Discrete time mathematical SIR model for disease transmission

Isaac Olusola 70 PAGES (12993 WORDS) Project
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ABSTRACT: Infectious  disease  has  become  a  source  of  fear  and  superstition  since  the first  ages  of  human  civilization.  In  this  study,  we  consider  the  Discrete  SIR  model for  disease  transmission  to  explain  the  use  of  this  model  and  also  show  significant explanation  as  regard  the  model.  We  discuss  the  mathematics  behind  the  model and  various  tools  for  judging  effectiveness  of  policies  and  control  methods. The  model  has  two  equilibrium  states,  namely  the  disease  free  equilibrium  and the  endemic  equilibrium  state.  The  stability  of  each  equilibrium  state  is  discussed and  the  endemic  equilibrium  has  been  found  to  be  stable  while  that  of  the  disease free  equilibrium  was  unstable.  The  basic  reproduction  number  was  computed  and gotten  to  be  1.2.  The  disease  was  found  to  persist  with  Ro  >  1,  whenever  the natural  death  rate  is  reduced,  the  death  rate  caused  by  the  disease  and  the transmission  rate is  increased  but  the  disease  dies  out  with  Ro  < 1, whenever  the transmission  rate  and  the  birth  rate  is  reduced  and  recovery rate  is  increased. The  data  used  in  the  study  was  extracted  from  a  journal  stated  in  chapter 4  and  were  analysed  using  the  one-way  sensitivity  analysis,  and  also  used  matlab 2013  to  run  respectively  simulations  for  the  change  in  each  parameter  extracted. The  results  of  the  sensitivity  analysis  showed  that  the  birth  rate,  the  transmission rate  reduced  and  increasing  the  recovery  rate  will  make  the  disease  die  out  of  the population. 

TABLE  OF CONTENTS:

Title page

Undertaking

Certification

Dedication

Acknowledgement 

Table  of content

Abstract 

CHAPTER 1: INTRODUCTION                                                                              1.1  Background  of  the Study

1 1.1.1  Infectious  Disease………………………………………………..4  1.1.1.1  Notes  on  Infectious  Disease……………………………………4 1.1.1.2  Modelling  of  Infectious  Disease………………………………6 

1.2  Significance  of  Study…...….……………………………………………………8           1.3  Objectives  of  Study…………………………………………………………….8                                                                                                                            1.4  Scope  of  Study………………………………………………………………..9 

1.5  Definition  of  Terms………………..……….…………………………………9

1.6 Organization  of Study……....……………...………………………………......11

CHAPTER 2: LITERATURE REVIEW  2.1  Review  of  various  SIR  Models…………………………………………………12

2.2 Difference Equation….………………......…………………………………….17   2.2.1  Classification  of  Difference Equation…………………………………17 

 2.3 Description  of  some Deterministic Discrete  Time Epidemic Models…………18   

2.3.1  The SI  Model  …..………………………………………………... 19           

2.3.1.1  The Reproduction  Number of  an SI  Model………  20                             

 2.3.2  The SI  Model  …..………………………………………………... 21           

2.3.3  The SEIR Model…………………………………………...22     

2.4 Linear and  Nonlinear Models……………….………………………23 

2.5 Equilibrium  States……………………………………………..24

2.6 Uncertainty  and  Sensitivity  analysis  in  Modelling………………………

CHAPTER 3: METHODOLOGY 3.1 Introduction……..…………………………………………………….. 27 

3.2 Preliminaries………….……………………………………………..……28 

3.3 Model  Formulation……….……………………………………………………28 

3.3.1  Model  Assumptions……………………………………………………..28 

3.3.2  Description  of  the  Discrete  SIR Model………………………………….29  

3.4  Model  Equations……………………………………………………………….31

3.5Equilibrium  Points………………………..……………………………………36

3.5.1Disease-Free  Equilibrium  Point…………………………………………...37                                                    3.5.2Endemic  Equilibrium Point………………………………………………...38   

3.6   Stability Analysis  of  the  Equilibrium  Points………………………...................40                                                                    3.6.1  Stability Analysis  of  the  Disease-free  Equilibrium……………………….41

3.6.2  Stability  Analysis  of the EndemicEquilibrium………………..........43

 3.7  The  Basic  Reproduction Number …………..……………………………...46 

CHAPTER 4: ANALYSIS AND  NUMERICAL SIMULATION 4.0  Introduction…..………………….……………………………………………..49

4.1 Equilibrium  Points………………………………………………..50 


4.2 Stability  Analysis…………………………………….……..51

4.2.1  Stability  Analysis  of the Disease-Free  Equilibrium Point……………51   

4.2.2  Stability  analysis  of  the Endemic Equilibrium  Point………………....53

4.3 Sensitivity  Analysis………………...………………………………………….55 

4.3.1  Sensitivity  Analysis  using  the  Basic  Reproduction  Number………….55   

4.3.2  Sensitivity  Analysis  of Disease transmission  by Simulation……56    

4.4 Summary..................................................................................................62

CHAPTER FIVE: SUMMARY AND CONCLUSION  5.0  Introduction……….……………………………………………………………66 

5.1 Summary  and  Conclusion……………………………………………………...67  References………………………………………………………68 

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APA

Isaac, O (2019). Discrete time mathematical SIR model for disease transmission. Afribary.com: Retrieved December 15, 2019, from https://afribary.com/works/discrete-time-mathematical-sir-model-for-disease-transmission

MLA 8th

Olusola, Isaac. "Discrete time mathematical SIR model for disease transmission" Afribary.com. Afribary.com, 19 Nov. 2019, https://afribary.com/works/discrete-time-mathematical-sir-model-for-disease-transmission . Accessed 15 Dec. 2019.

MLA7

Olusola, Isaac. "Discrete time mathematical SIR model for disease transmission". Afribary.com, Afribary.com, 19 Nov. 2019. Web. 15 Dec. 2019. < https://afribary.com/works/discrete-time-mathematical-sir-model-for-disease-transmission >.

Chicago

Olusola, Isaac. "Discrete time mathematical SIR model for disease transmission" Afribary.com (2019). Accessed December 15, 2019. https://afribary.com/works/discrete-time-mathematical-sir-model-for-disease-transmission