Ratio Mathematica Volume 46, 2023

A dynamical analysis of a mathematical
model on type-2 diabetic from obestiy

Gnana Priya G. *
A.Sabarmathi †

Naga Soundarya Lakshmi V.S.V‡

Abstract

A model for type-2 diabetic from obesity with two control variables
diet with physical activity and medication is formulated.The disease
free and endemic equilibrium points of the model are obtained. The
existence of optimal controls is verified through Pontryagainaximum
principle. The local stability is analyzed using Routh-Hurwitz cri-
teria. The global stability is studied using Lyapunov function. The
parameters are chosen based on the female population in India. The
aim of this research is to construct a model for type-2 diabetic from
obesity using parameters based on the female population in India. We
have introduced two control variables as diet with physical activity
and medication. The positive endemic equilibrium is obtained. The
local and global stability of the model are analyzed with some specific
conditions. Numerical simulations are carried out to exhibit the flow
of variables with controls. Our study mainly highlights the awareness
of metabolic risk by healthy diet, physical activities and medications.
Keywords: Obesity, Diabetic, Pontryagain’s maximum principle, Lya-
punov function, Stability.
2020 AMS subject classifications: 34D20, 34D23,81T80 1

*Department of Mathematics, Auxilium college, (Affiliated to Thiruvalluvar
University,Serkadu,Vellore-632115) ; priya.prakasam51@gmail.com

†Department of Mathematics, Auxilium college,(Affiliated to Thiruvalluvar
University,Serkadu,Vellore-632115.

‡Department of Mathematics, Auxilium college, (Affiliated to Thiruvalluvar
University,Serkadu,Vellore-632115. san9sak14@gmail.com
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1072. ISSN: 1592-7415. eISSN: 2282-8214. ©G. Gnana Priya
et al. This paper is published under the CC-BY licence agreement.

156



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

1 Introduction

Obesity and Diabetes are two epidemics that are wreaking havoc around the
world. Obesity puts you at a much increased chance of having diabetes through-
out the course of your life. Obesity is more than a cosmetic issue. It is a long-term
medical condition that can progress to diabetes. Obese people account for over
90 percent of people with type 2 diabetes which is characterised by the body’s
failure to regulate and control blood sugar levels. Obesity allows diabetes to take
up residence in the body. Obesity increases the risk of developing diabetes which
is characterised by an excess of glucose in the bloodstream. Obesity is linked to
a higher risk of non communicable diseases such as diabetes, heart disease and
hypertension. Obesity is one of the primary causes of early death and 4.7 million
deaths worldwide was recorded in 2017 due to obesity. The key to understanding
the link between obesity and diabetes is to look at insulin resistance. Physical in-
activity, overeating, heredity, medications and psychological factors are the most
prevalent causes of obesity. Type 2 diabetes has no known treatment however, it
can be in control. Inactivity, family history, high blood pressure and obese are
the most common causes of type 2 diabetics. Obesity reduction is a significant
objective for the treatment and prevention of type 2 diabetes. Many authors stud-
ied a mathematical models for obesity and diabetic. A Boutayed Boutayed et al.
[2010] developed a mathematical model for the burdern of diabetes and its com-
plications. Wiam Boutayeb et al. [2014] introduced a mathematical model for
the impact of obesity on predisposed people to type 2 diabetes. Nikhil Kumar
et al. [2017] studied a diabetsity an epidemic with its causes, prevention and con-
trol with special focus on dietary regime. Miguel A. Ortega Ortega et al. [2020]
illustrated a technique for type 2 diabetes mellitus associated with obesity. Ab-
delfatah Kouidere K.Abdelfatah et al. [2020] developed a mathematical model for
the dynamics of population of diabetics and its complications with effect of be-
havioral factors. Wellars Banzi et al. [2021] studied a mathematical model for
Glucose-Insulin System and test of abnormalities of type 2 diabetic patients. In
this paper, we formulated and anlaysed a mathematical model for type-2 diabetic
from obesity with control variables.

2 Formulation of model

The system of Ordinary Differential Equation represents the Obesity and Dia-
betic model as follows::

157



A dynamical analysis of a mathematical model on type-2 diabetic

dS

dt
= ωN − (β + µ)S

dO

dt
= βS − (α + µ + u)O

dD

dt
= αO − (µ + u + v)D

dR

dt
= (u + v)D + uO −µR

(1)

With the initial conditions as S(t),O(t),D(t),R(t) ≥ 0. Also β,µ,ω,α,u,v >
0.

Where S(t),O(t),D(t),R(t) are the Susceptible, Infected by Obesity, In-
fected by Diabetic and Recovery state respectively and ω - Average birth rate of
female, µ - Average death rate of female, β - Obesity rate , α - Diabetic rate, N(t)
- Female Population. The control variables are u - Diet with Physical Activity and
v - Medication.
The following figure shows the considered Obesity and Diabetic model:

Figure 1: Obesity and Diabetic model

3 Equilibrium analysis

The steady states are G0(0, 0, 0, 0), G1(S, 0, 0, 0), G2(S̃, Õ, 0, 0), G3(S′,O′,D′, 0)
and G4(S∗,O∗,D∗,R∗).

158



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

Case(1): G0(0, 0, 0, 0) exists always.
Case(2): In G1(S, 0, 0, 0),
Let S be the positive solution of dS

dt
= 0

From (1),

S =
ωN

(β + µ)
> 0

S =
ωN

(β + µ)
> 0

∴ G1(S, 0, 0, 0) = (
ωN

(β+µ)
, 0, 0, 0)

Case(3): In G2(S̃, Õ, 0, 0) ,
Let S̃, Õ be the positive solutions of dS

dt
= 0,dO

dt
= 0.

From (1),

S̃ =
ωN

(β + µ)
> 0

Õ =
βωN

(β + µ)(α + µ + u)
> 0

∴ G2(S̃, Õ, 0, 0) = G2
(

ωN
(β+µ)

, βωN
(β+µ)(α+µ+u)

, 0, 0
)

Case(4): In G3(S′,O′,D′, 0),
Let S′,O′ and D′ be the positive solutions of dS

dt
= 0,dO

dt
= 0,dD

dt
= 0

From (1),

S′ =
ωN

(β + µ)

O′ =
βωN

(β + µ)(α + µ + u)

D′ =
αβωN

(β + µ)(α + µ + u)(µ + u + v)

∴ G3(S
′,O′,D′, 0) =

(
ωN
β+µ

, βωN
(β+µ)(α+µ+u)

, αβωN
(β+µ)(α+µ+u)(µ+u+v)

, 0
)

Case(5): For G4(S∗,O∗,D∗,R∗),
Let S∗, O∗, D∗ and R∗ be the positive solutions of dS

dt
= 0,dO

dt
= 0,dD

dt
= 0, dR

dt
= 0

From (1),

S∗ = ωN

(
1

β + µ

)

O∗ = ωN

(
β

(α + µ + u)(β + µ)

)

159



A dynamical analysis of a mathematical model on type-2 diabetic

D∗ = ωN

(
αβ

(α + µ + u)(β + µ)(µ + u + v)

)
R∗ = ωN

[
1

µ

(
αβ(u + v)

(α + µ + u)(β + µ)(µ + u + v)
+

βu

(α + µ + u)(β + µ)

)]
∴ the diabesity equilibrium is given by

G4(S
∗,O∗,D∗,R∗)

= (ωN

(
1

β + µ

)
,ωN

(
β

(α + µ + u)(β + µ)

)
,

ωN

(
αβ

(α + µ + u)(β + µ)(µ + u + v)

)
,

ωN

[
1

µ

(
αβ(u + v)

(α + µ + u)(β + µ)(µ + u + v)
+

βu

(α + µ + u)(β + µ)

)]
)

4 Pontryagain’s maximum principle
We compute the optimal controls of the control variables by using Pontrya-

gain’s maximum principle. We prove the following four properties :
I: The set of controls and state variables that correspond to them is non empty.
II: U is a closed and convex control set.
III: The state system’s R.H.S is bounded by a linear function in the state and

control variables.
IV: The integrand of the objective functional is convex on U and is bounded

below by k2 + k1|U|
η, with k1 > 0, k2 > 0, η > 1.

To Prove I:
From (1),

F1 = ωN − (β + µ)S
F2 = βS − (α + µ + u)O
F3 = αO − (µ + u + v)D

F4 = (u + v)D + uO −µR

(2)

From (2) ∣∣∂F1
∂S

∣∣ < ∞, ∣∣∂F1
∂O

∣∣ < ∞,∣∣∂F1
∂D

∣∣ < ∞, ∣∣∂F1
∂R

∣∣ < ∞∣∣∂F2
∂S

∣∣ < ∞, ∣∣∂F2
∂O

∣∣ < ∞,∣∣∂F2
∂D

∣∣ < ∞, ∣∣∂F2
∂R

∣∣ < ∞∣∣∂F3
∂S

∣∣ < ∞, ∣∣∂F3
∂O

∣∣ < ∞,∣∣∂F3
∂D

∣∣ < ∞, ∣∣∂F3
∂R

∣∣ < ∞∣∣∂F4
∂S

∣∣ < ∞, ∣∣∂F4
∂O

∣∣ < ∞,∣∣∂F4
∂D

∣∣ < ∞, ∣∣∂F4
∂R

∣∣ < ∞

160



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

As it is continuous and bounded then there exist a unique solution of (1) by Exis-
tence and uniqueness theorem.
Hence, I is satisfied.
To prove II:
The control set is given by,

U = {(u(t),v(t))/0 ≤ u ≤ 1, 0 ≤ v ≤ 1, t ∈ [0, 1]}

U is closed, from our definition.
Let u,v ∈ U, α ∈ [0, 1]
Thenαu + (1 −α)v ≥ 0
As u ≤ 1 and v≤ 1,
0 ≤ αu + (1 −α)v ≤ (1 −α), for all u,v ∈ U
Hence II is satisfied.
To prove III:
From (2),

F1 ≤ ωN
F2 ≤ βS −uO

F3 ≤ αO − (u + v)D
F4 ≤ (u + v)D + uO

It can be rewritten as follows:

F(t,X,U) ≤




0 0 0 0
β 0 0 0
0 α 0 0
0 0 0 0


 .


S
O
D
R


 +




0
−O
−D
D + O



[
u
v

]

Which can be written as the linear combination of controls∣∣F(t,X,U)∣∣ ≤‖m̄‖ ∣∣X̄∣∣ + ‖D + O‖|u(t),v(t)|
≤ K

[∣∣X̄∣∣ + |u(t),v(t)|]
As S,O,D are bounded and K is upper bound.
∴ III is satisfied.

To Prove IV:
The control variables and state variables are non negative u,v ∈ U.
u,v is convex and closed.
Then from (3),

J = AI(t) +
ω1u

2 + ω2v
2

2

161



A dynamical analysis of a mathematical model on type-2 diabetic

≥−AI(t) +
ω

2
(u2 + v2)

= −k2 + k1(u,v)2

where ω = ω1 + ω2, k1 =
ω
2

.
Here k2 > 0 , k1 > 0 and η = 2 > 1
Hence IV is satisfied.
Hence there exists an optimal control variables which is found in the following
section.

5 Existence of optimal control
The objective functional is defined as in (2) as follows:

J = minAI(t) +
1

2

∫ T
0

(ω1u
2 + ω2v

2)dt (3)

Where AI(t) is the number of individuals, ω1, ω2 are the weight parameters for the
cost of diet with physical activity and the cost of medication respectively.
From previous section, our system (1) converts into a problem of minimizing.
An Hamiltonian H is defined as,

H(S,O,D,R) = AI(t) +
ω1u

2

2
+
ω2v

2

2
+ λS (ωN − (β + µ)S)

+ λO (βS − (α + µ + u)O) + λD (αO − (µ + u + v)D)
+ λR ((u + v)D + uO −µR) (4)

The adjoint system is given by

dλS
dt

= λS(β + µ) −λOβ
dλO
dt

= λO(α + µ + u) −αλD −λRu
dλD
dt

= λD(µ + u + v) −λR(u + v)
dλR
dt

= λRµ

with the final condition λS(T) = λO(T) = λD(T) = λR(T)=0 for free problem.
The optimality is found from ∂H

∂u
= 0 and ∂H

∂v
= 0.

∴ The optimal controls are,

u∗ = min
(

1,max
(

0,
D(λD−λR)−OλR

ω1

))
, v∗ = min

(
1,max

(
0,

D(λD−λR)
ω2

))
Here u∗, v∗ are positive when λD > λR and D(λD −λR) > OλR.

162



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

6 Local stability

The Jacobian matrix for the system (1) is



−(β + µ) 0 0 δ

β −(α + µ + u) 0 0
0 α −(µ + u + v) 0
0 u u + v −µ


 (5)

At the interior equlilibrium (5) becomes



−ωN

S
0 0 0

β −βS
O

0 0
0 α −αO

D
0

0 u u + v −(u+v)D+uO
R


 (6)

The characteristic equation of (6) is given by

∣∣∣∣∣∣∣∣
−ωN

S
−λ 0 0 0

β −βS
O
−λ 0 0

0 α −αO
D
−λ 0

0 u (u + v) −(u+v)D+uO
R

−λ

∣∣∣∣∣∣∣∣ = 0

λ4 +

(
ωN

S
+
βS

O
+
αS

D
+
uO

R
+

(u + v)

R

)
λ3

+ (
ωNβ

O
+

(u + v)DωN

SR
+
uOωN

SR
+
ωNα

D
+

(u + v)DβS

R
+
uβS

R
+
βαS2

DO

+
(u + v)αS

R
+
uOαS

RD
)λ2 + (

(u + v)αωNDβ

R
+
uβωN

R
+
ωNβαS

DO

+
(u + v)αωN

R
+
ωNuOα

RD
+

(u + v)αβS2

RO
+
uαβS2

RD
)λ

+
(u + v)αβωNS

RO
+
uαβωNS

RD
= 0 (7)

163



A dynamical analysis of a mathematical model on type-2 diabetic

Comparing (4) with S4 + AS3 + BS2 + CS + D = 0
where

A =
ωN

S
+
βS

O
+
αS

D
+
uO

R
+

(u + v)

R
,

B =
ωNβ

O
+

(u + v)DωN

SR
+
uOωN

SR
+
ωNα

D
+

(u + v)DβS

R

+
uβS

R
+
βαS2

DO
+

(u + v)αS

R
+
uOαS

RD
,

C =
(u + v)αωNDβ

R
+
uβωN

R
+
ωNβαS

DO
+

(u + v)αωN

R

+
ωNuOα

RD
+

(u + v)αβS2

RO
+
uαβS2

RD
,

D =
(u + v)αβωNS

RO
+
uαβωNS

RD

Here A > 0; D > 0; AB −C > 0; C(AB −C) −A2D > 0 when 1 −S > 0 and
the system (1) is locally stable by the Routh-Hurwitz criteria.

7 Global stability
We establish the Lyapunov function given below,

V (S,O,D,R) =

(
(S −S∗) −S∗ln

S

S∗

)
+ l1

(
(O −O∗) −O∗ln

O

O∗

)
+l2

(
(D −D∗) −D∗ln

D

D∗

)
+ l3

(
(R−R∗) −R∗ln

R

R∗

) (8)
Differentiate (8) with respect to t,

dV

dt
= (

S −S∗

S
)
dS

dt
+ (

O −O∗

O
)
dO

dt
+ (

D −D∗

D
)
dD

dt
+ (

R−R∗

R
)
dR

dt

From the model equations (1),

dV

dt
= (

S −S∗

S
) (ωN − (β + µ) S) + (

O −O∗

O
) (βS − (α + µ + u)O)

+(
D −D∗

D
) (αO − (µ + u + v)D) + (

R−R∗

R
) ((u + v)D + uO −µR)

= (S −S∗)
(
ωN

S
− (β + µ)

)
+ l1(O −O∗)

(
βS

O
− (α + µ + u)

)

164



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

+l2(D −D∗)
(
αO

D
− (µ + u + v)

)
+ l3(R−R∗)

(
(u + v)D + uO

R
−µ

)
At (S∗,O∗,D∗,R∗), we have

dV

dt
= (S −S∗)[

ωN

S
− (

ωN

S∗
)] + l1(O −O∗)[

βS

O
− (

βS∗

O∗
)]

+l2(D−D∗)[
αO

D
−(

αO∗

D∗
)]+l3(R−R∗)[(u+v)(

D + uO

R
−(

(u + v)D∗ + uO∗

R∗
)]

= (S −S∗)ωN(
1

S
−

1

S∗
) + l1(O −O∗)β(

S

O
−
S∗

O∗
)

+l2(D −D∗)α(
O

D
−
O∗

D∗
) + l3(R−R∗)[(u + v)(

D

R
−
D∗

R∗
) + u(

O

R
−
O∗

R∗
)]

Choosing l1 =
1
β
, l2 =

1
α
, l3 =

1
(u+v)u

dV

dt
= (S −S∗)ωN

(
1

S
−

1

S∗

)
+

(O −O∗)β
β

(
S

O∗
−
S∗

O∗

)
+

(D −D∗)α
α(

O

D
−
O∗

D∗

)
+

(R−R∗)
(u + v)u

(
(u + v)D

R
−
D∗

R∗

)
+

(R−R∗)
(u + v)u

(
u(
D

R
−
D∗

R∗
)

)
= −

ωN(S −S∗)2

SS∗
+

1

OO∗
[SOO∗ −S∗O2 −SO∗2 + S∗O∗O]

+
1

DD∗
[OD∗D −D2O∗ −OD∗2 + D∗O∗D] +

1

uRR∗
[DR∗R−R2D∗ −R∗2D

+ R∗RD∗] +
1

(u + v)RR∗
[DR∗R−R2D∗ −R∗2D + R∗RD∗]

= −(S −S∗)ωN
(
S −S∗

SS∗

)
+ (O −O∗)

(
SO∗ −S∗O

OO∗

)
+ (D −D∗)(

OD∗ −DO∗

DD∗

)
+
R−R∗

u

(
DR∗ −RD∗

RR∗

)
+
R−R∗

u + v

(
DR∗ −RD∗

RR∗

)
=
−ωN(S −S∗)2

SS∗
+

(
S −

SO∗

O
−
S∗O

O∗
+ S∗

)
+

(
O −

OD∗

D
+
DO∗

D∗
+ O∗

)
+

1

u

(
R−

DR∗

R
+
D∗R

R∗
+ R∗

)
+

1

u + v

(
R−

DR∗

R
+
D∗R

R∗
+ R∗

)

∴ dV
dt
< 0, when R

S
< R

∗

S∗
, S
O
< S

∗

O∗
, O
D
< O

∗

D∗
,D
R
< D

∗

R∗
,O
R
< O

∗

R∗
Here the system (1) is globally asymptotically stable by Lyapunov theorem.

165



A dynamical analysis of a mathematical model on type-2 diabetic

8 Numerical simulations
We have choose the values of parameters as ω = 0.0234, µ = 0.008, α =

0.2792, β = 0.23 based on a female population.
Figure (2) shows the flow of variables with control parameters. The obesity rate

Figure 2: Obesity and Diabetic with control parameters

Figure 3: Obesity and Diabetic without control parameters

and diabetic rate are stable with high recovery rate with respect to time . Figure
(3) shows the flow of variables without control parameters. The diabetic rate is
high due to no recovery rate.
Figure (4) shows the flow of individuals of susceptible class for various values of
β. If β values are increasing, the number of individuals of susceptible class move

166



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

Figure 4: Flow of individuals in Susceptible Class for various values of β

towards the infected by obesity class which becomes stable with respect to time.

Figure 5: Infected by Obesity class for different values of α with control parame-
ters

Figure(5) and figure(6) show the flow of individuals of infected by obesity
class for various values of α with and without control parameters respectively. If
we use the control parameters by giving different values to α, then the number of
individuals in obesity class will be stable with respect to time. Otherwise it will
increase the diabetic rate. While comparing, figures (5) and (6) the obesity rate of
figure (5) is higher than the obesity rate of figure(6). So we can understand that
the obesity rate can be managed with control parameters. Which gives that the

167



A dynamical analysis of a mathematical model on type-2 diabetic

Figure 6: Infected by Obesity class for different values of α without control pa-
rameters

obesity rate is controlled with the control parameters.

Figure 7: Infected by Diabetic class for different values of α with control param-
eters

Figure(7) and figure(8) show the flow of individuals in infected by diabetic
class for various values of α with and without control parameters respectively. If
we use the control parameters by giving different values to α, then the number of
individuals in diabetic class will be stable with respect to time. Otherwise it will
reduce the recovery rate. While comparing, figures (7) and (8) the diabetic rate of
figure (7) is higher than the diabetic rate of figure(8). So we can understand that

168



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

Figure 8: Infected by Diabetic class for different values of α without control pa-
rameters

the diabetic rate can be managed with control parameters. Which gives that the
diabetic rate is controlled with the control parameters.

Figure 9: Recovered class for various values of control parameters

Figure(9) shows the flow of individuals in recovered state for the various values
of control parameters. We can observe the following cases:
Case 1 : The recovery rate increases when both the control parameters exist.
Case 2 : The recovery rate is slightly decrease when the control parameter diet
with physical activity is involved.
Casse 3 : The recovery rate decreases from case 2, when the control parameter
medication is involved.
Case 4 : There is no recovery rate, when no control parameters are involved.

169



A dynamical analysis of a mathematical model on type-2 diabetic

Hence, both the control parameters are required to control the obesity and
diabetic rates and to increase the recovery rate.

9 Conclusions
This paper presents a mathematical model for type 2 diabetic from obesity

with controls diet with physical activity and medication. By the Pontryagin’s
maximum principle, the optimal levels of two control variables are attained with
some specific conditions. An optimal control approach is proposed in order to re-
duce the burden of obesity to diabetic. The positive equilibrium points have been
found. The system is locally asymptotically stable with a condition on susceptible
population. The system is globally asymptotically state in the identified paramet-
ric domain. The numerical simulations show the effectiveness of proposed control
strategies. In the numerical simulations, we can observe that the flow of diabetic
state was dropped down, when the two control parameters are involved. We can
also observe that the obesity rate can be managed at moderate levels when the
controls parameters are at optimal levels. Hence the two introduced control pa-
rameters healthy diet with physical activity and medication are effectively reduce
the rates of obesity and diabetic and increase the recovery rate.

References
W. Banzi, I. Kambutse, V. Dusabejambo, E. Rutaganda, F. Minani, J. Niy-

obuhungiro, L. Mpinganzima, and J. M. Ntaganda. Mathematical modelling
of glucose-insulin system and test of abnormalities of type 2 diabetic patients,.
International Journal of Mathematics and Mathematical Sciences, 2021, 2021.

W. Boutayeb, M. E. Lamlili, A. Boutayeb, and M. Derouich. Mathematical mod-
elling and simulation of β-cell mass, insulin and glucose dynamics: Effect of
genetic predisposition to diabetes. Journal of Biomedical Science and Engi-
neering, 2014, 2014.

A. Boutayed, E. Twizell, K.Achouayb, and A.Chetouani. On the solution of
fourth-order rational recursive sequence. Advanced Studies in Contemporary
Mathematics, 20:525– 545, 2010.

K.Abdelfatah, L. Abderrahim, F. Hanane, B.Omar, and R.Mostafa. A new mathe-
matical modeling with optimal control strategy for the dynamics of population
of diabetics and its complications with effect of behavioral factors. Journal of
Applied Mathematics, Hindawi, 2020.

170



G. Gnana Priya, A.Sabarmathi and V.S.V. Naga Soundarya Lakshmi

N. Kumar, N. Puri, F. Marotta, T. Dhewa, S. Calabrò, M. Puniya, and J. Carter.
New insights into the noise reduction wiener filter. audio, speech, and language
processing. Functional Foods in Health and Disease, 7:1 –16, 2017.

M. A. Ortega, O. Fraile-Martı́nez, I. Naya, N. Garcı́a-Honduvilla, M. Álvarez-
Mon, J. Buján, Á. Asúnsolo, and B. de la Torre. Type 2 diabetes mellitus
associated with obesity (diabesity). the central role of gut microbiota and its
translational applications. Nutrients, 12:2749, 2020.

171