Lecture 6 - Predator-Prey and Epidemic Models | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 1 
 
 
 
 
 
 
 
 
Paper: Differential Equations-I  
Lesson :  Predator-Prey and Epidemic Models 
Course Developer :  Gurudatt Rao Ambedkar 
Department/College  :  Department of Mathematics, 
Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Page 2


Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 1 
 
 
 
 
 
 
 
 
Paper: Differential Equations-I  
Lesson :  Predator-Prey and Epidemic Models 
Course Developer :  Gurudatt Rao Ambedkar 
Department/College  :  Department of Mathematics, 
Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 2 
 
 
 
 
 
Table of Contents:  
 Chapter : Predator-Prey and Epidemic Models 
 
? 1. Learning Outcomes  
? 2. Introduction 
? 3. An epidemic model for influenza 
? 4. Predators and prey 
? 5. Model of a Battle 
? 6. Interpretation of parameters 
? 7. Equilibrium points 
? 8. Trajectories and phase-plane diagram 
? 9. Review of the battle model 
? Exercise 
? Summary 
? References 
 
 
 
 
 
 
 
Page 3


Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 1 
 
 
 
 
 
 
 
 
Paper: Differential Equations-I  
Lesson :  Predator-Prey and Epidemic Models 
Course Developer :  Gurudatt Rao Ambedkar 
Department/College  :  Department of Mathematics, 
Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 2 
 
 
 
 
 
Table of Contents:  
 Chapter : Predator-Prey and Epidemic Models 
 
? 1. Learning Outcomes  
? 2. Introduction 
? 3. An epidemic model for influenza 
? 4. Predators and prey 
? 5. Model of a Battle 
? 6. Interpretation of parameters 
? 7. Equilibrium points 
? 8. Trajectories and phase-plane diagram 
? 9. Review of the battle model 
? Exercise 
? Summary 
? References 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 3 
 
 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Model’ and develop the skills of 
modeling 
? How to relate a general life situation to a mathematical model 
? How to prepare a mathematical model with the help of differential 
equations 
? Understand the concept of phase-plane and its analysis 
? Understand the concept of equilibrium points  
 
 
2. Introduction 
 
    We face many problems in our day to day life. These problems are 
sometime become very small and sometime become very serious. Everybody 
wants a better future and mathematics help us to get it. We can model a life 
situation with mathematics and the results of this model help us to predict 
the future. To prepare a model, we need to convert the problem into word 
equation and then associate mathematical equations. In this chapter we use 
differential equations to develop a model. A single model can be used for 
many similar situations. Here we develop models for spread of disease, 
interaction between two species and model for battle. 
I.Q. 1 
 
3. An epidemic model for influenza 
Here a model is developed to describe the spread of disease in population 
and apply it to describe the influenza in city. To do so the population is 
divided into three groups: those susceptible to catching the disease, those 
infected with disease and capable of spreading it and those who have 
recovered and are immune from the disease.  A system of two coupled 
differential equations is obtained by modelling these interacting groups. 
3.1. Model assumptions 
Page 4


Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 1 
 
 
 
 
 
 
 
 
Paper: Differential Equations-I  
Lesson :  Predator-Prey and Epidemic Models 
Course Developer :  Gurudatt Rao Ambedkar 
Department/College  :  Department of Mathematics, 
Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 2 
 
 
 
 
 
Table of Contents:  
 Chapter : Predator-Prey and Epidemic Models 
 
? 1. Learning Outcomes  
? 2. Introduction 
? 3. An epidemic model for influenza 
? 4. Predators and prey 
? 5. Model of a Battle 
? 6. Interpretation of parameters 
? 7. Equilibrium points 
? 8. Trajectories and phase-plane diagram 
? 9. Review of the battle model 
? Exercise 
? Summary 
? References 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 3 
 
 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Model’ and develop the skills of 
modeling 
? How to relate a general life situation to a mathematical model 
? How to prepare a mathematical model with the help of differential 
equations 
? Understand the concept of phase-plane and its analysis 
? Understand the concept of equilibrium points  
 
 
2. Introduction 
 
    We face many problems in our day to day life. These problems are 
sometime become very small and sometime become very serious. Everybody 
wants a better future and mathematics help us to get it. We can model a life 
situation with mathematics and the results of this model help us to predict 
the future. To prepare a model, we need to convert the problem into word 
equation and then associate mathematical equations. In this chapter we use 
differential equations to develop a model. A single model can be used for 
many similar situations. Here we develop models for spread of disease, 
interaction between two species and model for battle. 
I.Q. 1 
 
3. An epidemic model for influenza 
Here a model is developed to describe the spread of disease in population 
and apply it to describe the influenza in city. To do so the population is 
divided into three groups: those susceptible to catching the disease, those 
infected with disease and capable of spreading it and those who have 
recovered and are immune from the disease.  A system of two coupled 
differential equations is obtained by modelling these interacting groups. 
3.1. Model assumptions 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 4 
 
In the case of considering a disease, the population can be categorized into 
different classes; susceptible ) (t S and infectious infectives ) (t I , where t 
denotes the time. The population liable to catch the disease is called the 
susceptibles, while the infectious infectives are those infected with the 
diseases that are capable to transfer it to a susceptible. The last category is 
of those who have recovered from the disease and who are now safe from 
further infection of the disease. 
Initially, some assumptions are made to build the model, which are as 
follows: 
? To ignore the random differences between individuals, we assume the 
populations of susceptibles and infectious infectives are large. 
? We assume that the disease is spread by contact only and ignore the 
births and deaths in this model. 
? We set the latent period for the disease equal to zero. 
? We assume all those who recover from the disease are then safe (at 
least within the time period considered). 
? At any time, the population is mixed homogenously, i.e. we assume 
that the susceptibles and infectious infectives are always randomly 
distributed over the area in which the population lives. 
3.2. Formulating the differential equations 
The rate of change in the number of susceptibles and infectious infectives 
describe in word equations with the help of an input-output compartment 
diagram. This process is illustrated in the following example. 
Example 1: Create a compartmental diagram for the model and develop 
appropriate word equation for the rate of change of susceptibles and 
infectious infectives. 
Solution: Since births are ignored in the model and infectious infectives 
cannot become susceptibles again i.e. the loss of those who become infected 
is the only way to change the number of susceptibles. The number of 
infectives decreases due to those infectives who die, become safe or are 
isolates and changes due to the susceptibles becoming infected. 
The appropriate word equations are 
? ? ? ?
? ?
rate of change in no.of susceptibles inf
inf
rate of change in no.of infectives (1)
inf cov
. cov
rate of susceptiblesbecome ected
rateof susceptiblesbecome rateof ectives
ected have re ered
rate of change in no of re e
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ? inf cov red rateof ectiveshave re ered ?
Page 5


Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 1 
 
 
 
 
 
 
 
 
Paper: Differential Equations-I  
Lesson :  Predator-Prey and Epidemic Models 
Course Developer :  Gurudatt Rao Ambedkar 
Department/College  :  Department of Mathematics, 
Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 2 
 
 
 
 
 
Table of Contents:  
 Chapter : Predator-Prey and Epidemic Models 
 
? 1. Learning Outcomes  
? 2. Introduction 
? 3. An epidemic model for influenza 
? 4. Predators and prey 
? 5. Model of a Battle 
? 6. Interpretation of parameters 
? 7. Equilibrium points 
? 8. Trajectories and phase-plane diagram 
? 9. Review of the battle model 
? Exercise 
? Summary 
? References 
 
 
 
 
 
 
 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 3 
 
 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Model’ and develop the skills of 
modeling 
? How to relate a general life situation to a mathematical model 
? How to prepare a mathematical model with the help of differential 
equations 
? Understand the concept of phase-plane and its analysis 
? Understand the concept of equilibrium points  
 
 
2. Introduction 
 
    We face many problems in our day to day life. These problems are 
sometime become very small and sometime become very serious. Everybody 
wants a better future and mathematics help us to get it. We can model a life 
situation with mathematics and the results of this model help us to predict 
the future. To prepare a model, we need to convert the problem into word 
equation and then associate mathematical equations. In this chapter we use 
differential equations to develop a model. A single model can be used for 
many similar situations. Here we develop models for spread of disease, 
interaction between two species and model for battle. 
I.Q. 1 
 
3. An epidemic model for influenza 
Here a model is developed to describe the spread of disease in population 
and apply it to describe the influenza in city. To do so the population is 
divided into three groups: those susceptible to catching the disease, those 
infected with disease and capable of spreading it and those who have 
recovered and are immune from the disease.  A system of two coupled 
differential equations is obtained by modelling these interacting groups. 
3.1. Model assumptions 
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 4 
 
In the case of considering a disease, the population can be categorized into 
different classes; susceptible ) (t S and infectious infectives ) (t I , where t 
denotes the time. The population liable to catch the disease is called the 
susceptibles, while the infectious infectives are those infected with the 
diseases that are capable to transfer it to a susceptible. The last category is 
of those who have recovered from the disease and who are now safe from 
further infection of the disease. 
Initially, some assumptions are made to build the model, which are as 
follows: 
? To ignore the random differences between individuals, we assume the 
populations of susceptibles and infectious infectives are large. 
? We assume that the disease is spread by contact only and ignore the 
births and deaths in this model. 
? We set the latent period for the disease equal to zero. 
? We assume all those who recover from the disease are then safe (at 
least within the time period considered). 
? At any time, the population is mixed homogenously, i.e. we assume 
that the susceptibles and infectious infectives are always randomly 
distributed over the area in which the population lives. 
3.2. Formulating the differential equations 
The rate of change in the number of susceptibles and infectious infectives 
describe in word equations with the help of an input-output compartment 
diagram. This process is illustrated in the following example. 
Example 1: Create a compartmental diagram for the model and develop 
appropriate word equation for the rate of change of susceptibles and 
infectious infectives. 
Solution: Since births are ignored in the model and infectious infectives 
cannot become susceptibles again i.e. the loss of those who become infected 
is the only way to change the number of susceptibles. The number of 
infectives decreases due to those infectives who die, become safe or are 
isolates and changes due to the susceptibles becoming infected. 
The appropriate word equations are 
? ? ? ?
? ?
rate of change in no.of susceptibles inf
inf
rate of change in no.of infectives (1)
inf cov
. cov
rate of susceptiblesbecome ected
rateof susceptiblesbecome rateof ectives
ected have re ered
rate of change in no of re e
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ? inf cov red rateof ectiveshave re ered ?
Predator-Prey and Epidemic Models 
Institute of Lifelong Learning, University of Delhi                                                                                                pg. 5 
 
 
 
        Infected               recovered  
S 
Figure 1: Compartmental diagram for the epidemic model of influenza in a city, where there 
is no reinfection. 
 Let us first consider to model the total rate of susceptibles infected 
that only a single infective spread the infection in susceptibles. It is clear 
that the growth in the number of infectives will be greater due to greater the 
number of susceptibles. Thus, the rate of susceptibles diseased by a single 
infective will be an increasing function of the number of susceptibles. For 
ease, let us assume that the rate of susceptibles infected by a single 
infective is directly proportional to the number of susceptibles. Let ) (t S be 
the number of susceptibles at time t and ) (t I
 
be the number of infectives at 
time t
, 
then  
 
 
Where, constant ? is called the transmission coefficient or infection rate 
(Proportionality constant).  
Hence, ) (t S ? will be the rate of susceptibles infected by a single infective and 
if we multiply ) (t S ? to the number of infectives, we will get total rate of 
susceptibles infected by infectives. Hence  
 ? ? ? ? 2 ) ( ) ( inf t I t S ected e susceptibl of rate ? ? .    
   
We must also account for those who have recovered from disease. In 
general, those infectives who died due to disease, those who become 
protected to the disease and those who become isolated will be counted as 
removed. The number of infectives removed in the time interval should 
depend only on the number of infectives, but not upon the number of 
susceptibles. Let the rate of infectives recovered from the disease is directly 
proportional to the number of infectives. We write 
Susceptibles Recovered 
 
Infectives 
? ?
? ?
inf ( )
inf ( )
rate of susceptible ected S t
rate of susceptible ected S t ?
?
??