1. Population Forecasting using Geometric Increase Method
Presented by:
Bhantooa Luvindraj ( )
Boyjoo Manoj ( )
Bundhoo Deepshika ( )
Ramgoolam Oomeshnathsingh ( )
Seburn Indra ( )

2. Population Forecasting
Population Forecasting consists of mathematical models which are used to analyse changes in population numbers.
There are several factors affecting changes in population:
Increase due to births
Decrease due to deaths
Increase/Decrease due to migration
Increase due to annexation
All the above data can be obtained from the census population records. In Mauritius, these information can be obtained at the Central Statistics Office (CSO)

3. …Population Forecasting
The various mathematical methods available are generally classified in two categories: Short term methods and Long term methods
Short term methods (1-10 years)
Arithmetic progression
Geometric progression
Incremental increase method
Decreasing rate of growth
Simple graphical method
Long term methods (10-50 years)
Comparative graphical method
Ratio method
Logistic curve method

4. Why is population forecasting important?
Population forecasting is an integral part of design. It is essential to take into account the population at the end of the design period.
Fundamental to planning (Assumptions and estimates used in determining sewage flow have a permanent effect on planning decisions and outcomes)
Premature and excessive investments in works
System failure and hence increasing customer complaints
Environmental impact
Essential to service provider so as to know the spare capacity of the system
Identification of weak links of system Ability to accept new/unexpected demands

5. When can projections be carried out?
Projections are likely to be carried out for the design of a system. A service provider should have knowledge of current demand/flow and anticipated future projections at all times. Projections should be determined:
Once the needs of the service are already known and the objectives determined
Stakeholder requirements have been identified
Adequate raw data on existing flows/demands is available

6. Geometric Increase Method
The basic model for geometric change in population size is:
P = Po λt
which is based on the hypothesis that rate of change of population is proportional to the population. According to this, method it is assumed that the rate of increase of population growth in a community is proportional to the present population.
Po denotes initial size,
P denotes population at time t
t denotes time (measured in decades)
λ is the ‘finite population multiplier’ which can be interpreted as λ = ℮i for continuous change or λ = 1+ i for discrete (constant) ‘compound interest’ or ‘birth-pulse’ populations.

7. Population: (thousands)
Example Predict the population for the years 2023, 2033, and 2043 from the following census figures of a town using geometric method.
Year
1943
1953
1963
1973
1983
1993
2003
2013
Population: (thousands) 60 65 63 72 79 89 97
120

8. Solution 1. USING DISCRETE METHOD (RATE OF CHANGE IS CONSTANT)
P = Poλt
λ = (1+ i) for discrete change
Therefore P = Po (1+i)t where,
P0 : Initial population size
P: Population size at time t
i: Average percentage increase per decade
t: Number of decades

9. …Solution (using discrete method)
Year
Population: (thousands)
Increment per Decade
Percentage Increment per Decade
1943 60 -
1953 65 +5
(5÷60) x 100 = +8.33
1963 63 -2
(2÷65) x 100 = -3.07
1973 72 +9
(9÷63) x 100 =
1983 79 +7
(7÷72) x 100 = +9.72
1993 89
+10
(10÷79) x 100 =
2003 97 +8
(8÷89) x 100 = +8.98
2013
120
+23
(23÷97) x 100 =
Net values
+60
+74.61
Averages
8.57
10.66

10. …Solution (using discrete method)Solution
Population for 2023 = Population 2013 x (1+i/100) t
= 120 x ( /100), where i = 10.66, t = 1
= 120 x /100 = 132.8
Population for 2033 = Population 2013 x (1+i/100) t
= 120 x ( /100)2, where i = 10.66, t = 2
= 120 x  = 146.95
Population for 2043 = Population 2013 x (1+i/100) t
= 120 x ( /100)3, where  i = 10.66, t = 3
= 120 x 1.355 = 162.60

11. 2. CONTINOUS METHOD (RATE OF CHANGE IS INCREASING)
𝑑𝑃 𝑃 =𝑖 . 𝑑𝑡
𝑑𝑃 𝑃 = 𝑖 . 𝑑𝑡
ln 𝑃=𝑖𝑡+𝑐
(When t = 0, P = Po, therefore c = ln Po)
ln 𝑃=𝑖𝑡+ ln 𝑃 𝑜
ln 𝑃 − ln 𝑃 𝑜 =𝑖𝑡
( ln 𝑃− ln 𝑃 𝑜 = ln 𝑃 ln 𝑃 𝑜 )
ln 𝑃 ln 𝑃 𝑜 =𝑖𝑡
𝑃 𝑃 𝑜 = 𝑒 𝑖𝑡
𝑃= 𝑃 𝑜 𝑒 𝑖𝑡

12. …Solution (using continuous method)
P = Poλt
λ = ℮ i for continuous change
P0 : Initial population size
P: Population size at time t
i: Average percentage increase per decade
t: Number of decades
The average rate of increase ‘i’ is calculated in the same way as for the discrete change.

13. …Solution (using continuous method)
Population for 2023 = Population 2013 x e it
= 120 x e(10.66/100 *1), where i = 10.66, t = 1 =
 Population for 2033 = Population 2013 x eit
=120 x e(10.66/100 *2), where i = 10.66, t = 2
=148.52
Population for 2043 = Population 2013 x e it
=120 x e(10.66/100 *3), where i = 10.66, t = 3 =

14. …solution (comparison of results)
Year
Forecasted Population
Discrete Method
Continuous Method
2023
132.80
133.50
2033
146.95
148.52
2043
162.60
165.22

15. …solution (comparison of results)

16. …solution (comparison of results)

17. …solution (comparison of results)
In the graph, we can conclude that values obtained from the continuous method are higher than those obtained from the discrete method. This is because in the discrete method, the rate of increase of population is constant whereas the continuous method has an increasing rate of increase of population.
In order to calculate the population number for any other specific year within the decade, the same graph can be used.

18. Advantages of Geometric Progression
Geometric extrapolation is desirable for short intervals
Simple method
When forecasting for a new city
Geometric rates are preferable to arithmetic rates for the extrapolation of decreases in population over a series of years

19. Limitations of Geometric Progression
When the geometric rate of increase is high and the period of time is long
If the accuracy of the basic census figures is subject to considerable doubt
Where the death rate is declining while the birth rate remains nearly constant

20. Population Forecasting in the Design of Waste Water system
Quantity of sewage at the end of a design period
= Per capita production of sewage x Forecasted population at the end of the design period
The quantity of wastewater generated per capita is estimated to be 80% of the water consumption per capita.
The water consumption per occupant per day, for different institutions, can be obtained from the table

21. …Population Forecasting in the Design of Waste Water system

22. Conclusion
In the light of the above, we can see that the Geometric Increase Method is a simple realistic population model based on past information. This method tends to give a higher estimate than normal since it behaves exponentially. It more accurately describes the continuous and cumulative nature of population growth. In normal practice, an average of the arithmetic method and geometric method is performed to get a more accurate estimate.

23. THANK YOU FOR YOUR ATTENTION