Lecture 5 - Compartmental Model | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Compartmental Models 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: Compartmental Models 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
Page 2


Compartmental Models 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: Compartmental Models 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
Compartmental Models 
 
 
Table of Contents 
 Chapter: Compartmental Models 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Compartmental Model 
• 4. Balance Law 
• 5. Exponential Decay Model and Radioactivity 
• 6. Lake Pollution Model 
• 7. Seasonal Flow Rate 
• 8. Case Study: Lake Burley Griffin 
• 9. Drug Assimilation Model 
• 10. Exponential Growth Model (Population Growth) 
• 11. Density Dependent Population Growth Model 
• 12. Limited Population Growth Model with Harvesting 
• Exercise 
• Summary 
• References 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Page 3


Compartmental Models 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: Compartmental Models 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
Compartmental Models 
 
 
Table of Contents 
 Chapter: Compartmental Models 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Compartmental Model 
• 4. Balance Law 
• 5. Exponential Decay Model and Radioactivity 
• 6. Lake Pollution Model 
• 7. Seasonal Flow Rate 
• 8. Case Study: Lake Burley Griffin 
• 9. Drug Assimilation Model 
• 10. Exponential Growth Model (Population Growth) 
• 11. Density Dependent Population Growth Model 
• 12. Limited Population Growth Model with Harvesting 
• Exercise 
• Summary 
• References 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Compartmental Models 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of a compartmental model. 
• Concept of compartmental diagram. 
• Balance law and Word equation of the model. 
• Exponential Decay Model: Radioactive Decay 
• Lake Pollution Model. 
• Drug Assimilation Model : Case of single pill and course of pills. 
• Exponential Growth Model. 
• Density Dependent Growth. 
• Limited Growth Harvesting. 
 
2. Introduction: 
Compartmental models  provides a means to formulate models for 
processes which have inputs and/or outputs over time. In this chapter, we 
will study modelling of  radioactive decay processes, pollution levels in 
lake systems and the absorption of drugs into the bloodstream, 
exponential growth model, density dependent growth, limited growth 
harvesting using compartmental techniques. 
3. Compartmental Model: 
Definition: Compartmental Model is a model in which there is a place 
called compartment which has amount of substance in and amount of 
substance out over time. One example of compartmental model is the 
pollution into and out of a lake where lake is the compartment. Another 
example is the amount of carbon-di-oxide in the Earth’s atmosphere. The 
compartment is the atmosphere where the input of CO
2
 occurs through 
many processes such as burning and output of CO
2
 occurs through plant 
respiration. It can be shown in the form of a diagram called 
compartmental diagram which is shown below. 
 Input CO
2  
Output CO
2
 
 
  Fig. 1: Input – output compartmental diagram for CO
2
 
4. Balance Law: 
Statement: The rate of change of quantity of substance is equal to ‘Rate 
in’minus ‘Rate out’ of the compartment.  
Atmosphere 
(Compartment) 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
Page 4


Compartmental Models 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: Compartmental Models 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
Compartmental Models 
 
 
Table of Contents 
 Chapter: Compartmental Models 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Compartmental Model 
• 4. Balance Law 
• 5. Exponential Decay Model and Radioactivity 
• 6. Lake Pollution Model 
• 7. Seasonal Flow Rate 
• 8. Case Study: Lake Burley Griffin 
• 9. Drug Assimilation Model 
• 10. Exponential Growth Model (Population Growth) 
• 11. Density Dependent Population Growth Model 
• 12. Limited Population Growth Model with Harvesting 
• Exercise 
• Summary 
• References 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Compartmental Models 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of a compartmental model. 
• Concept of compartmental diagram. 
• Balance law and Word equation of the model. 
• Exponential Decay Model: Radioactive Decay 
• Lake Pollution Model. 
• Drug Assimilation Model : Case of single pill and course of pills. 
• Exponential Growth Model. 
• Density Dependent Growth. 
• Limited Growth Harvesting. 
 
2. Introduction: 
Compartmental models  provides a means to formulate models for 
processes which have inputs and/or outputs over time. In this chapter, we 
will study modelling of  radioactive decay processes, pollution levels in 
lake systems and the absorption of drugs into the bloodstream, 
exponential growth model, density dependent growth, limited growth 
harvesting using compartmental techniques. 
3. Compartmental Model: 
Definition: Compartmental Model is a model in which there is a place 
called compartment which has amount of substance in and amount of 
substance out over time. One example of compartmental model is the 
pollution into and out of a lake where lake is the compartment. Another 
example is the amount of carbon-di-oxide in the Earth’s atmosphere. The 
compartment is the atmosphere where the input of CO
2
 occurs through 
many processes such as burning and output of CO
2
 occurs through plant 
respiration. It can be shown in the form of a diagram called 
compartmental diagram which is shown below. 
 Input CO
2  
Output CO
2
 
 
  Fig. 1: Input – output compartmental diagram for CO
2
 
4. Balance Law: 
Statement: The rate of change of quantity of substance is equal to ‘Rate 
in’minus ‘Rate out’ of the compartment.  
Atmosphere 
(Compartment) 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
Compartmental Models 
Symbolically, if X(t) is the amount of quantity in the compartment, then 
 
???????? ???????? = ???????????????? ???????? - ???????????????? ???????????? 
4.1. Compartmental Diagram:  
  Rate In  Rate Out 
 
4.2. Word Equation: 
In words, balance law can be written as :
 { } { }
  of change 
Rate Rate out
of a substance
Net Rate
in
??
= -
??
??
 
This Equation is known as WORD EQUATION of the model. 
5. Exponential Decay Model and Radioactivity: 
Radioactive elements are those elements which are not stable and emit a- 
particles, ß- particles or photons while decaying into isotopes of other 
elements. Exponential decay model for radioactive decay can be 
considered as a compartmental model with compartment being the 
radioactive material with no input but output as decay of radioactive 
sample over time. 
    Emitted Particles 
    
  
Fig. 2: Input – output compartmental diagram for radioactive nuclei 
5.1. Word equation: By Balance Law, word equation can be written as : 
 of change  amount of 
of radioactive material radioactive material
at time t decayed
Rate Rate ? ?? ?
? ?? ?
=
? ?? ?
? ?? ?
? ?? ?
 
5.2. Assumptions for the radioactive Decay Model: 
1. Amount of an element present is large enough so that we are 
justified in ignoring random fluctuations. 
2. The process is continuous in time. 
3. We assume a fixed rate of decay for an element. 
4. There is no increase in mass of the body of material.  
5.3. Formulating the differential equation: 
Let N(t) be the number of radioactive nuclei at time t 
Compartment  
Radioactive material 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
Page 5


Compartmental Models 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: Compartmental Models 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
Compartmental Models 
 
 
Table of Contents 
 Chapter: Compartmental Models 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Compartmental Model 
• 4. Balance Law 
• 5. Exponential Decay Model and Radioactivity 
• 6. Lake Pollution Model 
• 7. Seasonal Flow Rate 
• 8. Case Study: Lake Burley Griffin 
• 9. Drug Assimilation Model 
• 10. Exponential Growth Model (Population Growth) 
• 11. Density Dependent Population Growth Model 
• 12. Limited Population Growth Model with Harvesting 
• Exercise 
• Summary 
• References 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Compartmental Models 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of a compartmental model. 
• Concept of compartmental diagram. 
• Balance law and Word equation of the model. 
• Exponential Decay Model: Radioactive Decay 
• Lake Pollution Model. 
• Drug Assimilation Model : Case of single pill and course of pills. 
• Exponential Growth Model. 
• Density Dependent Growth. 
• Limited Growth Harvesting. 
 
2. Introduction: 
Compartmental models  provides a means to formulate models for 
processes which have inputs and/or outputs over time. In this chapter, we 
will study modelling of  radioactive decay processes, pollution levels in 
lake systems and the absorption of drugs into the bloodstream, 
exponential growth model, density dependent growth, limited growth 
harvesting using compartmental techniques. 
3. Compartmental Model: 
Definition: Compartmental Model is a model in which there is a place 
called compartment which has amount of substance in and amount of 
substance out over time. One example of compartmental model is the 
pollution into and out of a lake where lake is the compartment. Another 
example is the amount of carbon-di-oxide in the Earth’s atmosphere. The 
compartment is the atmosphere where the input of CO
2
 occurs through 
many processes such as burning and output of CO
2
 occurs through plant 
respiration. It can be shown in the form of a diagram called 
compartmental diagram which is shown below. 
 Input CO
2  
Output CO
2
 
 
  Fig. 1: Input – output compartmental diagram for CO
2
 
4. Balance Law: 
Statement: The rate of change of quantity of substance is equal to ‘Rate 
in’minus ‘Rate out’ of the compartment.  
Atmosphere 
(Compartment) 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
Compartmental Models 
Symbolically, if X(t) is the amount of quantity in the compartment, then 
 
???????? ???????? = ???????????????? ???????? - ???????????????? ???????????? 
4.1. Compartmental Diagram:  
  Rate In  Rate Out 
 
4.2. Word Equation: 
In words, balance law can be written as :
 { } { }
  of change 
Rate Rate out
of a substance
Net Rate
in
??
= -
??
??
 
This Equation is known as WORD EQUATION of the model. 
5. Exponential Decay Model and Radioactivity: 
Radioactive elements are those elements which are not stable and emit a- 
particles, ß- particles or photons while decaying into isotopes of other 
elements. Exponential decay model for radioactive decay can be 
considered as a compartmental model with compartment being the 
radioactive material with no input but output as decay of radioactive 
sample over time. 
    Emitted Particles 
    
  
Fig. 2: Input – output compartmental diagram for radioactive nuclei 
5.1. Word equation: By Balance Law, word equation can be written as : 
 of change  amount of 
of radioactive material radioactive material
at time t decayed
Rate Rate ? ?? ?
? ?? ?
=
? ?? ?
? ?? ?
? ?? ?
 
5.2. Assumptions for the radioactive Decay Model: 
1. Amount of an element present is large enough so that we are 
justified in ignoring random fluctuations. 
2. The process is continuous in time. 
3. We assume a fixed rate of decay for an element. 
4. There is no increase in mass of the body of material.  
5.3. Formulating the differential equation: 
Let N(t) be the number of radioactive nuclei at time t 
Compartment  
Radioactive material 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
Compartmental Models 
 ???? 0
 = initial radioactive nuclei present at time t
0
  
 Since the rate of change of nuclei is directly proportional to the number 
of nuclei at the start of time period therefore, ()
in
C t C =  
? 
dN
KN
dt
= - ,  where K is the constant of proportionality indicating rate 
of decay per nucleus in unit time. 
At initial condition, number of radioactive nuclei is ???? 0
 therefore, N(0) = ???? 0
 
Hence initial value problem corresponding to exponential decay model is 
given by: 
0
   ; N(0)= n ; K > 0
dN
KN
dt
= -
 
 
5.4. Solution of the differential equation of Exponential Decay 
Model: 
We have 
dN
KN
dt
= - 
? 
dN
Kdt
N
= - 
Integrating both sides, we get 
 
dN
K dt
N
= -
??
 
? ln lnC N Kt =-+ , where C is the constant of integration. 
? ln
N
Kt
C
??
= -
??
??
 
? 
Kt
N
e
C
-
??
=
??
??
 
? 
Kt
N Ce
-
=     ………………………(1) 
Put initial condition, N(0) = ???? 0
 i.e., at t = 0, N =  ???? 0
 we get 
 
(0)
0
K
n Ce
-
= 
? 
0
0
  e 1 nC = = ? 
Institute of Lifelong Learning, University of Delhi                                                 pg. 5