Mathematical Model of Ghana s Population Growth

International Journal of Modern Management Sciences, 2013, 2(2): 57-66 International Journal of Modern Management Sciences ISSN: 2168-5479 Florida, US...
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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. Copyright © 2013 by Modern Scientific Press Company, Florida, USA

Int. J. Modern Mgmt. Sci.2013, 2(2): 57-66

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(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. Copyright © 2013 by Modern Scientific Press Company, Florida, USA

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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

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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 Copyright © 2013 by Modern Scientific Press Company, Florida, USA

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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