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This paper examines the spread and control of Avian Influenza. A non-linear mathematical model for the problem is formulated and analyzed. For the prevalence of the disease and the ease of analysis, the model was considered in proportions of susceptible, infectious and recovered compartments. The analysis of the stability show that the system will be stable if there is a bound on the growth of infected birds in the community. This means that the disease will die out after enough time if there is a bound on the growth rate of infected birds. The endemic flu state showed that the disease will persist if there is a bound on the infection transition rate.
1.1 Background of Study
1.2 Influenza (Flu)
1.2.1 Types of Influenza
1.2.2 Genetic Drift and Shift of Influenza
1.3 Avian Influenza
1.3.1 Symptoms of Avian Influenza
1.3.2 Spread and Control of Avian Influenza
220.127.116.11 Spread of Avian Influenza
18.104.22.168 Control of Avian Influenza
1.4 Aims and Objectives
1.5 Limitations of the Study
2.0 Literature Review
2.1 Etymology of Model
2.1.1 Mathematical Models
2.1.2 Essence of Mathematical Modelling
22.214.171.124 Mathematical Modelling and the Scientific Method
126.96.36.199 Mathematical Modelling and the Practice of Engineering
188.8.131.52 Principles of Mathematical Modelling
2.1.3 Epidemic Model
184.108.40.206 Types of Epidemic Models
2.1.5 Deterministic Compartmental Models
220.127.116.11 The SIR Model
18.104.22.168.2 The SIR Model with births and deaths
22.214.171.124 The SIS Model with births and deaths
126.96.36.199 The SIRS Model
2.1.6 Models with more Compartments
188.8.131.52 The SEIS Model
184.108.40.206 The SEIR Model
220.127.116.11 The MSIR Model
18.104.22.168 The MSEIR Model
22.214.171.124 The MSEIRS Model
2.1.7 Reproduction Number
2.2 Models and Analysis from Previous Works
3.1 Model Formulation (SIRS)
3.2 Schematic Diagram for the SIRS Model
3.3 Model Equations
3.4 Variables and their Descriptions
3.4.1 Parameters and their Descriptions
3.5 Equilibrium State
3.5.1 Disease – Free Equilibrium State
3.5.2 Endemic Equilibrium State
3.6 Stability Analysis of the Endemic Equilibrium
4.0 Results and Discussions
Disease, like wellness, is an unavoidable aspect of life. It may be the fairly containable kinds like cold, catarrh etc or the more fatal kinds like Cancer, Ebola, AIDS etc. It could be contagious, infectious, transmissive or otherwise. Irrespective of the type of disease, it always has a negative effect on the individual(s) and the society at large; financially, physically, economically, mentally or socially. An economic effect of disease could be an increase in mortality rate such as the loss of about 800,000 birds to bird flu (Avian Influenza) from 18 northern states in Nigeria as reported by The Guardian with more than 70,000 from Kano State alone. With a high consumption of poultry and poultry products in Nigeria, such as chicken, turkey, eggs and even manure obtained from the waste from the poultry farm, the spread of Avian Influenza, which is highly fatal, a great population of Nigerians are at risk of infection. It is safe to say that Avian Influenza affects the layman not only on the health level, but also on the economic grand scale. Reports of highly pathogenic Avian Influenza epidemics in poultry, such as A(H5N1), can seriously impact local and global economies and international trade.
Thus, a suitable means of predicting the spread and eventually, the control of this influenza is of extreme importance. Such predictions can be obtained mathematically, facilitating the management of such diseases with respect to public health. Mathematical representation of disease parameters is a data-reliant process and this prediction is often based on the implementation of mathematical models. The important feature is bridging the gap between mathematics (models) and the real world(data). These models are theoretically developed but applied to real life scenarios represented by a given data. Such data may contain the signature of social effect; hence, a comprehensive understanding of the phenomenon of disease involves a variety of mathematical tools, from model creation to the determination of solutions to differential equations to statistical analysis.
1.1 BACKGROUND OF STUDY
The earth as a whole is prone (susceptible) to Influenza with Africa and Asia being the leading continents with this traits. Hence, the need to expatiate on the effects globally is necessary.
1.1.1 NIGERIA: A Brief Introduction
Nigeria, a nation of people with diverse beliefs, cultures, practices and preferences, often referred to as The Giant of Africa owing to her large population and economy, occupies an area of 923,768km2 and an estimated population of over 174.5 million. She has been described “a pulsating powerhouse being the most populous nation on the continent[Africa]” and the seventh most populous country in the world with one of the largest population of youths in the world.
Agriculture used to be the principal foreign exchange earner in Nigeria before the advent of Crude Oil. Crop farming as well as Animal farming were and still are the major agricultural practices in Nigeria cutting across rice, groundnut, cocoa, palm fruit, cattle rearing, poultry farming etc.
The top five(5) major cities in Nigeria along with their estimated populations include;
ü Lagos (7,937,932)
ü Kano (3,848,885)
ü Ibadan (3,078,400)
ü Kaduna (1,652,844)
ü Port-Harcourt (1,320,214)
From the data given, it can be observed that the economic and industrial capitals of the south-western region and the north-western region (Lagos and Kano States respectively) have very high populations resulting in an estimated total of about 11.7 million forming about 6.75% of the entire national population. These states have thriving poultry farming business that extends to interstate distribution of frozen products obtained from such farms. These poultry farms can suffer setbacks ranging from the more mundane financial problems to climatic problems to diseases such as Avian Influenza. The most recent cases of bird flu (Avian Influenza) in Nigeria were recorded in Lagos and Kano States. The bird flu also known as Avian Influenza outbreak in Nigeria was first recorded in 2006 and lingered till 2008 before it was contained.  Recent outbreaks occurred in Abia, Enugu and Lagos states.
1.2 INFLUENZA (FLU)
Influenza, also called flu, is a contagious respiratory infection caused by a variety of influenza viruses. Its symptoms include muscle aches, soreness, headache, and fever.