Age-Structured Population Projection of Bangladesh by Using a Partial Differential Model with Quadratic Polynomial Curve Fitting

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.


Introduction
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.

Age-Structured Population Model
For the prediction of age-structured population, various differential equation models have been formulated by different Mathematicians in different time.In [1], Murray studies the linear age structure population model where he used time-independent death rate.An extensive study of linear and nonlinear age-dependent population dynamics can be found in the works of Webb [2], Gurtin [3] and Iannelli [4].
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, ( )

Numerical Scheme for the Model
The age-structured population model can be written as with initial condition Based on [5] and [6] we discretize the time derivative by a forward difference formula, and the age derivative with a backward difference on a discrete mesh , and the backward difference approximation for u x 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.

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

Curve Fitting
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.

Quadratic Polynomial Models
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 , and a a a to zero, that is, Setting those equations in matrix form gives The above are solved for 0 1 2 , , a a a .

Incorporation of Data into Explicit Upwind Difference Scheme
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 ( ) , u t x is the total population distribution function rather than a density distribution func- tion.
• We have used age distributed population of 2001 as initial data ( ) 0 u x and we have assumed the population of the (0 -4) age-group as the newborns baby (population) and the population of this first age-group of the years of 2001 to 2012 of data (Table 1) [7] is the basis to formulate the birth rate ( ) b t .We have estimated the birth rate by fitting quadratic polynomials.To evaluate birth rate ( ) b t , we have been fitting quadratic polynomials

Total Population Projection
Figure 1 shows our estimated total population projection marked by "Age-Structured Population Model (By Quadratic Polynomial)" for the years 2001 to 2050.Here it is observed that the initial population in our model is 131 million for the year 2001 and the predicted population for the year 2050 is 246.3280 million.In the year 2021, in our model the predicted population will be 170.2044 million whereas the total population was 75 million in 1971.So the total population in 2021 will be 2.26 multiple of the total population of 1971.
Figure 2 shows the comparison of the predicted population of Bangladesh from the years 2001 to 2050 with the projection of Bangladesh Bureau of Statistics (BBS) which is made by Md.Kabir [9], the predicted population by Dutta and Andallah [10] and the predicted population by Md.Minarul Haque, Faruque Ahmed, Sayedul Anam, Md.Rashed Kabir [11].Here, the population of Bangladesh is calculated by three mathematical methods.Dutta and Andallah have used "Linear Equation", Md.Minarul Haque, Faruque Ahmed, Sayedul Anam, Md.   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 Figure 2 one can observe that the inclination of the increasing population of our Age-Structured Population Model is more consistent than the demographical method of BBS and the population model by Linear Equation.From the comparison it is also observed that the prediction by the Quadratic Polynomial Curve Fitting method is much closer to the prediction by Logistic Population Model.

Population Projection for Different Age-Groups
Figure 3 is showing the population projection for different age-groups of Bangladesh from the years 2001 to 2050.It is clearly observed that, the age of the people in the scale (0 -15) will increase rapidly.The middle aged people will also increase but the increasing rate will be partially sloth and the increasing rate of aged people will be very sloth.
Figure 4 shows the comparison of our estimated population per-square kilometer projection marked by "Age-Structured Population Density (By Quadratic Polynomial)" with that of marked by "Bangladesh Bureau of Statistics Density (by Md.Kabir)" for the years 2001 to 2050.Here it is observed that the initial population persquare kilometer is 888 in 2001 in our projection whereas it is 881 according to the projection of BBS.It is also observed that, for the year 2050 our predicted population per-square kilometer is 1669, but it is 1995 according to BBS.In 2021, in our model the predicted population per-square kilometer will be 1153 whereas the predicted population per-square kilometer will be 1271 according to the projection of BBS.
The total predicted population for the years 2001 to 2050 is presented in Table 3 and the predicted population per-square kilometer is presented in Table 4.

Conclusion
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    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 [7] and [8] respectively.The results have showed a "very good agreement" with the age distribution population size up to 2012 given in [7].These have justified the correctness of our implementation of the explicit upwind finite difference scheme for the age-structured population projection in Bangladesh and in other countries as well.We have predicted the total population and population density up to 2050 and these results have showed a very good agreement with the results according to the projection by Logistic Population Model [11] which have been predicted up to 2035.
We have observed that, in the year 2050, our population will be 246.3280 million whereas it is 294.38 million according to Kabir [9] and 283.8058 million according to the projection by Linear Equation [10].In [10], the calculation is based on a linear modeling of birth and death rate and our calculation is based on a quadratic polynomial curve fitting of birth and death rate.We have estimated the birth rate and death rate by a non-linear profile.So, the results are more realistic.This motivates us for further study of the model.
nonnegative function, and a boundary condition at 0 x = (the inflow boundary of the domain) ; must be obtained from the boundary condition (4) at 0 x = and initial value 0 for i U i N + ∈ , obtain from the initial condition (3) at 0 t = .

Figure 1 .
Figure 1.Example of a figure caption (figure caption).

Figure 2 .
Figure 2. Comparison of our population projection with different methods.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

Figure 3 .
Figure 3. Population projection of different age-groups.

Figure 4 .
Figure 4. Comparison of our population density projection with Bangladesh Bureau of Statistics for the years 2001 to 2050.

Table 1 .
Mid-year population distribution in percentage by age-group from 2001 to 2012.For this we have used data of birth rate from Table 1 [7].• We have estimated death rate, which has been parameterized by µ in the age specific population model, by fitting quadratic polynomials.To evaluate death rate ( ) For this we have used data of death rate from Table 2 [8].Using theses age and time dependent death rate, initial value and boundary condition on Explicit Upwind Finite Difference Scheme, we can forecast age specific population distribution.
a u t .

Table 2 .
Death rate by age-group and year (per 1000 population).

Table 3 .
Population projection of Bangladesh from 2001 to 2050 (in millions).

Table 4 .
Population per-square kilometer of Bangladesh from 2001 to 2050.