic tests.
The modeling of natural phenomena is a very successive task to deal with
any phenomenon [40]. In the first modeling of the phenomenon of diabetes, two
types of population were considered, pre-diabetic and diabetic. The authors of [5]
modeled this natural phenomenon as a system of linear ordinary differential
equations of the first degree. Then, in 2007 [10], a modification was made to the
above system. This time, another type of population was considered, namely
diabetic patients with complications.
Several studies were performed for this model, they show that the system is
well defined, also the stability and determination of equilibrium points was done.
Then in the year 2014 [6], the authors is thought to try to reduce the neg- ative
effect of the phenomenon of diabetes, for this they proposed an approach of
optimal control that help to minimize as much as possible the spread of the
phenomenon, focusing on the study of the diabetic population. Three types of
diabetic patients are considered, patients who become diabetic for different rea-
sons, which may be genetic or related to a negative lifestyle. The other two types
are diabetic patients without complications and diabetic patients with complica-
tions. The modeling of the mathematical model is well explained in [6], and the
existence and uniqueness of solution is well shown, also the existence of control.
In the ten years, the studies carried out with this control have shown their ef-
fectiveness in reducing the number of diabetic patients, which shows the success
of this strategy of searching for an optimal control to control this phenomenon. A
dynamic 6-compartment control system was proposed for the study of the
diabetes phenomenon in [11], they divided the population in general as follows:
healthy people H, pre-diabetic patients due to genetics which is denoted by P, pre-
diabetic patients due to lifestyle denoted by E. The other three types are di- abetic
patients without complications and diabetic patients with complications, which
are denoted by D and C respectively. In [11], they paid attention to the fact that
there are several types of consciousness that are directly related to diabetes, there
are genetic influences and bad lifestyle, on the other hand there is the psychology
of the person. For this reason, they proposed optimal controls that take these
influences into account.
In [12], another reformulation of dynamic model of diabetic population was
proposed. They resulted that the factors most related to diabetes or to the fact that
a person becomes diabetic are the genetic factors and a bad lifestyle. To protect
diabetic patients, an awareness program was proposed using media and education;
psychological follow-up was also considered and medical treatment, this time
they grouped patients according to their age through a continuous dynamic
system. They divided the general population into 4 different compartments given
in the following form: pre-diabetic patients due to genetics, pre-diabetic patients
due to lifestyle, diabetic patients with complication and without complication.
In this work we propose a very sample and performance strategy based on
non-linear models and genetic algorithm (GA)s that processes into three steps: 1)
discretization of the considered non-linear model using classical numerical
methods (trapezoidal rule and Euler–Cauchy algorithm); 2) estimation of the
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 136
optimal control, in several points, based on GA with appropriate fitness function
and suitable genetic operators (mutation, crossover, and selection); 3) construc-
tion of the optimal control using interpolation model.
This paper is organized as follows: The second section provides an intro-
duction on the importance of using genetic algorithms to solve optimal control
problems. The third section presents two dynamic control systems, one is the
kernel linear system and the other is the nonlinear system proposed in recent
years, we have given the theoretical summary that has been done on both models.
The fourth section presents the minimization of the objective functions of each
control system of the diabetic population by the genetic algorithm and we
compare the results found by the results of the classical method. Finally, we end
the paper with a general conclusion.
GENETIC ALGORITHM
In [9] Boutayeb et al., in recent years artificial intelligence is invading all fields, it
has also become an essential part of human life. For example, in this work, we
will use the artificial intelligence methods to treat the optimal control problems
associated with a diabetic population [29; 14].
In this article, we deal with a very complicated phenomenon, because several
effects come into play, for example genetic influence, age, lifestyle and others.
Genetic algorithms are artificial intelligence methods and are heuristic
search techniques that are very simple to handle [30; 31]. Genetic algorithms
(GAs) are search optimization algorithms based on three essential operators,
selection, crossover and mutation [21]. In general, GAs were first developed by
Holland and are derived from Darwin’s theory of evolution [20]. In the first step,
a population of initial solutions, called chromosomes, is randomly selected, then
these solutions are evaluated by the objective function or fitness function, if the
solution has a very high performance then this is the solution we are looking for,
otherwise, we move to the crossover and mutation step which generates a new
solution from the initial solution, likewise this solution generated and evaluated by the
objective function, and so on until we find the best solution. Genetic algorithms are
very effective in complex optimization problems more than simple methods,
which sometimes fail to achieve a certain problem due to its complexity [32].
They are used in many research areas and are very useful in real world
applications, because they are very simple and give the best solutions. Classical
optimization methods, which are purely computational methods, start with a
single initial solution and then search for the optimal solution, but the genetic
algorithm starts with an initial population of candidates and then searches for the
best optimal solution in the search space [28].
Optimal control problems are also among the optimization problems that
have been solved by the GA. The real beginning of the use of GA for dynamic
control systems is in the years 1992 by Krishnakumar and Goldberg [22] who
gave a start to GA in a very important discipline of applied mathematics [23; 26; 27].
The main objective of our work is to achieve a significant reduction in the number
of diabetics without complications and with treatable complications, which
involves the search for optimal control to achieve this goal. The dynamic PEDC
model is a non-linear mathematical model representing the evolution of the
diabetic population and the influence of uncomplicated diabetic patients on pre-
diabetics. The advantage of using the GA to solve this type of optimal control
Intelligent optimal control of nonlinear diabetic population dynamics system using …
Системні дослідження та інформаційні технології, 2024, № 1 137
problem is that we do not need to determine the adjoint values or characterize the
control in any specific way. We only need to treat the problem well and specify
the objective function we need to minimize and its various constraints.
THE MODELS
In this section, we will focus on different dynamic and controlled systems for di-
abetic patients found in the literature that minimize the negative impact of this
phenomenon. First, we will start with the dynamically controlled Kernel system,
which allows the study of diabetic patients without entering their bodies.
The EDC model
In [13] Boutayeb et al. have proposed an optimal control approach modeling the
progression from pre-diabetes to diabetes with and without complications. Three
types of diabetic patients are considered, patients who become diabetic for
different reasons, which may be genetic or related to a negative lifestyle, this type
is noted by E. The other two types are diabetic patients without complications and
diabetic patients with complications, which are denoted by D and C respectively.
We conclude that the diabetic population N is divided into three types such that
)()()()( tDtCtEtNN at time t :
.)()()()(
)(
;)()()()(
)(
;)()(
)(
23
21
13
tCtDtE
dt
tdC
tCtDtE
dt
tdD
tEI
dt
tdE
The modeling of the model is explained in [13], the model contains eight
parameters: ,,,,,, 321I and , which were estimated in [10]. The
control of this phenomenon is the subject of several research works, due to the
fact that this phenomenon has social burdens that should be reduced. For this
purpose the authors proposed to control the transition rate from diabetes with
complications to diabetes without complications.
The controlled model is given by the following system:
,)()()())(1()())(1(
)(
;)()()))(1(()())(1(
)(
;)()))(1()((
)(
23
21
13
tCtDtutEtu
dt
tdC
tCtDtutEtu
dt
tdD
tEtuI
dt
tdE
(1)
where u is a control. The objective function we are trying to minimize is given by:
dttAutCtDuJ
T
))()()(()( 2
0
,
where A is a positive weight that balances the size of the terms. U is the control
set defined by:
]},0[,1)(0,bleuismeasura/{ TttuuU .
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 138
The goal is to find a control *u in U that minimizes the objective function:
)(min)( * uJuJ
Uu
.
The system (1) is well defined, moreover theorems 3.2, 3.3, 3.4 and 3.5 in
[13] have shown respectively the existence and uniqueness and the positivity of
the solution, the existence of an optimal control and the characterization of this
control for this system (1).
Characterisation of the control optimal. The principle of maximum
Pontryagin and Hamiltonian has been used to characterize the optimal control,
and the adjoint variables. After mathematical calculations and demonstrations that
can be found in [13], we find that the adjoint variables are defined by:
)()(1
;)1)((1
;)()1()(
3233
22
*
322
13311
*
211
u
u
or and , 321 are the adjoint variables With conditions: ) ( ) ( 21 TT
0 ) ( 3 T and ** , DE and * C are the solutions of system (1) and the control is
defined as :
)]()()([
2
1
,0max,1min 232
*
133
*
121
** DEE
A
u .
This study, which was conducted over a period of ten years, summarizes that
the optimal control approach is very effective in reducing the effect of this
phenomenon, which is demonstrated in the experimental results found.
The CHP EDC TS model
An improvement of the EDC model was proposed in 2020 by Kouidere et al. [11]
noted ST CHPEDC . This time by a nonlinear controlled dynamic system. Six
types of populations are considered, people who are healthy H, pre-diabetic
patients due to genetics which is denoted by P , pre-diabetic patients due to
lifestyle denoted by E. The other three types are diabetic patients without com-
plications and diabetic patients with complications, which are denoted by D and C
respectively. We conclude that the diabetic population N is divided into six types
such that )( ) ( ) ( ) ( ) ( ) ( ) ( tCtCtDtEtPtHtNN TS at time t :
);()(
)()(
)()(
)(
;)()(
)()()()(
)()(
)(
;)()(
)()(
)()(
)(
;)()()(
)(
;)()()(
)(
;)()(
)(
1212
12123
2211
2
311
21
tC
N
tEtC
tCtD
dt
tdC
tC
N
tEtC
N
tEtD
tDtP
dt
tdC
tD
N
tEtD
tEtP
dt
tdD
tEtH
dt
tdE
tPtH
dt
tdP
tHI
dt
tdH
S
T
T
S
T
TT
(2)
Intelligent optimal control of nonlinear diabetic population dynamics system using …
Системні дослідження та інформаційні технології, 2024, № 1 139
with 0)0(,0)0(,0)0(,0)0(,0)0( TCDEPH , and 0)0( SC .
The modeling of the model is explained in [11], the model contains twelve
parameters: and , ,, , , , , , , , , ,, 321212121µI which were
estimated in [11]. The objective remains to control the transition rate from
prediabetes to diabetes with complications and transition from diabetes with
complications to diabetes without complications.
The controlled model is given by the following system:
,)()(
)()(
))(1()()(
)(
;)())((
)()(
))((
)()(
))(1()()(
)(
;)()()()(
)()(
))(1()())(1()(
)(
;)()))(1(()(
)(
;)()()(
)(
;)()(
)(
12312
11211
1223
122
1241
42
311
21
tC
N
tEtC
tutCtD
dt
tdC
tCtu
N
tEtC
tu
N
tEtD
tutDtP
dt
tdC
tCtutD
N
tEtD
tutEtutP
dt
tdD
tEtutH
dt
tdE
tPtH
dt
tdP
tHI
dt
tdH
S
T
T
S
T
T
T
T
where the controls )(),(),( 321 tututu , and C are the proposed controls.
The objective function we are trying to minimize is given by :
)()()(),,,( 4321 fffT TETDTCuuuuJ
dttu
G
tu
F
tu
B
tu
A
tEtDtCT
T f
)(
2
)(
2
)(
2
)(
2
)()()( 2
4
2
3
2
2
2
1
0
,
where 0,0,0 FBA , and 0G .
The system (4) is well defined, moreover theorems 1, 2, 3, 4, and 5 in [11]
have shown respectively the existence and uniqueness and the positivity of the
solution, the existence of an optimal control and the characterization of this
control for this system (4):
.],0[ / 1)(0
;1)(0
;1)(0
;1)(
0
),,,(
444
333
222
111
4321
fmaxmin
maxmin
maxmin
maxmin
Ttutuu
utuu
utuu
utuu
uuuu
(3)
Characterisation of the control optimal. The principle of maximum
Pontryagin and Hamiltonian has been used to characterize the optimal control,
and the adjoint variables. After mathematical calculations and demonstrations that
can be found in [11], we find that the adjoint variables are defined by:
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 140
23122111 )( ,
35143122 )( ,
N
tD
tututu
)(
))(1())(1()))(1((1 144433
N
tC
tu
N
tC
tu
N
tD
tu TT )(
))(1(
)(
))(1(
)(
))(1( 32632215 ,
)(
)(
))(1(1 222144 N
tE
tu
262125
)(
))(1(
N
tE
tu , (4)
))((
)(
))(1()(1 11325145 tu
N
tE
tutu
N
tE
tu
)(
))(1( 3216 ,
)(66 .
With the transversality conditions at time ,0)(,0)(, 21 fff TTT
1)(,1)(,1)( 543 fff TTT and 0)(6 fT .
Furthermore, for ],0[ fTt , the optimal controls *
3
*
2
*
1 ,, uuu and *
4u are given by:
)(
)(
,0max,1min 45*
1 tC
A
u T ,
N
tEtD
B
u
)()(
,0max,1min 45
1
*
2 ,
N
tEtC
F
u T )()(
,0max,1min 56
2
*
3 ,
)(
)(
,0max,1min 34*
4 tE
G
u .
EXPERIMENTAL RESULTS
In this section, we represent a comparison between the classical numerical
method used to solve the mathematical models proposed in the literature and the
intelligent genetic algorithm method. In the first part we will start with the kernel
model (1) and in the second part we will make the comparison in the nonlinear
model (4).
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In this research paper, we used the genetic algorithm to adjust the different
controls and the different parameters of the system, knowing that it is a powerful
evolutionary algorithm.
The advantage of using the genetic algorithm to solve this type of optimal
control problem is that we do not need to determine the adjoint values or charac-
terize the control in any specific way. We only need to treat the problem well and
specify the objective function we need to minimize and its various constraints.
We present the results obtained by solving numerically the optimality systems (1)
and (4) using the genetic algorithm method. In this control problems, we used the
(GA) to find the most accurate initial conditions for the state variables.
In both models, which we will study, the quality of the controls of each of them,
we have used the same values of the parameters that were used in the work [11; 13].
MatLab was used to write and compile the code, and the following data was used:
Genetic algorithm configuration:
Crossover : multiple;
Crossover : 0.8;
Initialization : random;
Number of iteration: 100 dim ;
Mutation : gaussian;
Population size : 200;
Selection function : stochastic(uniform).
The choice of the parameters of the genetic algorithm operators (selection,
crossover, mutation) is a difficult task. In our work, our configuration is based on
the following information:
a) Crossover: this step allows the production of new solutions, from the
previous individuals. Each bit is chosen from either parent with the same
probability [35; 36].
b) Mutation: it is sometimes found that all solutions are weak, which means
the crossover operator does not lead to a new gene. The mutation causes random
changes of rate pm in the genes, but this rate should not be larger, to avoid loss of
the principle of selection and evolution [37].
c) Population size: the size of the population has a direct influence on the
optimal control capability of our problem. Our choice of this parameter is based
on [38; 39].
Linear EDC model
Let’s start with the linear model (1). The use of single-objective optimization
methods is not obvious on all this type of problem, first of all the understanding
of the phenomenon is essential to put the pseudo-code simple to execute. Sec-
ondly, for a very good choice of the initial population that an important step in the
method of genetic algorithm.
The Boutayeb model (1) parameters: ; 05.0 ; 08.0 ; 02.0 ;2000000 µI
; 5.0 ; 1.0 ; 5.0 ; 05.0 321 .3550000 ; 10 : 1 : .0 Adim
The Boutayeb model (2) compartiments initialization: ;1056.66 ) 0( E
.105 55 ) 0(; 105 102 ) 0( CD
As the linear case [33; 34], we used least square linear method to estimate
the values of the parameters of the dynamic system of population diabetic and we
found values close to those of the research paper [13; 34], which was based on the
values of the World Diabetes Federation [1; 4].
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 142
From the Fig. 1, a, we can notice that the use of the Gauss–Seidel type
implicit finite difference method developed by Gumel [19] is give an estimate and
does not give an optimal control *u .
For that we propose to solve this problem with the genitic algorithm (GA),
we can see in the Fig. 2, a, b, c and Fig. 1, b that GA gives an optimal control but
does not give a good effect.
Fig. 1. The comparison between the control given by the GA and Gumel methods: a — the control
u given by system (1) using Gumel method; b — the control u using GA
a b Time t
Fig. 2. The compartment E, D and C with and without control: a — the com-
partment D with and without control using GA; b — the compartment E with
and without control using GA; c — the compartment C with complications
with and without control using GA
1 — C without control
2 — C with control
2
1
c Time t
b Time t
1 — E without control
2 — E with control
1
2
1 — D without control
2 — D with control
2
1
a Time t
1 — D without control
2 — D with control
Intelligent optimal control of nonlinear diabetic population dynamics system using …
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It can be noticed from Fig. 1, a that the control given by Gumel’s method
takes almost a fixed value which is exactly 1. Of course, this allows a com- plete
control of the dynamics of the phenomenon, throughout the five years, but
consumes the totality of human and material resources. The proposed solution,
based on a nonlinear model and a genetic algorithm, produces reasonable con-
trols with very low cost (in fact, we never need all the resources in a given period
to achieve our goal) (see Fig. 2). Furthermore, the genetic algorithm shows a great
ability to produce controls in only a few iterations.
A nonlinear CHP EDC TS model
We now turn to the non-linear model (4) which contains 6 compartments. We
have seen that four compartments have been controlled by four different controls.
Our objective is to learn about the controls given by the intillegent method and
their quality.
The model 4 shows that the number of pre-diabetic population by the nega-
tive effect of behavioral factors on diabetic patients and other diabetics E and D
without complications are those who control blood glucose by diet and exercise
before it is too late, CT diabetics with treatable complication and CS diabetics with
severe complications is decreased by the use of controls 321 ,, uuu and 4u
respectively. This can be seen clearly in the Figs. 3, a, b, 4, a, b, 5, a, b.
In genetic algorithm, there exist several types of convergences: a) the con-
structed control sequence becomes very close to the optimal control sought; b) the
constructed control sequence becomes stagnant; c) the cost sequence of the
constructed control sequence becomes stagnant; d) the maximum number of
iterations, set by the user, is exhausted. It is impossible to refer to the optimal
control sought because it is not known; therefore the first type of convergence
remains impractical. The second and third type of convergence is directly
influenced by the size of the population, the crossover rate, and the mutation rate;
indeed, a bad choice of these can lead the genetic search to very bad local
minima. The fourth type of convergence is called artificial convergence because
the user is satisfied with a reasonable improvement of the quality of the initial
population especially when the studied phenome is very complex and calls for
several algorithmic components. We opted for the artificial convergence because
the studied system is very complex and the control estimation is part of a very
large research project involving several components. Moreover, we can notice
Fig. 3. The controls of non- linear system: a — objective function of GA; b — the
controls of non- linear system
3
1
24 1 – u1
2 – u2
3 – u3
4 – u4
a bGentration Time t
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 144
that a few iterations were able to achieve a very good estimation; of course, we
can increase the number of iterations to improve the obtained control even if the
one obtained is very satisfactory (see Fig. 3, a).
In this work, we present two types of models. A linear dynamic model and a
non-linear model. In the first model, we considered a single control on both
behaviors, which does not correspond to human nature, because each diabetic
patient has his own needs. But the second non-linear model takes into account 6
different behaviors, that is, the most possible cases of diabetes, moreover four
controls were considered, which helps us to treat each behavior with its needs. In
conclusion, the dynamic controlled model ST CHPEDC gives us more infor- mation
because of the diversity of diabetes types, and the control strategy for each type.
In general, we introduced a comparison between the following systems: a)
Gumel and linear model; b) Gumel and non-linear model; and c) Genetic Al-
gorithm and non-linear model.
Solved by Gumel method, the linear and non-linear models, the produced
control recommends always the implementation of all disponible resources be-
cause of the formula of the control that implements the operators max-min which
leads to the value 1. Solved by genetic algorithm, the linear and non-linear mod-
els, the produced control is capable to reduce significantly the number of diabetic
and diabetics with complications using less than 70% (mean) of resources (see
Fig. 4. The compartment (a) CT and ( b ) CS with and without control given by
system (7) and (10) using GA
Time t
1 — CT with control
2 — CT without control
1
2
1 — CS with control
2 — CS without control
1
2
Time ta b
Fig. 5. The compartment E (a) and D ( b ) with and without control given by sys-
tems (2) and (3) using GA
Time ta b
1 — E with control
2 — E without control
1
2
1 — E with control
2 — E without control
1
2
Time t
Intelligent optimal control of nonlinear diabetic population dynamics system using …
Системні дослідження та інформаційні технології, 2024, № 1 145
Figs. 1 and 3). In addition, the non-linear system is more suitable to understand
and control the phenomenon under study.
The estimated control are in the hand of diabetologists and it will be
traduced in terms of medication, exercise, and diet.
ACKNOWLEDGMENT
This work was supported by Ministry of National Education, Professional
Training, Higher Education and Scientific Research (MENFPESRS) and the
Digital Development Agency (DDA) and CNRST of Morocco (Nos.
Alkhawarizmi/2020/23).
CONCLUSION
In this paper we have elaborated a state of the art on the different dynamic
systems with economic function proposed in the literature controlling diabetes in
order to alleviate the socio-economic damage caused by it. Then, we have used of
the best performing local search artificial intelligence methods (metaheuristics) to
solve the kernel model dealing with the common compartments between all these
models. Controlling diabetes is a major challenge, and studies have highlighted
the lim- itations of linear models in describing the complexity of this chronic
disease. Non-linear models, by introducing more realistic terms, have enabled a
better understanding of the dynamics of the diabetic population. However, tradi-
tional methods of order estimation, such as Pantryagain’s maximum principle and
Gumel’s numerical method, have proved costly and difficult for diabetologist
professionals to implement. In this paper, we have proposed an innovative strat-
egy based on nonlinear models and the use of GAs. This three-step approach
discretizes the nonlinear model, estimates the optimal control using GAs, and
constructs the final control using an interpolation model. The results obtained
demonstrated the success of this approach, with a significant reduction in the
number of cases of diabetes and complications in diabetic patients, while using
less than 70% of available resources. Despite the success of GAs in diabetes
control, there are a number of limitations to consider. Firstly, model resolution
time can be a challenge, not least due to the complexity of the non-linear models
used. In addition, the use of integer derivatives may restrict access to certain
relevant information. In future work, it is proposed to explore two avenues to
overcome these limitations, 1) the use of parallelism techniques could speed up
model solution time; 2) the use of fractional equations rather than ordinary
differential equations could be considered.
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Received 11.11.2022
El Ouissari Abdellatif, El Moutaouakil Karim
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 148
INFORMATION ON THE ARTICLE
Abdellatif El Ouissari, ORCID: 0000-0003-0956-2029, Sidi Mohamed Ben Abdellah
University; Higher School of Engineering in Applied Sciences, Fez, Morocco, e-mail:
abdellatif.elouissari@usmba.ac.ma
Karim El Moutaouakil, ORCID: 0000-0003-3922-5592, Sidi Mohamed Ben Abdellah
University, Fez, Morocco, e-mail: karim.elmoutaouakil@usmba.ac.ma
ІНТЕЛЕКТУАЛЬНЕ ОПТИМАЛЬНЕ КЕРУВАННЯ НЕЛІНІЙНОЮ
СИСТЕМОЮ ПОПУЛЯЦІЙНОЇ ДИНАМІКИ ХВОРИХ НА ДІАБЕТ ІЗ
ВИКОРИСТАННЯМ ГЕНЕТИЧНОГО АЛГОРИТМУ / Абделлатиф Ель Уіссарі,
Карім Ель Мутауакіль
Анотація. Цукровий діабет є хронічним захворюванням, яким страждають міль-
йони людей у всьому світі. Виконано кілька досліджень, спрямованих на конт-
роль проблеми діабету, із використанням як лінійних, так і нелінійних моде-
лей. Однак складність лінійних моделей не в змозі глибинно описати динаміку
діабетичного населення. Щоб отримати більше деталей про цю динаміку, до
математичних моделей уведено нелінійні терміни, що призвело до більш скла-
дних моделей, які повністю відповідають реальності (здатні відтворювати спо-
стережувані дані). Найбільш часто використовуваними методами для оціню-
вання контролю є принцип максимуму Пантрягейна та числовий метод
Гумеля. Однак ці методи призводять до дуже дорогої стратегії з точки зору
матеріальних і людських ресурсів; крім того, діабетологи не в змозі викорис-
товувати формули, реалізовані запропонованими елементами контролю. За-
пропоновано вибіркову стратегію та продуктивність, засновану на нелінійних
моделях і генетичних алгоритмах (GA), яка виконується в три етапи: 1) дис-
кретизація розглянутої нелінійної моделі за допомогою класичних числових
методів (правило трапеції та алгоритм Ейлера–Коші); 2) оцінювання оптима-
льного контролю в кількох точках на основі GA з відповідною функцією при-
стосованості та відповідними генетичними операторами (мутація, схрещу-
вання та відбір); 3) побудова оптимального керування за допомогою
інтерполяційної моделі (сплайнів). Отримані результати показали, що викори-
стання GA для нелінійних моделей було успішно вирішено, що призвело до
контрольного підходу, який демонструє значне зменшення кількості випадків
діабету та діабетиків з ускладненнями. Примітно, що цей результат досягаєть-
ся з використанням менше ніж 70% доступних ресурсів.
Ключові слова: оптимальне керування, диференціальне рівняння, діабет, ге-
нетичні алгоритми, штучний інтелект, інтелектуальний локальний пошук.
|
| id | journaliasakpiua-article-304622 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:32Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/59/2b913af574bbd2a50ba125f08412a459.pdf |
| spelling | journaliasakpiua-article-3046222024-05-23T07:09:36Z Intelligent optimal control of nonlinear diabetic population dynamics system using a genetic algorithm Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму El Ouissari, Abdellatif El Moutaouakil, Karim оптимальне керування диференціальне рівняння діабет генетичні алгоритми штучний інтелект інтелектуальний локальний пошук optimal control differential equation diabetes genetic algorithms artificial intelligence intelligent local search Diabetes is a chronic disease affecting millions of people worldwide. Several studies have been carried out to control the diabetes problem, involving both linear and non-linear models. However, the complexity of linear models makes it impossible to describe the diabetic population dynamic in depth. To capture more detail about this dynamic, non-linear terms were introduced into the mathematical models, resulting in more complicated models strongly consistent with reality (capable of re-producing observable data). The most commonly used methods for control estimation are Pantryagain’s maximum principle and Gumel’s numerical method. However, these methods lead to a costly strategy regarding material and human resources; in addition, diabetologists cannot use the formulas implemented by the proposed controls. In this paper, the authors propose a straightforward and well-performing strategy based on non-linear models and genetic algorithms (GA) that consists of three steps: 1) discretization of the considered non-linear model using classical numerical methods (trapezoidal rule and Euler–Cauchy algorithm); 2) estimation of the optimal control, in several points, based on GA with appropriate fitness function and suitable genetic operators (mutation, crossover, and selection); 3) construction of the optimal control using an interpolation model (splines). The results show that the use of the GA for non-linear models was successfully solved, resulting in a control approach that shows a significant decrease in the number of diabetes cases and diabetics with complications. Remarkably, this result is achieved using less than 70% of available resources. Цукровий діабет є хронічним захворюванням, яким страждають мільйони людей у всьому світі. Виконано кілька досліджень, спрямованих на контроль проблеми діабету, із використанням як лінійних, так і нелінійних моделей. Однак складність лінійних моделей не в змозі глибинно описати динаміку діабетичного населення. Щоб отримати більше деталей про цю динаміку, до математичних моделей уведено нелінійні терміни, що призвело до більш складних моделей, які повністю відповідають реальності (здатні відтворювати спостережувані дані). Найбільш часто використовуваними методами для оцінювання контролю є принцип максимуму Пантрягейна та числовий метод Гумеля. Однак ці методи призводять до дуже дорогої стратегії з точки зору матеріальних і людських ресурсів; крім того, діабетологи не в змозі використовувати формули, реалізовані запропонованими елементами контролю. Запропоновано вибіркову стратегію та продуктивність, засновану на нелінійних моделях і генетичних алгоритмах (GA), яка виконується в три етапи: 1) дискретизація розглянутої нелінійної моделі за допомогою класичних числових методів (правило трапеції та алгоритм Ейлера–Коші); 2) оцінювання оптимального контролю в кількох точках на основі GA з відповідною функцією пристосованості та відповідними генетичними операторами (мутація, схрещування та відбір); 3) побудова оптимального керування за допомогою інтерполяційної моделі (сплайнів). Отримані результати показали, що використання GA для нелінійних моделей було успішно вирішено, що призвело до контрольного підходу, який демонструє значне зменшення кількості випадків діабету та діабетиків з ускладненнями. Примітно, що цей результат досягається з використанням менше ніж 70% доступних ресурсів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-03-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/304622 10.20535/SRIT.2308-8893.2024.1.10 System research and information technologies; No. 1 (2024); 134-148 Системные исследования и информационные технологии; № 1 (2024); 134-148 Системні дослідження та інформаційні технології; № 1 (2024); 134-148 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/304622/296453 |
| spellingShingle | оптимальне керування диференціальне рівняння діабет генетичні алгоритми штучний інтелект інтелектуальний локальний пошук El Ouissari, Abdellatif El Moutaouakil, Karim Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title | Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title_alt | Intelligent optimal control of nonlinear diabetic population dynamics system using a genetic algorithm |
| title_full | Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title_fullStr | Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title_full_unstemmed | Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title_short | Інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| title_sort | інтелектуальне оптимальне керування нелінійною системою популяційної динаміки хворих на діабет із використанням генетичного алгоритму |
| topic | оптимальне керування диференціальне рівняння діабет генетичні алгоритми штучний інтелект інтелектуальний локальний пошук |
| topic_facet | оптимальне керування диференціальне рівняння діабет генетичні алгоритми штучний інтелект інтелектуальний локальний пошук optimal control differential equation diabetes genetic algorithms artificial intelligence intelligent local search |
| url | https://journal.iasa.kpi.ua/article/view/304622 |
| work_keys_str_mv | AT elouissariabdellatif intelligentoptimalcontrolofnonlineardiabeticpopulationdynamicssystemusingageneticalgorithm AT elmoutaouakilkarim intelligentoptimalcontrolofnonlineardiabeticpopulationdynamicssystemusingageneticalgorithm AT elouissariabdellatif íntelektualʹneoptimalʹnekeruvannânelíníjnoûsistemoûpopulâcíjnoídinamíkihvorihnadíabetízvikoristannâmgenetičnogoalgoritmu AT elmoutaouakilkarim íntelektualʹneoptimalʹnekeruvannânelíníjnoûsistemoûpopulâcíjnoídinamíkihvorihnadíabetízvikoristannâmgenetičnogoalgoritmu |