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In this paper, the age-specific population of Bangladesh based on a linear first order (hyperbolic) partial differential equation which is known as Von-Foerster Equation is studied. Applying quadratic polynomial curve fitting, the total population and population density of Bangladesh are projected for the years 2001 to 2050 based on the explicit upwind finite difference scheme for the age-structured population model based on given data (source: BBS & ICDDR, B) for initial value in the year 2001. For each age-group, the future birth rates and death rates are estimated by using quadratic polynomial curve fitting of the data for the years 2001 to 2012. Quadratic polynomial curve fitting is also used for the boundary value as the (0 - 4) age-group population based on the population size of the age-group for the years 2001 to 2012.

The fast growth of population during the past decades has frustrated the development efforts in Bangladesh. In 1971 the population of the country was around 75 million. According to the 5th census of Bangladesh Bureau of Statistics (BBS) in 2011, the total population of Bangladesh is 150 million. The area of Bangladesh is 147,570 square kilometers only and it is one of the most densely populated countries all over the world. To handle such mass population in such short land is a huge problem for the government to take any developing steps. Due to scarcity of resources, it is not possible to provide educational, health, medical, transport and housing facilities to the entire population. A rapidly increasing population plugs the economy into mass unemployment and under employment. As a result, the actual development is just getting being delayed a hampered a lot. With the help of population model, we can predict what the number of population will be by the year of 2050. Therefore, the government should take immediate steps to keep the population under control and the people themselves should adopt family planning for their own benefit. In order to make an efficient planning for the demands of different age-group, it is important to predict the age-structured population of the country. Therefore, in this paper we project the future age-structured population of Bangladesh based on a partial differential equation model. The information we get from age-specified population group can help us in future planning of social and economical development. For example, if we can predict the population of children of 0 - 4 years old, we can provide necessary medical care and baby food to reduce the mortality rate and keep children healthy and nourished. After 50 years of liberation of Bangladesh, the government of Bangladesh is going to celebrate the year 2021 as “apotheosis of liberation”. The government has already declared the year as “vision 2021”. To educate the people of Bangladesh within 2021, we can afford necessary support for children of 4 - 15 years old. We can provide human resource development program for young people to reduce the unemployment problem and also proper health care to elder people.

For the prediction of age-structured population, various differential equation models have been formulated by different Mathematicians in different time. In [

In this paper, we study a linear first order hyperbolic partial differential equation to predict age?dependent population. We project the future age-structured population in Bangladesh based on this linear model of population which is known as Von-Foerster equation, given as follows:

with

where,

The age-structured population model can be written as

with initial condition

Based on [

approximation for

and the backward difference approximation for

We consider uniform grid spacing with step size h and k for space and time respectively,

which is the explicit upwind difference scheme for the age-structured population model.

Here

and initial value

We implement the explicit upwind difference scheme and introduce quadratic polynomial curve fitting procedure for the age-structured population model.

Curve fitting is the process of constructing a curve, or mathematical function that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a “smooth” function is constructed that approximately fits the data.

Given n data points

The residual at each data point is given by

The sum of the square of the residuals is given by

To find the constants of the polynomial regression model, we put the derivatives successively with respect to

Setting those equations in matrix form gives

The above are solved for

To predict the Age Distributed Population we incorporate the initial and boundary data into the Explicit Upwind Scheme with respect to the assumptions and considerations below:

We assume that

・ We have considered the age of people of Bangladesh in between 0 to 85+ years. We divide this age-group (0 to 85 years) into 18 sub-groups, each sub-group contains 5 years interval i.e. 0 - 4, 5 - 9, 10 - 14, ・・・, 80 - 84, 85+.

・ We have used age distributed population of 2001 as initial data

Age | Population in percentage in the years 2001-2012 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | |

0 - 4 | 11.8 | 12.1 | 12.1 | 12.0 | 12.1 | 11.8 | 11.5 | 11.4 | 11.2 | 11.0 | 10.7 | 10.8 |

5 - 9 | 11.4 | 11.2 | 11.2 | 11.2 | 11.1 | 11.3 | 11.6 | 11.7 | 11.7 | 11.6 | 11.4 | 11.2 |

10 - 14 | 12.8 | 12.2 | 11.8 | 11.3 | 11.1 | 10.7 | 10.7 | 10.7 | 10.7 | 10.6 | 10.7 | 11.0 |

15 - 19 | 10.7 | 10.8 | 10.6 | 10.7 | 10.6 | 10.4 | 10.0 | 9.7 | 9.3 | 9.1 | 8.9 | 8.9 |

20 - 24 | 8.6 | 8.2 | 8.2 | 8.3 | 8.0 | 8.1 | 8.2 | 8.0 | 8.1 | 8.2 | 8.0 | 7.7 |

25 - 29 | 6.6 | 6.9 | 6.8 | 6.5 | 6.8 | 7.0 | 6.7 | 6.6 | 6.7 | 6.6 | 6.7 | 6.6 |

30 - 34 | 6.7 | 6.5 | 6.5 | 6.5 | 6.2 | 6.1 | 6.2 | 6.1 | 5.9 | 6.2 | 6.4 | 6.2 |

35 - 39 | 6.7 | 6.7 | 6.5 | 6.5 | 6.5 | 6.4 | 6.2 | 6.1 | 6.1 | 5.9 | 5.8 | 5.9 |

40 - 44 | 5.9 | 6.2 | 6.4 | 6.5 | 6.6 | 6.4 | 6.4 | 6.3 | 6.2 | 6.2 | 6.1 | 5.9 |

45 - 49 | 4.1 | 4.3 | 4.7 | 5.0 | 5.3 | 5.7 | 5.9 | 6.1 | 6.3 | 6.3 | 6.2 | 6.2 |

50 - 54 | 3.3 | 3.4 | 3.5 | 3.6 | 3.8 | 3.9 | 4.1 | 4.5 | 4.8 | 5.1 | 5.5 | 5.8 |

55 - 59 | 3.4 | 3.4 | 3.2 | 3.2 | 3.2 | 3.2 | 3.3 | 3.4 | 3.5 | 3.6 | 3.8 | 4.0 |

60 - 64 | 2.7 | 2.8 | 3.0 | 3.0 | 3.1 | 3.1 | 3.1 | 3.1 | 3.0 | 3.0 | 2.9 | 3.0 |

65 - 69 | 2.4 | 2.3 | 2.3 | 2.4 | 2.3 | 2.3 | 2.4 | 2.6 | 2.6 | 2.7 | 2.7 | 2.7 |

70 - 74 | 1.4 | 1.5 | 1.6 | 1.6 | 1.7 | 1.9 | 1.8 | 1.8 | 1.9 | 1.9 | 1.9 | 1.9 |

75 - 79 | 0.9 | 0.8 | 0.8 | 0.9 | 1.0 | 1.0 | 1.0 | 1.1 | 1.1 | 1.2 | 1.3 | 1.3 |

80 - 84 | 0.4 | 0.4 | 0.4 | 0.4 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.6 | 0.6 | 0.6 |

85+ | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.3 |

(Source: ICDDR, B).

Then we have used

・ We have estimated death rate, which has been parameterized by

For this we have used data of death rate from

Using theses age and time dependent death rate, initial value and boundary condition on Explicit Upwind Finite Difference Scheme, we can forecast the age specific population distribution.

Age | Death rate (per 1000 population) in the years 2001-2012 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | |

0 - 4 | 54.4 | 55.6 | 48.7 | 47.0 | 43.5 | 37.9 | 36.6 | 31.1 | 31.7 | 32.4 | 30.6 | 22.4 |

5 - 9 | 1.3 | 1.1 | 0.8 | 0.9 | 0.9 | 0.7 | 0.9 | 0.8 | 0.5 | 0.5 | 0.5 | 0.3 |

10 - 14 | 0.8 | 0.6 | 0.8 | 0.7 | 0.7 | 0.4 | 0.6 | 0.4 | 0.2 | 0.4 | 0.6 | 0.3 |

15 - 19 | 0.9 | 0.7 | 1.1 | 1.0 | 0.9 | 0.9 | 0.8 | 1.0 | 0.8 | 0.8 | 0.7 | 0.7 |

20 - 24 | 0.8 | 1.1 | 1.1 | 1.1 | 0.9 | 0.9 | 0.5 | 1.1 | 1.0 | 1.0 | 1.1 | 0.9 |

25 - 29 | 1.0 | 1.0 | 1.0 | 0.8 | 0.8 | 0.8 | 0.8 | 1.0 | 0.9 | 0.6 | 1.1 | 1.5 |

30 - 34 | 1.2 | 1.9 | 1.1 | 1.3 | 1.4 | 0.7 | 1.5 | 0.9 | 1.2 | 1.4 | 1.2 | 0.4 |

35 - 39 | 1.9 | 1.3 | 1.9 | 2.2 | 2.2 | 2.0 | 2.1 | 1.8 | 1.5 | 1.4 | 1.2 | 0.7 |

40 - 44 | 3.1 | 3.5 | 2.8 | 3.0 | 2.6 | 2.6 | 1.8 | 2.0 | 2.3 | 2.2 | 1.7 | 1.4 |

45 - 49 | 3.9 | 3.7 | 3.5 | 4.3 | 4.1 | 4.1 | 3.6 | 4.0 | 3.6 | 3.5 | 2.7 | 4.8 |

50 - 54 | 6.0 | 8.0 | 5.5 | 5.7 | 6.0 | 6.9 | 6.7 | 6.2 | 7.7 | 4.3 | 5.1 | 7.0 |

55 - 59 | 9.5 | 9.9 | 11.0 | 8.7 | 9.2 | 7.9 | 10.3 | 8.1 | 10.4 | 10.9 | 9.6 | 10.3 |

60 - 64 | 19.9 | 22.7 | 17.2 | 18.7 | 18.9 | 17.2 | 19.4 | 17.7 | 15.0 | 18.8 | 15.7 | 15.1 |

65 - 69 | 27.2 | 28.4 | 35.5 | 35.5 | 33.7 | 29.4 | 30.2 | 34.5 | 30.2 | 28.3 | 23.2 | 24.5 |

70 - 74 | 51.4 | 49.3 | 43.7 | 53.6 | 54.6 | 46.2 | 56.6 | 53.0 | 47.3 | 46.9 | 41.5 | 47.2 |

75 - 79 | 84.3 | 92.6 | 87.2 | 98.6 | 93.9 | 84.1 | 84.4 | 67.8 | 78.9 | 77.8 | 74.8 | 91.7 |

80 - 84 | 105.7 | 142.5 | 148.5 | 128.6 | 127.2 | 94.1 | 124.4 | 128.9 | 125.0 | 119.2 | 99.8 | 108.2 |

85+ | 166.7 | 161.0 | 204.3 | 196.1 | 194.2 | 162.2 | 190.5 | 222.4 | 165.1 | 169.7 | 166.9 | 176.2 |

(Source: ICDDR, B).

Rashed Kabirhave used “Logistic Population Model” and we have used “Quadratic Polynomial Curve Fitting”. Our curve is going to most nearer to the curve of Logistic Population Model with the increased period of time. Here it is observed that the initial population is 131 million in 2001 in our projection as well as same in the projection by Linear Equation where as it is 130.02 in the projection of BBS and 129.090 million in the projection by Logistic Population Model. It is also observed that, for the year 2050 our predicted population is 246.3280 million, but it is 294.38 million corresponding to BBS and 283.8058 corresponding to the projection by Linear Equation.

In 2035, in our model the predicted population will be 204.2385 million whereas the predicted population will be 235.67 million corresponding to the projection of BBS, 232.5276 million corresponding to the projection by Linear Equation and 205.979 million corresponding to the projection by Logistic Population Model. By

The total predicted population for the years 2001 to 2050 is presented in

We have considered a continuous and deterministic mathematical model known as Von-Foerster model, which is a linear first order partial differential equation used to predict population distribution by age at any time, given the initial distribution and the variation of birth and death rates with age and time. Although this is a linear equation, it is not easy to solve the difficulty of enforcing boundary condition. For this, we have used finite difference

Year | Projection by Quadratic Polynomial Curve Fitting | Projection by BBS | Projection by Linear Equation | Projection by Logistic Population Model |
---|---|---|---|---|

2001 | 131.0000 | 130.02 | 131.0000 | 129.090 |

2002 | 132.6495 | 132.60 | 133.6691 | 130.927 |

2003 | 134.3484 | 135.12 | 136.3539 | 132.784 |

2004 | 136.0868 | 137.54 | 139.0453 | 134.662 |

2005 | 137.8620 | 139.90 | 141.7436 | 136.559 |

2006 | 139.6718 | 142.21 | 144.4503 | 138.478 |

2007 | 141.5142 | 144.75 | 147.1678 | 140.416 |

2008 | 143.3875 | 147.34 | 149.8988 | 142.376 |

2009 | 145.2902 | 150.00 | 152.6461 | 144.355 |

2010 | 147.2211 | 152.73 | 155.4129 | 146.356 |

2011 | 149.1792 | 155.53 | 158.2017 | 148.378 |

2012 | 151.1639 | 158.41 | 161.0150 | 152.484 |

2013 | 153.1747 | 161.37 | 163.8548 | 154.569 |

2014 | 155.2113 | 164.41 | 166.7226 | 156.676 |

2015 | 157.2739 | 167.53 | 169.6193 | 158.804 |

2016 | 159.3624 | 170.73 | 172.5453 | 160.953 |

2017 | 161.4771 | 173.99 | 175.5003 | 163.124 |

2018 | 163.6183 | 177.31 | 178.4838 | 165.316 |

2019 | 165.7864 | 180.68 | 181.4949 | 167.530 |

2020 | 167.9816 | 184.08 | 184.5324 | 169.766 |

2021 | 170.2044 | 187.49 | 187.5950 | 172.024 |

2022 | 172.4549 | 190.91 | 190.6813 | 174.304 |

2023 | 174.7334 | 194.34 | 193.7901 | 176.606 |

2024 | 177.0399 | 197.76 | 196.9201 | 178.931 |

2025 | 179.3745 | 201.18 | 200.0703 | 181.277 |

2026 | 181.7373 | 204.60 | 203.2396 | 183.646 |

2027 | 184.1280 | 208.01 | 206.4274 | 186.037 |

2028 | 186.5467 | 211.42 | 209.6328 | 188.451 |

2029 | 188.9930 | 214.83 | 212.8554 | 190.887 |

2030 | 191.4668 | 218.25 | 216.0947 | 193.345 |

2031 | 193.9678 | 221.69 | 219.3503 | 195.827 |

2032 | 196.4957 | 225.14 | 222.6218 | 198.331 |

2033 | 199.0503 | 228.62 | 225.9087 | 200.857 |

2034 | 201.6313 | 232.13 | 229.2108 | 203.407 |

2035 | 204.2385 | 235.67 | 232.5276 | 205.979 |

2036 | 206.8715 | 239.27 | 235.8587 | |

2037 | 209.5302 | 242.91 | 239.2037 | |

2038 | 212.2142 | 246.59 | 242.5621 | |

2039 | 214.9235 | 250.32 | 245.9336 | |

2040 | 217.6576 | 254.10 | 249.3177 | |

2041 | 220.4165 | 257.93 | 252.7142 | |

2042 | 223.2000 | 261.81 | 256.1225 | |

2043 | 226.0078 | 265.74 | 259.5426 | |

2044 | 228.8397 | 269.71 | 262.9743 | |

2045 | 231.6957 | 273.73 | 266.4174 | |

2046 | 234.5753 | 277.78 | 269.8719 | |

2047 | 237.4786 | 281.88 | 273.3378 | |

2048 | 240.4053 | 286.01 | 276.8153 | |

2049 | 243.3552 | 290.18 | 280.3045 | |

2050 | 246.3280 | 294.38 | 283.8058 |

Year | Our density | Density by BBS |
---|---|---|

2001 | 888 | 881 |

2002 | 899 | 899 |

2003 | 910 | 916 |

2004 | 922 | 932 |

2005 | 934 | 948 |

2006 | 947 | 964 |

2007 | 959 | 981 |

2008 | 972 | 998 |

2009 | 985 | 1016 |

2010 | 998 | 1035 |

2011 | 1011 | 1054 |

2012 | 1024 | 1073 |

2013 | 1038 | 1094 |

2014 | 1052 | 1114 |

2015 | 1066 | 1135 |

2016 | 1080 | 1157 |

2017 | 1094 | 1179 |

2018 | 1109 | 1202 |

2019 | 1123 | 1224 |

2020 | 1138 | 1247 |

2021 | 1153 | 1271 |

2022 | 1169 | 1294 |

2023 | 1184 | 1317 |

2024 | 1200 | 1340 |

2025 | 1216 | 1363 |

2026 | 1232 | 1386 |

2027 | 1248 | 1410 |

2028 | 1264 | 1433 |

2029 | 1281 | 1456 |

2030 | 1298 | 1479 |

2031 | 1314 | 1502 |

2032 | 1332 | 1526 |

2033 | 1349 | 1549 |

2034 | 1366 | 1573 |

2035 | 1384 | 1597 |

2036 | 1402 | 1621 |

2037 | 1420 | 1646 |

2038 | 1438 | 1671 |

2039 | 1456 | 1696 |

2040 | 1475 | 1722 |

2041 | 1494 | 1748 |

2042 | 1513 | 1774 |

2043 | 1532 | 1801 |

2044 | 1551 | 1828 |

2045 | 1570 | 1855 |

2046 | 1590 | 1882 |

2047 | 1609 | 1910 |

2048 | 1629 | 1938 |

2049 | 1649 | 1966 |

2050 | 1669 | 1995 |

method for the numerical solution of the age-structured population model. For the numerical experiment, the year 2001 is used as the initial time. We have predicted the age distribution population up to the year 2050. In this experiment we have provided the data for birth rate and death rate up to 2012 from [

ShirinSultana,MahmudulHasan,Laek SazzadAndallah, (2015) Age-Structured Population Projection of Bangladesh by Using a Partial Differential Model with Quadratic Polynomial Curve Fitting. Open Journal of Applied Sciences,05,542-551. doi: 10.4236/ojapps.2015.59052