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Mixed Effects Analysis of Height Growth in Ethiopian Children Aged 1-12 Years: A Cohort Study

Dereje Danbe Debeko*1 and Ayele Taye Goshu2

1 Department of Statistics, Hawassa University, P.O. Box 05, Hawassa, Ethiopia
2 Department of statistics, Cotebe Teaching University, Addis Ababa, Ethiopia

KEYWORDS:

Height growth;

Mixed effects Models;

Children;

Ethiopia

ABSTRACT

Modelling physical growth is a key component to examine and identify defining
characteristics in the growth process. The goal of this study was to model and capture
known features of height growth in Ethiopian children aged 1–12 years. Height
measurements of 1760 children followed from 1 to 12 years at Young Lives Ethiopia, a
younger cohort, used in the study. The mixed effects method was used to estimate the rate
of change within and between subjects over time and to identify defining covariates. Adult
height and rate of change over time were individual-specific resulting individual-level
growth differences. There was a negative relationship between individual-specific adult
height and rate of change over time. The decelerated rate of change was observed from
childhood to the onset of puberty in both sexes. Boys were taller than girls between the
ages of 3 and 7 years. Mother’s educational status, access to quality drinking water, age,
and sex had a significant effect on height growth. Children who had a decelerated rate of
growth change during the childhood period become taller later in life. Adult height could
be determined by an individual-specific rate of change over time.

INTRODUCTION

Growth is a continuous and dynamic process
influenced by different unknown factors.
Modelling this dynamic process to understand,
estimate, and capture the defining characteristics
such as initial level, rates of change, periods of
acceleration and deceleration when the process
enters and leaves different developmental
phases (Grimm et al., 2011; Howa et al., 2016).
A common practice of child growth is to
measure the increase in body mass, to control
and modify the external conditions that affect

growth gain (Oliveira et al., 2000; Gómez et al.,
2008; Aggrey, 2009).

There have been different modeling approaches
applied to the growth measurement to identify
defining characteristics in growth process. An
example is the construction of the curve-fitting
models that relate age with height and estimate
age at which an individual attain maximum
growing by associating features in various
growth phases (Laird, 1965; Grossman et al.,
1985; Grossman and Koops, 1988; Galeano-
Vasco et al., 2014). Studying growth in one
phase may have an important influence or

East African Journal of Biophysical and Computational Sciences

Journal homepage : https://journals.hu.edu.et/hu-journals/index.php/eajbcs
Hawassa University

College of Natural & Computational Sciences

Year 2021

Volume xx No xx

*Corresponding author:
Email: dalbrayii@gmail.com +251913927596 https://dx.doi.org/10.4314/eajbcs.v5i2.1S

 
Research article



2

association with subsequent phases since
individual growth is monitored as a sentinel
indicator of overall well-being (Tanner, 1981;
Gold stein et al., 2002; Richard et al., 2014).

Many growth modelling approaches are used to
obtain descriptions of change in growth
processes accounting individual specific effects
observed over time, average change, between-
individual differences in change and to identify
determinants factors (Grimm et al., 2011).
However, modelling growth trajectory is
difficult process due to the model parameters
that could not be possible to elaborate from
biological perspectives (Aggrey, 2002; Aggrey,
2009; Galeano-Vasco et al., 2014). Many
scholars have been used cross-sectional data to
model features in growth process. Despite of
cost effectiveness and easy access to data, cross-
sectional centiles for example, only offer a
cross-sectional coverage, and hence the growth
path of an individual monitored longitudinally
in time is unknown since these types of
modelling do not describe the dynamic aspect of
growth process over time (Grajeda et al., 2016).

The mixed effects modeling approaches are
among the commonly used methods to capture
growth process. This modelling approach is
capable of incorporating subject specific rate of
change over time and difference across the
subjects in the linear predictor expression form
(Bates et al., 2015). Mixed effects models have
been applied to the longitudinal data in different
settings (Devidian and Giltinan, 1995; Pan and
Goldstein, 1998; Grimm et al., 2011; Richard et
al., 2014; Chirwaet al., 2014). In this approach,
mixed effects refer to the population mean of
the parameter and random effects that indicate
the differences between the mean value of the
parameter and the adjusted value for each

East  Afr.  J.  Biophys.  Comput.  Sci. (2024), Vol. 5, No. 2, 1-12

subject (Littell  et al., 2000; Wang and  Zuidhof,
2004).  The  mixed  effects  methods  quantify
variability  between  and  within  individuals
letting  a  flexible  covariance  structure  (Pinheiro
and  Bates,  1995;  Aggrey,  2009)  to
accommodate  time  dependent  and  time
independent  covariates  within  individual
residual terms (Pan and Goldstein, 1998). Many
studies have been used the mixed effects models
in  linear  and  nonlinear  approaches  (Pan  and
Goldstein,  1998;  Craig  and  Schinckel,  2001;
Schinckel  et al., 2005; Aggrey, 2009; Grimm  et
al., 2011). However, both modelling  approaches
have  not  been  applied  to  the  longitudinally
collected  data  from  low  income  countries’
settings.   Thus,  the  main   objective   of  the
current  study  was  to  model  height   growth  in
Ethiopian  children   aged   1-12   years   and
to   identify  determinant   factors   using
mixed   effects  modelling   approaches   in
linear  and  nonlinear forms.

MATERIALS AND METHODS

The study design and source of data

Data  on  growth  measurement  gathered  over
time  by  Young  Lives  Ethiopia,  a  younger
cohort,  was  used  in  this  study.  Young  Lives  is
an  international  collaborative  research  project
supported  and  coordinated  by  a  team  based  at
Oxford  University,  UK.  The  cohort  has  been
studying the lives of children in Ethiopia aiming
to  reduce  childhood  poverty.  This  project  built
up  lives  of  3,000  children  living  in  20  sites
across Addis Ababa (the capital) and four other
regions  (Amhara,  Oromia,  Former  Southern
Nations Nationalities and Peoples (SNNPs), and
Tigrai). The Young Lives Ethiopia cohorts  have
been  aimed  to  follow  children  in  two  age
groups:  a  younger  cohort  following  of  2,000



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

3

children who were 0.5 to 1.5 years old and an
older cohort following 1,000 children aged 7.5
to 8.4 years at the baseline (first round) in 2002.
The rest three rounds of surveys were carried
out in 2006, 2009, and 2013, respectively, for
both cohorts. Details of the cohort studies have
been referred via the official website of the
project (www.younglivesethiopia.org). Based on
the inclusion criteria, a total of 1760 children
were included in the study. Height growth
measurements observed from each subject in
four survey rounds were used as outcome
variable (see Figure 1). Age and other socio-
economic, demographic and health related
covariates were included in the study.

Statistical Methods

Two modeling approaches were used in the data
analysis as described in the next section. The
linear mixed effects modeling approach was
used to identify determinant factors associated
with height growth difference between and

within subjects. The nonlinear mixed effects
modeling approach was used to capture growth
trajectories and to estimate relationship between
rate of change over time and maximum (adult)
height.

Linear Mixed Effects Models

Linear mixed models (LMMs) may be
expressed in different but equivalent forms. It is
common to express such a model in hierarchical
form, or just as a mixed model, including
additional random-effect terms and associating
variance and covariance components (John and
Sanford, 2015). When the levels that we
observed represent a random sample from the
set of all possible values, the random effects can
be incorporated in the model (Bates, 2010). This
approach decomposes the outcome of an
observations as fixed effect (population mean)
and random effect (subject specific change over
time), and it account for the correlation structure
of variations among subjects.

Figure 1: Individual height growth measurements plotted on measurement time by sex (male: left
and females right).



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

4

Model description

LMMs used in this study have only two-levels,

across and within individual variations. For ijy

is height measurements of ith subject taken on jth

measurement occasion, where j = 4; i = 1,2,…,n.
The extended form of linear model with random
components is described as follows:

ijqijiqijipijpijij zbzbxxy   ...... 1111

The matrix form of this model is equivalent and
considerably simpler to write as:

,0 ijiiijij bZXY  

where
ijY is the × 1 vector of response

observations in the thi subject at thj

measurement occasion,
ijX is the pni  model

matrix of fixed-effect regressors,  is the 1p

vector of fixed-effect coefficients which is

invariant across groups,
iZ is the qni  matrix of

regressors for the random effects of
observations in subject i ,

ib is the q × 1 vector

of random effects for group i , potentially
different in different groups and ij is the × 1

vector of errors for thj measurement in thi

subject.

Model assumptions

The linear mixed effects analysis was performed
with some assumptions: random effects are
different across the subject and normally
distributed with mean zero and variance co-

variance structure, covariates are uncorrelated
with each other (no multicollinearity), time
variant covariates are a subset of the time
invariant covariates, error terms are assumed to
have a multivariate normal distribution and
within subject measurement errors are auto
correlated.

Under certain conditions, physical growth does
not follow linear pattern over time. Modeling
growth spurts using linear form of models could
lead us into wrong conclusion and may have
weak prediction power. Thus, the nonlinear
models are alternative ways of modeling growth
spurts. This modelling approach uses mixed or
fixed effects form based on the objectives of
underlying study.

Non-Linear Mixed Effects Models

Nonlinear mixed effects models refer to the
population mean of the parameter and random
effects that indicates the differences between the
mean value of the parameter and the adjusted
value for each individual growth over time
(Wang and Zuidhof, 2004).The nonlinear mixed
effects growth curve models used in this study
are Logistic and Gompertz which are most
commonly used growth curve models due to
their mathematical tractability with biologically
meaningful parameters.

Logistic Model (Nelder, 1961)

ij
ijii

i
ij tbb

b
y 










))(exp()(1 3322

11



5

Gompertz Model (Winsor, 1932)

ijijiiiij tbbby   ))(exp()exp(()( 332211

where, ijy is height of thi subject at thj

measurement occasion, ijt is age of thi subject

at thj measurement time, 1 is asymptotic or

maximum height (adult height), 2 is scaling

parameter and 3 is growth rate (between

subjects); ib1 is random effects for 1 , ib2 is

random effects for i2 , ib3 is random effect for

i3 and ij is error term.

kik Nb ,0(~ ), ),0(~ 2 Nij for 3,2,1k ,























333231

232221

131211

bbb

bbb

bbb

k

Since our sample data is repeated measurements
taken on the same subject, the expression for the
within-subject variance-covariance matrix can
be formed in the following way:

inii IR  2
, inI = An identity matrix ( ii nn  ),

i = Correlation structures and 2 = Residual

variance of the model.

East  Afr.  J.  Biophys.  Comput.  Sci. (2024), Vol. 5, No. 2, 1-12

Methods of parameter estimation

Maximum  likelihood  estimation  method  was
implemented  to  estimate  underlying  model
parameters  using  R-package  “lme4”  and
“nlme”.  Statistical  test  was  done  at  5%  level  of
significance.  Further  details  of  mixed  effects
model  parameter  estimation  has  been  described
by  Lindstrom  and  Bates  (1990),  Lindstrom  and
Bates  (1995),  Davidian and Giltinan  (1995) and,
Sedigheh and Debasis (2012).

Model adequacy checking

The  model  assumptions  were  checked  using
residual  plots  versus  fitted  values,  QQ  and  P-P
plots.  The  goodness  of  fit  was  tested  based  on
the  Bayesian  Information  Criteria  (BIC)  and
Akaikie  Information  Criteria  (AIC). However,
some  scholars  argue  that  several  measures  of
model  fit, even  the  likelihood  ratio  chi-square,
that  often  appear  to  reflect  relatively  poor  fit  –
when a model fits data very closely  –  that is, if
residual  variances  are  quite  small  (Browne  et
al.,  2002).

RESULTS

Descriptive statistics

Descriptive  statistics  of  height  measurements
taken  on  each  survey  visit  presented  below  in
Table  1.  More  height  growth  variation  was
observed  in  girls  than  in  boys  after  period  I.
Growth  variation  became  higher  in  the  fourth
period for both sexes



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

6

Table 1: Summary statistics of height in each measurement periods by sex.

Measurement

Period in years

Female Male

N Min Max Mean Std. D N Min Max Mean Std. D

Period I (age 1) 841 56.00 89.50 70.425 5.498 947 55.30 89.50 71.45 5.227

Period II (age 5) 843 86.50 124.00 103.563 5.512 947 80.90 124.90 103.95 5.291

Period III (age 8) 843 99.10 157.00 120.628 6.469 947 102.0 146.00 120.66 6.129

Period IV (age 12) 843 118.2 178.00 142.205 7.880 947 120.0 161.50 139.80 6.628

Figure 2: Smooth line curve of height growth over measurement time by sex

Boys and girls had almost the same height
between 12-84 months (1-7 years). After 84
months girls become taller than boys (Figure 2).
However, the smooth curve plot in Figure 2
doesn’t show where the rate of change was
accelerated or decelerated. In order to identify
where rapid and slow rate of change lies we
have used height growth velocity curves
presented in Figure 3.

Height growth Velocity

Height growth velocity was calculated to
investigate time of accelerated and decelerated
growth periods. To assess these curvatures, the

following velocity formula used (Grajeda et al.,
2016):

)( ijtheight = height change for thi individual at
thj measurement period.

Between 60 to 84 months, boys and girls had
shown very similar rate of growth change. After
84 months (7 years) rate of change in girls
became accelerated than boys. The growth
difference in this time could be due to various
factors. Thus, we have used linear mixed effects
models to identify factors associated with rate of
change for both sexes. On the other hand,

,

)()(
)(

)1(

)1()1(

where

tt

theighttheight
theight

jiij

jiji
ij












East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

7

nonlinear mixed effects models were used to
estimate height at maximum growth and rate of

change over time within and between subjects.

Figure 3:  Smooth line and growth velocity curves by sex

Linear fixed and mixed effects models

Table 2 presents the goodness of fit test for the
linear models. Linear mixed effects model better
fit the data than fixed effects model.

Table 2: Assessing the goodness of fit of
linear models
Fit statistics Linear fixed Linear mixed

AIC 44485.57 43010.67
Log.Lik -24546.60 -21446.34

The estimated values of the mixed effects model
are presented in Table 3. Within an individual
growth variation over time was around 0.03 cm.
There was a positive relationship between
within-individual rate of change (slope) and
height at the baseline (random intercept). This
points that a child who

was taller at the baseline shown faster growth
over time (r = 0.728). There was a positive
correlation between two consecutive
measurements taken on the same subjects at
different measurement occasions (AR (1): =
0.2) (Table 3).

Table 3: Estimated values of random effects
model

Random
Components

Estimates Serial
correlation
Continuous
AR(1)

St. Dev. Correlation

(Intercept) 1.893 (Intr) = 0.2
Age in
months

0.0299 0.728

Residual = 4.664

Access to quality drinking water, mother’s
educational status and sex had a significant
effect on height growth. Children who had
access to quality drinking water were 3.8 cm



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

8

taller than children who had no access to quality drinking water (Table 4).

Table 4: Estimated values of different covariates on height growth of children aged 1-12 years
based on the linear mixed effects model
Covariates Categories β Std.Error DF t-value p-value
(Intercept) - 66.97115 2.618180 5067 25.57927 0.0000
Age - 0.49914 0.007732 5067 64.55163 0.0000
Sex Male (ref.)

Female -1.30927 1.348283 1698 -0.97106 0.3317
Birth order - -0.01189 0.042960 1698 -0.27685 0.7819
BCG status Yes (ref.)

No -0.21590 0.244046 1698 -0.88469 0.3765
Had quality drinking
water:

No (ref.)
Yes -3.86616 0.748315 5067 -5.16649 0.0000

Household size - 0.24943 0.132781 5067 1.87850 0.0604
Had ANC visit No (ref.)

Yes 0.20970 0.225077 5067 0.93170 0.3515
Breast feeding duration Never fed (ref.)

fed for 1-3 months 0.48811 2.516145 1698 0.19399 0.8462
fed for 4-6 months 2.07767 2.539510 1698 0.81814 0.4134
fed for > 6 months 1.48964 2.419669 1698 0.61564 0.5382

Mother’s age at birth - -0.00786 0.019595 1698 -0.40107 0.6884
Father’s education: No education ref.

Elementary (1-8) 1.70812 0.210840 5067 8.10153 0.0000
0ther -1.52816 3.234500 5067 -0.47246 0.6366

Mother’s education: No education ref.
Elementary (1-8) 0.83562 0.233946 5067 3.57187 0.0004
>= High school 1.93923 0.290056 5067 6.68570 0.0000

Region Addis Ababa ref.
Amhara -1.60553 1.437172 5067 -1.11714 0.2640
Oromia -4.24911 1.382365 5067 -3.07380 0.0021
SNNP -1.19146 1.384214 5067 -0.86074 0.3894
Tigrai -2.45187 1.403808 5067 -1.74658 0.0808

Area of residence Urban ref.
Rural -0.72144 0.436144 5067 -1.65413 0.0982

Interaction effects
Age*sex Male (ref.)

Female 0.02708 0.007462 5067 3.62902 0.0003
Quality drinking
water*region

A.A (No ref.)
Amhara 2.17289 0.919798 5067 2.36236 0.0182
Oromia 6.04705 0.928457 5067 6.51301 0.0000
SNNP 2.99661 0.911643 5067 3.28704 0.0010
Tigrai 2.14791 0.907350 5067 2.36724 0.0180

Age*Had quality
drinking water

(No ref.)
Yes 0.03497 0.008313 5067 4.20725 0.0000



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

9

Mother’s educational status also plays a
significant role on height growth of children.
Children whose mother had elementary (1-8)
and high school plus educational status were
0.84cm and 1.9 cm taller than children whose
mother had no formal education, respectively.
As age increased by a month, girls’ height
increased by 0.27 cm compared to boys holding
other covariates constant in the model. The
interaction effect between age and sex was
higher in girls compared to boys. Height in girls
increased by 0.035 cm than boys as age
increased by one month. On the other hand,

access to quality drinking water had a
significant effect on height growth of children
(Table 4). Children those who had access to
quality drinking was had 0.035 cm increased
height compared to children those who had no
access to quality drinking water.

Nonlinear Models

After identifying the best model fit to the data,
the average rate of change over time and
maximum (adult height) estimated height for
both sexes separately.

Table 5: Goodness of fit test for nonlinear mixed effects models by sex
Models Fit statistics Fixed effects models Random effects

Male Female Male Female

Gompertz AIC 23759.5 21781.9 4243.28 2902.96
BIC 23799.2 21798.4 4280.23 2956.96

Logistic AIC 23784.4 21772.6 4216.93 2921.48

Table 6: Estimated values and fit statistics based on nonlinear mixed effects growth curve models
by sex

Female
Nonlinear

mixed
models

Fixed estimate Random components Fit  statistics
paramete

r
Estimated Std.err Ó Ó Ó AIC BIC

Gompertz b1 179.44 2.398 -0.889 10.582 0.049 0.374 0.718 2901.48 2934.96
b2 1.054 0.0108
b3 0.0105 0.0003

Logistic
b1 166.703 1.716 -0.802 9.750 0.103 0.758 0.752 2922.96 2956.44
b2 1.623 0.0215
b3 0.0156 0.0003
b3 0.0087 0.0003

Male
Gompertz b1 171.652 1.3951 -0.879 7.9496 0.039 0.227 0.717 4216.93 4253.88

b2 0.98945 0.0066
b3 0.0111 0.0002

Logistic
b1 161.340 1.034 -0.802 7.2219 0.047 0.340 0.995 4243.28 4280.23
b2 1.4912 0.0125
b3 0.01618 0.0002



10

Model comparisons

Incorporating an individual-specific rate of
change over time in the nonlinear models had
dramatically reduced the estimation error and
increased the fitting performance of the model.
For instance, when adult height, rate of change
and scaling (point of growth change) allow to
varying across individuals, the fitting
performance of the models had improved (Table
6).

Random effects parameters were selected based
on their capability to map with theoretical and
physical meanings. When adult height and rate
of change vary across individual over time,
Logistic models better fitted the data for both
sexes. The mean adult height was estimated to
be 166.7 cm and 171.6 cm in girls and boys,
respectively. Adult height and rate of change
had inverse relationship for both sexes (for girls
r13, = -0.802; for boys r13 = -0.879).

DISCUSSION

This study was aimed to model individual
specific growth spurt over time in children aged
1-12 years old. The mixed effects models were
applied to the height growth measurements to
interpolate growth spurt within and between
subjects over time. The effect size of different
covariates on height growth was estimated using
the linear mixed effects models. Decelerated
and accelerated growth periods were identified
using growth velocity curves.

The fitting performance of the models increased
when adult height and rate of change were
allowed to as individual specific. Random
effects were partitioned into between (σ) and

East  Afr.  J.  Biophys.  Comput.  Sci. (2024), Vol. 5, No. 2, 1-12

within  (Σ)  subject  variations.  The  consistent
result  by  other  study  (Spyrides  et  al.,  2008)
reported  that  considering  random  effects  in  the
proposed  models  increase  the  precision  of
estimated  parameters  since  parameters  vary
from  individual  to  individual  that  might  be
demanded  using  margin  of  errors  in  the  fixed
effects  model  set-up.  However,  most  of  the
mixed effects models applied to human physical
growth  data  had  no  clear  identification  towards
random and fixed effects parameters that should
be  included  in  the  models.  Thus,  one  of  the
objects  of  this  study  was  to  identify  model  that
best fits height growth in the defined time points
when individual specific effects were taken in to
account.  In  addition,  this  study  was  concerned
to  identify  mixed  effects  parameters  that  best
map  on  theoretical  basis  having  biologically
meaningful  interpretations.  Of  course,  here  the
study  was  concerned  to  use  some  notations
suggested by Browne and co-authors (Browne  et
al.,  2002)  to  identify  those  parameters  and  to
select best fit to the data. The authors discussed
their  arguments  that  many  measures  of  model
fit, even  the  likelihood  ratio  chi-square, and
often  appear  to  reflect  relatively  poor  fit:  This
suggest  a  poor  fit  despite  the  model  closely
fitting  the  data.   Therefore,  this   study  had
carefully  examined  and  identified  that  adult
height  and  rate  of  change  over  time  were
individual  specific  and  had  random  effect  on
height growth.

The  present  study  found  that  rapid  growth  was
observed  in  children  between  1  to  3  years
followed  by  decelerated  rate  of  change.  Other
consistent  study  reported  that  children  of  both
sexes grow at approximately the same rate until
the adolescent growth spurt (Rogol  et al., 2000).
Adult  height  and  rate  of  change  had  inverse
relationship  for  both  sexes.  The  consistent



East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12

11

findings reported that the adolescents with a
later APHV tended to have a higher height later
in life (Chen et al., 2022). The interaction effect
of age on rate of change in growth was higher in
girls compared to boys. This may be due to girls
experience adolescence earlier than boys.

CONCLUSIONS

The logistic mixed effects models best fit and
captured height growth pattern for both sexes.
Boys were taller than females until onset of
puberty. The most decelerated rate of growth
was observed in early childhood period for both
sexes. Mother’s educational status, access to
quality drinking water, age and sex could be one
of the determinant factors of height growth in
children aged 1- 12 years. Rate of change at the
base line had no a significant effect on adult
height. Child whose growth was accelerated
during childhood period attains adult height
later in life.

Competing interest

Authors have no conflict of interest

Acknowledgement

We, the authors, would like to thank Oxford
University and Young Live Ethiopia for the data
access.

References

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Aggrey S. E. 2009. Logiatic Nonlinear Mixed Effects
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.


	INTRODUCTION
	MATHEMATICAL MODEL FORMULATION
	The modified mathematical model
	Model formulation

	QUALITATIVE ANALYSIS OF THE MODIFIED MODEL
	Well-posedness
	Steady state
	Basic reproduction number
	Local stability of disease free equilibrium
	Global stability of disease free equilibrium
	Endemic equilibrium point
	Local stability of endemic equilibrium
	Bifurcation analysis
	Sensitivity analysis

	NUMERICAL SIMULATIONS AND DISCUSSION
	EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL
	Optimal protection and hospitalization using modified model
	Existence of an optimal control
	The Hamiltonian and optimality system

	Numerical simulations of optimal control problem
	Optimal control comparisons and strategies

	CONCLUSION
	INTRODUCTION
	PRELIMINARIES
	Intuitionistic Fuzzy Set
	Operations over Intuitionistic Fuzzy Sets

	Multi-objective programming problem

	MATHEMATICAL FORMULATION OF PROBLEM
	Intuitionistic Fuzzy Multi-objective Linear Decision-Making Problem
	Intuitionistic Fuzzy Goal Model of IF-MOLDMP

	PROPOSED SOLUTION METHOD
	 Extended Yager-membership function in the IFDE
	Develop a new intuitionistic fuzzy (IF) aggregation operator
	The intuitionistic fuzzy goal programming method
	Interactive Penalty Function Method (IPFM)

	ALGORITHM FOR PENALIZED IFGP METHOD
	NUMERICAL EXAMPLE
	RESULTS AND DISCUSSION
	CONCLUSION
	INTRODUCTION
	MATHEMATICAL MODEL FORMULATION
	The modified mathematical model
	Model formulation

	QUALITATIVE ANALYSIS OF THE MODIFIED MODEL
	Well-posedness
	Steady state
	Basic reproduction number
	Local stability of disease free equilibrium
	Global stability of disease free equilibrium
	Endemic equilibrium point
	Local stability of endemic equilibrium
	Bifurcation analysis
	Sensitivity analysis

	NUMERICAL SIMULATIONS AND DISCUSSION
	EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL
	Optimal protection and hospitalization using modified model
	Existence of an optimal control
	The Hamiltonian and optimality system

	Numerical simulations of optimal control problem
	Optimal control comparisons and strategies

	CONCLUSION

