# A Mathematical Model For The Spread And Control Of Avian Influenza (Bird Flu)

A Mathematical Model For The Spread And Control Of Avian Influenza (Bird Flu)

## Abstract

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.

## Table of Content

### Chapter One

• 1.0 Introduction
• 1.1 Background of Study
• 1.1.1 Nigeria
• 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
• 1.3.2.1 Spread of Avian Influenza
• 1.3.2.2 Control of Avian Influenza
• 1.4 Aims and Objectives
• 1.5 Limitations of the Study

### Chapter Two

• 2.0 Literature Review
• 2.1 Etymology of Model
• 2.1.1 Mathematical Models
• 2.1.2 Essence of Mathematical Modelling
• 2.1.2.1 Mathematical Modelling and the Scientific Method
• 2.1.2.2 Mathematical Modelling and the Practice of Engineering
• 2.1.2.3 Principles of Mathematical Modelling
• 2.1.3 Epidemic Model
• 2.1.3.1 Types of Epidemic Models
• 2.1.3.1.1 Stochastic
• 2.1.3.1.2 Deterministic
• 2.1.4 Terminology
• 2.1.5 Deterministic Compartmental Models
• 2.1.5.1 The SIR Model
• 2.1.5.1.2 The SIR Model with births and deaths
• 2.1.5.2 The SIS Model with births and deaths
• 2.1.5.3 The SIRS Model
• 2.1.6 Models with more Compartments
• 2.1.6.1 The SEIS Model
• 2.1.6.2 The SEIR Model
• 2.1.6.3 The MSIR Model
• 2.1.6.4 The MSEIR Model
• 2.1.6.5 The MSEIRS Model
• 2.1.7 Reproduction Number
• 2.2 Models and Analysis from Previous Works

### Chapter Three

• 3.0 Introduction
• 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 â€“ Free Equilibrium State
• 3.5.2 Endemic Equilibrium State
• 3.6 Stability Analysis of the Endemic Equilibrium

### Chapter Four

• 4.0 Results and Discussions
• 4.1 Results
• 4.2 Discussions

### Chapter Five

• 5.0 Conclusion
• 5.1 Summary
• 5.2 Recommendations
• Reference

## Chapter One

### 1.0 Introduction

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.[2] 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.[8]

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.[1] 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]

#### 1.1 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 â€śa pulsating powerhouse being the most populous nation on the continent[Africa]â€ť[4] and the seventh most populous country in the world with one of the largest population of youths in the world.[3][4]

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.[3][4]

The top five(5) major cities in Nigeria along with their estimated populations include;[3]

• Lagos (7,937,932)
• Kano (3,848,885)
• 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. [5] 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.

A proper characterization of the INFLUENZA viruses, and most especially the Avian Influenza, is needed for a better understanding of the type, class, morbidity and fatality of the various Influenza viruses circulating in the environment.

##### 1.2.1 Types of Influenza

There are three (3) different types of influenza. They are Type A, Type B, and Type C. Types A and B causes the annual influenza epidemics that have up to 20% of the population sniffling, aching, coughing, and running high fevers. Type C also causes flu; however, its flu symptoms are much less severe.

###### Type A Influenza Virus:

Type A flu or influenza A viruses are capable of infecting animals, although it is more common for people to suffer the ailments associated with this type of flu. Wild birds commonly act as the hosts for this flu virus.

Type A flu virus is constantly changing and is generally responsible for the large flu epidemics. The influenza A2 virus (and other variants of influenza) is spread by people who are already infected. The most common flu hot spots are those surfaces that an infected has touched and rooms where recent contact has been made.

###### Type B Influenza Virus:

Unlike type A flu viruses, type B flu is found only in humans. Type B flu may cause a less severe reaction than type A flu virus, but occasionally, type B flu can still be extremely harmful. Influenza type B viruses are not classified by subtype and do not cause pandemics.

###### Type C Influenza Virus:

Influenza C viruses are also found in people. They are, however, milder than either type A or B. People generally do not become very ill from the influenza type C viruses. Type C flu viruses do not cause epidemics.

##### 1.2.2 Genetic Drift and Shift of Influenza

The influenza virus is antigenically unstable and new strains and variants are constantly emerging. Each year one or two subtypes of influenza A may be in circulation and one type of influenza B.

###### Antigenic drift:

Here, when thereâ€™s a minor change in the amino acid sequence of the Haemagglutinin (HA) molecules in the virus envelope, it results in a genetic drift. These minor changes occur all the time, hence, making these changes almost like a continuous process. Haemagglutinin is the main antigen associated with immunity. Neuraminidase (NA), the second main antigen in the virus plays a minor role in immunity. The drifted strains of influenza may infect partially immune people who have been previously exposed. Influenza A drifts more than influenza B.

###### Antigenic shift:

Here, as opposed to minor changes resulting in drifts, major changes in the Haemagglutinin (HA) or Neuraminidase (NA) take place resulting in the emergence of a virus which contains a Haemagglutinin different from those of previously circulating viruses. When this happens it gives rise to major epidemics or pandemics in populations throughout the world that have no immunity to the new strains. Examples include; Influenza H1N1 (2009), Spanish flu (1918), Asian flu (1957) and Hong Kong flu (1968/69).

#### 1.3 Avian Influenza

Bird flu, (Avian Influenza), is caused by a type of influenza virus (RNA virus in the family of Orthomyxoviridae) that is hosted by birds but may infect several species of mammals. It was first identified in Italy in the early 1900s and is now known to exist worldwide.[6] The first case of Avian Influenza in humans occurred in Hong-Kong in the year 1997 and was referred to as A H5N1.

â€śBird fluâ€ť is a phrase similar to â€śswine fluâ€ť, â€śdog fluâ€ť, â€śhorse fluâ€ť or â€śhuman fluâ€ť in that it refers to an illness caused by any of many different strains of influenza viruses that have adapted to a specific host. All known viruses that cause influenza in birds belong to the species INFLUENZA A VIRUS.[7] The subtypes H5 and H7 are the most deadly, responsible for the deaths and losses in humans and poultry with recorded deaths in humans at about 60%.[8] while the H9 subtype is usually of low risk as it does not infect humans directly.

Avian influenza is an infection caused by avian (bird) influenza (flu) A viruses. This influenza A viruses occur naturally among birds. Wild birds worldwide get flu A infections in their intestines, but usually do not get sick from the flu infections. Avian Influenza is very contagious among birds and some of these viruses can make certain domesticated bird species, including chickens, ducks, and turkeys very sick and in some cases, may be fatal. The infected birds can shed influenza virus in their saliva, nasal secretions, and faeces.

Susceptible birds become infected when they have contact with contaminated secretions or excretions or with surfaces that are contaminated with secretions or excretions from infected birds. Domesticated birds may become infected with avian influenza virus through direct contact with infected waterfowl or other infected poultry, or through contact with surfaces (such as dirt or cages) or materials (such as water or feed) that have been contaminated with the virus.

Avian Influenza is a very complex disease. The pathogen mutates at a high rate, allowing it to jump species barriers and expand its host range. Various strains of Avian Influenza have been known to infect a large number of wild bird species, a number of species of domestic birds, a number of species of mammals (such as pigs, dogs and horses) as well as humans. The multi-species conglomerate of hosts, that Avian Influenza creates, poses serious difficulties for tracing and controlling the disease. Because of this complexity, the efforts are directed at reducing the circulation within the poultry population, as the main animal population responsible for the transmission of the disease to humans. The control strategies currently in place target the domestic bird populations and the humans. In the early 2000s only culling was applied in an attempt to reduce the spread of the disease. Large numbers of chickens were destroyed, causing significant hardship and economic loss. Nowadays, multiple control strategies are in place: culling, vaccination of poultry, increasing biosecurity[17] .

Avian Influenza viruses can be classified into two(2) distinct groups based on their propensity to cause diseases in poultry. They include;

Low Pathogenic Viruses: The Low Pathogenic Viruses cause outbreaks in poultry but are not fatal. They may go undetected and usually causes only mild symptoms (such as ruffled feathers and a drop in egg production).
An example is the H7N9 virus subtype of the Avian Influenza. Its first victims were numbered at about three(3) humans in the city of Shanghai and Anhui Province(2013). No cases of H7N9 virus have been recorded outside of China since. Closure of live bird markets which lasted for months affected the agriculture sectors of the economy of the affected places as well as international trade. Continued surveillance is recommended to aid early detection and control of the virus.

High Pathogenic Viruses: The High Pathogenic Viruses, on the other hand, results in high death rates and spreads more rapidly through flocks of poultry. This form may cause disease that affects multiple internal organs and has a mortality rate that can reach (90-100)% often within 48 hours.

An example is the H5N1 virus subtype. It is a highly pathogenic AI virus, first infected humans in 1997 during a poultry outbreak in Hong Kong SAR, China. Since its widespread re-emergence in 2003 and 2004, this avian virus has spread from Asia to Europe and Africa and has become entrenched in poultry in some countries, resulting in millions of poultry infections which are often fatal. Outbreaks in poultry are seriously impacting livelihoods, the economy and international trade in affected countries. It is also fatal in human infected case.

The strain of most importance is the H5N1 and to an extent, the H7N9 strains. It is so alarming due to its ability to pass from wild birds to poultry and then on to people. This is the most causative strain of interest in Nigeria. While wild birds are commonly immune to the devastating and possibly deadly effects of H5N1, the virus has deadly effects to a large percentage of the people infected with it. The risk of Avian Influenza is generally low in most people because the virus does not typically infect humans. Infections have occurred as a result of contact with infected birds. Spread of this infection from human to human has been reported, though in extremely rare cases.[9][10]

The Avian Influenza virus, current strain under consideration, H5N1, causes two distinctly different forms of disease. The first one is â€ślow pathogenic avian influenzaâ€ť (LPAI) which is common and mild and has only signs such as ruffled feathers, reduced egg production, or mild effects on respiratory system in birds. The outbreaks can be so mild that the virus may not be detected without any regular testing. In contrast, the second form, â€śhigh pathogenic avian influenzaâ€ť (HPAI) is rare but highly lethal and it can be characterized by sudden onset of severe disease, rapid contagion and a mortality rate that can approach 100% within 48hours (WHO 2006a).

Moreover, H5N1 virus can be easily transmitted by the movement of live birds, people (especially with shoes and other clothing) and contaminated vehicles, equipment, feed and cages. As a characteristic of HPAI virus, it can survive up to 35days in the environment due to low temperature. (WHO 2006a) When we look at the agents who carry the avian influenza virus, we can categorize them such as migratory/resident wild birds, domestic water birds, terrestrial poultry, and live bird markets (FAO 2005).

The virus carriers to different regions are â€śmigratory wild birdsâ€ť which are seen to be the natural reservoir of the low pathogenic avian influenza (LPAI) virus and they are not affected by the disease. In different regions, wild birds get into contact with â€śdomestic water birdsâ€ť, such as ducks. As a result of these contacts â€śdomestic water birdsâ€ť contract the LPAI. Then the LPAI passes to the â€śterrestrial poultryâ€ť as a result of contacts with â€śdomestic water birdsâ€ť. A direct contact between â€śmigratory wild birdsâ€ť and â€śpoultryâ€ť is also possible but the contact rate is extremely low as compared to the contact rate between â€śpoultryâ€ť and â€śdomestic water birdsâ€ť. The LPAI virus turns into the high pathogenic avian influenza (HPAI) virus in â€śpoultryâ€ť, via mutation. HPAI spreads very rapidly among poultry flocks causing deaths. Then HPAI can pass back to â€śdomestic wild birdsâ€ť through contact with â€śpoultryâ€ť.

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