In the previous section, we saw the mathematical model involving inequalities. In this section, we will see another example.
Example 4:
•
The situation is:
A population control unit wants to find out the population of a country after 10 years.
•
This situation can be converted into a mathematical problem. For that,
we engage in the process of mathematical modeling. The final result can
be obtained in 4 steps:
Step 1:
First we
study the situation. We will write it in order:
(i) The population changes with time.
For example, population next year will be different from the present population.
(ii) Population increases with births.
(iii) Population decreases with deaths.
(iv) Based on the above three information, we can write:
time, number of births and number of deaths are the parameters.
•
We can write:
Step 1 involves studying the situation and identifying the parameters. Essential parameters should be carefully identified.
Step 2:
This can be written in order:
(i) Let us denote time by the letter ‘t’. It is time in years.
♦ So when t = 0, it indicates the present year 2023.
♦ When t = 1, it indicates the next year 2024
♦ When t = 2, it indicates the year 2025 so on . . .
(ii) Let us denote the population at any time by p(t).
• So when t = 0, the population will be p(0).
♦ It is the population as on first of January 2023
• When t = 1, the population will be p(1).
♦ It is the population as on first of January 2024 so on . . .
(iii) We can find p(1) as follows:
p(1) = population as on first of January 2024
= population as on first of January 2023 [this is p(0)]
+ Number of births in the year 2023
- Number of deaths in the year 2023
(iv) In general, we can write:
p(t+1) = Population as on the first of January of year t [this is p(t)]
+ Number of births in the year t
- Number of deaths in the year t
• Let us denote,
♦ Number of births in the year t as B(t)
♦ Number of deaths in the year t as D(t)
• Now the equation becomes:
p(t+1) = p(t) + B(t) - D(t)
(v) Now we consider the term birth rate.
• It is the number of births per 1000 of the population per year.
• For example, suppose that the birth rate is 7 and p(t) is 1500000.
• Number of “thousands” in p(t) = $\frac{1500000}{1000}$ = 1500.
• There will be 7 new births in each of those thousands.
• So the number of new births in the year t = 1500 × 7 = 10500.
• Let us denote birth rate by the letter ‘b’. Then for our present example, b = $\frac{7}{1000}$.
• We see that, number of births can be easily calculated. All we need to do is, multiply the population by b.
• That means, B(t) = p(t) × b
(vi) Similarly, if d is the death rate, then number of deaths can be calculated as:
D(t) = p(t) × d
(vii) Now the equation that we wrote in (iv) becomes:
p(t+1) = p(t) + b p(t) - d p(t)
⇒ p(t+1) = (1 + b - d) p(t)
• So if we put t = 9, we will get the population in the tenth year. But for that, we need to know the population in the ninth year.
• To know the population of the ninth year, we need to know the population of the eighth year. So on . . .
• This is a lengthy process. We do not have the populations in eighth or ninth years. All we have is p(0), which is the population of the present year.
(viii) Let us try to find an alternate method.
• Consider the equation: p(t+1) = (1 + b - d) p(t)
• Let us put t = 0. Then we get:
p(0+1) = (1 + b - d) p(0)
⇒ p(1) = (1 + b - d) p(0)
• Let us put t = 1. We get:
$\begin{array}{ll}{} &{p(2)} & {~=~} &{(1+b-d) p(1)} &{} \\
{} &{} & {~=~} &{(1+b-d) (1+b-d)p(0)} &{} \\
{} &{} & {~=~} &{(1+b-d)^2 p(0)} &{} \\
\end{array}$
• Let us put t = 2. We get:
$\begin{array}{ll}{} &{p(3)} & {~=~} &{(1+b-d) p(2)} &{} \\
{} &{} & {~=~} &{(1+b-d) (1+b-d)^2p(0)} &{} \\
{} &{} & {~=~} &{(1+b-d)^3 p(0)} &{} \\
\end{array}$
• We see a pattern. We can easily write:
p(t) = (1+b-d)t p(0)
(ix) Consider the term (1+b-d). All items inside the brackets are constants. So the term as a whole will be a constant. We denote it by the letter ‘r’. It is called the growth rate. It is also known as Malthusian parameter, in honor of Robert Malthus who first brought this model to popular attention.
• So the equation in (viii) can be modified as:
p(t) = rt p(0)
(x) Consider the equation p(t) = rt p(0).
• With the derivation of this equation, we have completed step 2.
• We see that population is a function of t. This is because, t is the only variable. r and p(0) are constants.
• p(t) is an exponential function. Any function of the form c rt, where c and r are constants is called an exponential function.
•
We can write:
Step 2 involves drawing the necessary diagrams and writing the relevant mathematical equations/inequalities. In short, we write a mathematical problem
in this step. Any mathematical problem will have a definite solution.
So in this step, it is necessary to recheck all the works done thus far.
Step 3:
In this step, we put the equation into actual use. This can be written in order:
(i) Suppose that, the present population p(0) is 250000000 and the rates b and d are 0.02 and 0.01 respectively.
(ii) We get r = (1+b-d) = (1+0.02-0.01) = 1.01
(iii) So the population in the tenth year will be:
p(10) = 1.0110 × 250000000 = 276,155,531.25
•
We can write:
Step 3 involves the actual application of the result obtained in step 2. Calculators or digital computers can be used for
lengthy problems.
Step 4:
This can be written in order:
(i) First we do the validation.
We have obtained the population in step 3. But it has decimal values. Population cannot be in the decimal form. It must be a whole number. We can say that, the equation derived in step 2 is not a very accurate equation.
• However, we can make an approximation. We can state that, the population in the tenth year will be 276,155,531 approximately.
(ii) Next we do the interpretation.
We state that the equation derived in step 2 is not completely dependable. This is because, we assumed b and d to be constant for all the years. In actual practice, this is not possible. Improvements or decline in health management systems in the society can alter b and d. Also, if there is migration into or out of the society, population will change considerably. To get an accurate mathematical model, we must take such factors also into consideration.
• We can write:
In step 4, we do validation and interpretation.
It is clear that, after validation and interpretation, we may need to go back to step 1 and look for additional factors and parameters that may be playing crucial roles. This is shown in the flow chart in fig.B.3 below:
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| Fig.B.3 |
In the next section, we will see some interesting situations where mathematical modelling can be used effectively.
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