A MATHEMATICAL MODEL ON THE DYNAMICS

Ebere Omeje

MARK ADAUGO NAOMI CHIDINMA

PG/M.Sc/12/64534

A MATHEMATICAL MODEL ON THE DYNAMICS OF EBOLA VIRUS DISEASES IN HUMAN

POPULATION

DEPARTMENT OF MATHEMATICS

FACULTY OF PHYSICAL SCI

Ebere Omeje Digitally Signed by: Content manager’s Name

DN : CN = Webmaster’s name

O= University of Nigeria, Nsukka

OU = Innovation Centre

i

MARK ADAUGO NAOMI CHIDINMA

A MATHEMATICAL MODEL ON THE DYNAMICS OF EBOLA VIRUS DISEASES IN HUMAN

DEPARTMENT OF MATHEMATICS

IENCE

Digitally Signed by: Content manager’s Name

DN : CN = Webmaster’s name

O= University of Nigeria, Nsukka

OU = Innovation Centre

ii

A MATHEMATICAL MODEL ON THE DYNAMICS OF EBOLA VIRUS DISEASES IN HUMAN

POPULATION

BY

MARK ADAUGO NAOMI CHIDINMA

PG/M.Sc/12/64534

A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE ACADEMIC REQUIREMENT FOR THE AWARD OF MASTER OF SCIENCE (M.Sc) DEGREE IN APPLIED MATHEMATICS FROM THE DEPARTMENT

OF MATHEMATICS, FACULTY OF PHYSICAL SCIENCES,

UNIVERSITY OF NIGERIA, NSUKKA

DECEMBER, 2015

iii

CERTIFICATION

This work is carried out by MARK, ADAUGO NAOMI CHIDINMA, PG/M.Sc/12/64534 under the supervision of Prof. G.C.E Mbah both of the Department of Mathematics, University of Nigeria, Nsukka. It is an original work and has not been submitted in part or full for any other degree to this University or any other University.

Mark Adaugo Naomi Chidinma Date

(Student)

Prof. G.C.E Mbah Date

(Supervisor)

Prof. M.O. Oyesanya Date

(Head of Department)

External Examiner Date

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DECLARATION

I declare that the contents of this thesis are original except where due

references has been made. It has not been submitted before for any other

degree to any other institution.

MARK ADAUGO NAOMI CHIDINMA DECEMBER 2015

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DEDICATION

To God Almighty

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ACKNOWLEDGEMENT

I am eternally grateful to God Almighty, who alone made this work possible. May His name alone be praised!

I deeply appreciate my Supervisor, Prof. G.C.E Mbah, for his unflinching support, patience and thorough guidance to me throughout this work. He is indeed a father. My sincere appreciation goes to all my lecturers in the Department of Mathematics UNN especially Prof. F.I. Ochor and Prof. M.O. Oyesanya who have been my fathers in my sojourn in this department.

I really appreciate Dr. & Mrs. Alhassan and their children for their encouragement, prayers and support they gave to me throughout this program. May the Lord reward your labour of love.

I am also grateful to my course mates and friends: Eze Sunday, Onah Sunday, Didiugwu Cornelius, Chuks, Chima, Vincent, Ijeoma Okafor, Joy Shaibu, Ruth Omachoko, to mention but a few who helped me in one way or the other. I really enjoyed working with you.

To my friends and colleagues at work: Mr. E.C. Ozor, Mr. Gberokoo, Mr. Ubi, Mr. Agbenla, Mrs. Anumodu, Mrs Anozie, Njoku Joy and Peter Anozie, you have really been a pillar of support to me.

To my parents Sir Barr.& Lady I.C. Mark and to my sibling Chimdiya Mark, your support has been unquantifiable. Thanks a million!

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ABSTRACT

Ebola Virus Disease (EVD) is a rare, acute and infectious diseases caused by

Ebola Zaire virus, one of the five strains of the Ebola virus (filovirus), which

is often fatal if untreated. In this research work, we developed a model that

describes the dynamics of Ebola Virus Disease (EVD) and the human

population compartments involved with vital dynamics (birth and death

rates), incorporating treatment of exposed individuals and quarantining of

infectious individuals which are influenced by public enlightenment

campaign, availability of isolation centres and surveillance coverage. A

system of nonlinear differential equations was formulated for the

transmission. Stability analysis of the model indicated that, the Disease Free

Equilibrium (DFE) where the contact rate between the infected and

infections individuals undergoing treatment and the susceptible individuals

is very negligible is stable and indeed, when the Basic Reproduction

Number Ro is 1, the Disease Free Equilibrium (DFE) and the Endemic

Equilibrium (EE) coincide. The model shows that with enhanced public

enlightenment and quarantining structures put in place, very serious

outbreak with high mortality rate can be better controlled.

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TABLE OF CONTENT

TITLE PAGE i

CERTIFICATION . . . . . . . . . ii

DECLARATION . . . . . . . . . iii

DEDICATION . . . . . . . . . iv

ACKNOWLEDGEMENT . . . . . . . v

ABSTRACT . . . . . . . . . vi

TABLE OF CONTENTS . . . . . . . . vii

CHAPTER ONE: INTRODUCTION

1.0 What is Ebola? . . . . . . . . 1

1.1 Signs and symptoms of Ebola . . . . . . 1

1.2 Aim and objectives of the study . . . . . 2

1.3 Scope of study . . . . . . . 3

1.4 Limitations of study. . . . . . . . 3

CHAPTER TWO: LITERATURE REVIEW

CHAPTER THREE: BACKGROUND STUDY

3.0 Ebola as a disease . . . . . . . . 8

3.1 Test and diagnosis of Ebola. . . . . . . 10

3.2 Mode of transmission . . . . . . . 11

3.3 Control of spread . . . . . . . 13

3.4 Compartmentalization in a population . . . . . 20

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CHAPTER FOUR: THE MODEL

4.0 Model parameters . . . . . . . . 22

4.1 Flow diagram . . . . . . . . 23

4.2 Model Equations . . . . . . . . 24

4.3 Equilibrium Analysis . . . . . . 25

4.3.0 Disease free Equilibrium . . . . . . 25

4.3.1 Endemic Equilibrium . . . . . . 26

4.4 Stability Analysis . . . . . . . . 32

4.5 Numerical Simulation . . . . . . . 36

CHAPTER FIVE: SUMMARY

5.0 Discussion of Result . . . . . . . 40

5.1 Conclusion . . . . . . . . . 41

5.2 Recommendation . . . . . . . . 42

REFERENCES . . . . . . . . . 43

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CHAPTER ONE

INTRODUCTION

1.0 What is Ebola?

Ebola is an acute and infectious disease marked by fever and severe internal

bleeding, spread through contact with infected body fluids by a filo virus (Ebola

virus), previously known as Ebola condition in humans and nonhuman primates

such as monkeys, gorillas and chimpanzees. Ebola is one of the several viral

hemorrhagic fevers (VHF) caused by infection with virus of the filoviridae family,

genus Ebola virus. The first cases of Ebola were reported simultaneously in 1976

in Yambuku and the surrounding area, near the Ebola River in Zaire, which is now

the Democratic Republic of the Congo and in Nzara, Sudan where it takes its

name.

1.1 Signs and Symptoms of Ebola

The time interval from infection with Ebola to the onset of symptoms is 2 to

21 days, although 8 to 10 days is said to be most common. Humans are not

infectious until they develop symptoms.

Ebola Virus Disease (EVD) is often characterized by the abrupt onset of

fever, intense weakness, muscle pain, headache and sore throat. These signs are

usually followed by vomiting, diarrhea, rash, impaired kidney and liver function,

and in some severe cases, both internal and external bleeding example, oozing

from the gums, blood in the stools).

In summary, the signs and symptoms of EVD may include;

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i. Fever

ii. Headache

iii. Joint and muscle aches

iv. Weakness

v. Diarrhea

vi. Vomiting

vii. Stomach pain

viii. Lack of appetite

Some patients may experience:

ix. A rash

x. Red eyes

xi. Hiccups

xii. Cough

xiii. Some throat

xiv. Chest pain

xv. Difficulty in Breathing

xvi. Difficulty in Swallowing

xvii. Bleeding inside and outside of the body

1.2 Aim And Objectives of the Study

The aim of this model is to model mathematically the transmission dynamics

of Ebola Virus Disease (EVD) and specifically to:

1. Carry out a detailed study of the Ebola Virus Disease;

2. Develop a mathematical model of the transmission dynamics of Ebola-Zaire

strain (Zaire ebolavirus (EBOV));

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3. Investigate the existence and stability of the equilibrium state of the model;

4. Interpret the results of the analysis;

5. Use the study in providing useful contribution that will aid in the awareness

of the dynamics and possible control measures of the disease.

1.3 Scope of the Study

This work covers a detailed study of Zaire Ebola Virus (EBOV) dynamics of

the disease and possible control measures for the spread of the disease. A

mathematical model was developed using a system of first order ordinary

differential equations. Equilibrium analysis of the disease was done using the

model equations, a test for stability of the Equilibrium and Numerical

simulation for the model was carried out.

1.4 Limitations of the Study

This work was carried out using secondary data and not experimental data.

Only the equilibrium analysis, stability analysis and numerical simulation were

carried out on the model equations to understand the behaviour of the

transmitted disease over time.

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CHAPTER TWO

LITERATURE REVIEW

In this chapter we review works done by other researchers on mathematical

modeling, especially the mathematical models on Ebola Virus Disease. According

to Logan(2010) a mathematical model is an equation, or set of equations, that

describes some physical problem or phenomenon that has its origin in science,

engineering, or some other areas while mathematical modeling is the process by

which we obtain and analyze the model. Mathematical models have been important

tools in analyzing the epidemiological characteristics of infectious disease since

the pioneer work of Kermack and Mckendrick (1927).

Some of the well known models for the transmission dynamics of some

diseases include: Ronald Ross model for the control of malaria (Ross, 1915);

Capasso and Parei-Fontana (1979) model for the 1973 Cholera epidemic and the

Hethcote and Yorke (1984) model for the spread and control of gonorrhoea.

Ebola virus disease (EVD) has erupted many times in some zones since it

was first found in 1976 and many models have been done to control the spread.

The outbreak of EVD in 2014 started from Guinea, then spread through West

Africa of which the most serious region is Liberia. Until November 14, 2014, the

World Health Organization had reported 14, 415 cases and 5, 506 cases died. The

WHO declared Nigeria EVD free on October 20, 2014, after no new cases had

been detected for 42 days (Who, 2014). Althaus et al (2015) fitted an EVD

transmission model to the reported daily numbers of incident cases and death

during the outbreak in Nigeria which allowed them to estimate the basic

reproduction number Ro, and to describe how the net reproduction number Rt

changed after control intervention were implemented. They then compared the

risks of an outbreak from a single undetected case in Nigeria and the other West

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Africa countries with ongoing EVD transmission. They applied an EVD

transmission model which followed SEIR susceptible-exposed-infections-

recovered dynamics that they used to estimate the reproduction number of EVD in

Guinea, Sierra Leone and Liberia (Althaus, 2014).

Webb. et al (2015) presented a model which consists of the populations at

time t of susceptible S(t) (capable of being infected), exposed E(t) (incubating

infected), I(t) (infectious infected), isolated infectious II(t) (exposed and infectious

infected) who have been identified and isolated from the susceptible population),

which they applied to Sierra Leone and Liberia by first fitting WHO data for each

country from outbreak in the Spring of 2014 to September 23, 2014. They then

simulated forward projections of the epidemic in each of these countries, based on

varied efficiencies in identifying, isolating, and contact tracing of infected

individuals. Their model predictions indicated that the containment of the epidemic

requires a high level of both the general identification and isolation process and the

contact training process for removing infectious individuals from the susceptible

population.

Attangana and Goufo (2014) constructed a model which followed SIRD

(susceptible-in-infected-recovered-total death population), transmission dynamic

describing the spread of Ebola hemorrhagic fever. Their model was first

constructed using the classical derivative and then converted to the generalized

version using the beta-derivative. They studied in detail the endemic equilibrium

points and provided the eigenvalues associated using the Jacobian method. They

first ahead with their investigation by solving the model numerically using an

interaction method. Their simulations were done in terms of time and beta. Their

study showed that, for small portion of infected individuals, the whole country

could die out in a very short period of time in case there is not good prevention.

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Amenagbawon and Aboubakary (2015) performed a study that simulated

the transmission dynamics of Ebola Zaire virus using two models: a modified SIR

model with the understanding that the recovered can become infected again and the

infected die at a certain rate and a quarantine model, which ascertained the effects

of quarantining the infected. They formulated system of Ordinary Differential

Equations (ODE) from the transmission and the method of linearized stability

approach which they used to solve the equations. Their stability analysis of both

models medicated that, the Disease Free Equilibrium (DFE) states of the models

were unstable of they exist. They results showed that, with the nature of Ebola

Zaire virus, uncontrolled transmittable contacts between the infected and the

susceptible can lead to a very serious outbreak with high mortality rate, but with

effective quarantining structures put in place such situation can be better managed

and out Break controlled.

Abdulrahman et al (2015) developed and analyzed a model that followed an

SLIR (susceptible-latent-infected-recovered) transmission dynamics for controlling

the spread of Ebola Virus Disease (EVD) in a population with vital dynamics (birth

and death rates not equal), incorporating quarantining of infectious individuals

which they said to be influenced by availability of isolation centres and

surveillance coverage. They also considered improved personal hygiene of the

susceptible population influenced by public enlightenment campaign. Their

numerical simulations showed that improved personal hygiene and quarantining of

infectious individuals are enough to control the spread of EVD, with improved

personal hygiene being the more effective and efficient of the two control

parameters.

Zhi-QiangXia, et al. (2015) developed and analyzed a model of EVD

transmission in Liberia with seven compartments SEIsIpHFR (susceptible,

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exposed, suspected infectious individuals, probable infected individuals,

hospitalized cases, dead but not yet buried, individuals removed from the chain if

transmission). They investigated the impact of different transmission routes on the

EVD outbreak in Liberia and estimate the basic reproduction number R0=2.012 in

the absence of effective control measures, based on the data released by World

Health Organization and the actual transmission situations. Their sensitivity and

uncertainty analysis revealed that the transmission coefficients of suspected and

probable cases have stronger correlations on the basic reproduction number. They

went ahead to study the influence of control measures (isolation and safe burial

measures) on EVD outbreak. They found that if combined control measures were

taken, the basic reproduction number will be less than one and thus EVD in Liberia

would be well contained.

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CHAPTER THREE

3.0 Ebola as a Disease

Ebola Virus Disease is a disease caused by a virus of genus Ebolavirus.

Genus Ebolavirus is one of the three members of the Filoviridae family (filovirus),

along with genus marburgvirus and genus cuevarirus. The genus ebolavirus is a

virological taxon included in the family filoriridae, order mononegavirales

according to Kuhn et al (2010). The members of this genus are called ebolaviruses.

The Genus Ebolavirus comprises of five distinct subspecies which are named after

the region where each was originally identified.

a. Bundibugyo Ebolavirus (BDBV)

b. Zaire Ebolavirus (EBOV)

c. Reston Ebolavirus (RESTV)

d. Sudan Ebolavirus (SUDV)

e. Tai Forest Ebolavirus (TAFV)

The virus causing the 2014 West African Outbreak belongs to the Zaire

species. The first cases of Zaire ebolavirus (EBOV) were reported to have

appeared in 1976 in Yambuku and the surrounding area near the Ebola River in

Zaire, which is now the Democratic Republic of the Congo and in Nzara, Sudan,

where it takes its name Ebola virus is the only member of the Zaire ebolavirus

species and the most dangerous (Leroy et al, 2007). The virus is most commonly

spread by personal contact, and it has incubation period of two to twenty-one days

of takes approximately eight hours for the virus to replicate, and can occur several

times before the onset of symptoms. Hundreds to thousands of new virus particles

are then released during periods of hours to a few days, before the cell dies

according to Healthlink USA (2015). Once the virus enters the body, it targets

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several types of immune cells that represent the first line of defense against

invasion. It infects the dendritic cells, which normally display signals of an

infection on their surfaces to activate T lymphocytes- the white blood cells that

could destroy other infected cells before the virus replicates further with defective

dendrotic cells failing to give the right signal, the T cells don’t respond to

infection, and neither do the antibodies that depend on them for activation. The

virus can start replicating immediately, and very quickly. According to a study

published in Cell Host & Microbe n 13th August, 2014, researchers found that one

of Ebola’s proteins, called VP24, binds to and blocks a transport protein on the

surface of immune cells that plays an important roe in the interferon pathway. The

interferon is a type of molecules that cells use to hinder further viral reproduction.

The white blood cells themselves don’t become infected with the virus, but a series

of other factors-a lack of stimulation from some cells and toxic signals from other-

prevent these primary immune cells from putting up a fight. As the virus travels in

the blood to new sites, other immune cells called macrophages eat it up. Once

infected, they release proteins that trigger coagulation, forming small clots

throughout the blood vessels and reducing blood supply to organs. They also

produce other inflammatory signaling proteins and nitric oxide, which damage the

living of blood vessels, causing them to leak. This damage leads to internal

hemorrhaging- bleeding from the eyes, nose, or other orifices, which is one of the

main symptoms of the infection, though not all patients exhibit external

hemorrhaging. Ebola wipes out cells required to produce coagulation proteins and

other important components of plasma, which affects the liver. Damaged cells in

the gastrointestinal tract lead to diarrhea that often puts patients at risk of

dehydration. And in the adrenal gland, the virus cripples the cells that make

steroids to regulate blood pressure and causes circulatory failure that can starve

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organs of oxygen. The damage to blood vessels leads to a drop in blood pressure,

and patients die from shock and multiple organ failure.

3.1 Test and Diagnosis of Ebola

It can be difficult to distinguish Ebola Virus Disease (EVD) from other

infectious diseases such as malaria, typhoid fever and meningitis. Before Ebola can

be diagnosed, other diseases should be ruled out such as:

a. Malaria

b. Typhoid fever

c. Shigellosis

d. Cholera

e. Leptospirosis

f. Plague

g. Rickettsiosis

h. Relapsing fever

i. Meningitis

j. Hepatitis

k. Other viral hemorrhagic fevers

Confirmation that symptoms are caused by Ebola virus infection are made

using the following investigations according to the World Health Organization:

i. Antibody-capture enzyme-linked immunosorbent assay (ELISA)

ii. Antigen-capture detection tests

iii. Reverse transcriptase polymerase chain reaction (RT-PCR) assay.

iv. Defection microscopy

v. Virus isolation by cell culture

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In the more advanced stages of the disease or after recovery, the diagnostic

test available is

i. IgM and IgG antibodies

Ebola can be diagnosed in diseased in deceased patients by

i. Immunohistochemistry testing

ii. PCR

iii. Virus isolation

According to the World Health Organization, samples from patients are

extreme biohazard risk., laboratory testing on non-inactivated samples should be

conducted under maximum biological containment conditions.

3.2 Mode of Transmission

In an outbreak or isolated case among humans, the manner in which the

virus is transmitted from the natural reservoir to a human is unclear. It is thought

that fruit bats of the pteropodidae family are natural Ebola virus hosts. Ebola is

introduced into the human population through close contact with the blood,

secretions, organs or other bodily fluids of infected animals such as chimpanzees,

gorillas, fruit bats, monkeys, forest antelope and porcupines found ill or dead or in

the rainforest.

Transmission of Ebola between humans can occur in several ways, including

through:

a. Direct contact through broken skin and mucus membranes with the blood,

secretions, organs or other bodily fluids of infected people.

b. Indirect contact with environments contaminated with such fluids.

c. Exposure to objects (such as needles) that have been contaminated with

infected secretions.

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d. Burial ceremonies in which mourners have direct contact with the body of

the deceased person can also play a role in the transmission of Ebola.

e. Men who have recovered from the disease can still transmit the virus

through their semen for up to seven (7) weeks after recovery from illness.

f. Health-care workers have frequently been infected while treating patients

with suspected or confirmed Ebola Virus Disease (EVD). This has occurred

through close contact with patients when infection control precautions are

not strictly practiced.

People remain infectious as long as their blood contains the virus. No formal

evidence exists of sexual transmission, but sexual transmission from convalescent

patients cannot be ruled out. There is evidence that live Ebola virus can be isolated

in seminal fluids of convalescent men for eight-two (82) days after onset of

symptoms. Evidence is not available yet beyond 82 days. There is no evidence of

live Ebola virus in vaginal secretions.

Ebola tends to spread quickly thorugh families and friends as they are

exposed to infectious secretions when caring for an ill individual. The virus can

also spread quickly within healthcare settings for the same reason, highlighting the

importance of wearing appropriate protective equipment, such as masks, gowns

and gloves.

There is no evidence that Ebola can be spread via insect bites.

• A WHO Ebola Situation assessment for October 6, 2014, states that the virus

is most easily transmitted through blood, faeces, and remit breast milk, urine

and semen have also been found to transmit the Ebola virus, and it is

believed that if may even be transmitted through tears and saliva according

to Alina Bradford, Live Science Contributor.

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3.3 Control of Spread

There is currently no licensed vaccine available for Ebola. Several vaccines

are being tested, but at this time none are available for clinical use.

We shall consider this subject in two (2) ways:

A. Control in Patients:

To improve survival, patients should undergo:

• Supportive care-rehydration with oral or intravenous fluids, thereby,

balancing the patient’s fluids and electrolytes.

• Maintenance of oxygen status and blood pressure.

• Treatment of specific symptoms.

There is as yet no proven treatment available for Ebola Virus Disease.

However, a range of potential treatments including blood products, immune

therapies and drug therapies and currently are being evaluated. No licensed

vaccines are available yet, but two potential vaccines are undergoing human

safety testing.

B. The Ebola virus can be eliminated from the environment with heat, alcohol-

based products, and sodium hypochlorite (bleach) or calcium hypochlorite

(bleaching powder) at appropriate concentrations. It is also susceptible to a wide

range of commonly used disinfectants, including aldehydes, halogens, peroxides,

phenolics, and quaternary ammonium compounds.

Good outbreak control relies on applying a package of interventions, namely

case management, surveillance, contact tracing, a good laboratory service, safe

burials and social mobilization. Raising awareness of risk factors for Ebola

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infection and protective measures that individuals can take is an effective way to

reduce human transmission.

Risk reduction messaging should focus on several factors:

• Reducing the risk of wildlife-to-human transmission from contact with

infected fruit bats or monkeys/ages and the consumption of their raw meat.

Hand gloves should be used to handle animals and their meats should be

thoroughly cooked before eating

• Reducing the risk of possible sexual transmission, because the risk of sexual

transmission cannot be ruled out. Men and women who have recovered from

Ebola should abstain from all after onset of symptoms. If sexual abstinence

is not possible, male or female condom is recommended and there should

not be any contact with body fluids.

• Outbreak containment measures, including prompt and safe burial of the

dead, identifying people who may have been in contact with someone

infected with Ebola and monitoring their health for twenty-one (21) days,

the importance of separating the healthy from the sick to prevent further

spread, and the importance of good hygiene and maintaining a clean

environment.

• Health-care workers should always take standard precautions when caring

for patients, regardless of their pressured diagnosis. These include basic

hand hygiene, respiratory hygiene, use of personal protective equipment

(ppe), to block splashes or other contact with infected materials, safe

injection practices, and safe burial practices.

We shall proceed by studying the control measures thus

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1. General Patient Care in any Health-Care Facility

Strengthen and carefully apply standard precautions when providing care to

All patients regardless of the signs and symptoms they present with. This is

especially important because the initial manifestations of HF may be non-specific.

Hand hygiene is the most important measure. Gloves should be worn for any

contact with blood or body fluid. Medical mask and goggles or face shield should

be used if there is any potential for splashes of blood or body fluids to the face and

cleaning of contaminated surfaces is burring Ebola virus out breaks, each health-

care facility in high-transmission affected areas should have a dedicated and well

equipped triage area at the entrance, to identify any potential Ebola virus case

seeking care is the facility. This area should be staffed with professional (e.g

doctor or nurse) trained on basic IPC principles and specific precautions for HF

(Hemorrhagic fever) and on the use of a standard algorithm to identify Ebola cases.

Staff in the triage area should wear a scrub suit, a gown, examination gloves and a

face shield. The area should be large enough to keep the patient at a 1-metre

distance at least and should be equipped with an easily accessible hand hygiene

facility (either alcohol-based handrub dispensers or a sink or a bucket with faucet

containing water, liquid soap and single-use towels), thermometer, bin with lid and

infectious waste plastic bags, a sharp’s container (if rapid diagnostic tests for

malaria or any other similar practice is meant to be performed here). The hand

hygiene technique posters and the standard triage algorithm to identify Ebola cases

should be clearly displayed in this area. Triage staff should follow a ‘no touch’

process when interviewing the patient. A distance of at least one metre (3 feet)

should be kept from the patient at all times, whenever possible.

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2. Direct Patient Care (for Suspected of Confirmed Patients with Ebola)

• Suspected or confirmed cases should be put in single isolation rooms with

adjoining dedicated toilet or latrine, showers, sink equipped with running

water, soap and single-use towels, alcohol-based handrub dispensers, stocks

of personal protective equipment (PPE), stocks of medicines, and

ventilation, screened windows, doors closed and restricted access, if

isolation rooms are unavailable, cohort these patients in specific confined

areas while rigorously keeping suspected and confirmed cases separate and

ensure the items listed here for isolation rooms are readily available. Patient

beds should be at least 1 metre (3 feet) apart.

• Clinical and non-clinical personnel should be assigned exclusively to Ebola

patient areas and members of staff should not move freely between the

Ebola isolation areas and other clinical areas during the outbreak.

• All non-essential staff should be restricted from Ebola patient care areas.

• Visitor access to the patient should be stopped, but if it is not possible, their

number should be limited to include only those necessary for the patient’s

well-being and care, such as child’s parents .

• Visitors wished to observe the patient should do so from an adequate

distance.

• Visitors should be screened before entering to see Ebola patients, for signs

and symptoms of Ebola.

• Visitors should perform hand hygiene and make use of personal protective

equipments (PPE) like double gloves, disposable gown, disposable apron

worm over the gown or coverall, fluid-resistant medical surgical mask with a

structured design that does not collapse against the month, eye protection,

water proof books, fluid-resistant particulate respirator.

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• While exiting, visitors should carefully remove and dispose of PPE into

waste containers.

• Single-use disposable PPE should not be recycled, but re-useable equipment

(like goggles and face shields) should be carefully cleaned and

decontaminated.

• Dedicated equipment (e.g stethoscopes) should be rigorously used for each

patient.

• Prevention of needle stick and injuries from other sharp instruments.

• Respiratory hygiene and cough etiquette.

• Environmental clearing by using adequate procedures for the routine

cleaning and disinfection of environmental and other frequently touched

surfaces.

• Management of linens to avoid transfer of pathogens to other patients and or

the environment.

• Proper waste disposal.

• Patient care equipment.

3. Non-patient care activities (for suspected or confirmed patients with Ebola

virus)

A. Diagnostic Laboratory Activities

• All laboratory sample processing must take place under a safety cabinet or at

least a fume cabinet with exhaust ventilation.

• Activities such as micro-pipetting and centrifugation can mechanically

generate fine aerosols that might pose a risk of transmission of infection

through inhabitation as well as the risk of direct exposure.

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• Laboratory personnel handling potential Ebola clinical specimens should

wear full set of PPE

• When removing PPE, avoid any contact between the soiled items (e.g

gloves, gowns) and any area of the face (that is, eyes, nose, or mouth).

• Apron or gown should not be reused, they should be discarded immediately.

• Hand hygiene should be performed immediately after the removal of PPE

used during specimen handling and after any contact with potentially

contaminated surfaces even when PPE is worm.

• Specimens should be placed in clearly-labeled, non-glass, leak-proof

containers and delivered directly to designated specimen handling areas.

• All external surfaces of specimen containers should be disinfected

thoroughly (using an effective disinfectant) prior to transport.

B. Movement and Burial of Human Remains

The handling of human remains should be kept to a minimum. The

following recommendations should be adhered to in principle, but may need some

adaptation to take account of cultural and religious concerns.

• Wear the full set of PPE.

• PPE should be put on at the site of collection of human remains, worm

during the process of collection and placement in body bags, and should be

removed immediately after. Hand hygiene should be performed immediately

following the removal of PPE.

• Remains should not be sprayed, washed or embalmed; any practice of

washing the remains in preparation for “clean burials” should be

discouraged.

• Only trained personnel should handle remains during the outbreak.

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• PPE is not required for individuals driving or riding a vehicle to collect

human remains, provided that drivers or riders will not be handling a dead

body of a suspected or confirmed case of Ebola.

• After wrapping in sealed, leak-proof bags, human remains should be placed

inside a coffin if possible, and buried promptly.

C. Post-Mortem Examinations

• Post-mortem examination of Ebola patient remains should be limited to

essential evaluations only and should be performed by trained personnel.

• Personnel examining remains should wear full set of PPE.

• In addition, personnel performing autopsies of known or suspected Ebola

patients should wear a particulate respirator.

• When removing PPE, contact between soiled gloves or equipment and the

face (eg eyes, nose or mouth) should be avoided.

• Hand hygiene should be performed immediately following the removal of

PPE.

• Tissue or body fluids for disposal should be carefully marked sealed

containers for incineration.

D. Managing exposure to virus through body fluids including blood .

• people including health workers, with per-coetaneous or mucus-coetaneous

exposure to blood, body fluids, secretions, or excretions from a patient with

suspected or confirmed Ebola should immediately and safely stop any

current tasks, leave the patient care area, safety remove PPE and wash the

affected skin surfaces with soap and water.

• Exposed people should be medically evaluated twice daily for 21 days after

the incident.

29

• People suspected to having been infected should be cared for, isolated, and

the same recommendations already given must be applied until a negative

diagnosis is confirmed.

• Contact tracing and following of family, friends, co-workers and other

patients, exposed to Ebola virus in essential.

3. 4 Compartmentalization of the transmission in a population

In this study, we shall be looking at the transmission of Ebola Virus Disease

(EVD) in a population (N) with seven (7) compartments:

i Susceptible class (S): This is a class of individuals very likely to be

influenced by the disease due to lack of personal hygiene and exposure to

infected/infectious individuals.

ii. Exposed class (E): This class is made of individuals that have contracted the

disease based on interaction of susceptible with infected persons but are not

yet capable of transmitting the disease, because the disease is at its

incubation period.

iii. Exposed under treatments class (Et): This class is a sub-class of the exposed

class. Here, individuals that are exposed, based on public enlightenment

about the disease, quickly seek for good and urgent treatment. If they are

properly treated, they do not get infectious.

iv. Infected and infectious class (I): This class is made up of individuals that

have already started developing symptoms of the disease, because the

disease has exceeded its incubation period. They are prone to die as a result

of the disease. Persons in this class are infectious to those in the susceptible

class.

30

v. Treated infected and infectious class (IT): This class is a subclass of the (I)

class. Individuals in this class are those that are infected and infectious, but

are undergoing treatment.

vi. Quarantined infected and infectious class (Iq): This class is also a subclass of

the (I) class. Individuals in this class are those that are infected and

infectious but based on public enlightenment have been quarantined in good

isolation centers.

vii. Recovered class (R): this class is made up of individual from the exposed

class under treatment and the infected and infectious individuals under

treatment, who have successfully received and respond treatment.

According to the Centres for Disease Control and Prevention (CDC),

research shows that patients who recover from Ebola can develop antibodies

that will prevent them from the virus for at least ten years (10) or possibly

even longer.

31

CHAPTER FOUR

THE MODEL

4.0 FORMULATION OF MODEL

The total population (N) is divided into seven (7) classes of Susceptible (S),

Exposed (E), Exposed Treated (��), Infected (I), Infectious under treatment (��),

Infectious Quarantined (��), and Recovered (R) individuals.

The model parameters are defined in Table 4.0.

Table 4.0 Model Parameters

Parameter Description

� Recruitment rate into the susceptible class, and those that do not

protect themselves from contact with infectious individuals due to lack

of enhanced personal hygiene.

� Rate of recruitment of susceptible into the exposed population based

on the contact rates with infectious individuals.

Rate of maturity from exposed to the infected and infectious class.

Rate at which exposed people receive treatment based on public

enlightenment

� Rate at which the recovered people join the susceptible class again

� Rate at which treated members of the exposed class recover due to

quick response to good treatment

�� Rate at which treated members of the exposed individuals get

infectious.

� Rate at which infectious individuals join the infectious class under

32

treatment.

�� Rate at which infectious individuals join the class of infectious

individuals under quarantine.

� Rate at which treated infectious individuals recover from the disease

due to timely and proper treatment.

� Natural death rate

� Contact rate between the susceptible class and infectious class not

treated or quarantined.

�� Contact rate between the susceptible and infectious under treatment.

� Rate at which untreated infectious individuals die due to the infection.

�� Rate at which the infected and quarantined individuals die as a result of

the disease.

FIG 4.1 EBOLA TRANSMISSION DYNAMICS FLOW DIAGRAM

� � �

��

�� ��

��

�

�

�

�� �

��

�

�

�

� + ��

� + � � �

�

�

33

MODEL EQUATIONS

���� = � − (� + �)� + �� - - - - - (4.1)

���� = �� − (� + + )� - - - - - - (4.2)

����� = � − (� + � + ��)�� - - - - - - (4.3)

� �� = � + ��� − !� + �� + (� + � )"� - - - - - (4.4)

� ��� = � � − (� + �)�� - - - - - - (4.5)

� #�� = ��� − (� + ��)�� - - - - - - (4.6)

�$�� = � �� + ��� − (� + �)� - - - - - - (4.7)

where � = � � + ����

In this model we shall make some useful assumptions

1. Rate at which recovered individuals become susceptible � is negligible.

2. Every infectious individual under treatment does not die as a result of the

disease.

3. Every dead individual is buried properly and timely to avoid contamination.

Hence, since � = 0, equations (4.1) and (4.7) become

���� = � − (� + �)� - - - - - (4.1*)

�$�� = � �� + ��� − �� - - - - - - (4.7*)

34

4.3 EQUILIBRIUM ANALYSIS

4.3.1 DISEASE FREE EQUILIBRIUM

At equilibrium, equations (1*) – (7*) are set equal to zero.

That is;

���� = ��

�� = ����� = �

�� = � ��� = � #

�� = �$+�� = 0

We define

!�, �, �� , �, �� , �� , �" = (�-, �-, ��-, �-, ��-, ��-, �-) in equations (1*) – (7*).

Consequently,

� − (� + �)� = 0

�- = ./01 = .

/023 024 � - - - - - - (i)

��- − (� + + )�- = 0

�- + ����- − !� + �� + (� + � )"�- = 0

�- = 5�6074��6830840(/093) - - - - - - (ii)

� �- − (� + �)��- = 0

��- = 83 6:0/ - - - - - - (iii)

���- − (� + ��)��- = 0

35

��- = 84 6/094 - - - - - - (iv)

� ��- + ��- − ��- = 0

�- = 73��60; �6/ - - - - - - (v)

In a disease free equilibrium,

�- = ��- = 0 - - - - - (+)

Substituting (+) in (i), (ii), (iii), (iv), and (v), we have

�- = ./ , �- = ��- = ��- = �- = 0

!�-, �-, ��-, �-, ��-, ��-, �-" = (./ , 0, 0, 0, 0, 0, 0) - - - (++)

Equation (++) is the disease free equilibrium (DFE).

4.3.2 ENDEMIC EQUILIBRIUM

We define !�, �, �� , �, �� , �� , �" = (�∗, �∗, ��∗ , �∗, ��∗ , ��∗, �∗) and set equations (1*)

– (7*) equal zero respectively

� ��� = 0 ⇒ � �∗ = (� + �)��∗

⇒ ��∗ = 83 ∗:0/ - - - - - - (a)

� = � �∗ + ����∗

= � �∗ + �� 83 ∗:0/ - - - - - - (b)

36

����� = 0 ⇒ �∗ = (� + � + ��)��∗

⇒ ��∗ = ;�∗/073074 - - - - - - (c)

� �� = 0 ⇒ �∗ + ���� = !� + �� + (� + � )"�∗

�∗ + 74;�∗930/073074 = !� + �� + (� + � )"�∗

�∗ > + 74;�∗930/073074? =

!830840(/093)" ∗@

Let � + � = �∗

And A = + 74;/∗073074

⇒ �∗ = (830840/∗) ∗@ - - - - (d)

���� = 0 ⇒ ��∗ = (� + + )�∗

�∗ = B(/0;05)�∗ 1

= (/0;05) .>(D3ED4EF∗)G∗H ?

∗>230 I4JEF?

�∗ = (/0;05) .>(D3ED4EF∗)H ?

230 I4JEF - - - - - (4.8)

���� + ��

�� = � − ��∗

= (� + + )�∗

37

⇒ �∗ = .K/�∗/0;05 = .K/�∗

@3 LℎNON A = � + +

�∗ = � − � @3H4H230 I4JEF

LℎNON A� = � + � + ��

�∗ = � − � @3@4@>230 I4JEF? - - - - (4.9)

We substitute (++) in (c)

��∗ = ;�∗/073074

= (� − /@3@4@>230 I4JEF?

��∗ = >� − /(:0/)@3@4@(:0/)(23024)? - - - - - (4.10)

From (d)

�∗ = (830840/∗) ∗@

⇒ �∗ = @�∗830840/∗

We substitute (++), to get

�∗ = @830840/∗ >� − /(:0/)@3@4

@(:0/)(23024)? - - - - (4.11)

� #�� = 0 From equation (6), we have

��∗ = 84 ∗/094 = 84 ∗

/∗∗ LℎNON �∗∗ = � + ��

We substitute (4*) in (6) to get

38

��∗ = 84/∗∗ P @

830840/∗ >� − /(:0/)@3@4@(:0/)(23024)?Q - - - - (4.12)

� ��� = 0 ⇒ ��∗ = 83 ∗

:0/

We substitute the value of �∗ to get

��∗ = 83:0/ P @

830840/∗ >� − /(:0/)@3@4@(:0/)(23024)?Q -- - - - (4.13)

�$�� = 0 ⇒ �∗ = 73��0; �

/

�∗ = / P� >� − (:0/)/;@3@4

@(:0/)(23024)?Q

�∗ = / (� � − 73.;@3@4(:0/)

@(:0/)(23024) + ;83:0/ P R

830840/∗ >� − /(:0/)@3@4@(:0/)(23024)?Q - (4.14)

Therefore at endemic equilibrium,

!�, �, �� , �, �� , �� , �" = (�∗, �∗, ��∗ , �∗, ��∗ , ��∗, �∗)

Is given by equations (4.9) – (4.14)

4.4 BASIC REPRODUCTION NUMBER

The basic reproduction number denoted by R0 is a parameter used to determine

how long a disease will prevail in a particular population.

To derive the basic reproduction number R0 of the DFE, we employ the next

generation operator technique described by Diekmann and Heesterbeek (2000),

39

and which was subsequently analysed by Vanden and Watmough (2002). This is

given as

�- = (S) , (S) denotes the spectral radius of the next generation matrix k.

S = TUK

Rewriting equations (4.1) to (4.7) starting with the infected compartments for the

population:

�, �� , �, �� , �� and then followed by the uninfected classes: �, �, then the model

becomes.

���� = �� − (� + + )� - - - - - - (4.15)

����� = � − (� + � + ��)�� - - - - - - (4.16)

� �� = � + ��� − !� + �� + (� + � )"� - - - - - (4.17)

� ��� = � � − (� + �)�� - - - - - - (4.18)

� #�� = ��� − (� + ��)�� - - - - - - (4.19)

���� = � − (� + �)� + �� - - - - - (4.20)

�$�� = � �� + ��� − (� + �)� - - - - - - (4.21)

40

From the equations above � VWX � are defined as

��

0

� = 0

0

0

0 0 � �- ���- 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

(� + + )�

−� + (� + � + ��)��

−� − ���� + (� + �� + �∗)�

−� � + (� + �)��

−��� + �∗∗��

Let S ∗ = � + + ,

� =

U =

41

S�∗ = � + � + ��,

SY∗ = � + �� + �∗, SZ∗ = � + � .

Taking the partial derivatives of V with respect to (�, �� , �, �� , ��)

S ∗ 0 0 0 0

− S�∗ 0 0 0

− −�� SY∗ 0 0

0 0 −� SZ∗ 0

0 0 −�� 0 �

�- = N[\NW�V]^N_ `� (TUK )

�- = 23�6(5R40;74)R3R4Ra + 2a�683(5R40;74)

R3R4RaRb

U =

42

4.5 STABILITY ANALYSIS OF DISEASE FREE EQUILIBRIUM

To analyse the local stability of the disease-free equilibrium we obtain the Jacobian

matrix of the malaria model (4.1) to (4.7) at the disease free equilibrium point, and

also use our basic reproduction number of the model.

Theorem 4.1

The disease free equilibrium point for system (4.1) to (4.7) is locally

asymptotically stable if �- < 1 and unstable if �- > 1.

Rewriting (4.1) to (4.7) and getting the Jacobian matrix (J) of the malaria model

(4.1) with � = f − (� + �� + � + �� + �� + �)

−� 0 0 −� �- −���- 0 0

0 −S 0 � �- ���- 0 0

0 −S� 0 0 0 0

0 �� −SY 0 0 0

0 0 0 � −SZ 0 0

0 0 0 �� 0 −� 0

0 0 � 0 0 −�

g- =

43

If �� = 0

−� 0 0 −� �- 0 0 0

0 −S 0 � �- 0 0 0

0 −S� 0 0 0 0

0 �� −SY 0 0 0

0 0 0 � −SZ 0 0

0 0 0 �� 0 −� 0

0 0 � 0 0 −�

From the Jacobian matrix of J00, we obtain the eigenvalues

� = −�

�� = −S

�Y = −S�

�Z = −SY

�h = −SZ

�i = −�

�j = −�

Since all the eigenvalues are negative, we conclude that the disease free

equilibrium (DFE) is locally stable.

g-- =

44

For

�� = 0, LN XNkNOl[WN �- VWX �WXNl[m �n^[][oO[^l VWX _ℎ`L kℎVk Vk �- =1, �WXNl[m �n^[][oO[^l VWX p[_NV_N TONN �n^[][oO[^l m`[Wm[XN

�∗ = (/0;05)(D3ED4EF)q23 = R3Ra

R23

o^k S = + 74;/073074

= + 74;R4 = R45074;

R4

⇒ �∗ = R3RaR r

23

= R3Ra(R45074;) s R4

23

= R3R4Ra23(R45074;)

�∗ = �6$6 - - - - - - (*)

�� �- = 1 ⇒ �∗ = �-

�∗ = .K/t6u6

/0;05

�� �- = 1 �∗ = .K/�6

/0;05 �- = ./

= /P>vFK�6?Q

/0;05

⇒ �∗ = 0 = ��∗ = �∗ = ��∗ = ��∗ = �∗

45

Vk �- = 1 ⇒ pT� = ��

[� �- > 1 �∗ = �6

$6 [� �- < 1, �∗ < 0, Lℎ[mℎ X`N_ W`k Nr[_k

�∗ = .K/t6u6

/0;05 �`O �- = 2

�∗ = .K/t64/0;05 = /xv

FKt64 y

/0;05 = � >�- − �6� ? > 0

[��- < 1, �- = �

�∗ = .K�/�6/0;05 ≈ /(�6K��6)

/0;05 < 0

⇒ �- < 1, �� X`N_ W`k Nr[_k

We have shown that

1. �`O �� = 0, VWX �- = 1, pT� [_ _kVo]N. {]_` pT� VWX �� m`[Wm[XN

2. T`O �- < 1, kℎN �WXNl[m �n^[][oO[^l X`N_ W`k Nr[_k. 3. T`O �- > 1, kℎN �WXNl[m �n^[][oO[^l Nr[_k_.

�- [_ [WXNNX V o[�^OmVk[`W n^VWk[k|.

46

4.5 NUMERICAL SIMULATION

For the purpose of the model validation, numerical simulation was undertaken

using Runge-Kutta method with the aid of MATLAB, using the data provided in

Table 4.1 and varying values of the control parameters, , � , ��, � , �. The results

are displayed in fig 1 - fig 3.

TABLE 4.1

PARAMETER VALUES

SYMBOL VALUE

� 986.3

� 0.07479

� 0.0000236

�� 0.0000118

�� 0.06225

� 0.025

�� 0.025

0.083

Fig 1: Dynamics of the control model for

� = 0.005, �� = 0.009, � =

Fig 1 A

Fig 1: Dynamics of the control model for �- = 1.1432, with control parameters 0.27.

Fig 1 B Fig 1 C

47

= 0.1245, � = 0.09,

Fig 1 C

Fig 2: Dynamics of the model without control for

Fig 2 A

Fig 2: Dynamics of the model without control for �- = 1.6405.

Fig 2 B Fig 2 C

48

Fig 2 C

Fig 3: Dynamics of the model for

� = 0.09, � = 0.005, �� = 0.009Fig 3: Dynamics of the model for �� = 0, �- = 1.1349 with control parameters =

009, � = 0.27

49

= 0.1245,

50

CHAPTER FIVE

5.0 DISCUSSION OF RESULT

In our work, we have derived and analyzed a mathematical model for the spread of

Ebola virus disease in a population. A flow diagram showing the transmission and

control strategies was drawn. We completed the basic reproduction number �- for

the model. We were able tos show that when �- = 1, the disease free equilibrium

and the endemic equilibrium coincide. When the �- > 1 the endemic equilibrium

exist. When �- < 1 does not exist while the disease free equilibrium is locally

stable. Numerical simulations were done to ascertain the validity of the model in

real life.

Fig 1 is divided into three subfigures fig1A, fig 1B and Fig 1C.

Fig 1A shows that the rate of susceptible individuals reduce over time as the

disease goes out of the population.

Fig 1B reveals that with control measures over time, the population of the

susceptible individual decreases sharply. It also shows the exposed and exposed-

under- treatment classes when control parameters are implemented. The rate of

exposed individuals increases and comes down over time. This means that With

early detection and treatment of individuals who have had contact with infectious

individuals, the exposed individuals phase out over time.

Fig 1C shows the infected/infectious, infected/infectious – under- treatment and

infected/infectious -quarantined classes when control parameters are implemented.

We see a slight but sharp increase of infected individuals at the beginning of the

outbreak, but with control parameters implemented, the population of infected

individuals reduces. This means that with good isolation centers and proper

treatment, the disease dies out of the population within a short period of time.

51

Fig 2 shows the susceptible, exposed, exposed-under-treatment,

infected/infectious, infected/infectious-under-treatment, infected/infectious –

quarantined classes without control parameters. We see from figures 2B and 2C

that the population of the exposed and under treatment, infected and under

treatment, infected/infectious quarantined individuals phase out of the population

with time. This means that when infected/infectious individuals are neither

quarantined nor treated and the exposed individuals are not given any form of

treatment as a result of poor public awareness of the disease, lack of good isolation

centers and early detection of interaction of susceptible individuals and infectious

individuals, Ebola virus disease persists in the population for a longer period of

time.

Fig 3 shows the transmission of the disease over the population when there is no

interaction between the susceptible class and the infected/infectious and under

treatment class but with control parameters the rate of the exposed individuals

reduces over time after the onset of the outbreak. This means that when there is

strict compliance to zero patient-visitor contact, the transmission of the disease in

the population reduces over time.

5.1 CONCLUSION

An Ebola outbreak in a human population can be catastrophic. Given the result

obtained from the analysis of the model, an uncontrolled transmittable contact

between the infected and the susceptible can increase the spread of the disease in

the population and cause the disease outbreak to linger for a longer period of time.

This implies that timely implementation of the control parameters would go a long

way in controlling the spread of the disease in a population ravaged by the Ebola

virus disease. Our work revealed that good public enlightenment, aggressive

52

quarantine system and effective treatment would ultimately reduce the

transmittable contacts.

5.2 RECOMMENDATION

In the light of this work and its scope, where we considered cases of proper

disposal of contaminated dead individual and no contact rate with the contaminated

corpse, we would recommend a further research where there is a contact between

the susceptible population and the contaminated population due to poor

handling/disposal of contaminated population, and the rate at which the

contaminated individual join the susceptible class. Timely identification of an

outbreak should be taken seriously, as this is of paramount importance in

controlling the spread of the disease. Outbreak containment measures like prompt

and safe burial of the dead, identification of people who may have had contact with

infected individual and monitoring their health for 21 days, isolating the sick from

the healthy to avoid spread, good hygiene and maintaining a clean environment can

help in controlling the spread of Ebola virus disease in human population.

53

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