International Journal of Modern Management Sciences, 2013, 2(2): 57-66
International Journal of Modern Management Sciences
ISSN: 2168-5479
Florida, USA
Journal homepage:www.ModernScientificPress.com/Journals/IJMGMTS.aspx
Article
Mathematical Model of Ghana’s Population Growth
T. Ofori1, *, L. Ephraim2, and F. Nyarko3
1
Institute of Petroleum Engineering, Heriot-Watt University, (HWU) Edinburgh, UK
2
University of Mines and Technology, Faculty of Engineering, Tarkwa
3
University of Mines and Technology, Academic and Students Affairs Office, Tarkwa
* To whom correspondence should be addressed; E-Mail:[email protected]
Article history: Received 18 March 2013, Received in revised form 19 April 2013, Accepted 23 April
2013, Published 25 April 2013.
Abstract: The purpose of this paper is to use mathematical models to predict the
population growth of Ghana. Ghana is a small country located in West Africa. It borders
Burkina Faso, Ivory Cote and Togo and the Gulf of Guinea. The Exponential and the
Logistic growth models were applied to model the population growth of Ghana using data
from 1960 to 2011. The Exponential model predicted a growth rate of 3.15% per annum
and also predicted the population to be114.8207 in 2050. We determined the carrying
capacity and the vital coefficients
and
are
and
, respectively.
Thus the population growth of Ghana according to the logistic model is
and
predicted Ghana’s population to be 341.2443 in 2050. The MAPE of was computed as
16.31% for the Exponential model and 95.21 for the Logistic model.
Keywords: Exponential growth model, Logistic growth Model, Population growth, MAPE,
Carrying Capacity, Vital Coefficient.
1. Introduction
Projection of any country’s population plays a significant role in the planning as well as in the
decision making for the socio-economic and demographic development. Today the major issue of the
world is the tremendous growth of the population especially in the developing countries like Ghana.
A mathematical model is a set of formulas or equations based on quantitative description or
real world phenomenon and created in the hope that the behavior it predicts will resemble the real
behavior on which it is based (Glenn Ledder, 2005). It involves the following processes.
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
58
(1) The formulation of a real-world problem in mathematical terms: thus the construction of
mathematical model.
(2) The analysis or solution of the resulting mathematical problem.
(3) The interpretation of the mathematical results in the context of the original situation.
A model can be in many shapes, sizes and styles. It is important to emphasize that a model is
not real-world but merely a human construct to help us better understand real-world system. One
uses models in all aspect of our life, in order to extract the important trend from complex processes
to permit comparison among systems to facilitate analysis of causes of processes acting on the
system and to make a prediction about the future. In this paper we model the population growth of
Ghana using the Exponential and the Logistic growth models.
2. Materials and Methods
A research is best understood as a process of arriving at dependent solutions to the problems
through the systematic collection, analysis and interpretation of data. In this paper, secondary
population data was taken from World Development Indicator and Global Development Finance –
Google Public Data Explorer (www.google.com.gh/publicdata/explore). The Exponential and Logistic
growth mathematical models were used to compute the projected population values employing Maple.
The Goodness of fit of the models is assessed using the Mean Absolute Percentage Error (MAPE).
3. The Exponential Growth Model
In 1798 Thomas R. Malthus proposed a mathematical model of population growth. He
proposed by the assumption that the population grows at a rate proportional to the size of the
population. This is a reasonable assumption for a population of a bacteria or animal under ideal
conditions (unlimited environment, adequate nutrition, absence of predators, and immunity from
disease). Suppose we know the population P0 at some given time
projecting the population P, at some future time
function ( )
satisfying ( )
, and we are interested in
, In other words we want to find a population
.
Then considering the initial value problem
( )
( )
Integrating by variable separable in (1)
∫
∫
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(1)
Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
( )
or
( )
{ (
59
)}
(2)
where k is a constant called the Malthus factor, is the multiple that determines the growth rate.
Equation (1) is the Exponential growth model with (2) as its solution. It is a differential equation
because it contains an unknown function
and it derivative
⁄ . Having formulated the model, we
now look at its consequences. If we rule out a population of 0, then ( )
equation shows that
⁄
for all
So if
then
for all . This means that the population is always increasing. In fact,
as ( ) increases, equation (1) shows that
⁄
becomes larger. In order words, the growth rate
increases as the population increases. Equation (1) is appropriate for modeling population growth
under ideal conditions, thus we have to recognize that a more realistic must reflect the fact a given
environment has a limited resources.
4. The Logistic Growth Model
This model was proposed by the Belgianmathematical biologist Verhulst in the 1840s as model
for world population growth. His model incorporated the idea of carrying capacity. Thus the
population growth not only on how to depends on the population size but also on how far this size is
from the its upper limit i.e. (maximum supportable population. He modified Malthus’s Model to make
a population size proportional to both the previous population and a new term
( )
where
(3)
and
are the vital coefficients of the population. This term depicts how far the population is
from its maximum limit. Now as the population value gets closer to , this new term will become very
small and tend to zero, providing the right feedback to limit the population growth. Thus the second
term models the competition for available resources, which tends to limit the population growth. So the
modified equation using this new term is:
( )(
( ))
(4)
( )
This equation is known as the Logistic Law of population growth. Solving (4) applying the initial
conditions, the (4) become
(5)
By the application of separation of variables and integrating, we obtain ∫ (
(
(
))
)
∫
(6)
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
At
60
and
(
(
))
Substituting c into (6) and solving for P yields
(7)
(
)
Now taking the limit as
of (7)
(
Putting
and
)
(8)
the values of
are
and
respectively, then we obtain from (7) the
following.
(
)
(
(9)
)
(10)
Dividing (10) by (9) we have
(11)
Hence solving for
(
)
(
)
Substituting
we have
(12)
into the first equation (9) we obtain
(
Therefore the limiting value of
(
(13)
)
is giving by
)
(14)
5. Mean Absolute Percentage Error (MAPE)
It is an evaluation statistic which is used to assess the goodness of fit of different models in
national and sub national population projections. This statistic is expressed in percentage. The concept
of mean absolute percentage error (MAPE) seems to be the very simple but of great importance in the
selecting a parsimonious model than the other statistics. A model with smaller MAPE is preferred to
the others models.
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
61
The mathematical form of MAPE is given under
∑
where
̌ and
̌
(15)
are the actual, fitted and number of observation of the (dependent variable)
population respectively.
Lower MAPE values are better because they indicate that smaller percentages errors are
produced by the forecasting model. The following interpretation of MAPE values was suggested by
Lewis (1982) as follows: Less than 10% is highly accurate forecasting, 10% to 20% is good
forecasting, 21% to 50%is reasonable forecasting and 51% and above is inaccurate forecasting.
6. Results and Discussion
To estimate the future population of Ghana, we need to determine growth rate of Ghana using
the Exponential Growth model in (2). Using the actual population of Ghana in million on table 1 below
with t  0 corresponding to the year 1960, we have
the fact that
. We can solve for the growth rate ,
when
(
)
Hence the general solution
( )
(16)
This suggests that the predicted rate of Ghana population growth is
with the Exponential
growth model. With this we projected the population of Ghana to 2050.
Again, based on table 1, let
Then
correspond to the years 1960, 1961 and 1962 respectively.
also correspond 6.7421, 8.559313 and 10.784734.
Substituting the values of
and
into (14) we get
This is the
predicted carrying capacity of the population of Ghana.
From equation (12), we obtain
(
).
Therefore the value of
population growth is approximately
From
hence
. This also implies that the predicted rate of Ghana
with the Logistic growth model.
and equation (15), we obtained
Substituting the values of
into equation (7) we obtain
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
( )
(
62
(17)
)
As the general solution and we use this to predict population of Ghana to 2050. The predicted
populations of Ghana with both models are presented on the table 1 below.
Table 1. Projection of Ghana’s Population using Exponential and Logistic Growth Models
Year
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
Actual
Population
(in millions)
6.7421
6.9584
7.1769
7.3997
7.6408
7.8078
7.9866
8.1504
8.3107
8.4841
8.6818
8.9113
9.1678
9.4357
9.6922
9.9227
10.1190
10.2907
10.4614
10.6043
10.9227
11.2460
11.6247
12.0397
12.4623
12.8720
13.2619
13.6387
14.0110
14.3926
14.7934
15.2161
Projected Population (in millions)
Exponential Model
6.7421
6.9579
7.1805
7.4103
7.6474
7.8922
8.1447
8.4053
8.6744
8.9519
9.2384
9.5341
9.8392
10.1540
10.4789
10.8143
11.1604
11.5175
11.8861
12.2665
12.6590
13.0641
13.4822
13.9137
14.3589
14.8184
15.2926
15.7820
16.2871
16.8082
17.3462
17.9012
Logistic Model
6.7421
7.0932
7.4624
7.8506
8.2588
8.6878
9.1389
9.6131
10.1115
10.6353
11.1858
11.7643
12.3722
13.0108
13.6317
14.3865
15.1267
15.9042
16.7205
17.5770
18.4776
19.4221
20.4135
21.4538
22.5453
23.9604
24.8914
26.1508
27.4712
28.8550
30.3061
31.8261
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
15.6558
16.1055
16.5549
16.9969
17.4292
17.8553
18.2011
18.7157
19.1655
19.6323
20.1144
20.6109
21.1199
21.6398
22.1706
22.7124
23.2642
23.8244
24.3918
24.9658
63
18.4741
19.0653
19.6754
20.3050
20.9549
21.6255
22.3175
23.0317
23.7687
24.5294
25.3143
26.1244
26.9461
27.8232
28.7136
29.6325
30.5808
31.5594
32.5693
33.6116
34.6872
35.7976
36.9428
38.1251
38.3451
40.6042
41.9036
43.2446
44.2845
46.0566
47.5305
49.0515
50.6213
52.2412
53.9129
55.6382
57.4188
59.2563
61.1525
63.1095
33.1839
35.0860
36.8321
38.6598
40.5725
42.5735
44.6662
46.8543
49.1411
51.5304
54.0259
56.6313
59.3504
62.1869
65.1447
68.2275
71.4392
74.7835
78.2640
81.8845
85.6486
89.5595
93.6207
97.8354
102.2066
106.7370
111.4293
116.2857
121.3084
126.4990
131.8589
137.3893
143.0906
148.9630
155.0066
161.2203
167.6031
174.1531
180.8681
187.7452
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
Mean
Error
Absolute
Percentage
64
65.1291
67.2133
69.3642
71.5839
73.8748
76.2389
78.6786
81.1960
83.7948
86.4764
89.2438
92.0997
95.0460
98.0886
101.2276
104.4670
107.8101
111.2602
114.8207
16.3106%
194.7814
201.9716
209.3122
216.6976
224.4222
232.1795
240.0624
248.0636
256.1749
264.3875
272.6929
281.0809
289.5420
298.0656
306.6415
315.2585
323.9055
332.5712
341.2443
95.2082%
Fig 1 depicts that from 1960 the population of Ghana has increased throughout. This may be
attributed to the improvement in the education, agricultural productively, water and sanitation and
health services. There was a belief in Ghana that the more children one had, one would have a higher
social and economic status, have higher work force in their farms and receive better care in old age.
This coupled with other factors had an overall effect on the increase in population. The exponential
model predicted Ghana’s population to be 114.8207 in 2050 whereas the Logistic model projected it to
be 341.2443. This is presented on figure 2. From equation (14) we calculated the Mean Absolute
Percentage Error (MAPE) of both models. The MAPE for Exponential and the Logistic model are
16.3106% and 95.2083% respectively.
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
65
Fig. 1: Graph of actual population from 1960 to 2011
Fig. 2: Graph of predicted population values
Figure 2 above shows the graph of the predicted population of Ghana with both models. The
Logistic Model is in blue and it deviate far from the actual population. The green line represents the
forecast of the exponential model which is quiet similar to the actual population graph.
7. Conclusion
In conclusion the Exponential Model predicted a growth rate of approximately 3% and
predicted Ghana’s population to be 114.8207 million in the year 2050 with a MAPE of 16.3106%. The
Logistic Model on the other hand predicted a carrying capacity for the population of Ghana to be
. Population growth of any country depends on the vital coefficients. Here we found out that
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Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66
the vital coefficients
and
are
66
and
respectively. Thus the
population growth rate of Ghana according to this model is approximately 5% per annum. It also
predicted the population of Ghana to be 341.2443 million in 2050 with a MAPE of 95.2082%.Based
on Lewis (1982) we can conclude that the Exponential Model gave a good forecasting result as
compared to the Logistic model.
Appendix
Map of Ghana
Fig. 3: Map of Ghana
References
Glen Ledder, (2005), Differential Equations: A modeling Approach. McGraw-Hill Companies Inc.
USA.
Lewis, C.D (1982). International and business forecasting method; A practical guide to exponential
smoothing and curve fitting. Butterworth Scientific, London.
Malthus T.R, (1987). An Essay on the Principle of Population (1st edition, plus excepts 1893 2nd
edition), Introduction by Philip Appeman, and assorted commentary on Malthus edited by
Appleman, Norton Critical Edition, ISBN 0-393-09202-X.
Verhulst P. F., (1838). Noticesur la loique la population poursuitdans son Accroissement,
Correspondance, athematiqueet physique, 10.
World Development Indicators and Global Development Finance (WDIGDF): Google Public Data
Explorer. http://www.google.comgh/publicdata/explore.
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