American Journal of Applied Science and Technology
61
https://theusajournals.com/index.php/ajast
VOLUME
Vol.05 Issue 04 2025
PAGE NO.
61-68
10.37547/ajast/Volume05Issue04-15
Calculation and Evaluation of Tourism Indicators Using
Mathematical Models
Jabbarov Nasriddin
Tashkent University of Economics, professor, Uzbekistan
Yusupova Shakhlo
Joint Belarusian-Uzbek Inter-Sectoral Institute of Applied Technical Qualifications in Tashkent, basic doctoral student, Uzbekistan
Received:
26 February 2025;
Accepted:
22 March 2025;
Published:
25 April 2025
Abstract:
Tourism plays an important role in the economy of many countries, developing infrastructure, creating
new jobs and contributing to budget revenues. In this article, various mathematical models, including the Lotka-
Volterra model, the Cobb-Douglas production function, the multiplier effect model, the Markov chain model, and
dynamic game models, are adapted for tourism resources. For each model, estimated values are provided, and
tourism indicators are calculated.
Keywords:
Mathematical model of tourism, tourist flow, Lotka-Volterra, Markov chain, Cobb-Douglas function,
multiplier effect, dynamic games.
Introduction:
In the Republic of Uzbekistan, various mathematical
models can be employed to analyze the sustainable
development of tourism. These models integrate the
economic, environmental, and social dimensions of
the tourism. Mathematical modeling serves as an
effective tool for analyzing, forecasting, and
managing tourism.
Famous scholars who have studied economic and
mathematical models of tourism: Raffaella
Casagrandi and Sergio Rinaldi
–
Developed
mathematical models to analyze the dynamic
interactions between tourism, environment, and
sustainability. Their models help assess the long-term
impact of tourism activities on natural resources.
John Tribe
–
A leading researcher in tourism
economics. His studies focus on economic growth,
investment, and sustainability aspects of tourism.
Richard W. Butler
–
Famous for the "Tourism Area Life
Cycle"
(TALC)
model,
which
analyzes
the
development stages of tourist destinations. Edward
Inskeep
–
Developed models for tourism planning and
regional development. His work is focused on
sustainable tourism development. Geoffrey Wall
–
Created models for assessing the environmental and
economic
impacts
of
tourism.
He
used
multidimensional evaluation methods. Zokirjon
Khudoykulov and Bakhodir Sattarov
–
Scholars
specializing in mathematical models for tourism in
Uzbekistan. They are known for their research on
modeling and evaluating tourism's impact on the
economy.
METHODS
Data analysis: The model results were analyzed using
statistical software such as Python and MATLAB.
Identified trends and variations were presented in
graphical and tabular formats.
Validation and verification: The obtained results were
compared with previous studies.
The model results
were validated against statistical data to assess
accuracy.
By applying these mathematical models in tourism
sectors, it is possible to contribute to the
development of the following areas of tourism:
Economic and mathematical models are effectively
used to evaluate and manage tourism indicators. The
Cobb-Douglas function helps optimize investment in
tourism infrastructure and labor resource allocation,
American Journal of Applied Science and Technology
62
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
while the Lotka-Volterra model prevents excessive
exploitation of natural resources and supports
sustainable tourism. Markov chains predict tourist
movements, and differential equations analyze
changes in tourism demand over time. The multiplier
model assesses the impact of tourism investments on
the local economy, while the dynamic game model
helps balance short-term profits with long-term
sustainability. To ensure ecological stability, the
Lotka-Volterra model is applied, whereas the socio-
economic model evaluates the effects of tourism on
local income and employment. Additionally,
the multiplier model measures the broader economic
impact of tourism on the state budget. Proper
resource allocation and investment planning
contribute to the effective development of the
tourism sector, improving service quality and
increasing tourist flows.Below are the main
mathematical models, their essence, and their
potential applications (note that the existing models
have been adapted to tourism factors).
RESULTS AND ANALYSIS
Lotka-Volterra Model (for Ecotourism Management)
[1,2]
. The Lotka-Volterra model (originally describing
the relationship between predator and prey
populations, adapted here for tourism resources)
represents the interrelationship between natural
resources and tourism demand for sustainable
development.
Equations:
1
dN
N
rN
aNP
dt
K
dP
bNP
dP
dt
=
−
−
=
−
Here:
N
–
the ecological state of historical
monuments,
P
–
tourist flow,
r
–
the natural regeneration rate of historical
monuments,
K
–
the maximum capacity of historical
monuments,
a
–
the harmful impact of tourists on resources,
b
–
the economic benefit of tourists,
d
–
the rate of decrease in tourist flow.
This model is primarily used in ecotourism
management. This model can be used to develop
ecotourism or manage the preservation of natural
resources in the Kashkadarya region.
0
4000
N
=
,
0
1000
P
=
,
0.05
r
=
,
5000
K
=
,
0.001
a
=
,
0.0005
b
=
,
0.01
d
=
.
We will analyze the model for a 10-year period.
During the project, the following issues will be
addressed:
1.
How will the ecological state
N
of the monuments
and the tourist flow
P
change each year?
2.
When will stability be achieved?
We will perform these calculations using Python. For
these calculations, we will use tools for solving
systems of ODEs (ordinary differential equations),
such as scipy.integrate.odeint.
This graph shows the impact of tourism on natural
resources based on the Lotka-Volterra model:
American Journal of Applied Science and Technology
63
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
🟢
Green line
–
Ecological state of historical
monuments (initially decreases, then stabilizes).
🔵
Blue line
–
Tourist flow (initially increases, then
decreases as resources deplete and stabilizes).
Summary:
✓
Initially, the increase in tourist flow leads to a
decline in resources.
✓
Over time, natural regeneration helps achieve
stability.
✓
If the damage coefficient
a
is reduced, ecological
sustainability is restored more quickly.
This model describes the complex interrelationship
between natural resources and tourist flow in Uzbek
tourism.
Based
on
the
graphical
analysis,
management strategies can be developed to keep
resources in balance.
Cobb-Douglas Production Function (For Tourism and
Economic Development) [3].
The Cobb-Douglas
production function is a mathematical model used in
the analysis of economics and production processes,
which expresses the relationship between the volume
of production (for example, income or output) and
the main resources - capital and labor.
Main applications:
a) production analysis in macroeconomics: to study
the dependence of the volume of production in the
economy on the main factors (capital and labor), b)
determining production efficiency: the contribution
of capital and labor to production is measured by
elasticity coefficients, c) helping politicians and
entrepreneurs in decision-making. For example, to
determine how much production can be increased by
increasing investment in capital or labor.
We want to use the Cobb-Douglas production
function to assess the economic impact of tourism.
This function expresses the dependence of the
economy on three main factors - capital, labor and
technology:
Y
A K
L
=
Where:
Y
- total income in the tourism sector (volume
of output),
A
- technological coefficient (innovations),
K
- investments in tourism infrastructure,
L
- labor resources (number of employees),
,
- elasticity coefficients of capital and labor
(
1
+
).
This model can be used to analyze investments in
tourism infrastructure in cities such as Tashkent,
Samarkand, Bukhara, Kashkadarya.
Example:
In Shahrisabz, total income in the tourism sector
Y depends on investments in infrastructure
K
and
labor resources
L
. Using the Cobb-Douglas
production function, it is necessary to find the total
income in the tourism sector
Y
.
Here:
0.5
A
=
,
200
K
=
(million soums),
100
L
=
(thousand people),
0.6
=
elasticity coefficient of investment.
0.4
=
elasticity coefficient of labor.
We need to find the total revenue of the tourism
sector
Y
.
1. We put values into the function:
6 0.6
0.4
0.5 (200 10 )
100000
79.19
Y
A K
L
=
=
The total revenue of the tourism sector is
79.19
million soums.
This graph illustrates economic growth in the tourism
sector based on the Cobb-Douglas production
function:
American Journal of Applied Science and Technology
64
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
🔴
Red line
–
Total revenue in the tourism sector.
Summary:
✓
As investment in tourism infrastructure
increases, economic revenue grows.
✓
The rate of revenue growth is high initially but
slows down over time (diminishing returns).
✓
Increasing labor force
L
and technological
efficiency
A
boosts overall economic output.
These findings highlight the significance of efficient
resource management in the tourism sector.
Multiplier Effect Model [4].
The multiplier effect
model actually measures the recurring cost of
spending in the economy, when applied to tourism,
this model represents the recirculation of tourist
spending in the local economy:
1
1
M
MPC
=
−
Where:
M
-is the tourism multiplier,
MPC
-is the marginal propensity to consume
(the proportion of the amount spent by tourists
that affects the local economy).
Example:
This model can be used to calculate the impact of
tourism activities in Kashkadarya on the local
economy. For example, if a tourist spends an average
of
$500
and the marginal propensity to consume of
the local population
0.8
MPC
=
, the total impact
can be calculated as:
1
5
1 0.8
M
=
=
−
Therefore,
the
economic
benefit
is:
$500 5
$2500
=
. If one tourist had
$2500
,
then it can be assumed that he will increase the
number of tourists.
This graph demonstrates the economic impact of
tourism based on the Multiplier Effect model:
American Journal of Applied Science and Technology
65
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
🟢
Purple line
–
The total economic impact of tourism.
Summary:
✓
The more tourists spend, the greater the
economic impact.
✓
MPC
(Marginal Propensity to Consume)
determines the size of the multiplier effect.
✓
If
0.6
MPC
=
, then each unit of spending has
a 2.5x economic impact on the local economy.
Markov Chain Model (For Forecasting Tourist Flow)
[5,6].
Markov chains are mathematical models used
to analyze probabilistic transition processes,
representing the probability of moving from one state
to another. In this model, the future depends only on
the current state and not on the past. Markov chains
are used to analyze the relationship between states
that occur over time. Each state transitions to another
state with a certain probability, and these
probabilities are expressed in the form of a matrix.
Markov chains are used in tourism:
1. Forecasting tourist flows:
The Markov model is used to estimate the
probability of a tourist flow from one region to
another. For example, to determine the probability of
a flow of tourists from one region to another.
2. Analyzing tourist behavior:
Using the Markov model, the probability of tourists
choosing different services or changing travel routes
is predicted.
3. Tourism infrastructure management:
It is used to study the tendencies of tourists to
stay in hotels, use transport vehicles or visit tourist
attractions.
Markov chain model:
1
t
t
P
P
T
+
=
Here:
t
P
–
tourist flow (vector) at a given moment in
time,
T
–
transition probability matrix,
1
t
P
+
–
tourist flow (vector) in the next period.
Conditions: The sum of each row must be equal to
1
,
and the transition probabilities must be between
0
and
1
.
This model can be used to forecast tourist flows in
different regions of Uzbekistan.
Example:
Suppose that tourists in Uzbekistan visit the cities
of Tashkent, Samarkand, and Kashkadarya. The
following transition probability matrix can be
obtained:
0.6
0.3
0.1
0.4
0.4
0.2
0.2
0.5
0.3
T
=
Here:
60%
of tourists staying in Tashkent stay in the
city,
30%
move to Samarkand, and
10%
move to
Kashkadarya. Each row represents the probability of
moving from one city to another.
If the initial state is
0
[0.5,0.3,0.2]
P
=
(
50%
of
tourists are in Tashkent,
30%
in Samarkand, and
20%
in Kashkadarya), the next state is:
1
0
P
P
T
=
i.e.
1
[0.46,0.37,0.17]
P
=
American Journal of Applied Science and Technology
66
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
This means that the probabilities in the first period
are:
Probability of being in the first state:
0.46
Probability of being in the second state:
0.37
Probability of being in the third state:
0.17
This graph forecasts the future distribution of tourist
flows using the Markov Chain model:
🔵
Blue line
–
Percentage of tourists in Tashkent.
🔴
Red line
–
Percentage of tourists in Samarkand.
🟢
Green line
–
Percentage of tourists in Kashkadarya.
Summary:
✓
Initially, Tashkent had
50%
, Samarkand
30%
,
and Kashkadarya
20%
of tourists.
✓
Over time, tourist flows stabilize at a steady-state
distribution.
✓
Samarkand retains its share of tourists most
consistently.
✓
The share of tourists visiting Kashkadarya is
expected to increase in the long run.
Dynamic
Game
Model
(For
Sustainable
Management) [7].
The dynamic game model is used
to optimize resource use and ensure sustainable
development in the tourism sector. This model
represents a decision-making process that takes into
account multiple interests. Interdependent parties
(e.g., tourism operators, local authorities, and nature
conservation organizations) develop a strategy for
action.
Application of the dynamic game model:
1. Ecological sustainability: reducing environmental
damage by optimizing the level of resource use. For
example: developing a strategy to limit the use of
natural resources in the region along rivers and water
bodies and their restoration.
2. Economic development: optimizing investments in
infrastructure to increase the economic benefits of
tourism.
3. Social sustainability: improving the living standards
of local populations and optimizing the social impact
of tourism.
The dynamic game model is used to optimize the use
of tourism resources and their conservation:
( )
0
( )
max
( ( ))
( ( ))
T
x t
F x
U x t
C x t
dt
=
−
This model can be used to ensure environmental
sustainability in the regions of the Republic of
Uzbekistan. Kashkadarya is a developing region in the
field of tourism, where the following can be taken
into account according to this model:
Here:
( ( ))
U x t
–
utilitarian functions reflecting the
benefits of tourism,
( ( ))
C x t
–
the function of tourism costs,
T
–
a limited time interval (duration of the
game\tourist season\several years),
( )
x t
–
the level of resource utilization.
( )
F x
- maximum profit.
The goal of the power is maximum profit:
Where:
100 ( )
x t
: Tourist income.
2
50 ( )
x t
: Costs of environmental damage.
American Journal of Applied Science and Technology
67
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
We calculate the following optimal control: We need
to choose
( )
x t
as the maximum, that is, according to
the optimal control condition, we take the derivative
of the function with respect to
( )
x t
and set it equal
to 0.
2
100 ( )
50 ( )
0
d
x t
x t
dt
−
=
100 100 ( )
0
x t
−
=
( ) 1
x t
=
Now we plug the value of
( )
x t
into the utility
function:
0
( )
100 50
50
T
F x
dt
T
=
−
=
Optimal tourist flow rate:
( ) 1
x t
=
Total profit:
( )
50
F x
T
=
over time, profits
increase.
This graph shows the optimal use of tourism
resources and sustainable management based on the
Dynamic Game model:
🟢
Brown line
–
Overall profit (tourism revenue
–
environmental damage).
Summary:
✓
As the number of tourists increases, revenue also
rises, but so does environmental damage.
✓
At the peak point (optimal tourist flow), economic
profit is maximized.
✓
If tourist numbers exceed this optimal level,
environmental damage outweighs the benefits,
leading to economic losses.
✓
A sustainable tourism strategy should maintain
tourist numbers at the optimal level.
This means that the balance between maximum
profit and environmental damage is maintained by
maintaining the tourist flow in Kashkadarya at a
stable level.
The model of dynamic games serves as an important
tool for managing the use of resources, ensuring
environmental stability and increasing economic
efficiency in the tourism sector of Uzbekistan. For
this, long-term strategies will be developed, taking
into account the interests of all parties.
DISCUSSION
The above models can be used together to analyze
the sustainable development of tourism in
Uzbekistan. For example, the Lotka-Volterra model is
used to manage environmental factors, the Cobb-
Douglas model is used for economic analysis, and the
Markov chain model helps predict tourist flows.
These models serve as the basis for the formation of
a short-term and long-term strategy for the
development of tourism in Uzbekistan and effective
resource management. Through strategic planning of
tourism, it is possible to determine the necessary
measures to open new routes, improve the quality of
services and develop regional tourism.
Using mathematical modeling, you can deeply
analyze all aspects of the tourism sector and
implement: Efficient use of tourist resources,
managing and balancing tourist flows, improving
investment
efficiency,
ensuring
social
and
environmental stability, development of strategic
development plans.
American Journal of Applied Science and Technology
68
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
In short, these mathematical models serve the
sustainable
and
effective
development
of
Uzbekistan
’
s tourism sector. To be more specific:
Lotka-Volterra Model: Ecotourism management
–
stability
is
achieved
in
6
–
7
years.
Cobb-Douglas Model: Tourism investments and
revenue
–
efficient resource allocation is essential.
Multiplier Effect Model: Tourist spending increases
economic
impact
by
2.5
times.
Markov Chain Model: Tourist flows across Uzbek
cities
stabilize
over
time.
Dynamic Game Model: Optimal tourist flow ensures a
balance between economic and environmental
factors.
REFERENCES
Lotka, Alfred J., \Elements of Physical Biology\
https://archive.org/details/elementsofphysic017171
mbp
Williams and Wilkins
, 1925, page 125-131.
Volterra, Vito, Fluctuations in the Abundance of a
Species
Considered
Mathematically\
https://www.nature.com/articles/118558a0
Nature
1926, Vol. 118, page 558-560.
Cobb, Charles W., Douglas, Paul H., \A Theory of
Production,\
https://digamo.free.fr/cobbdoug28.pdf
\
American Economic Review
, 1928, Vol. 18, N0. 1,
page 139-165.
Archer, Brian, \Tourism Multipliers: The State of the
Art,\
https://books.google.com/books/about/Tourism_M
ultipliers.html?id=mREKAQAAIAAJ&
Progress in
Tourism, Recreation and Hospitality Management
,
1982, Vol. 3, page 125-145.
Feller, William, \An Introduction to Probability Theory
and
Its
Applications\
\John Wiley and Sons, 1968, Vol.1,
2-Editio, page 302-310
Kemeny, John G., Snell, J. Laurie, \Finite Markov
Chains\ 1976
Nash,
John,
\Non-Cooperative
Games\
https://www.cs.upc.edu/~ia/nash51.pdf
Annals of
Mathematics
, 1951, Vol. 54, N0.2, page 286-295.
Butler, R. W. 1980. The concept of a tourist area cycle
of evolution: implications for management of
resources.
Canadian Geographer
24:5-12.
Wall,
G. 1997.
Is
ecotourism
sustainable?
Environmental Management
21:83-491
Casagrandi R., Rinaldi S.\A Theoretical Approach to
Tourism Sustainability\
Theoretical Approach to Tourism Sustainability
Ecology& Society
, 2002, Vol. 6, No. 1
Ahmedova Q., Yusupova Sh., Mirzoodilov B.\
Mathematical model for ensuring stability of tourism
growth in Kashkadarya region \
“
QarDU xabarlari
”
,
2024(2) 2, page 62-68.
