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A MATHEMATICAL MODEL OF TUBERCULOSIS (TRANSMISSION DYNAMICS AND CONTROL)

BY

OGUCHE UNEKU ADUOJO
MATRIC NO: 11MS1131





A PROJECT WORK SUBMITTED TO THE DEPARTMENT OF MATHEMATICAL SCIENCES, FACULTY OF NATURAL SCIENCES,
KOGI STATE UNIVERSITY, ANYIGBA.
IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF BACHELOR OF SCIENCE DEGREE (B.SC HONS) IN MATHEMATICS.




JANUARY, 2015.

DECLERATION
I OGUCHE UNEKU ADUOJO an undergraduate student of the department of Mathematical Sciences (Mathematics) with Matriculation Number 11ms1131, declare that this project work represent my original work and is considered adequate in scope and quality as meeting the requirement in fulfillment for the award of Bachelor of Sciences degree (Mathematics) of Kogi State University, Anyigba has not been part of any presentation for any other degree. It is the result of my research except where references are made to published literature which is duly acknowledge.

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OGUCHE UNEKU ADUOJO DATE










CERTIFICATION
This is to certify that this project work titled "Analysis Of A Mathematical Model Of Tuberculosis Transmission Dynamics" by Oguche Uneku Aduojo (11ms1131) meets the requirement in Fulfillment of Award of Bachelor of Science (B.Sc) Degree in Mathematics in the Department of Mathematical Sciences, Kogi State University, Anyigba and is accepted for its contribution to knowledge and literary presentation.

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Mr. Omale David Date
Supervisor


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Prof. S.E Uwamusi Date
Head of Department



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Prof. J.I Omada Date
Dean, Faculty of Natural Sciences



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Prof. O.M Bamigbola Date
External Examiner




DEDICATION
This project work is dedicated to the Almighty God, my source of wisdom and understanding who kept me from all odds from the beginning up to this moment without Him i am nothing. He has been my fortitude and help throughout my course of study and making this research work a huge success.
This work is also dedicated to my Late Father Pastor Michael. A. Oguche for his passion to see his Children become learned but were call home early and my wonderful and lovely siblings for their immeasurable love, prayer and financial support that has made me a successful. May God bless and reward you mightily.













ACKNOWLEDGEMENT
My appreciation goes to the almighty God for his love, mercies, kindness, guidance and protection throughout my years in school; for he is my source of inspiration and knowledge and the bedrock behind me in making this project a reality. All praise and adoration to his Holy name.
Am so grateful to my wonderful supervisor, Mr. Omale David who took his time to give me useful constructive comments, suggestion and encouragement moral and professional advice and patience from inception to completion. You are just like a father to me May God Almighty blesses you and grant you your heart desires.
My sincere appreciation goes to the head of department Prof. S.E Uwamusi for his fatherly advice and guidance God bless you sir and to other lecturers in mathematics, statistic, and computer option for their support academically, morally and their encouragement. May God bless them all in Jesus name.
My appreciation goes to my one in million parents Mr. and Mrs. M.S Oguche for their prayers, love and support through them my needs are always met they are the best parent in the whole universe. May you live long to eat the fruit of your labour.
I also appreciate the help, care, encouragement, support and love of my brothers and sisters Matthew, Divine, Ezekiel, Joy, Deborah, Joseph, and James . I can't thank you enough you mean so much to me my prayer is that God will bless you abundantly and take you to a greater height for sky is your starting point.
I wish to appreciate My Aunty Martha Oguche, my grandmother Mrs. Rebecca Omale Mr. John Omale, Mr. Audu Omale, e.t.c for their prayers and support. May God reward and surprise you wonderful in Jesus name.
Also I appreciate my Miss Glory Elijah for her care; Support and understanding may God bless and keep you dear. I love you.
My sincere appreciation goes to my lovely Mentor and Father Mr. and Mrs. S. O. Agada for their support morally, physically, spiritually and financially for their accommodation am so grateful sir/ma may God reward you and take you to greater height in the assignment He has commission into your Hand.
I also want to extend my gratitude to my amiable Brother and Sister in Faith Olaolu Muyiwa, Adetola Albert, Esther Sule, Simeon Abraham, Meshach Salifu, Toluju Wonders Olumide Adeluwoye, Seyi Fumi, Josuha Samuel, Sule Tosin, Isaac Sule, Success Peter, and as many I could not mention and The Redeemed Christian Fellowship (RCF) Anyigba and the entire family of Body of Christ Ksu Chapter Anyigba and Prayer Unit (RCF) their encouragement and prayers may God reward you all.
And to my best friend Moses Meliga Ogwu and Isaac Idegu Oboni who has really helped me morally, physically and financially thanks for your care and encouragement may God bless you and take you to a greater height.
Finally I want to thank my dear and wonderful friends and Classmate Adejoh Patrick, Agada David, Femi fred, Omale Martins, Samson Paul, Thamos Moses, Kayode John, Olatunji John, Funsho, Juwon,Oreyemi,Segun, Friday Ajayi, Noah Adamu, Idris Bashir, Adekunle Ohse, Omolara Helen, Ojo Blessing, Juliet Philip, Friday Ajayi, Kayode David,. Am grateful for your support and cooperation throughout my staying in school and to make this program an easy one for me God bless you.








TABLE OF CONTENT
CONTENT PAGE
Title page ………………………………………………………i
Declaration ……………………………………………………...ii
Certification ……………………………………………………..iii
Dedication ……………………………………………………....iv
Acknowledgement …………………………………………….....v
Table of content …………………………………………………vii
Abstract ……………………………………………………….viii

CHAPTER ONE
1.0 Introduction ………………………………………………...1
1.1 Classification of infectious diseases………………………...….1
1.2 The concept of mathematical modeling………………………..2
1.3 The basic reproduction number ……………………………….6
1.4 Definition of terms …………………………………………… 7
1.5 Causes of tuberculosis ……………………………………….10
1.6 Mode of transmission …………………………………………11
1.7 Symptoms of tuberculosis …………………………………… 12
1.8 Effect of tuberculosis ………………………………………….. 13
1.9 Diagnosis of tuberculosis …………………………………… ….14
1.10 Treatment of tuberculosis ……………………………………..14
1.11 Control of spread of tuberculosis ……………………………… 15
1.12 Aims and objective ……………………………………………. 15
1.13 Scope of the study ………………………………………….……15
1.14 Limitation of the study ……………………………………..…… 16
CHAPTER TWO
2.0 Literature review …………………………………………............ 18
CHAPTER THREE
3.0 Methodology ……………………………………………………… 24
3.1 Model of formulation………………….........................................24
3.2 Assumption of the model …………………………………………. 25
3.3 Definition of variables and parameter ……………………………….. 26
3.4 The Seir model …………………………………………………………. 27
3.5 Model of equations …………………………………………………….. 28
CHAPTER FOUR
4.0 Solution of models ………………………………………………………. 32
4.1 Solution of the Seir model ………………………………………………….32
4.2 Positivity of the model …………………………………………………… 33
4.3 Existence of steady states of the system ……………………………………36
4.4 Disease-free equilibrium state ……………………………………………..36
4.5 Basic reproductive number, r0 …………………………………………….. 38
4.6 The herd immunity threshold, h1 . ………………………………………….39
4.7 Analysis of the model ……………………………………………………..39
4.8 Equilibrium of the model …………………………………………………. 39
4.9 Stability of the equilibrium ………………………………………………….42
4.10 Numerical analysis and results …………………………………………….45
4.11 Estimation of the basic reproductive number, ro ………………………….. 47
4.12 Estimation of the herd immunity threshold, h1……………………………. 48
4.13 Estimation of the equilibrium points ……………………………………….48
4.14 Stability analysis of the equilibrium points …………………………………49
4.14 Sensitivity analysis ………………………………………………………….51
4.15 Sensitivity analysis by simulation …………………………………………..54
4.16 Discussion of results ……………………………………………………. 58
CHAPTER FIVE
Conclusion and Recommendations
5.0 Introduction ……………………………………………………………. 62
5.1 Conclusion…………………………………………………………………62
5.2 Recommendation …………………………………………………………63
5.3 References …………………………………………………………………65
















ABSTRACT
In this Research, a Susceptible - Exposed - Infected - Recovered (SEIR) epidemiological model is formulated to determine the transmission of tuberculosis. The equilibrium points of the model were analysis and their stability was investigated. By analyzing the model, a threshold parameter Ro was found which the basic reproductive number is. It is noted that when Ro < 1 the disease will fail to spread and when Ro > 1 the disease will persist in the population and become endemic. The model has two non–negative equilibrium namely the disease free equilibrium and the endemic equilibrium.