MATHEMATICAL MODEL FOR THE DYNAMIC SPREAD AND CONTROL OF POLIO IN NIGERIA
ABSTRACT
In this project, I presented a nonlinear mathematical model for the spread of Polio in a population with variable size structure including the role of vaccination. Using an expanded SIR model, the present contribution takes into account the e?ects of a rapidly growing population on the e?ectiveness of various vaccination protocols and on the burden of disease in Nigerian communities. Necessary and su?cient conditions were found for elimination of Polio in such a population in the event of an outbreak.
CHAPTER 1
1.0    Introduction
1.1    Background of Study
1.2    Aims and objective
1.3    Significance of the study
1.4    Scope of study
CHAPTER 2
2.0 Literature Review
CHAPTER 3
3.0 Material and methods
3.1 Model Formulation
3.2 Assumption of the Model
3.3 Disease Free State
CHAPTER 4
4.0 Result and Discussion
4.1Results
4.2 Discussion
CHAPTER 5
5.0 Summary, Conclusions and recommendations
5.1 Summary
5.2 Conclusion
5.3 Recommendations
References
CHAPTER ONE
1.0                                    INTRODUCTION
1.1    Background of Study
  Polio (also called poliomyelitis) is a contagious, devastating disease that was virtually eliminated from the Western hemisphere in the second half of the 20th century. Although polio has plagued humans since ancient times, its most extensive outbreak occurred in the first half of the 1900s before the vaccination, created by Jonas Salk in 1952, became widely available in 1955. Poliomyelitis (polio) is a highly infectious disease caused by Poliovirus. It invades the nervous system, and can cause total paralysis in a matter of hours. It can strike at any age, but affects mainly children under three (over 50% of all cases). The virus enters the body through the mouth and multiplies in the intestine. Poliovirus mainly passes through person-to- person contact. Initial symptoms are fever, fatigue, headache, vomiting, and stiffness in the neck and pain in the limbs. However, immune and or partially immune adults and children can still be infected with poliovirus and carry the virus for long enough to take the virus from one country to another, infecting close contacts and contaminating sanitation systems. There is no cure for polio; it can only be prevented through immunization. Polio vaccine, given multiple times, usually protects a child for life.   In 1988, the World Health organisation(WHO) launched the Global Polio Eradication Initiative, which aimed to use large-scale vaccination with the oral vaccine to eradicate polio worldwide by the year 2000. Although important progress has been made, polio remains endemic in several countries. Also, the current control measures will likely be inadequate to deal with problems that may arise in the post-polio era. A panel convoked by the National Research Council concluded that the use of antiviral drugs may be essential in the polio eradication strategy (WHO, 1997, 2000; Abdulraheem and Saka, 2004; GPE, 2007; Agbeyegbe, 2007).
On April 12, 2005, we celebrated the 50-year anniversary of the publication of the largest and first clinical trial for a vaccine.  In 1955, researchers demonstrated the effectiveness of the Salk polio vaccine (Francis et al., 1955) and the news media that day exclaimed exciting themes: “The vaccine works.
Thompson and DuintjerTebbens (2005) provide a retrospective analysis of the polio vaccination
In most of the models, it is usually assumed that infection spread in susceptible population due to infective population only but some diseases like polio are contagious during incubation period so interaction of exposed population with susceptible population also has its role in the spread of disease. We, therefore, have considered interaction between susceptible and exposed population in our model. The spread of communicable diseases not only depend on interaction of population but also on the immunity of the individual. The immunity to the specific disease in the individuals can be artificially developed with the help of vaccination. But if vaccine is administered during incubation period (as symptoms of polio are not visible during incubation period) it can be produce hazardous effects also.  
1.2Aim and Objectives
Mathematical modeling can be used for a number of di?erent reasons. How well any particular objective is achieved depends on both the state of knowledge about a system and how well the modeling is done. Examples of the range of objectives are:
1. Developing scienti?c understanding - through quantitative expression of current knowledge of a system (as well as displaying what we know, this may also show up what we do not know); 2. Test the e?ect of changes in a system; 3. Aid decision making, including; (i) tactical decisions by managers, (ii) strategic decisions by planners.
 1.3    Significance of the Study
The important of the different institution in the development of pragmatic and programs on polio, which include the following purpose.
1.     The polio vaccine saved countless lives and ended the crippling paralyzation that sometimes came with the virus.
2.     It also opened the door for the discovery of other cures and vaccinations.
3.     Help in deciding whether there is need to streamline or enhance existing health promotion efforts designed for vulnerable segments, as the study would identify key variables that need to be addressed and invested on.
1.4Scope and Limitation of the Study
This project cover the use of mathematical model of the spread and treatment of polio virus in Nigeria and the limitation include learning and familiarization with modeling in solving physical problem, report with the interest in modeling as a tool for solving problem among others.