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The goal of this project, is to study using Mathematical Modelling, the impact of immunization with a partially effective vaccine on the transmission dynamics of influenza infection. To determine condition under which an epidemic occur. If it occur, what fraction of a even-mixed population get infected. To predict future spread of disease. To come up with plan for containment and eradication.

Table of Contents
Abstract ii
Declaration and Approval iv
Dedication vii
Acknowledgments x

Chapter 1: Introduction
1.1 Influenza 1
1.1.1 Definition and meaning of INFLUENZA 1
1.1.2 Signs and Symptoms 1
1.1.3 The Pathogen (Causes and Types) 1
1.1.4 Global effect of influenza 2
1.2 Influenza pandemics 2
1.2.1 Case Study 3
1.3 Vaccination 4
1.3.1 Meaning 4
1.3.2 Types of vaccines 4
1.4 Epidemiology and Target Group 4
1.4.1 Expectant mothers 4
1.4.2 Children 4
1.4.3 Elderly People 5
1.4.4 People with special health conditions 5
1.4.5 HealthCare Workers 5
1.5 Aim of the Project/objectives 5

2.1.1 The SI Model 8
2.1.2 The SIR Model 8
2.1.4 The SIRV model. 9
2.1.5 The SEIRD Model 10


4.1 Non-dimensionalization 18
4.2 The Positive Invariant Compact Set 20
4.3 The Basic Reproductive Number 20
4.3.1 The Disease Free Equilibrium 22

5.1 Numerical Simulation 29

6.1 Discussion 34
6.2 Recommendation 34
Bibliography 36

Chapter 1

1.1 Influenza

1.1.1 Definition and meaning of INFLUENZA

Influenza, also called flu or grippe, an acute viral infection of the upper or lower respiratory track that is marked by fever, chills, and a generalized feeling of weakness and pain in the muscles, together with varying degrees of soreness in the head and abdomen. The most common symptoms include: high fever, runny nose, sore throat, muscle and joint pain, headache, coughing, and feeling tired[5]. These symptoms typically manifest two days after exposure to the virus and at most last less than a week. The cough, how- ever, may last for more than two weeks[5]. In children, there may be diarrhea and vomiting, but not common in adults[6]. Diarrhea and vomiting occur commonly in gastroenteritis, which is an unrelated disease and sometimes inaccurately referred to as "stomach flu" or the "24-hour flu"[6].

1.1.2 Signs and Symptoms

Seasonal influenza is manifested by a sudden fever, headache, dry coughing, muscle and joint pain, severe malaise, sore throat and a runny nose. The cough can last at least 14 days. Majority recuperate within seven days without requiring medical attention. This disease can lead to illness or death especially in people at high risk. Illnesses range from mild to severe and to even death. High risk groups has high chance of getting infected and die. The annual epidemics are approximated to result about 3 to 5 million cases of severe illness, and about 290 000 to 650 000 respiratory deaths. Most deaths caused by influenza happens among people age 65 or older[5]. Epidemics leads to worker/school absenteeism and loss in productivity. Health facilities are overwhelmed greatly.
Research conduct estimated 99% of fatality is among the children under 5 years with influenza related lower respiratory tract infections are found in developing countries[7]. The effects of seasonal influenza epidemics in developing countries are not fully known

1.1.3 The Pathogen (Causes and Types)

Influenza is caused by viruses and transmitted through air. Influenza A and B viruses circulate and are responsible for seasonal epidemics of disease.[7], [8]

 Figure 1. FLU VIRUS

Influenza A viruses are classified into subtypes on the basis of two surface antigens (foreign proteins) combinations — hemagglutinin (HA) and the neuraminidase (NA), the proteins on the surface of the virus. Currently circulating in humans are subtype A(H1N1) and A(H3N2) influenza viruses known to have caused pandemics. Influenza A virus can be transmitted from wild birds to other species. This causes outbreaks in domestic poultry which spread into human influenza pandemic.

Influenza B viruses are not classified into subtypes, but can be broken down into lineages. Currently circulating influenza type B viruses belong to either B/Yamagata or B/Victoria lineage.

Influenza D viruses primarily affect cattle and are not known to infect or cause illness in people.

1.1.4 Global effect of influenza

Social impacts include intangible costs such as, domestic violence, behavoiral change, social interaction, pain, depression, suffering and impaired quality of life. Locally, mis- diagnosis of influenza viruses and other respiratory viruses contributes to misuse of an- timalarial and an, behavoiral change, social interaction, pain, depression, suffering and impaired quality of life. Locally, misdiagnosis of influenza viruses and other respiratory viruses contributes to misuse of antimalarial and antibiotic drugs. Europe and America research shows that if immunization programs target children under two years and the elderly over 65 years, the cost of influenza infection could be reduced by over 50%.

1.2 Influenza pandemics

Influenza pandemics are approximated to occur in intervals of half milenium. Epidemics are more frequent, and occurs in most parts of the world per year. Pandemic affecting the world can happen within a matter of months due to an antigenic shift. The influenza virus has led to repeated epidemics of great febrile syndrome every 1 to 4 years during the recent centuries. The initial epidemic report of an influenza-like illness was experienced in 1173–74, [?] but the first definite epidemic was reported in 1694 [9].
1.2.1 Case Study

The following list discussed but not least, shows some of the pandemics experienced in history.

The Spanish flu pandemic of 1918 – 19.

This was the most severe outbreak of the 20th century with greatest numbers of mortality, among the devastating pandemics in human history. This affected mostly children and elderly with low immunity. This killed approximate 40–50 million deaths according to recent research.

The Asian flu pandemic of 1957

This outbreak of influenza was first identified in February 1957 in East Asia and that in sequence spread to countries worldwide. It was the second major influenza pandemic to occur in the 20th century. The 1957 outbreak was caused by influenza A subtype H2N2, or Asian flu virus [1, 3]. Research has indicated that this virus was a reassortant strain, originating from strains of avian influenza and human influenza viruses. In the 1960s the human H2N2 strain underwent a series of minor genetic modifications, a process known as antigenic drift. The little alterations produced occasional epidemics. Months later, several cases of infection were reported, especially in young children, the elderly, and pregnant women as a results of second pandemic wave of illness that struck the Northern Hemisphere in November 1957 and was also already widespread in the United Kingdom. By December a total of some 3,550 deaths had been reported in England and Wales. The second wave was particularly devastating, and by March 1958 an estimated 69,800 deaths had occurred in the United States[1, 2, 4].

Table 1 refers to the AntigenShift and the Pandemics involved.



Viral Strain

Death Toll





Spanish flu Pandemic

H1N1 (A/Brevig Mission/1/18; A/South Carolina/1/18; and A/New York/1/18)

50 - 100 million


Asian flu Pandemic

H2N2 (A/Singapore/1/57)

1 million


Hong Kong flu Pandemic H3N2

H3N2 (A/NT/60/68; and A/Hong Kong/1/68)

1 million


Pandemic in children and young adults (Russian flu)

H1N1 (A/USSR/90/77)


USA, Canada, Singapore, Malaysia, Egypt, Europe.

H1N2 (A/New Caledonia/20/99 H1 and A/Moscow/10/99 N2)


Northern Italy

Influenza B (B/Victoria/2/87)


St. Elisabeth Hospital, Tilburg

H3N2 (A/Sydney/5/97)


Poultry Farm in Netherlands

H7N7 (Avian Influenza A)


USA, Canada, Europe, Japan

H3N2 (A/Fujian/411/2002)

1.3 Vaccination

1.3.1 Meaning

This is the vaccine inoculation to protect from a particular strain of disease. It is de- pendent on the immune status of the recipient. The ideal way to protect people from influenza is to administer vaccinations annually.

1.3.2 Types of vaccines

Each is designed to train the immune system how to combat certain types of germs — and the serious effects they cause. Depending on a list of factors, researchers chose the kind of vaccine they will produce. The main types of vaccines:

Live-attenuated vaccines.
Inactivated vaccines.
Subunit, recombinant, polysaccharide, and conjugate vaccines.
Toxoid vaccines 2.

1.4 Epidemiology and Target Group

Influenza is easily spread, mainly in areas with many people including institutions and nursing places. Frequent hand washing, covering mouth and nose with face masks can prevent transmission. Seasonal epidemics spread mainly during winter in temperate climate, and throughout the year in tropical regions, leading to irregular outbreaks.

1.4.1 Expectant mothers

This is the group with high risk of contaminating with the disease. They require immunization to avoid the negative impact to the unborn. They are also very weak at this season hence exposing them into high risk of getting sick. Conclusion: Expectant mothers should be frequently checked and immunized regularly. This is to protect the mother and the infants.

1.4.2 Children

Young children are susceptible to disease infections leading to burden associated with influenza with frequent clinic visits, hospitalizations and deaths compared to non-elderly adults. The vaccine effectiveness among children varies. Inactivated vaccines are the best
vaccination programs below two years old. Those older than two years, best suit for either inactivated vaccine (IV) or live-attenuated vaccine (LAIV).

1.4.3 Elderly People

Influenza is a primary cause to high deaths among the elderly. High income countries have lower risk for elderly deaths than low income countries. The risk of morbidity and mortality in the elderly can be reduced with the use of inactivated vaccines. However the effectiveness reduces with ageing and among those with underlying medical conditions.

1.4.4 People with special health conditions

People with unhealthy conditions, such as cardiac disease, morbid obesity, chronic respi- ratory and compromised immune status, are more prone to develop fatal sickness due to influenza infection than healthy people of the same age group. Influenza vaccine effec- tiveness has been demonstrated among individuals with underlying health conditions in a number of settings. This group has been targeted for influenza vaccination, and con- tinue to be an appropriate targeted group for vaccination. Identification of individuals is difficult due to stigma related issues.

1.4.5 HealthCare Workers

This group suffers high risk due to exposure, compared to the general population. Vacci- nating this group is ideal and effective and increases work attendance. Healthcare work- ers vaccination not only protects the individual, but also sustain healthcare services dur- ing epidemics and protection of vulnerable patients.

1.5 Aim of the Project/objectives

Vaccination being as an effective strategy against infection, does not guarantee due to different immune status of individual. Seasonal drift in the viral genome, annual vac- cination against the influenza virus strains anticipated to be in circulation during the upcoming season is necessary to prevent new infections and subsequent outbreaks. Fail- ures: present influenza vaccine to protect all vaccine recipient warrant the determination. Aim

To unveil through Mathematical Modelling, the influence of a partially effective vac- cine on the influenza infection transmission dynamics.

To determine condition under which an epidemic happen.

If it occur, what fraction of a uniform population get ill.
Disease spreading forecasting.

For strategic development to control and elimination.

This address question whether such vaccine can permanently terminate the disease spread. While Mathematical Model establish the strength of imperfect vaccine to tame influenza, having several models suggested for transmission dynamics of influenza, the impact of an imperfect vaccine hasn’t been fully analyzed. The model aim to recruit susceptible individuals and infected individuals into the population.

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