FIX utama
International Journal of Modern Management Sciences Journal homepage:www.ModernScientificPress.com/Journals/IJMGMTS.aspx
ISSN: 2168-5479
Florida, USA Article
Mathematical Model of Gh
ana’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:adetunde@googlemail.com
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.
(2)
(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 , and we are interested in
projecting the population P, at some future time , In other words we want to find a population function satisfying .
Then considering the initial value problem
(1)
Integrating by variable separable in (1)
∫ ∫
(3)
or
{ } (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 for all So if then equation shows that ⁄ 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
(3)
where 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)
(4)
At and
Substituting c into (6) and solving for P yields
(7)
Now taking the limit as of (7)
(8)
Putting and 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 we have
(12)
Substituting into the first equation (9) we obtain
(13)
Therefore the limiting value of 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.
(5)
The mathematical form of MAPE is given under
∑ ̌ (15)
where ̌ and 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 0corresponding to the year 1960, we have . We can solve for the growth rate , the fact that 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 correspond to the years 1960, 1961 and 1962 respectively. Then 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 hence
.
Therefore the value of . This also implies that the predicted rate of Ghana population growth is approximately with the Logistic growth model.
From and equation (15), we obtained Substituting the values of into equation (7) we obtain
(6)
(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 Actual
Population (in millions)
Projected Population (in millions) Exponential Model Logistic Model
1960 6.7421 6.7421 6.7421
1961 6.9584 6.9579 7.0932
1962 7.1769 7.1805 7.4624
1963 7.3997 7.4103 7.8506
1964 7.6408 7.6474 8.2588
1965 7.8078 7.8922 8.6878
1966 7.9866 8.1447 9.1389
1967 8.1504 8.4053 9.6131
1968 8.3107 8.6744 10.1115
1969 8.4841 8.9519 10.6353
1970 8.6818 9.2384 11.1858
1971 8.9113 9.5341 11.7643
1972 9.1678 9.8392 12.3722
1973 9.4357 10.1540 13.0108
1974 9.6922 10.4789 13.6317
1975 9.9227 10.8143 14.3865
1976 10.1190 11.1604 15.1267
1977 10.2907 11.5175 15.9042
1978 10.4614 11.8861 16.7205
1979 10.6043 12.2665 17.5770
1980 10.9227 12.6590 18.4776
1981 11.2460 13.0641 19.4221
1982 11.6247 13.4822 20.4135
1983 12.0397 13.9137 21.4538
1984 12.4623 14.3589 22.5453
1985 12.8720 14.8184 23.9604
1986 13.2619 15.2926 24.8914
1987 13.6387 15.7820 26.1508
1988 14.0110 16.2871 27.4712
1989 14.3926 16.8082 28.8550
1990 14.7934 17.3462 30.3061
(7)
1992 15.6558 18.4741 33.1839
1993 16.1055 19.0653 35.0860
1994 16.5549 19.6754 36.8321
1995 16.9969 20.3050 38.6598
1996 17.4292 20.9549 40.5725
1997 17.8553 21.6255 42.5735
1998 18.2011 22.3175 44.6662
1999 18.7157 23.0317 46.8543
2000 19.1655 23.7687 49.1411
2001 19.6323 24.5294 51.5304
2002 20.1144 25.3143 54.0259
2003 20.6109 26.1244 56.6313
2004 21.1199 26.9461 59.3504
2005 21.6398 27.8232 62.1869
2006 22.1706 28.7136 65.1447
2007 22.7124 29.6325 68.2275
2008 23.2642 30.5808 71.4392
2009 23.8244 31.5594 74.7835
2010 24.3918 32.5693 78.2640
2011 24.9658 33.6116 81.8845
2012 34.6872 85.6486
2013 35.7976 89.5595
2014 36.9428 93.6207
2015 38.1251 97.8354
2016 38.3451 102.2066
2017 40.6042 106.7370
2018 41.9036 111.4293
2019 43.2446 116.2857
2020 44.2845 121.3084
2021 46.0566 126.4990
2022 47.5305 131.8589
2023 49.0515 137.3893
2024 50.6213 143.0906
2025 52.2412 148.9630
2026 53.9129 155.0066
2027 55.6382 161.2203
2028 57.4188 167.6031
2029 59.2563 174.1531
2030 61.1525 180.8681
(8)
2032 65.1291 194.7814
2033 67.2133 201.9716
2034 69.3642 209.3122
2035 71.5839 216.6976
2036 73.8748 224.4222
2037 76.2389 232.1795
2038 78.6786 240.0624
2039 81.1960 248.0636
2040 83.7948 256.1749
2041 86.4764 264.3875
2042 89.2438 272.6929
2043 92.0997 281.0809
2044 95.0460 289.5420
2045 98.0886 298.0656
2046 101.2276 306.6415
2047 104.4670 315.2585
2048 107.8101 323.9055
2049 111.2602 332.5712
2050 114.8207 341.2443
Mean Absolute Percentage Error
16.3106% 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.
(9)
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
(10)
the vital coefficients and are 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.
(1)
The mathematical form of MAPE is given under
∑ ̌ (15)
where ̌ and 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 0corresponding to the year 1960, we have . We can solve for the growth rate , the fact that 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 correspond to the years 1960, 1961 and 1962 respectively. Then 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 hence
.
Therefore the value of . This also implies that the predicted rate of Ghana population growth is approximately with the Logistic growth model.
From and equation (15), we obtained Substituting the values of into equation (7) we obtain
(2)
(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 Actual
Population (in millions)
Projected Population (in millions)
Exponential Model Logistic Model
1960 6.7421 6.7421 6.7421
1961 6.9584 6.9579 7.0932
1962 7.1769 7.1805 7.4624
1963 7.3997 7.4103 7.8506
1964 7.6408 7.6474 8.2588
1965 7.8078 7.8922 8.6878
1966 7.9866 8.1447 9.1389
1967 8.1504 8.4053 9.6131
1968 8.3107 8.6744 10.1115
1969 8.4841 8.9519 10.6353
1970 8.6818 9.2384 11.1858
1971 8.9113 9.5341 11.7643
1972 9.1678 9.8392 12.3722
1973 9.4357 10.1540 13.0108
1974 9.6922 10.4789 13.6317
1975 9.9227 10.8143 14.3865
1976 10.1190 11.1604 15.1267
1977 10.2907 11.5175 15.9042
1978 10.4614 11.8861 16.7205
1979 10.6043 12.2665 17.5770
1980 10.9227 12.6590 18.4776
1981 11.2460 13.0641 19.4221
1982 11.6247 13.4822 20.4135
1983 12.0397 13.9137 21.4538
1984 12.4623 14.3589 22.5453
1985 12.8720 14.8184 23.9604
1986 13.2619 15.2926 24.8914
1987 13.6387 15.7820 26.1508
1988 14.0110 16.2871 27.4712
1989 14.3926 16.8082 28.8550
1990 14.7934 17.3462 30.3061
(3)
1992 15.6558 18.4741 33.1839
1993 16.1055 19.0653 35.0860
1994 16.5549 19.6754 36.8321
1995 16.9969 20.3050 38.6598
1996 17.4292 20.9549 40.5725
1997 17.8553 21.6255 42.5735
1998 18.2011 22.3175 44.6662
1999 18.7157 23.0317 46.8543
2000 19.1655 23.7687 49.1411
2001 19.6323 24.5294 51.5304
2002 20.1144 25.3143 54.0259
2003 20.6109 26.1244 56.6313
2004 21.1199 26.9461 59.3504
2005 21.6398 27.8232 62.1869
2006 22.1706 28.7136 65.1447
2007 22.7124 29.6325 68.2275
2008 23.2642 30.5808 71.4392
2009 23.8244 31.5594 74.7835
2010 24.3918 32.5693 78.2640
2011 24.9658 33.6116 81.8845
2012 34.6872 85.6486
2013 35.7976 89.5595
2014 36.9428 93.6207
2015 38.1251 97.8354
2016 38.3451 102.2066
2017 40.6042 106.7370
2018 41.9036 111.4293
2019 43.2446 116.2857
2020 44.2845 121.3084
2021 46.0566 126.4990
2022 47.5305 131.8589
2023 49.0515 137.3893
2024 50.6213 143.0906
2025 52.2412 148.9630
2026 53.9129 155.0066
2027 55.6382 161.2203
2028 57.4188 167.6031
2029 59.2563 174.1531
2030 61.1525 180.8681
(4)
2032 65.1291 194.7814
2033 67.2133 201.9716
2034 69.3642 209.3122
2035 71.5839 216.6976
2036 73.8748 224.4222
2037 76.2389 232.1795
2038 78.6786 240.0624
2039 81.1960 248.0636
2040 83.7948 256.1749
2041 86.4764 264.3875
2042 89.2438 272.6929
2043 92.0997 281.0809
2044 95.0460 289.5420
2045 98.0886 298.0656
2046 101.2276 306.6415
2047 104.4670 315.2585
2048 107.8101 323.9055
2049 111.2602 332.5712
2050 114.8207 341.2443
Mean Absolute Percentage Error
16.3106% 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.
(5)
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
(6)
the vital coefficients and are 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.