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Mathematical Modelling and Numerical Simulation
with Applications, 2024, 4(4), 514–543
https://dergipark.org.tr/en/pub/mmnsa
ISSN Online: 2791-8564 / Open Access
https://doi.org/10.53391/mmnsa.1555670
RESEARCH PAPER
A study of fractional optimal control of overweight and
obesity in a community and its impact on the diagnosis of
diabetes
Erick Manuel Delgado Moya ID 1,*,‡,Ranses Alfonso Rodriguez ID 2,‡,Alain
Pietrus ID 3,‡ and Séverine Bernard ID 3,‡
1School of Technology, University of Campinas (UNICAMP), R. Paschoal Marmo, 1888 - Jd. Nova
Itália, 484-332 - Limeira, SP, Brazil, 2Department of Applied Mathematics, Florida Polytechnic
University, Lakeland, FL 33805, USA, 3Department of Mathematics and Computer Sciences,
University of Antilles, LAMIA (EA 4540), BP 250, 97159, Pointe-a-Pitre, Guadeloupe, France
*Corresponding Author
‡erickdelgadomoya@gmail.com (Erick Manuel Delgado Moya); ranses.alfonso@gmail.com (Ranses Alfonso
Rodriguez); alain.pietrus@univ-antilles.fr (Alain Pietrus); severine.bernard@univ-antilles.fr (Séverine Bernard)
Abstract
Obesity and diabetes are diseases that are increasing every year in the world and their control is an
important problem faced by health systems. In this work, we present an optimal control problem based
on a model for overweight and obesity and its impact on the diagnosis of diabetes using fractional
order derivatives in the Caputo sense. The controls are defined with the objective of controlling the
evolution of an individual with normal weight to overweight and that overweight leads to chronic
obesity. We show the existence of optimal control using Pontryagin’s maximum principle. We perform
a study of the global sensitivity for the model using Sobol’s index of first, second and total order using
the polynomial chaos expansion (PCE) with two techniques, ordinary least squares (OLS) and least
angle regression (LAR) to find the polynomial coefficients, and two sampling methods, Monte Carlo
and Sobol’. With the obtained results, we find that among the parameters with the greatest influence
are those we used in the definition of the control system. We have that the best results are achieved
when we activate the three controls. However, when we only activate two controls, it shows better
results in preventing a person with normal weight from becoming overweight by controlling weight
gain due to social pressure and the evolution from overweight to obesity. All strategies significantly
reduce the number of cases diagnosed with diabetes over time.
Keywords: Optimal control; diabetes; global sensitivity analysis; obesity; overweight
AMS 2020 Classification: 49J15; 26A33; 37N25; 37N35; 92D30
➤Received: 25.09.2024 ➤Revised: 06.12.2024 ➤Accepted: 23.12.2024 ➤Published: 30.12.2024
514
Moya et al. |515
1 Introduction
The prevalence of overweight and obesity increased worldwide in recent decades. According to
the World Obesity Atlas 2023, the estimated value of overweight and obesity (body mass index
[BMI ]≥25 kg
m2
) in 2020 among the global population aged more than 5 years was 38%, which is
projected to reach 51% in 2035. In the United States (US), the prevalence of obesity (
[BMI ]≥30 kg
m2
)
increased from 30.5% in 1999-2000 to 41.9% in 2017-2020. By 2030, 48.9% of US adults may have
obesity, with 24.2% falling into the severely obese category ([BMI]≥35 kg
m2) [1–3].
More than 500 million people are living with diabetes worldwide, and it is predicted to more
than double to 1.3 billion people in the next 30 years. Almost all global cases (96%) diagnosed
with diabetes are type 2 diabetes (T2D). High body mass index (BMI) was the primary risk
for T2D – accounting for 52.2% of T2D disability and mortality – followed by dietary risks,
environmental/occupational risks, tobacco use, low physical activity, and alcohol use [4].
To determine body weight state, we use the body mass index (BMI), which is defined as [5]:
BMI =weight
height2.
Then individuals are considered of normal weight when
BMI ∈[18
.
6
,
24
.
9]
, overweight individ-
uals are when
BMI ∈[25
,
29
.
9]
, obese individuals are when
BMI ∈[30
,
40]
and in complicated
situations over 40. We know that the body mass index can be high for people with high muscle
mass but we assume that these cases are not included because before calculating the BMI we do a
preliminary analysis. The root cause of obesity and overweight is an energy imbalance between
consumed and expended calories.
Over the last few years, the study of mathematical models of prey-predator and those of general
structures such as SIR (susceptible-infected-recovered), SIS (susceptible-infected-susceptible),
SVIR (susceptible-vaccinated-infected-recovered) has continued its development and elements
such as delay and non-local dispersion have been incorporated [
6
–
9
]. We also note a growing use
of fractional derivatives in the modeling of epidemics such as HIV [
10
], COVID [
11
], Cancer [
12
],
Cholera [13], Influenza [14], and Hepatitis B [15].
Recently, the use of mathematical models to study the behavior of overweight and obesity has
increased [
16
–
22
]. Ejima et al. [
16
] presented a mathematical model that studies the genetic and
nongenetic effects leading to obesity and among the results they obtained that homozygous indi-
viduals are more susceptible to the risk of social contagion and the risk of spontaneous weight gain.
Kim and So-Yeun Kim [
17
] proposed a mathematical model with the inclusion of psychological
and social factors for the study of obesity. Paudel [
18
] presented a model for the dynamics of
obesity with a SIR structure and analyzed the effect of social network on the spread of obesity.
Al-Tuwairqi and Matbouli [
19
] proposed two mathematical models to study the impact of fast
food on the increase of obesity and the role of physical activity. Pietrus et al. [
20
] developed a
mathematical model to study the impact of media on the dynamics of obesity in a population.
Moya et al. [
21
] presented a model with ordinary differential equations that studies overweight,
obesity and the impact on the diagnosis of diabetes in a population, taking into account the
negative impact that an overweight or obese individuals can have on an individual of normal
weight, and also the impact of social pressure on the increase of overweight cases. Based on article
[
21
], Moya et al. [
22
] incorporated into the model the positive effect of the interaction between
a normal weight individual and overweight or obese individuals can have and transformed the
model for fractional order equations in the Caputo sense, taking advantage of the memory effect.
This model is used in the formulation of the control problem addressed in this paper taking into
516
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
account the results obtained in [22].
The aim of this work is to propose and solve an optimal control problem focused on reducing
overweight and obesity in a community and its impact on diabetes, taking advantage of a model
that uses fractional order derivatives in the Caputo sense.
This paper is organized as follows: In Section 2, we present the definitions used in the paper. In
Section 3, we introduce the model with fractional order derivatives in the Caputo sense. Section 4
contains the definition of controls, formulation of the control problem and demonstration of its
basic properties. Section 5 is devoted to the study of the global sensitivity analysis of the model
and the numerical simulations. We finish the paper with some conclusions in Section 6.
2 Theorical background
The following definitions are used to formulate and study the fractional order derivative model.
We assume that
α∈R+
,
b>0
,
f∈ACn[a
,
b]
(absolutely continuous) and
n= [α]
(integer part of
α
). We define the left-sided and right-sided fractional integral Riemann-Louville for
f:R+−→R
and α>0 as:
aIα
tf(t):=1
Γ(α)Zt
a
f(s)ds
(t−s)1−α, (Left)
tIα
bf(t):=1
Γ(α)Zb
t
f(s)ds
(s−t)1−α, (Right)
where Γis Gamma function, and we define Iα
tf(t) = 0Iα
tf(t).
The left-sided and right-sided Riemann–Liouville fractional derivatives are defined as [23,24]:
aDα
tf(t) = dn
dtn1
Γ(n−α)Zt
a
(t−s)n−α−1f(s)ds, (Left)
tDα
bf(t) = dn
dtn(−1)n
Γ(n−α)Zb
t
(s−t)n−α−1f(s)ds, (Right)
and we denote Dα
tf(t) = 0Dα
tf(t).
The left-sided and right-sided fractional derivatives proposed by Caputo are given by [23,24]:
c
aDα
tf(t) = 1
Γ(n−α)Zt
a
(t−s)n−1−αf(n)(s)ds, (Left)
c
tDα
bf(t) = (−1)n
Γ(n−α)Zb
t
(s−t)n−1−αf(n)(s)ds, (Right)
and we define cDα
tf(t) = c
0Dα
tf(t).
In the order fractional derivatives, we find the memory effect which is an important factor in
epidemic modeling [25–27].
In recent decades, works have been presented where models with fractional orders are used and
compared with real data and it has been obtained that they can capture real behaviors, see [
28
–
31
].
Now, we will present a general formulation of the fractional order optimal control problem (FOCP)
and obtain the necessary conditions for the optimality of the FOCP: Find the optimal control
u(t)
that minimizes the functional Jdefined as:
J(u) = Zb
0f(t,x,u)dt, (1)
Moya et al. |517
subject to the controlled model
cDα
tx(t) = g(t,x,u), (2)
with initial condition
x(0) = xI, (3)
where
x(t)
and
u(t)
are the state and control variables respectively,
f
and
g
are differential
functions and 0 <α≤1.
Theorem 1
If
f(x
,
u)
is a minimizer of
(1)
that satisfies the constraint
(2)
and the initial condition
(3)
,
then there is a function λ∈C1[0, b]such that the triplet (x,u,λ)satisfies:
i. the co-state and state systems, respectively
cDα
tx(t) = ∂H
∂λ (t,x(t),u(t),λ(t)),
c
tDα
bλ(t) = ∂H
∂x(t,x(t),u(t),λ(t)),
ii. the stationary condition
∂H
∂u(t,x(t),u(t),λ(t)) = 0,
iii. and the condition of transversality
tI1−α
bλ
t=b=λ(b) = 0,
where the Hamiltonian H is defined by
H(t,x,u,λ) = f(t,x,u) + λ(t)·g(t,x,u).
Lemma 1 The following equations are equivalent:
c
tDα
bλ(t) = ∂H
∂x(t,x(t),u(t),λ(t)),
cDα
tλ(b−t) = ∂H
∂x(b−t,x(b−t),u(b−t),λ(b−t)),
where α∈(0, 1].
The proof of Theorem 1 and Lemma 1 can be found in [24] and its applications in [24,32–34].
3 Model construction
In this section, we present the model that we will use in the definition of the optimal control
problem. The current model is based on those with ordinary differential equations presented in
[
20
] and [
21
] and the current version with fractional equations in Caputo’s sense found in [
22
],
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
where we increase the positive and negative impact of the interactions between overweight, obese
and normal weight individuals.
Based on the body mass index, we define the model compartments as: normal weight individuals,
S
, overweight individuals,
Ow
, obese individuals,
Ob
and diabetic individuals,
D
. We assume that
for the diagnosis of overweight and obesity in atypical cases (where the physical examination
shows that the patient has other factors that alter the BMI), medical examinations are necessary
because there are cases where the body mass index may be increased by muscle, inflammations in
the body and other factors.
We define the negative impact rate as
λO= (α∗)α(Ow+ϵOb)
N,
and positive impact rate as
λS=(β∗)αS
N,
where
α∗
and
β∗
are the effective contact rate and
N
is the total population (
N=S+Ow+Ob+D
).
The rates
MS
and
MD
represent the entry of individuals with normal weight and diabetes respec-
tively.
We assume that the impact of overweight and obese on a person with normal weight is different,
for this, we use the modification parameter ϵsuch that ϵ>1.
The rate
α1
characterizes cases diagnosed with diabetes that are not directly associated with weight
gain including genetic, racial, hereditary and other factors. Parameters
α2
and
α3
are the rates
of diabetes diagnoses associated with overweight and obesity, respectively. The rate of death
associated with diabetes is denoted by µD.
The rate
δ
characterize individuals who are overweight but improve to normal weight and
η
is for
obese individuals who become overweight. These two parameters are not related to the interaction
with a normal weight individual. Individuals who increase in body weight from overweight to
obese are defined by the rate γ.
The rate
β1
represents the social pressure that causes an individual with normal weight to become
obese. This rate is characterized by stress, lack of time for healthy eating and physical exercise,
sedentary lifestyle, etc.
The mortality rate from natural causes in the population is defined as
µ
and we assume that it is
the same from any compartment. We define
d
as the mortality rate associated with overweigth
and obesity.
Table 1 shows the definition of the parameters and their reference values and Figure 1 presents the
dynamics diagram of the model.
The fractional derivative operator in the Caputo sense,
cDα
t
has a
α
dimension, then on the
right-hand side of the model all parameters will have power dimension
α
except the modification
parameters [
35
]. The model that studies the behavior of overweight and obesity and its impact
on the diagnosis of diabetes in a population using the fractional derivatives in the Caputo sense,
Moya et al. |519
found in [22], is:
cDα
tS=Mα
S+ (δα+λS)Ow−(µα+αα
1+βα
1+λO)S,
cDα
tOw= (λO+βα
1)S+ηαOb−(λS+γα+µα+dα+δα+αα
2)Ow,
cDα
tOb=γαOw−(ηα+µα+tHdα+αα
3)Ob,
cDα
tD=Mα
D+αα
1S+αα
2Ow+αα
3Ob−(µα
d+µα+tDdα)D, (4)
with initial conditions
S(0) = S0≥0, Ow(0) = Ow0≥0, Ob(0) = Ob0≥0, D(0) = D0≥0 and α∈(0, 1].
Table 1. Parameters description of model (4)
Parameter Description Value Reference
MSRecruitment rates for normal weight 667.685 [36]
MDRecruitment rates for diabetic individuals 4.1 [21,22]
α∗Effective contact rates (negative impact) 2 [21,22]
β∗Effective contact rates (positive impact) 0.2 [21,22]
ddeath rate associated with weight gain 0.07 [21,22]
µNatural death rate 1
70.5 [21,22,37]
µdDiabetes death rate 0.013 [22]
ηRate of weight reduction from obese to overweight 0.1 [20]
γRate of weight gain from overweight to obese 0.0015 [20]
δRate of weight reduction from overweight to normal weight 0.002 [20]
α1Rate of diagnosis of diabetes not associated with body weight 0.1 [21,22]
α2Rate of diabetes diagnosis in overweight individuals 0.35 [21,22,38,39]
α3Rate of diagnosis of diabetes in obese individuals 0.4 [21,22,38,39]
β1Rate of weight gain associated with social factors 0.25 [21,22]
ϵ,tD,tHModification parameters 1.02, 1.03 Assumed
For model (4), we proved the existence and non-negativity of the solution and found the biologi-
cally feasible region where the model makes biological sense and we calculated and studied the
basic reproduction number in [22].
4 Definition of the controls and its policy
The objective of the controls is to reduce the number of overweight and obese individuals in the
population and as a consequence avoid new diagnoses of diabetes. The controls are divided into
three categories, the first control regulates negative relationships, the second the effect of social
pressure and the third the evolution from overweight to obese. Controls are defined as:
•u1
: Control that regulates the negative impact of the influence that overweight and obese
individuals have on individuals with normal weight in order to prevent an individual with
normal weight from changing their lifestyle and becoming overweight.
•u2
: Social pressure leads individuals to give up a healthy life, whether due to overwork,
sedentary lifestyle, or poor diet. This control focuses on regulating the impact of social pressure
on weight gain in individuals with normal weight. This would include performing other
activities, mainly physical exercise, walking in wooded areas and encouraging the individual to
change their diet by continually avoiding fast food, time management, Visit to nutritionists and
nutritionists, etc.
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|
Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
𝑺
𝑶𝑾
𝑶𝒃
𝑫
𝝁
𝝁 + 𝝁𝒅+ 𝒕𝑫𝒅
𝝁 + 𝒕𝑯𝒅
𝝁 + 𝒅
𝜼
𝜶𝟏
𝜶𝟐
𝜶𝟑
𝜸
𝑴𝑫
𝑴𝑺
Figure 1. Flow chart of Model (4)
•u3
: The objective of this control is focused on preventing the disease from developing and
leading to obesity. It consists of encouraging a gradual change in lifestyle, doing physical
exercise, visiting a nutritionist to follow a healthy diet and eating habits and doing activities,
avoiding stress and controlling schedules, etc.
All controls have economic and socio-psychological factors associated with them. For exam-
ple gyms and many physical activities, nutritional consultations, maintaining a diet, etc. may
have an economic cost and already the interest to practice this healthy lifestyle is related to
psychological-social factors. We assume that the difficulty and cost of the
u3
control is higher than
the others because an individual who is already overweight and obese needs more economic and
psychological-social elements to improve his lifestyle and overcome his disease1[40].
Optimal control problem formulation
Now, we present the optimal control problem with fractional derivatives in the Caputo sense. The
problem is theoretically focused on the control of new cases of overweight and obesity.
The functional objective to be minimized is:
J(u1,u2,u3) = Ztf
t0Ow(t) + Ob(t) + 1
2(B1u2
1(t) + B2u2
2(t) + B3u2
3(t))dt.
The coefficients
B1
,
B2
and
B3
represent the weight constants associated with the implementation of
the controls over a finite time horizon
[t0
,
tf]
(
t0
is the initial and
tf
the final moment of time) with
Bi>0
,
i=1
,
2
,
3B1≤B2<B3
. The cost involved in the application of control in compartments
S,Owand Obare given by Ztf
t0
B1u2
1(t)
2dt,Ztf
t0
B2u2
2(t)
2dt and Ztf
t0
B3u2
3(t)
2dt.
1
Obesity is a chronic, complex disease defined by excessive fat deposits that can impair health. Obesity can increase
the risk of type 2 diabetes, heart disease and certain types of cancer
Moya et al. |521
Then, your goal is to find the optimal controls u∗
1,u∗
2,u∗
3that satisfy:
J(u∗
1,u∗
2,u∗
3) = min
Uad
J(u1,u2,u3), (5)
where
Uad ={(u1
,
u2
,
u3): Lebesgue measurable
,
0≤ui≤1
,
i=1
,
2
,
3
,
∀t∈[t0
,
tf]}
. The
constraints of the optimal control problem is model
(4)
with the incorporation of the controls as
follows:
cDα
tS=fα
1=Mα
S+ (δα+λS)Ow−(µα+αα
1+ (1−u2)βα
1+ (1−u1)λO)S,
cDα
tOw=fα
2= ((1−u1)λO+ (1−u2)βα
1)S+ηαOb−(λS+ (1−u3)γα+µα+dα+δα+αα
2)Ow,
cDα
tOb=fα
3= (1−u3)γαOw−(ηα+µα+tHdα+αα
3)Ob,
cDα
tD=fα
4=Mα
D+αα
1S+αα
2Ow+αα
3Ob−(µα
d+µα+tDdα)D, (6)
with initial conditions
S(0) = S0≥0, Ow(0) = Ow0≥0, Ob(0) = Ob0≥0, D(0) = D0≥0 and α∈(0, 1].
For the given optimal control problem the Hamiltonian is defined as:
Hα(t) = Ow(t) + Ob(t) + 1
2(B1u2
1(t) + B2u2
2(t) + B3u2
3(t)) +
4
X
n=1
λnfα
n,
where λ1,λ2,λ3and λ4are the adjoint variables.
We have the following important theorem:
Theorem 2
If
u∗
1
,
u∗
2
and
u∗
3
are the controls associated to
(5)
,
S∗
,
O∗
w
,
O∗
b
and
D∗
are corresponding
optimal paths them, there are co-state variables
λn
,
n=1
, ..,
4
, such that the control system
(6)
and, the
following conditions are satisfied:
•the co-state equations using the equivalence of Lemma 1:
cDα
tλ1(t′) = (λ2−λ1)[(1−u1)λO+ (1−u2)βα
1]−λ1(µα+αα
1),
cDα
tλ2(t′) = 1+ (1−u2)(λ3−λ2)γα+ (λ1−λ2)(λS+δα)−λ2(dα+µα+αα
2),
cDα
tλ3(t′) = 1+ (λ3−λ1)ηα−λ3(µα+tHdα+αα
3),
cDα
tλ4(t′) = −λ4(µα
d+µα+tDdα), (7)
with t′=tf−t,
•with transversality conditions:
λn(tf) = 0, n=1, ..., 4, (8)
•and the optimality conditions:
Hα(S∗,O∗
w,O∗
b,D∗,λn,u∗
k) = min
0≤uk≤1Hα(S∗,O∗
w,O∗
b,D∗,λn,u∗
k), (9)
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|
Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
where n =1, ..., 4 and k =1, 2, 3. Futhermore, the control functions u∗
k,k=1, 2, 3 are given by
u∗
1=min max 0, λO(λ2−λ1)S
B1, 1,
u∗
2=min max 0, βα
1(λ2−λ1)S
B2, 1,
u∗
3=min max 0, (λ3−λ2)γαOw
B3, 1,
with the stationary conditions ∂Hα
∂uk
=0, k=1, 2, 3.
Proof The existence of the optimal control
(u∗
1
,
u∗
2
,
u∗
3)
and associated to the optimal solution
(S∗
,
O∗
w
,
O∗
b
,
D∗)
comes from the convexity of the integrand of the functional
(5)
with respect to
the
uk
,
k=1
,
2
,
3
controls and the Lipschitz properties of the state system with respect to the state
variables
(S
,
Ow
,
Ob
,
D)
. According to Pontryagin’s maximum principle, if
uk∈Uad
,
k=1
,
2
,
3
is an optimal control associated to
(5)
-
(6)
with the initial conditions and
tf
fixed, there exists an
absolutely continuous non-trivial map
λ:[t0
,
tf]−→R4
,
λ(t) = (λ1(t)
,
λ2(t)
,
λ3(t)
,
λ4(t))
called
adjoint vector such that
c
tDα
tfλn(t) = ∂Hα
∂xl
,n=1, ..., 4,
and using the equivalence of Lemma 1, we have:
cDα
tfλn(t′) = ∂Hα
∂xl
,n=1, ..., 4,
with
xl= (S
,
Ow
,
Ob
,
D)
,
t′=tf−t
and
Hα
is the Hamiltonian defined in
(7)
. Moreover, the
transversality conditions λn(tf) = 0, n=1, ..., 4 are satisfied.
Optimality is when the equations ∂Hα
∂uk
=0, at u∗
k,k=1, 2, 3. Then,
∂Hα
∂u1=B1u1−λO(λ1−λ2)S=0,
if and only if
u∗
1=λO(λ2−λ1)S
B1,
to be taken on the set {t: 0 ≤u∗
1≤1}.
Analogously, we have for u2that
∂Hα
∂u2=B2u2−βα
1(λ1−λ2)S=0,
if and only if
u∗
2=βα
1(λ2−λ1)S
B2,
Moya et al. |523
to be taken on the set {t: 0 ≤u∗
2≤1}.
For u3, we have:
∂Hα
∂u3=B3u3−γα
1(λ2−λ3)S=0,
if and only if
u∗
3=γα
1(λ3−λ2)S
B3,
to be taken on the set {t: 0 ≤u∗
3≤1}.
Since the optimality conditions are only satisfied in the interior of the control set. An important
element regarding the behavior of the controls is that the second derivative concerning
uk
,
k=
1, 2, 3, are:
∂2Hα
∂u2
1
=B1>0, ∂2Hα
∂u2
2
=B2>0, ∂2Hα
∂u2
3
=B3>0.
Applications of this theorem to control problems in epidemiology can be found in [24,32–34]
5 Numerical simulations
Method and data
The method used to solve numerically fractional order system of nonlinear differential equations
(4)
can be found [
22
,
37
,
41
–
43
]. The algorithm has the structure of a PECE (Predict-Evaluate-
Correct-Evaluate) method and combines a fractional order algorithm with a classical method.
The method chosen is the Adams-Bashforth-Moulton method for both integrators. The key to
deriving the method in the fractional variant is to use the trapezoidal quadrature product formula.
This algorithm is independent of the
α−
value and behaves very similar to the classical Adams-
Bashforth-Moulton method. The stability properties are unchanged in the fractional version
compared to the classical algorithm.
To solve the control problem primitively, with PECE and using the initial conditions of the state
variables and an estimate of the controls in the time interval
[0
,
tf]
, we obtain the values of the state
variables. We solve the system
(7)
and the traversal conditions
(8)
with PECE and the values of the
adjoint variables
λi
,
i=1
, ...,
4
are obtained. The controls are updated by a convex combination
of the previous control and the value calculated in
(9)
. This procedure is repeated iteratively
until the stop condition that is when the values of the controls of the previous iteration are very
close to those of the current iteration. The programming was performed in Matlab software. The
parameter values used in the computational simulations are in Table 1 and the initial conditions
are:
S0=874
.
1400
,
Ow0=1
.
2000
,
Ob0=1
.
5000
and
D0=100
.
0000
on the scale of 10000
individuals.
Global sensitivity analysis
Global sensitivity analysis (GAS) can provide information on the dependence of the model output
on each of its input parameters [
44
,
45
,
51
]. An advantage of GSA methods is that they explore
screening or variance decomposition to cover the limitations of local analysis. In our work, we use
the Sobol’ indices to perform the GAS [
46
]. The use of Sobol’ indices for the study of biological
and epidemiological models is very widespread and can be found in references such as [
46
–
48
].
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The system is analyzed from a probabilistic perspective where the model input is a random vector
X
, with a joint probability density function
fX
with support
IX
. This version of the model can
then be rewritten as:
Y=M(X),
which has a probability density function
fY
that is unknown prior to uncertainty propagation [
49
].
Assuming that the quantity (or quantities if there are more than one) of interest is a scalar value
and that the random input parameters are composed of independent and identically distributed
uniform parameters (iid)
Xi
,
i=1
, ...,
n
, scaled to have support
[0
,
1]m
, then the Hoeffding-Sobol
decomposition is given by
Y=M0+
n
X
i=1
Mi(Xi) + X
1≤i≤j≤n
Mij (Xi,Xj) + ... +M1,...,m(Xi, ..., Xn),
where
M0=E[Y],
Mi(Xi) = E[Y|Xi]−M0,
Mij (Xi,Xj) = E[Y|Xi,Xj]−M0−Mi−Mj,
E
is the expected value,
M0
is the mean value and the terms of increasing order are conditional
expectations defined in a recursive way, that characterize a unique orthogonal decomposition of
the model response [46,50].
We can now decompose the total variance of the response as follows
Var(Y) = X
u
Var(Mu(Xu)), for ∅=u⊂{1, ..., n},
where
Var(Mu(Xu))
expresses the conditional variance for the subvector
Xu
, containing the
variables which indices are indicated by the subset u[
50
]. Thus, the Sobol index associated to
the subset uis defined as the ratio between the contribution given by the interaction between the
components of uto the variance of the model and the total variance is described as [51]:
Su=Var(Mu(Xu))
Var(Y).
As result of this equation, we have that for u⊂{1, ..., n},u=∅:
X
u
Su=
n
X
i=1
Si+X
1≤i<j≤n
Sij +....S1,...,m=1,
that is, by construction the sum of all the Sobol’ indices must be equal to the unit [46].
The terms
Si=Var(Mi(Xi))
Var(Y),i=1, ..., n,
Moya et al. |525
are defined as the first-order Sobol’ indices, and denote the individual effect of the varible Xifor
the total model variate. The terms
Sij =Var(Mij(Xi j))
Var(Y), 1 ≤i≤j≤n,
are defined as the second-order indices and denote the effect of interaction between the varibles
Xi
and
Xj
. We can construct the Sobol’ indices of all order until the
m
-th order indices,
S1,...,m
, but
in this work we study until the second-order indices.
To measure the total contribution of the
i
-th random variable
Xi
on the total variance, either by its
single effect or by its interaction with others, and we use the total Sobol’ indices defined by
S⊤
i=X
i∈u⊂{1,...,n}
Su,i=1, ..., n. (10)
To calculate the total Sobol’ indices
(10)
, we need the underlying variances and use the alternative
of surrogate models based on Polynomial Chaos Expansion (PCE) [
52
–
55
]. The polynomial chaos
expansion of the computational model response is a sum of orthogonal polynomials weighted by
coefficients to be determined which reads as:
Y=M(X) = X
α∈NM
yαΨα(X), (11)
where the
Ψα(X)
are multivariate polynomials orthonormal with respect to
fX
,
α∈NM
is a
multi-index that identifies the components of the multivariate polynomials
Ψα(X)
, and the
yα
are
the corresponding coefficients (coordinates) [
52
]. The sum in Eq.
(11)
needs to be truncated to a
finite sum, introducing the truncated polynomial expansion of chaos:
Y≈MPC(X) = X
α∈A
yαΨα(X),
where A⊂NMis the set of selected multi-indices of multivariate polynomials.
Note the quality of the PCE is directly dependent on the number of terms you have in the
expansion. The family of orthonormal polynomials to be used are chosen according to the input
distribution of the model, where the aim is to minimize the number of terms needed in the
expansion to build a good computational representation of the model. For example, for uniform
we use Legendre’s orthonormal polynomials, for Gaussian we use Hermite’s, for Gamma we use
Laguerre’s and for Beta we use Jacobi’s [48,49,56].
Given the orthogonal polynomials that are used, we define the total degree truncation scheme
which corresponds to all polynomials in the Minput variables of total degree less than or equal to
p:
AM,p={α∈NM:|α|≤p}cardAM,p=P=M+p
p.
Note that the total-degree basis grows exponentially with the degree p.
There are several methods for calculating the coefficients of the polynomial expansion of chaos
for a given base. The two principal strategies to calculate the polynomial chaos coefficients non-
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intrusively are projection and regression.
To estimate the surrogate error estimation, we use with the Leave-One-Out (
ϵLOO
) cross-validation
error [48,55], calculated by
ϵLOO =PNS
i=1(M(x(i))−MPC\i(x(i)))2
PNS
i=1((x(i))−y)2.
It consists in building
Ns
metamodels
MPC\i
each one created on a reduced experimental design
χ\
x
(i)={
x
(j)
,
j=1
, ...,
NS
,
j=i}
(where
χ={
x
(i)
,
i=1
, ...,
NS
,
}
called the experimental design),
and comparing its prediction on the excluded point x
(i)
with the real value
y(i)
and
y
is the sample
mean of the experimental design response [52].
Several methods exist to calculate the coefficients
yα
of the polynomial chaos expansion for a given
basis.
In our work, we use regression and two different techniques to estimate the coefficients.
A different approach to estimate the coefficients in Eq.
(12)
is to set up a least-squares minimization
problem. The infinite series in Eq.
(11)
can be written as a sum of its truncated version Eq.
(12)
and a residual:
Y=M(X) =
P
X
α=0
yαΨα(X) + ϵP=y⊤Ψ(X) + ϵP,
where
P=cardAM,p
,
ϵP
is the truncation error,
y={y0
, ...,
yP}⊤
is a vector containing the
coefficients and
Ψ(X) = {ψ0(X)
, ...,
ψP(X)}⊤
, is the vector that assembles the values of all the
orthonormal polynomials in Xand compute the model response for that samples
y(1)=M(x(1)),
y(2)=M(x(2)),
.
.
.
y(NS)=M(x(NS)).
The classic least squares regression problem is defined as:
y⊤Ψ(X)≈M(x),
and its general solution can be expressed as:
ˆ
y=arg min E(y⊤Ψ(X)−M(x))2. (12)
Now, we focused on simulating with the following methods:
Method I: Ordinary Least-Squares (OLS). A direct approach to solving Eq.
(12)
is given by
Ordinary Least-Squares (OLS). Given a sample
χ={
x
(1)
, ..., x
(N)}
of size
N
of
X
(the experimental
design) and the corresponding model responses Y
={
y
(1)
, ..., y
(N)}⊤
, the ordinary least-squares
solution of Eq. (12) reads:
ˆ
y= (A⊤A)−1A⊤Y,
Moya et al. |527
where
Aij =Ψ(x(i)),i=1, ..., n,j=0, ..., P−1.
The main advantage of the least squares minimization method over the projection method is
that an arbitrary number of points can be used to calculate the coefficients, as long as they
are a representative sample of the random input vector. This method theoretically, its error,
methodology and example can be found in [48,55,57,58].
Method II: Least Angle Regression (LARS). A complementary strategy to favor high-dimensional
sparsity is to directly modify the least-squares minimization problem
(12)
by adding a penalty
term of the form λ∥y∥1, i.e., solving:
ˆ
y=arg min E(y⊤Ψ(X)−M(X))2+λ∥y∥1,
where the regularization term
∥ˆ
y∥1=Pα∈A|yα|
forces the minimization to favour low-rank
solutions. The LARS algorithm used, in theoretical form, code and examples can be found in
[59–63].
In our method, we took into account a stopping criterion that consists of stopping adding regres-
sors after the error estimate is above its minimum value for at least 10% of the maximum number
of possible iterations [64–66].
For each method we will experiment with different sampling techniques (sampling methods)
[
67
–
73
]. In particular, we will use Monte Carlo (MC) and Sobol’ sequence sampling (Sobol’)
[70,71].
The code was implemented in MATLAB R2024A and we used the UQLab library [
55
,
74
,
75
] as an
implementation aid and based on the code presented in [
48
]. We used the time period years and
the maximum time period is 15 years.
The sensitivity study was carried out for the case (
α=1
.
0
) using the methods that are in the
reference [
48
,
52
,
53
] and we can observe the solution of the model for fractional orders studied
(
α=0
.
5
,
0
.
7
,
0
.
9
,
1
.
0
) the reported values change but the asymptotic behavior of the results does
not.
We use the independent uniform random variables for each of the parameters as the probabilistic
input model, with upper and lower bounds of 1.5% dispersion around the mean, and consider the
fixed nominal values presented in Table 1. To build the PCE surrogate model 1000 experimental
design samples were taken with a maximum polynomial degree is in
[10
,
13]
interval. For all the
cases studied, the leave-one-out error (
ϵLOO
) is in the range of
[8
.
615006e−07
,
5
.
658380e−06]
and the quality of the surrogate approximation showed an reasonable to the purpose.
Table 2 shows the first order Sobol’ indices and Table 3 shows the total Sobol’ indices for the OLS
and LARS methods with the MC and Sobol’ sampling techniques. Table 4 presents the second
order Sobol’ indices for the OLS and LARS methods with MC and Sobol’ sampling techniques.
Figure 2a and Figure 3a show the first order and total order Sobol’ indices for the OLS method and
the different sampling techniques. Figure 2b and Figure 3b present the first order and total order
Sobol indices for the LARS method and Figure 4a and Figure 4b show the second order Sobol’
indices for OLS and LARS, respectively.
The parameter that has the most influence is the rate of effective contact associated with the impact
that an overweight and obese person has on a person with normal weight (
α∗
). With the different
techniques studied and sampling methods to obtain the Sobol’ indices, the first and total order
Sobol’ indices average was 0.503448 and 0.502248, respectively.
In decreasing order, the parameter
δ
follows, that is the rate that leads an overweight person to a
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
normal weight. This parameter is important because its increase defines control of overweight
in the community. The average value of the first-order Sobol’ index is 0.171327 and for the total
index we have 0.171188.
The parameter
β1
at the third position with an average value of the Sobol’ index of first-order of
0.162849 and in the total-order of 0.163707. We can observe that the two ways for a normal weight
person to become overweight that we studied with the model are in the first and third order of
influence.
The other parameters with consecutive influence are the death rate associated with overweight
and obesity with an average first order Sobol’ index of 0.034274 and total 0.00342985, the param-
eter
γ
which is associated with people who manage to overcome obesity and reach the state of
overweight with average values of 0.025539 and 0.02535 of first and total order index.
Finally we have
MS
which is the recruitment rate which measures the entry of new individuals
to the dynamics and has an average value of 0.024462 and 0.024749, for the first and total order,
respectively.
The other parameters of the dynamics were studied but their first and total order Sobol’ indices
are below the threshold value 10−4.
It is important to mention that the influence of the parameters for the first order and for the total
order of the Sobol’ indices coincide. Another important factor is that the parameters associated
with weight gain from a normal weight state (
α∗
,
β1
) and weight loss from obesity to overweight
and from overweight to normal weight (
δ
,
γ
) are among the most influential parameters and have
an obvious importance in the control of overweight and obesity in the community.
Using as a reference the Sobol’s index of second-order with the different techniques and sampling
methods we have that the relationship between parameters that has more influence is between
MS
and
α∗
that are the recruitment rate and the effective contact rate of the influence of obese and
overweight are an individual with normal weight.
Continuing in decreasing order, the relationship that continues is between
α∗
and
β1
which rep-
resent the pathways for an individual to gain weight studied in the model, that is through the
influence of the obese and overweight and through social pressure.
Parameter
α∗
is also related to
α2
which is the rate of diabetes diagnosis in overweight individuals.
Next, we have the relationship between the recruitment rate
MS
and
β1
and
α2
which are the
weight gain of the individual to become overweight and the diagnosis of diabetes in overweight
individuals, so we can say that with the recruitment rate the social pressure can lead more people
to become overweight and with it a higher diagnosis of diabetes.
With the use of MC sampling method there appears a relationship which is not there when we
use Sobol’ sampling which is the relationship between
β1
and
α2
which both are related to
MS
and in the opposite case there appears the relationship between
α∗
and
d
which represents the
relationship between the effective contact rate of the impact of an obese and overweight on a
normal weight individual with the death associated with obesity and overweight. The use of
different PCE and sampling techniques showed us relationships that are important to keep in
mind that would appear with one methodology and not with the other, see Table 4 and Figure 4.
All parameters participating in the relationships characterized by second-order Sobol’ indices are
among the seven most influential in the first and total order Sobol’ indices.
The information provided by the global stability contributes both independently and jointly be-
tween parameters helping to make decisions for parameter estimation, in the qualitative analysis
of the model and in the construction of the computational simulations and we offer different
techniques that contribute to obtain this information between the model and the parameters.
Moya et al. |529
Table 2. Comparison of first-order Sobol’ indices for the peak duration using OLS and LARS for different
sampling methods
OLS LARS
Parameters MC Sobol’ MC Sobol’
MS0.024408 0.024807 0.024408 0.024287
MD0.000004 0.000000 0.000000 0.000000
µ0.000036 0.000014 0.000032 0.000007
µd0.000001 0.000000 0.000000 0.000000
d0.034268 0.034460 0.034492 0.033900
α∗0.503782 0.504101 0.502158 0.503751
β∗0.000004 0.000003 0.000000 0.000000
α10.000001 0.000005 0.000000 0.000000
α20.078682 0.078010 0.078719 0.077754
α30.000000 0.000000 0.000000 0.000000
β10.162075 0.161534 0.163478 0.163431
δ0.171080 0.171233 0.171233 0.171762
η0.000010 0.000002 0.000001 0.000000
γ0.025712 0.025855 0.025479 0.025110
Table 3. Comparison of total-order Sobol’ indices for the peak duration using OLS and LARS for different
sampling methods
OLS LARS
Parameters MC Sobol’ MC Sobol’
MS0.025017 0.025862 0.024073 0.024045
MD0.000010 0.000013 0.000016 0.000013
µ0.000027 0.000020 0.000040 0.000027
µd0.000004 0.000006 0.000003 0.000004
d0.034043 0.034105 0.034076 0.034970
α∗0.502332 0.503263 0.503175 0.500023
β∗0.000019 0.000006 0.000008 0.000016
α10.000024 0.000008 0.000013 0.000012
α20.078201 0.078304 0.078119 0.078095
α30.000027 0.000009 0.000006 0.000009
β10.161226 0.161132 0.164832 0.167639
δ0.173204 0.171339 0.170200 0.170009
η0.000028 0.000013 0.000023 0.000028
γ0.025642 0.025932 0.025416 0.025150
Table 4. Second-order Sobol’ indices. The table shows the influence of the relationships between model
parameters using the OLS and LARS techniques with the MC and Sobol’ sampling techniques
OLS LARS
Relationship MC Sobol MC Sobol Average
MS−α∗0.003929 0.003999 0.003816 0.003874 0.003904
β1−α∗0.003724 0.003601 0.003578 0.003641 0.003636
α∗−α20.001699 0.001737 0.001586 0.001784 0.001701
MS−β10.001686 0.001828 0.001570 0.001829 0.001782
MS−α20.000883 0.0.00074 0.000885 0.000915 0.000856
β1−α20.000730 0 0.000750 0 0.000370
α∗−d0 0.000807 0 0.000735 0.000385
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
First-Order Sobol' Indices (OLS)
MSMD d d* *
1231
Parameters
0
0.1
0.2
0.3
0.4
0.5
0.6
First-Order Indices
MC
Sobol'
(a) First-order Sobol’ indices using the OLS method
and the MC and Sobol’ sampling techniques
First-Order Sobol' Indices (LARS)
MSMD d d* *
1231
Parameters
0
0.1
0.2
0.3
0.4
0.5
0.6
First-Order Indices
MC
Sobol'
(b) First-order Sobol’ indices using the LARS method
and the MC and Sobol’ sampling techniques
Figure 2. First-order Sobol’ indices using the OLS and LARS methods and the MC and Sobol’ sampling techniques
Total-Order Sobol' Indices (OLS)
MSMD d d* *
1231
Parameters
0
0.1
0.2
0.3
0.4
0.5
0.6
Total-Order Indices
MC
Sobol'
(a) Total-order Sobol’ indices using the OLS method
and the MC and Sobol’ sampling techniques
Total-Order Sobol' Indices (LARS)
MSMD d d* *
1231
Parameters
0
0.1
0.2
0.3
0.4
0.5
0.6
Total-Order Indices
MC
Sobol'
(b) Total-order Sobol’ indices using the LARS method
and the MC and Sobol’ sampling techniques
Figure 3. Total-order Sobol’ indices using the OLS and LARS methods and the MC and Sobol’ sampling
techniques
Numerical simulations
For the simulations of the optimal control problem the time period is annual for a total time of 15
years and we assume the values for the weight constants associated with the implementation of
the controls are B1=150, B2=300 and B3=550.
Moya et al. |531
Second-Order Sobol' Indices (OLS)
MS-*
1-* *-2
MS-1MS-2 1-2
*-d
0
0.5
1
1.5
2
2.5
3
3.5
4
Second-Order Indices
10-3
MC
Sobol'
(a) Second-order Sobol’ indices using OLS and with
sampling using MC and Sobol’
Second-Order Sobol' Indices (LARS)
MS-*
1-* *-2
MS-1MS-2 1-2
*-d
0
0.5
1
1.5
2
2.5
3
3.5
4
Second-Order Indices
10-3
MC
Sobol'
(b) Second-order Sobol’ indices using LARS and with
sampling using MC and Sobol’
Figure 4. Sobol’s second order indices using OLS and LARS for PCE using MC and Sobol’ as sampling techniques
We constructed three possible strategies:
•
Strategy I:
u1=0
,
u2=0
,
u3=0
. This strategy focuses on controlling weight gain due to social
pressure and the progression from overweight to obese.
•
Strategy II:
u3=0
,
u1=0
,
u2=0
. In this strategy, we control the two ways taken into account
in the construction of the model for an individual with normal weight to evolve into overweight.
•
Strategy III:
ui=0
,
i=1
,
2
,
3
. All controls are active; we control the ways in which an individual
with normal weight becomes overweight and the evolution from overweight to obese.
The control strategies are based on the results of simulations and the study of the parameters
presented in [
22
]. Due to one of the objectives of the model, which is the impact of social pressure
on weight gain, and is presented in [
21
] and [
22
], the control over social pressure that causes an
individual with normal weight to become obese will always be active.
Now, let us study the impact of strategies for different fractional orders and compartments.
For the behavior of individuals with normal weight without application of controls, we can
observe that with the increase of fractional orders the number of individuals with normal weight
is reduced. For all control strategies and fractional orders, the number of cases with normal weight
increases with respect to the model without control, which implies that fewer people are gaining
weight and are able to maintain an adequate weight. The best results are achieved when all the
controls are activated followed by strategy II when only the controls are active in the negative
interactions of obese and overweight with overweight people and the control of weight gain by
social pressure on individuals with normal weight, see Table 5 and Figure 5a,Figure 5b,Figure 5c
and Figure 5d.
In the case of overweight individuals, there is a growth and then it stabilizes for the different
fractional orders and the higher the fractional order the higher the number of overweight indi-
viduals reported. For all the control strategies studied the number of individual overweights is
reduced but for order
α=0
.
5
the asymptotic behavior is analogous to the model without control
For constructed strategy and different fractional orders, we observed a decrease in the asymptotic
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
behavior which is relevant to strategy II and III. The strategy that showed the best results was
strategy III because it significantly reduced over time the number of overweight individuals
followed by strategy II. For higher fractional orders the impact of the controls is more significant,
see Table 6 and Figure 6a,Figure 6b,Figure 6c and Figure 6d. In the obese compartment, we have
that for all fractional orders we initially have a decrease, then a growth,h and then stabilization.
In this case, the higher fractional orders the higher the number of obese reported at the end of
the study for the model without controls. The maximum and minimum values for the different
fractional orders and control strategies can be found in Table 7. In this case, the impact of the
strategies is behaviorally more effective for lower fractional orders and strategies III and II show
the best results respectively. We can observe that with strategies III and II for fractional orders
α=0
.
5
and
α=0
.
7
, not only control growth during the study (happens for all orders) but also
cause a significant decrease in the number of obese reported, see Figure 7a,Figure 7b,Figure 7c
and Figure 7d. The diabetes compartment is directly related to the behaviors associated with body
weight. In this case, for all fractional orders, we have a growth behavior in the number of diabetic
individuals and it is maintained when applying the different control strategies. We can conclude
that the higher the fractional order, the higher the number of reported diabetics. All strategies
significantly reduce the number of reported diabetics over time, but the strategy with the best
results was Strategy III followed by Strategy II, see Table 8 and Figure 8a,Figure 8b,Figure 8c and
Figure 8d. The strategy that showed the best results was Strategy III, where all controls are active,
followed by Strategy II. All strategies are associated with the seven parameters with the highest
Sobol’ indices (specifically
α∗
,
β1
and
γ
). Strategy II, which showed the best results when controls
were not applied, is associated with parameters
α∗
and
β1
, which are in the three parameters with
the highest Sobol’ index of the first and total order, and they are also related by the Sobol’ index
of the second order, which does not show any significant relationship with parameter
γ
or any
other parameter of the model. We can conclude from the construction of the model, the control
strategies, and the values used in the simulations that if we manage to prevent an individual from
gaining weight, either by interaction with overweight and/or obese people, and reduce social
pressure (Strategy II), we manage to reduce overweight and obesity in the community and, as a
consequence, diabetes.
Table 5. Maximum and minimum values of the simulations of Model (4) of the normal weight individual
compartment without strategies and with the different control strategies (×10000).
Without Controls Strategy I Strategy II Strategy III
αMin Max Min Max Min Max Min Max
0.5 307.6150 1.1079e+03 356.9388 1.1401e+03 458.1019 1.1903e+03 560.4600 1.2250e+03
0.7 252.7330 1.8687e+03 301.7025 1.2094e+03 402.0024 1.2793e+03 503.0536 1.3312e+03
0.9 225.1468 1.2461e+03 274.0151 1.2974e+03 373.9049 1.3843e+03 474.0821 1.4214e+03
1.0 218.2975 1.2918e+03 266.4800 1.3494e+03 362.5733 1.4464e+03 462.3616 1.4859e+03
Table 6. Maximum and minimum values of the simulations of Model (4) of the overweight individual compart-
ment without strategies and with the different control strategies (×10000).
Without Controls Strategy I Strategy II Strategy III
αMin Max Min Max Min Max Min Max
0.5 1.2000 1.1050e+03 1.2000 875.8010 1.2000 601.5214 1.2000 465.5587
0.7 1.2000 1.2355e+03 1.2000 980.0019 1.2000 635.4550 1.2000 563.9612
0.9 1.2000 1.3707e+03 1.2000 1.0280e+03 1.2000 652.7550 1.2000 574.7420
1.0 1.2000 1.3857e+03 1.2000 1.0455e+03 1.2000 681.5733 1.2000 581.6812
Moya et al. |533
Table 7. Maximum and minimum values of the simulations of Model (4) of the obese individual compartment
without strategies and with the different control strategies (×10000).
Without Controls Strategy I Strategy II Strategy III
αMin Max Min Max Min Max Min Max
0.5 1.0100 1.5000 0.4176 1.5000 0.2177 1.5000 0.1828 1.5000
0.7 1.2544 1.5000 0.3746 1.5000 0.1554 1.5000 0.1128 1.5000
0.9 1.2953 1.5000 0.3644 1.5000 0.0929 1.5000 0.0807 1.5000
1.0 1.3245 1.5000 0.3595 1.5000 0.0806 1.5000 0.0642 1.5000
0 5 10 15
Time (years)
300
400
500
600
700
800
900
1000
1100
1200
1300
Number of individuals (x 10000)
Normal-Weight Individuals ( =0.5)
Without Control
Strategy I
Strategy II
Strategy III
(a) Behavior of normal weight individuals over
time with and without controls for α=0.5
0 5 10 15
Time (years)
200
400
600
800
1000
1200
1400
Number of individuals (x 10000)
Normal-Weight Individuals ( =0.7)
Without Control
Strategy I
Strategy II
Strategy III
(b) Behavior of normal weight individuals over
time with and without controls for α=0.7
0 5 10 15
Time (years)
200
400
600
800
1000
1200
1400
1600
Number of individuals (x 10000)
Normal-Weight Individuals ( =0.9)
Without Control
Strategy I
Strategy II
Strategy III
(c) Behavior of normal weight individuals over
time with and without controls for α=0.9
0 5 10 15
Time (years)
200
400
600
800
1000
1200
1400
1600
Number of individuals (x 10000)
Normal-Weight Individuals ( =1.0)
Without Control
Strategy I
Strategy II
Strategy III
(d) Behavior of normal weight individuals over
time with and without controls for α=1.0
Figure 5. Behavior of normal weight individuals over time with and without controls for different fractional
orders
534
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
Table 8. Maximum and minimum values of the simulations of Model (4) of the diabetic individual compartment
without strategies and with the different control strategies (×10000).
Without Controls Strategy I Strategy II Strategy III
αMin Max Min Max Min Max Min Max
0.5 100.0000 1.5063e+03 100.0000 681.5702 100.0000 195.1652 100.0000 175.6160
0.7 100.0000 2.6223e+03 100.0000 1.2649e+03 100.0000 366.8341 100.0000 199.8900
0.9 100.0000 4.2701e+03 100.0000 2.2035e+03 100.0000 653.1200 100.0000 296.6318
1.0 100.0000 5.3356e+03 100.0000 2.8479e+03 100.0000 854.9811 100.0000 385.9001
0 5 10 15
Time (years)
0
200
400
600
800
1000
1200
Number of individuals (x 10000)
Overweight Individualss ( =0.5)
Without Control
Strategy I
Strategy II
Strategy III
(a) Behavior of overweight individuals over time
with and without controls for α=0.5
0 5 10 15
Time (years)
0
200
400
600
800
1000
1200
1400
Number of individuals (x 10000)
Overweight Individualss ( =0.7)
Without Control
Strategy I
Strategy II
Strategy III
(b) Behavior of overweight individuals over time
with and without controls for α=0.7
0 5 10 15
Time (years)
0
200
400
600
800
1000
1200
1400
Number of individuals (x 10000)
Overweight Individualss ( =0.9)
Without Control
Strategy I
Strategy II
Strategy III
(c) Behavior of overweight individuals over time
with and without controls for α=0.9
0 5 10 15
Time (years)
0
200
400
600
800
1000
1200
1400
Number of individuals (x 10000)
Overweight Individualss ( =1.0)
Without Control
Strategy I
Strategy II
Strategy III
(d) Behavior of overweight individuals over time
with and without controls for α=1.0
Figure 6. Behavior of overweight individuals over time with and without controls for different fractional orders
Moya et al. |535
0 5 10 15
Time (years)
0
0.5
1
1.5
Number of individuals (x 10000)
Obese individuals ( =0.5)
Without Control
Strategy I
Strategy II
Strategy III
(a) Behavior of obese individuals over time with and
without controls for α=0.5
0 5 10 15
Time (years)
0
0.5
1
1.5
Number of individuals (x 10000)
Obese individuals ( =0.7)
Without Control
Strategy I
Strategy II
Strategy III
(b) Behavior of obese individuals over time with and
without controls for α=0.7
0 5 10 15
Time (years)
0
0.5
1
1.5
Number of individuals (x 10000)
Obese individuals ( =0.9)
Without Control
Strategy I
Strategy II
Strategy III
(c) Behavior of obese individuals over time with and
without controls for α=0.9
0 5 10 15
Time (years)
0
0.5
1
1.5
Number of individuals (x 10000)
Obese individuals ( =1.0)
Without Control
Strategy I
Strategy II
Strategy III
(d) Behavior of obese individuals over time with and
without controls for α=1.0
Figure 7. Behavior of obese individuals over time with and without controls for different fractional orders
6 Conclusions
In this work, we present an optimal control problem focused on the reduction of overweight
and obesity in a community and its impact on the diagnosis of diabetes. The model used in
the formulation of the control problem is published in [
22
] and its basic properties such as the
existence of solution, the biologically feasible region and a study of the basic reproduction number
were demonstrated. Three controls were defined centered on individuals moving from normal
and obese weight in the pathways studied in the model, by interaction with overweight and
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
0 5 10 15
Time (years)
0
200
400
600
800
1000
1200
1400
1600
Number of individuals (x 10000)
Diabetic Individuals ( =0.5)
Without Control
Strategy I
Strategy II
Strategy III
(a) Behavior of diabetic individuals over time with
and without controls for α=0.5
0 5 10 15
Time (years)
0
500
1000
1500
2000
2500
3000
Number of individuals (x 10000)
Diabetic Individuals ( =0.7)
Without Control
Strategy I
Strategy II
Strategy III
(b) Behavior of diabetic individuals over time with
and without controls for α=0.7
0 5 10 15
Time (years)
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Number of individuals (x 10000)
Diabetic Individuals ( =0.9)
Without Control
Strategy I
Strategy II
Strategy III
(c) Behavior of diabetic individuals over time with
and without controls for α=0.9
0 5 10 15
Time (years)
0
1000
2000
3000
4000
5000
6000
Number of individuals (x 10000)
Diabetic Individuals ( =1.0)
Without Control
Strategy I
Strategy II
Strategy III
(d) Behavior of diabetic individuals over time with
and without controls for α=1.0
Figure 8. Behavior of diabetic individuals over time with and without controls for different fractional orders
obese individuals and by social pressure, and on overweight individuals reaching the obese state.
We show the existence of optimal control using Pontryagin’s maximum principle. Using data
from the literature, we performed a global sensitivity study of the model and simulations of the
optimal control problem. For global sensitivity analysis, we use Sobol’s index in particular first,
second and total order. To find the Sobol’ indices, we use the exponentiation in chaos polynomials
using two methodologies the OLS and LARS and also two techniques of Monte Carlo and Sobol’
sampling. We also used different orders, first, second and total, and in the case of first and total
we matched the order and the influential parameters. The results showed that for a threshold
Moya et al. |537
value of
10−4
, we have seven influential parameters, in particular the most influential parameter
was the effective contact rate in the negative effect that an overweight or obese individual can
have on a normal weight individual. In addition, the other possible case of weight gain in a
normal-weight individual is in third place and all parameters that are involved with the controls
are among the most influential second Sobol’ indices. Three control strategies were developed,
focused on controlling social pressure and the evolution from overweight to obese, controlling
all the studied ways for a normal-weight individual to become overweight and the ways to go
from normal weight to overweight and from overweight to obese. All strategies were efficient
not only in reducing overweight and obesity in the community but also the number of cases of
diabetes. The strategy that showed the best results qualitatively and quantitatively was to control
all possible ways of going from normal weight to overweight and from overweight to obese (all
active controls). The second with the best results was to control only the possible cases addressed
in the model of going from normal weight to overweight. With the model and the data used, we
show that the most influential parameter in the model is the negative impact that an obese or
overweight individual can have on an individual with normal weight and also that by controlling
all possible evolutions from normal weight to overweight and from overweight to obese we reduce
not only overweight and obesity in the community but also diabetes and that if we control only
the ways in which an individual with normal weight evolves to overweight we would also obtain
good results.
Declarations
Use of AI tools
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this
article.
Data availability statement
Data sharing is not applicable to this article as no datasets were generated or analyzed during the
current study.
Ethical approval
The authors declare that this research complies with ethical standards. This research does not
involve human participants or animals.
Consent for publication
Not applicable
Conflicts of interest
The authors declare that they have no known competing financial interests or personal relation-
ships that could have appeared to influence the work reported in this paper.
Funding
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior – Brasil (CAPES) – Finance Code 13179.
538
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Mathematical Modelling and Numerical Simulation with Applications, 2024, Vol. 4, No. 4, 514–543
Author’s contributions
E.M.D.M.: Methodology, Conceptualization, Validation, Software, Data Curation, Writing the
Original Draft. R.A.R., A.P. and S.B.: Writing - Review & Editing, Supervision. All authors have
read and agreed to the published version of the manuscript
Acknowledgements
The authors would like to thank the reviewers and the editorial team of the journal.
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