Unexistence of limit cycle in an optimal control problem of a population of diabetics

Abstract

La diabetes, debido a sus complicaciones, es una de las enfermedades que más problemas plantean en la salud pública actual mundial. En este trabajo se parte de una población de diabéticos con y sin complicaciones y se asocia un problema de control optimal no lineal que describe la dinámica de la población. Para este modelo se prueba la existencia del estado de equilibrio y que es un punto de ensilladura. Además se obtuvo que no existen ciclos límite, lo que es un resultado importante, dado el problema que se describe. Se presentan ejemplos para los cuales el estado de equilibrio que se caracteriza no es necesariamente admisible.

Full text

REVISTA DE MATEMÁTICA: TEORÍA Y APLICACIONES 2018 25(2) : 239–259
CIMPA –UCR IS SN : 1409-2433 (PRINT), 2215-3373 (ONLINE)
DOI:https://doi.org/10.15517/rmta.v25i2.33692
UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL
CONTROL PROBLEM OF A POPULATION
OF DIABETICS
INEXISTENCIA DE CICLO LÍMITE EN UN
PROBLEMA DE CONTROL ÓPTIMO DE
UNA POBLACIÓN CON DIABETES
SÉVERINE BERNARD∗TÉNISSIA CÉSAR†
SILVÈRE P. NUIRO‡ALAIN PIÉTRUS§
Received: 21/Jun/2017; Revised: 7/May/2018;
Accepted: 16/May/2018
∗Laboratoire de Mathématiques Informatique et Applications, Université des Antilles, Guade-
loupe, France. E-Mail: severine.bernard@univ-antilles.fr
†Misma dirección que/Same address as: S. Bernard. E-Mail: tenissia.cesar@univ-antilles.fr
‡Misma dirección que/Same address as: S. Bernard. E-Mail: paul.nuiro@univ-antilles.fr
§Misma dirección que/Same address as: S. Bernard. E-Mail: alain.pietrus@univ-antilles.fr
239

240 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
Abstract
This paper deals with one of the most important public health problem
in the whole world that is diabetes, and more precisely its complications.
From a model examining the complications or not of a population of dia-
betics, we associate a nonlinear optimal control problem. Considering the
previous, we prove that the equilibrium state exists and is a saddle point.
Moreover, we claim the unexistence of limit cycle in such a population,
which is an interesting result concerning this world evil. Then we give
some examples for which we characterize the equilibrium state which is
not necessarily admissible.
Keywords: two-dimensional optimal control model; limit cycle; equilibrium
state; Hopf bifurcation theorem.
Resumen
La diabetes, debido a sus complicaciones, es una de las enfermedades
que más problemas plantean en la salud pública actual mundial. En este
trabajo se parte de una población de diabéticos con y sin complicaciones y
se asocia un problema de control óptimo no lineal que describe la dinámica
de la población. Para este modelo se prueba la existencia del estado de
equilibrio y que es un punto de ensilladura. Además se obtuvo que no
existen ciclos límite, lo que es un resultado importante, dado el problema
que se describe. Se presentan ejemplos para los cuales el estado de equi-
librio que se caracteriza no es necesariamente admisible.
Palabras clave: modelo de control optimal bi-dimensional; ciclo límite; estado
de equilibrio; teorema de bifurcación de Hopf.
Mathematics Subject Classification: 49J15, 34H05, 34H20, 90C46, 34C05.
1 Introduction
Diabetes is a chronic disease caused by a combination of hereditary and acquired
bad factors, like family antecedents, overweight, unhealthy diet, physical inac-
tivity. It occurs when the pancreas is not able to product insulin or when the
body can not use effectively the insulin it makes. Consequently, the glucose
from the diet stays in the blood instead of being transferred into the cells in or-
der to produce energy. But a constant high glucose level in the blood causes a
lot of damage to various organs and tissues, like kidney failure, blindness and
eyes problems, heart attack, lower limb amputation and many other ones. The
treatment is based on medication, healthy diet and physical exercises. Accord-
ing to the International Diabetes Federation, approximately 425 million adults
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 241
were living with diabetes in the whole world in 2017, which caused 4million
deaths and 727 bilion USD in health expenditure. Moreover, 352 million people
were at risk of developing diabetes. In particular, it concerned more than 3.5
million of persons in France in 2016 and more than 9percent of the population
of the archipelago of Guadeloupe, from the website of the French Federation of
Diabetics. Its progression is stagerring in developed countries and even more in
developing ones. Due to the mortality rate and the health expenditure caused by
this evil, it is important to understand the triggering factors of the disease and its
evolution.
There are in the literature many mathematical studies focused on this topic
and everything related to it [1, 2, 3, 4, 5, 6, 9, 11, 12]. The different existing
mathematical models are based on ordinary differential equations [4, 11, 12],
partial differential equations [11], delay differential equations [10, 11, 13, 14]
and stochastic differential equations [11]. An overview of some mathematical
models and software tools for the glucose-insulin regulatory system and diabetes
has been done in [11]. In the last decades, some models focused on the intra-
venous glucose tolerance test, which consists to observe how a body reacts after
the injection of a large amount of glucose [13], whereas other ones focused on
the subcutaneous injection of insulin (see [17] and [9] for an overview). Another
approach was to analyse the evolution of diabetes to the stage of complications
[4] and an interesting one was given in [5] where the authors attempt to model
mathematically the diabetes progression.
The blood glucose issue has also been seen as a control problem. In [6], J. R.
Faria showed, by using the Hopf bifurcation theorem adapted to optimal control
problem, that there is a cyclical behavior between the weight and the consump-
tion of a diabetic created by the medical treatment. In [3], the authors consid-
ered an optimal control problem for the evolution of numbers of pre-diabetics
and diabetics with and without complications and showed that the population of
diabetics with complications decreases when an optimal control is applied. In
[2], S. Bernard and A. Piétrus considered a new model of regulation adapted to
the one introduced in [12] and studied it in the framework of ordinary differen-
tial equations and optimal control theory. By controlling the external glucose
food intake, they proved that the plasma glycemia level can be minimized. In
our previous work [1], we considered a population of diabetics, divided into two
subcategories, one of diabetics with complications and another one of diabet-
ics without complications as in [4]. From the model examining the complica-
tions of individuals diagnosed with diabetes, we associated a nonlinear optimal
control problem, for which we proved that there is no cyclical behavior be-
tween the number of diabetics with complications, the one of diabetics without
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242 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
complications and the rate at which complications are healed. Moreover, as we
considered a nonlinear optimal control problem with a scalar control and two
states for which necessary conditions of an optimal control are well known, we
could characterize the equilibrium point by using an adaptation of the Hopf bi-
furcation theorem to optimal control models as it has been done in [7, 18, 19, 20].
This work can be seen as a generalization of the previous one [1]. Indeed,
we prove the existence of an optimal control and an equilibrium state whatever
the controlled parameter and for any concave performance index, whereas in
[1], it has been done in a particular case. Moreover, we are still able to claim
that there is no limit cycle between the number of diabetics with complications
and the one of diabetics without complications and that the equilibrium state is
always a saddle point, which is an encouraging result since it reflects the unex-
istence of a back and forth between these two subcategories of diabetics, when
the probability of developing complications or their cure rate are controlled.
Consequently, Section 2 is devoted to the study of the two dimensional non-
linear optimal control problem, that is we show the existence of an optimal con-
trol and characterize it via the Pontryagin’s maximum principle. In Section 3,
the stability analysis leads us to prove that the equilibrium state always exists
and is a saddle point. In Section 4, by studying some suitable examples, we
characterize the equilibrium state and see that it is not always admissible. We
finish by some concluding remarks and perspectives.
2 Optimal control problems in a population of diabetics
We begin with the mathematical model of [4] in which we choose to put all the
parameters depending on the time
D′(t) = I−(λ(t) + µ(t))D(t) + γ(t)C(t),
C′(t) = λ(t)D(t)−(γ(t) + µ(t) + ν(t) + δ(t))C(t),(1)
where:
•tis the time,
•D(.)the number of diabetics without complications,
•C(.)the number of diabetics with complications,
•Ithe incidence of diabetes,
•λ(.)the probability of developing a complication,
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 243
•µ(.)the natural mortality rate,
•γ(.)the rate at which complications are healed,
•ν(.)the rate at which patients with complications become severely dis-
abled,
•δ(.)the mortality rate due to complications.
Sometimes, we will omit the tin order to relieve the writing. According
to [4], this model has been constructed by considering a population in which I
cases are diagnosed with diabetes on a time interval. During this period of time,
the number of diabetics without complications decreases in order to naturally
death with rate µor develop complications with rate λ, and also increases with
the amount of diabetics with healed complications, with rate γ. In the same
period of time, the number of diabetics with complications decreases with this
last amount, the natural death, the mortality due to complications with rate δand
severely disabled patients with rate νand increases by the coming of diabetics
developing complications. The only one thing we add in this model is the fact
that all the rates depend on the time, which seems more realistic.
Theoretically, any parameter of the model, that is λ(.),µ(.),γ(.),ν(.)or
δ(.), can be chosen as control, but in practice, this is not possible since we can
not control the natural mortality rate for example. In this state of minds, we can
control the probability of developing complications by informing the diabetic
patients on a necessary good lifestyle with physical activity and healthy diet.
We can also control the rate at which complications are healed or the rate at
which patients with complications become severely disabled, with an increase
of health expenditure. Consequently, in the following, ucan be chosen as λ(.),
γ(.)or ν(.). The following result is a classical one and is just cited in order to
well pose the optimal control problem.
Theorem 1 For all fixed control u, there exists one and only one maximal solu-
tion [0, tm(u)], Du(.), Cu(.)of the Cauchy problem (1) with the initial condi-
tions Du(0) = D0∈IR+,Cu(0) = C0∈IR+and tm(u)∈IR+∪ {+∞}.
Remark 1 The values of Dand Cwill depend on the specific choice of the
control u. We underline this dependence by using Duand Cu, instead of Dand
Crespectively.
For a fixed discount rate r > 0, our aim is to maximize the objective function
+∞
0
exp(−rt)Fu(t), D(t), C (t)dt,
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244 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
where F(u, D, C )is a concave performance index. In practice, our aim would
be to minimize the complications in a population of diabetics. Consequently, F
can be a concave function, increasing with respect to u=γand/or Dand/or
decreasing with respect to u=λ,u=νand/or C. From these choices, we have
this second classical result.
Theorem 2 Let (D0, C0)be in IR2
+such that there is a control u(.)satisfying (1)
with the initial conditions D(0) = D0and C(0) = C0. There exists an optimal
control ¯udefined on [0,+∞[such that the associated trajectory (D¯u(.), C¯u(.))
satisfies (1), the initial conditions and which maximizes
+∞
0
exp(−rt)Fu(t), D(t), C (t)dt.
In order to characterize this optimal control, we apply as usual the
Pontryagin’s maximum principle. We refer the reader to [15] and [8, 16, 18]
for more details and different applications of the optimal control theory.
Theorem 3 With previous assumptions, there exists an application
P(.) = (PD(.), PC(.)) : [0,+∞[→IR2absolutely continuous called adjoint
vector, such that, for almost all t≥0,







P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t)−
∂F
∂D (u(t), D(t), C(t)),
P′
C(t) = (r+γ(t) + µ(t) + ν(t) + δ(t))PC(t)−γ(t)PD(t)−
∂F
∂C (u(t), D(t), C(t)),
with the limiting transversality conditions:





lim
t→+∞exp(−rt)PD(t)D(t) = 0,
lim
t→+∞exp(−rt)PC(t)C(t) = 0.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 245
And the optimal control ¯u, whose existence has been proved in previous theorem,
satisfies the following maximization’s condition:
H(¯u, D, C, PD, PC) = max
v∈L∞(IR+)H(v, D, C, PD, PC),
where His the associated Hamiltonian defined from the state equations and the
concave performance index F(u, D, C)as
H(u, D, C, PD, PC) = F(u, D, C) + PD[I−(λ+µ)D+γC]
+PC[λD −(γ+µ+ν+δ)C].
3 Stability analysis
This section is devoted to the existence proof of the equilibrium state and its
classification. In our previous work [1], we fixed a control, a cost function and
characterized the associated equilibrium state to see under which conditions it is
admissible. In the present one, we prove that the equilibrium state exists what-
ever the control and the objective function.
Theorem 4 Whatever the control uand whatever the concave performance in-
dex Fchosen as previously specified, there is an equilibrium state for the system























D′(t) = I−(λ(t) + µ(t))D(t) + γ(t)C(t),
C′(t) = λ(t)D(t)−(γ(t) + µ(t) + ν(t) + δ(t))C(t),
P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t)−
∂F
∂D (u(t), D(t), C(t)),
P′
C(t) = (r+γ(t) + µ(t) + ν(t) + δ(t))PC(t)−γ(t)PD(t)−
∂F
∂C (u(t), D(t), C(t)).
Proof. The equilibrium state (D∗, C∗, P ∗
D, P ∗
C), if it exists, is solution of the
following system























I−(λ(t) + µ(t))D(t) + γ(t)C(t) = 0,
λ(t)D(t)−(γ(t) + µ(t) + ν(t) + δ(t))C(t) = 0,
(r+λ(t) + µ(t))PD(t)−λ(t)PC(t)−
∂F
∂D (u(t), D(t), C(t))= 0,
(r+γ(t) + µ(t) + ν(t) + δ(t))PC(t)−γ(t)PD(t)−
∂F
∂C (u(t), D(t), C(t))= 0.
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246 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
In order to have existence of an equilibrium state, we evaluate the determi-
nant of this last system which is on the form
S=
−(λ+µ)γ0 0
λ−(γ+µ+ν+δ) 0 0
. . r +λ+µ−λ
. . −γ r +γ+µ+ν+δ

.
An easy computation gives
S= [λ(µ+ν+δ) + µ(γ+µ+ν+δ)]
·[(r+µ)(r+γ+µ+ν+δ) + λ(r+µ+ν+δ)] ,
which implies that S≥0and Sis clearly equal to zero if and only if all the
parameters of the model are equals to zero, since Sis a sum of non negative
terms. Consequently, Sis strictly non negative, which implies the existence of
an equilibrium state.
In order to characterize this equilibrium state, we have to choose the control
and the objective function. We will see in the following section some examples
for which we characterize it and note that it is not necessarily admissible.
Theorem 5 There is no limit cycle between the number of diabetics with compli-
cations, the one of diabetics without complications and the controlled parameter
chosen as previously specified.
Proof. The steps leading to this result are similar to the ones of [1]. Let us define
the Jacobian by
J=










∂D′/∂D ∂D′/∂C ∂D′/∂PD∂D′/∂PC
∂C ′/∂D ∂C′/∂C ∂C′/∂PD∂C′/∂PC
∂P ′
D/∂D ∂P ′
D/∂C ∂ P ′
D/∂PD∂ P ′
D/∂PC
∂P ′
C/∂D ∂ P ′
C/∂C ∂P ′
C/∂PD∂ P ′
C/∂PC










,
and a term Kby
K=





∂D′/∂D ∂D′/∂PD
∂P ′
D/∂D ∂P ′
D/∂PD






+





∂C ′/∂C ∂C′/∂PC
∂P ′
C/∂C ∂ P ′
C/∂PC






+2 





∂D′/∂C ∂D′/∂PC
∂P ′
D/∂C ∂ P ′
D/∂PC






.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 247
In order to study the existence of a limit cycle, it is necessary to know the
sign of the determinant of the Jacobian Jand of the term Kcalculated at the
equilibrium point. For our problem, we have
J=













−(λ+µ)γ0 0
λ−(γ+µ+ν+δ) 0 0
−∂2F
∂D2−∂2F
∂C ∂D r+λ+µ−λ
−∂2F
∂D∂C −∂2F
∂C 2−γ r +γ+µ+ν+δ













,
and
K=






−(λ+µ) 0
−
∂2F
∂D2r+λ+µ







+






−(γ+µ+ν+δ) 0
−
∂2F
∂C 2r+γ+µ+ν+δ







+2 






γ0
−
∂2F
∂C ∂D
−λ







.
Even if the matrix Jand the term Kare more general than the ones of our
previous study [1], we obtain exactly the same results for det Jand K, that is
det J= [λ(µ+ν+δ) + µ(γ+µ+ν+δ)]
·[(r+µ)(r+γ+µ+ν+δ) + λ(r+µ+ν+δ)] ,
and
K=−(λ+µ)(r+λ+µ)
−(γ+µ+ν+δ)(r+γ+µ+ν+δ)−2γλ.
It is clear that det J≥0and K≤0, which leads to the result.
Theorem 6 The equilibrium state defined previously is a saddle point.
Proof. In order to classify the equilibrium state, we have to know the sign of
Q= det(J)−1
2K2. Since det(J)and Kare exactly the same than those of [1],
we used a similar method.
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248 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
4 Characterization of the equilibrium state in
particular cases
In this section, we are going to characterize the equilibrium state, whose ex-
istence has been proved in the previous one, for different concave performance
index. We will see that it is not always easy to characterize it and even if we can,
it is not necessarily admissible taking into account the specificities of our prob-
lem. In the following, we consider the rate at which complications are healed as
control, that is u=γand for a fixed discount rate r > 0, the cost function we
want to maximize is
+∞
0
exp(−rt)Fu(t), D(t), C (t)dt,
where F(u, D, C )is the concave performance index that we will fix at each case.
In the first example, we choose the control u=γand the performance
index F(u, D, C) = ln u. In this case, our goal is to control the rate at which
complications are healed while maximizing it.
Proposition 1 If the control u=γand the performance index
F(u, D, C ) = ln uthen the associated equilibrium state is not admissible.
Proof. According to Section 2, we have the existence of the optimal control lead-
ing to maximize the rate at which complications are cured. In order to character-
ize it, we use the Pontryagin’s maximum principle which gives the existence of
an adjoint vector P(.)=(PD(.), PC(.)) : [0,+∞[→R2absolutely continuous
such that, for almost all t>0,



P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t),
P′
C(t) = −γ(t)PD(t)+(r+γ(t) + µ(t) + ν(t) + δ(t))PC(t),
with the associated Hamiltonian
H(γ, D, C, PD, PC) = ln γ+PD[I−(λ+µ)D+γC] + PC[λD −(γ+µ+ν+δ)C],
and the previously cited limiting transversality conditions. Consequently, the
optimal control is
¯γ=1
(PC−PD)C.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 249
Now, we try to characterize the associated equilibrium state of the system





















D′(t) = I−(λ(t) + µ(t))D(t) + γ(t)C(t),
C′(t) = λ(t)D(t)−(γ(t) + µ(t) + ν(t) + δ(t))C(t),
P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t),
P′
C(t) = −γ(t)PD(t)+(r+γ(t) + µ(t) + ν(t) + δ(t))PC(t),
by solving



















I−(λ+µ)D+ ¯γC = 0,
λD −(¯γ+µ+ν+δ)C= 0,
(r+λ+µ)PD−λPC= 0,
−¯γPD+ (r+ ¯γ+µ+ν+δ)PC= 0.
By replacing ¯γby 1
(PC−PD)C, we have



























I−(λ+µ)D+1
PC−PD
= 0,
λD −(µ+ν+δ)C−1
PC−PD
= 0,
(r+λ+µ)PD−λPC= 0,
(r+µ+ν+δ)PC+1
C= 0.
The use of the first, fourth and second equations gives successively
D=I(PC−PD)+1
(λ+µ)(PC−PD),
C=−1
(r+µ+ν+δ)PC
,
and λI
λ+µ+µ+ν+δ
(r+µ+ν+δ)PC
−µ
(λ+µ)(PC−PD)= 0.
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250 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
Moreover, with the third equation, we obtain
PD=λ
r+λ+µPCso PC−PD=r+µ
r+λ+µPC.
Consequently,
D=I
λ+µ+(r+λ+µ)
(r+µ)(λ+µ)PC
,
and λI
λ+µ+µ+ν+δ
(r+µ+ν+δ)PC
−µ(r+λ+µ)
(r+µ)(λ+µ)PC
= 0,
so
PC=r[µ(r+µ)−λ(ν+δ)]
λI(r+µ)(r+µ+ν+δ).
Thus, the equilibrium state is



































D∗=I
λ+µ+λI(r+λ+µ)(r+µ+ν+δ)
r(λ+µ)[µ(r+µ)−λ(ν+δ)],
C∗=−λI(r+µ)
r[µ(r+µ)−λ(ν+δ)],
P∗
D=r[µ(r+µ)−λ(ν+δ)]
I(r+µ)(r+λ+µ)(r+µ+ν+δ),
P∗
C=r[µ(r+µ)−λ(ν+δ)]
λI(r+µ)(r+µ+ν+δ).
Note that, if the term µ(r+µ)−λ(ν+δ)is equal to zero then PC=PD= 0
which is impossible since (PD, PC)has to be non trivial. It follows that
γ∗=−(r+λ+µ)(r+µ+ν+δ)
r+µ.
However, this equilibrium state is not admissible because γis a rate so has
to be non negative.
In the second example, we choose the control u=γand the performance
index F(u, D, C) = ln u+aln C, with a > 0. In this case, our goal is to
control the rate at which complications are healed while maximizing the number
of diabetics with cured complications.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 251
Proposition 2 If the control u=γand the performance index
F(u, D, C ) = ln u+aln C, with a > 0, then the associated equilibrium state
is admissible if and only if a > 1.
Proof. According to Section 2, we have the existence of the optimal control lead-
ing to maximize the number of diabetics with cured complications. In order to
characterize it, we use the Pontryagin’s maximum principle which gives the ex-
istence of an adjoint vector P(.) = (PD(.), PC(.)) : [0,+∞[→R2absolutely
continuous such that, for almost all t>0,





P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t),
P′
C(t) = −γ(t)PD(t) + (r+γ(t) + µ(t) + ν(t) + δ(t))PC(t)−a
C(t),
with the associated Hamiltonian
H(γ, D, C, PD, PC) = ln γ+aln C+PD[I−(λ+µ)D+γC ]
+PC[λD −(γ+µ+ν+δ)C],
and the previously cited limiting transversality conditions. Consequently, the
optimal control is
¯γ=1
(PC−PD)C.
We characterize the associated equilibrium state by solving





























I−(λ+µ)D+1
PC−PD
= 0,
λD −(µ+ν+δ)C−1
PC−PD
= 0,
(r+λ+µ)PD−λPC= 0,
(r+µ+ν+δ)PC+1−a
C= 0.
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252 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
Similar computations give the following equilibrium state:

































D∗=I
λ+µ[1 + λ(r+λ+µ)(r+µ+ν+δ)
(µ+ν+δ)(a−1)(r+µ)(λ+µ) + µ(r+λ+µ)(r+µ+ν+δ)],
C∗=λI(a−1)(r+µ)
(µ+ν+δ)(a−1)(r+µ)(λ+µ) + µ(r+λ+µ)(r+µ+ν+δ),
P∗
D=(µ+ν+δ)(a−1)(r+µ)(λ+µ) + µ(r+λ+µ)(r+µ+ν+δ)
I(r+µ)(r+µ+ν+δ)(r+λ+µ),
P∗
C=(µ+ν+δ)(a−1)(r+µ)(λ+µ) + µ(r+λ+µ)(r+µ+ν+δ)
λI(r+µ)(r+µ+ν+δ).
It follows that
γ∗=(r+λ+µ)(r+µ+ν+δ)
(r+µ)(a−1) ,
and this equilibrium state is admissible if and only if a > 1.
In the last example, we choose the control u=γand the performance index
F(u, D, C ) = αln u+aln C+K(D), with a, α > 0and Kan increasing
concave function of D. In this case, our goal is to control the rate at which
complications are healed while maximizing the number of diabetics with cured
complications and the number of diabetics without complications.
Proposition 3 Let us set σ=µ+ν+δand assume that the control u=γand
the performance index F(u, D, C) = αln u+aln C+K(D), with a, α > 0
and Kan increasing concave function of D. If µ= 0 and
•if K(D) = Dthen, the associated equilibrium state is admissible if and
only if α < a −σ−1Iand r > σI
σ(a−α)−I,
•or if K(D) = ln Dand a > λ
σthen, there is α∗satisfying
0< α∗<min a, aσ −λ
2σ+λ,
and r∗>0such that the associated equilibrium state is admissible.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 253
Proof. As previously, we have the existence of the optimal control leading to
maximize the number of diabetics with cured complications and the number of
diabetic without complications. In order to characterize it, we use the
Pontryagin’s maximum principle which gives the existence of an adjoint vec-
tor P(.)=(PD(.), PC(.)) : [0,+∞[→R2absolutely continuous such that, for
almost all t>0,





P′
D(t) = (r+λ(t) + µ(t))PD(t)−λ(t)PC(t)−K′(D(t)),
P′
C(t) = (r+γ(t) + µ(t) + ν(t) + δ(t))PC(t)−γ(t)PD(t)−a
C(t),
with the associated Hamiltonian
H(γ, D, C, PD, PC) = αln γ+aln C+K(D) + PD[I−(λ+µ)D+γC]
+PC[λD −(γ+µ+ν+δ)C],
and the previously cited limiting transversality conditions. Consequently, the
optimal control is
¯γ=α
(PC−PD)C.
We characterize the equilibrium state by solving





















I−(λ+µ)D+α
PC−PD= 0,
λD −(µ+ν+δ)C−α
PC−PD= 0,
(r+λ+µ)PD−λPC−K′(D)=0,
(r+µ+ν+δ)PC+α−a
C= 0.
In order to simplify the writing, let us set ρ=λ+µand T=PC−PD. The
use of the first and second equations gives
D=IT +α
ρT and C=λ
σD−α
σT .
Consequently,
C=λ(α+I T )−αρ
σρT .
Moreover, by using the fourth equation, we obtain
PC=a−α
(r+σ)Cso PC=(a−α)σρT
(r+σ)[λIT +α(λ−ρ)].
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254 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
Let us note that the quantity λIT +α(λ−ρ)is equal to zero if and only if
C= 0 which has no sense in this situation. Moreover, the third equation implies
that
PD=λT σρ(a−α)+(r+σ)[λI T +α(λ−ρ)]K′(D)
(r+σ)(r+ρ)[λIT +α(λ−ρ)] .
Thus
T=(r+ρ)(a−α)σρT −λT σ ρ(a−α)−(r+σ)[λIT +α(λ−ρ)]K′(D)
(r+σ)(r+ρ)[λIT +α(λ−ρ)] ,
which is equivalent to
λI(r+ρ)(r+σ)T2
+α(λ−ρ)(r+ρ)(r+σ)−(r+ρ−λ)(a−α)σρ +λIK′(D)(r+σ)T
+α(r+σ)(λ−ρ)K′(D) = 0.
At this stage, it is not easy to characterize Texplicitly. We use the fact that
µ= 0 that is λ=ρso
T=σ(a−α)r−I(r+σ)K′(D)
I(r+λ)(r+σ),
since Tis different from zero. Consequently, Tdepends on the derivative of
the function K. In order to finish the characterization, we consider two different
cases.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 255
The first one is K(D) = Dfor which we obtain
T=σ(a−α)r−I(r+σ)
I(r+λ)(r+σ).
Consequently

































D∗=I
λ+αI(r+λ)(r+σ)
λ[σ(a−α)r−I(r+σ)],
C∗=I
σ,
P∗
D=λ(a−α)σ+ (r+σ)I
(r+σ)(r+λ)I,
P∗
C=(a−α)σ
(r+σ)I.
It follows that
γ∗=ασ(r+λ)(r+σ)
σ(a−α)r−I(r+σ).
This equilibrium state is admissible if and only if σ(a−α)r−I(r+σ)>0. As
saying in [1], it occurs if and only if α < a −σ−1Iand r > σI
σ(a−α)−I.
The second case we consider is K(D) = ln Dfor which we obtain
T=σ(a−α)rD −I(r+σ)
I(r+λ)(r+σ)D.
Replacing Dby IT +α
ρT , we get
(r+λ)(r+σ)I2T2+[(r+σ)ρ+α(r+λ)(r+σ)−(a−α)rσ]I T −(a−α)αrσ = 0.
This is a polynomial equation on the form pT 2+qT +l= 0 with
p= (r+λ)(r+σ)I2,
q= [(r+σ)ρ+α(r+λ)(r+σ)−(a−α)rσ]I ,
l=rασ(α−a).
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256 S.BERNARD —T.CÉSAR —S.P.NUIRO -A.PIÉT RUS
If a>αthen the associated discriminant ∆ = q2−4lp is strictly non
negative and consequently, there are two solutions
T∗=1
2p−q±q2−4lp.
If q < 0then there is one non negative solution T∗, which ensures the existence
of an admisible equilibrium state. But qis negative if and only if
αr2+ [λ+α(σ+λ) + σ(α−a)]r+σλ(1 + α)<0,(2)
wich occurs only when
λ+α(σ+λ) + σ(α−a)<0,
and for this, we need that aσ > λ and 0< α < aσ −λ
2σ+λ. Moreover, we remark
that the discriminant of (2):
β(α) = [λ(1 + α)−aσ]2−4ασ2(a−α),
is such that
lim
α→0+β(α) = [λ−aσ]2>0.
So, there is α∗>0such that α∗<min a, aσ −λ
2σ+λand β(α∗)>0,since
βis a continuous function on ]0, a[. Therefore, it exists r∗>0solution of (2).
Finally, the equilibrium state is









































D∗=1
λI+2pα
−q+q2−4lp ,
C∗=I
σ,
P∗
D=λ(a−α)σ
I(r+σ)(r+λ)+
λ−q+q2−4lp
(r+λ)I(−q+q2−4lp)+2pα,
P∗
C=(a−α)σ
I(r+σ),
and it follows that
γ∗=α
T∗C∗=2pασ
I(−q+q2−4lp),
which is admissible.
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UNEXISTENCE OF LIMIT CYCLE IN AN OPTIMAL CONTROL . . . 257
5 Concluding remarks
This work is clearly a generalization of our previous one [1]. Considering a pop-
ulation of diabetics, we proved that there is no cyclical behavior between the
number of diabetics with complications and the one of those without complica-
tions, when an optimal control is applied; that the equilibrium state always exists
but is not always admissible and fortunately is always a saddle point. This result
is an interesting one and reassures us since it reflects the fact that if we control
the rate at which complications are healed, or the rate at which patients with
complications become severely disabled or the probability of developing com-
plications, by more health expenditure or a good lifestyle respectively, then there
is no back and forth between the group of diabetics with complications and the
one of diabetics without complications. In the following, it will be interesting
to see how external actions like medication, food consumption or physical ex-
ercises can impact on the development of complications for diabetics. The first
challenge would be to see how to include these parameters in the model and the
second one to measure their impact on the disease and its evolution.
Acknowledgments
We would like to thank the anonymous referees for their valuable suggestions
and remarks that enabled us to improve the presentation of this manuscript.
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