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**TIME SERIES ANALYSIS ON THE RATE OF TYPHOID FEVER**

TABLE OF CONTENTS

TITLE PAGE …………………………………………………………………………………… I CERTIFICATION……………………………………………………………………………… II DEDICATION…………………………………………………………………………………… III AKNOWLEDGEMENT……………………………………………………………………… IV ABSTRACT…………………………………………………………………………………… V TABLE OF CONTENT……………………………………………………………………… VI LIST OF FIGURES…………………………………………………………………………… XI LIST OF TABLES……………………………………………………………………………. XII CHAPTER ONE……………………………………………………………………………… 1

- INTRODUCTION 1
- DEFINITION OF TIME SERIES 1

- THE NATURE OF HEALTH TIME SERIES DATA 2

- SCOPE OF THE STUDY 2

- STATEMENT OF THE PROBLEM 3

- OBJECTIVE OF THE STUDY 4

- SOURCE OF DATA 4

- HISTORICAL BACKGROUND OF THE LAGOS

STATE UNIVERSITY TEACHING HOSPITAL (LASUTH) 1KEJA, LAGOS STATE 4

- PROBLEMS ENCOUNTERED DURING DATA COLLECTION 6
- LIMITATION OF THE STUDY 6

- ASSUMPTION OF THE STUDY 7

- DEFINITION OF TERMS 7

CHAPTER TWO

- LITERATURE REVIEW 10
- TI ME SERIES 10

2.2. 1 INTRODUCTION 10

- DEFINITION OF TIME SERIES 11
- LITERATURE REVIEW ON TYPHOID FEVER 12

- CAUSES AND SPREAD 13

- TYPHOID FEVER AND CANNED MEAT 13

- SYMPTOMS AND SIGNS 14

- LASORA TORY TEST FOR TYPHOID FEVER 15

- TREATMENT OF TYPHOID FEVER 16

- PREVENTION AND CONTROL 16

- TYPES OF TIME SERIES 17

- FORMS OF TIME SERIES 17

- CHARACTERISTICS OF TIME SERIES 18

- OBJECTIVES OF TIME SERI ES 18

- COMPONENT OF TIME SERIES DATA 19

- TEST FOR TREND 20

- SEASONAL COMPONENT 22

- TEST FOR SEASONALITY 22

2. 18. 1 FRIEDMAN’S TEST FOR SEASONALITY 22

- REASONSFOR EXAMININGSEASONALEFFECTS 23
- REMOVALOF SEASONAL(SERIAL)DEPENDENCY 23

- CYCLICAL COMPONENT 23

- IRREGULARVARIATION 24

- TYPES OF MODEL 24

- ESTIMATIONOF TREND 25

- AVERAGINGMETHODS 25

- EXPONENTIALSMOOTHING 26

- LEST SQUARES METHOD 27

- ESTIMATIONOF SEASONALVARIATION 27

- SEASONALLYADJUSTEDSERIES 28

- ESTIMATIONOF CYCLICALCOMPONENT 29

- ESTIMATION OF THE RANDOM OR IRREGULAR COMPONENT 29
- STATIONARITYOF TIME SERIES 29

- TYPES OF STATIONARITY 30

- VARIANCE STABILIZATION 31

- AUTOCOVARIANCEAND AUTOCORRELATION FUNCTION 31

- METHOD OF ESTIMATINGCOVARIANCE 32

- STANDARDERROR OF AUTOCORRELATIONESTIMATES 33

- STANDARD ERROR OF PARTIAL AUTOCORRELATION

ESTIMATE: 35

- FUNCTIONS OF AUTOCORRELATION FUNCTION {ACF} 35
- CORRELOGRAM 36

- STOCHASTIC AND DETERMINISTIC MATHEMATICAL MODELS 36

- STOCHASTIC MODELS 37

- SOME USEFUL STATIONARY STOCHASTIC PROCESSES 37

- LINEAR STATIONARY MODELS 39
- THE GENERAL LINEAR PROCESS (GLP) 39

- AUTOCOVARLANCE GENERATING FUNCTION OF A GLP 41

- STATIONARITY AND INVERTIBILITY CONDITION FOR GLP 41

- THE AUTOREGRESSIVE PROCESS 42

- THE STATIONARY CONDITION FOR AR (P) 43

- AUTOCORRELATION FUNCI10N OF AUTOREGRESSIVE PROCESS 43

- YULE-WALKER’S EQUATION: AUTOREGRESSIVE PARAMETERS

IN TERMS OF THE AUTOCORRELA TIONS 44

- THE MARKOV [FIRST-ORDER AUTOREGRESSIVE AR (1)1 45
- THE SECOND ORDER AUTOREGRESSIVE PROCESS (AR(2)) 46

- A RECURSIVE METHOD FOR CALCULATING ESTIMATES OF AUTOREGRESSIVE PARAMETERS (LEVISON-DURBIN

ALGORITHM) 48

VII

2.52 MOVING AVERAGE PROCESS (MA) 49

- INVERTIBILITY OF MOVING AVERAGE PROCESS 51
- MIXED AUTOREGRESSIVE MOVING AVERAGE

PROCESS ARMA(p,q) 52

- AUTOCORRELATION FUNCTION OF MIXED PROCESS 53
- PARTIAL AUTOCORRELATION FUNCTION 54

- THE LINEAR NON-STATIONARY MODELS 54
- THE GENERAL FORM OF THE AUTOREGRESSIVE

INTEGRA TED MOVIN AVERAGE PROCESS [ARIMA (p, d, q)] 54

- SPECIAL CASES OF ARIMA(p, d, q) 55

2.57 DIFFERENCE EQUATION FORM OF THE MODEL 56

2.58.1 FORECASTING 56

2.58. 2 TECHNIQUES FOR FORECASTING 57

CHAPTER THREE

- DESIGN AND METHODOLOGY 58
- TIME SERIES MODEL 58

- STATIONARITY OF TIME SERIES 59
- ESTIMATIONOF TREND 59

- SMOOTHING OR MOVING AVERAGE METHOD 60

- LEAST SQUARE METHOD 60

- EXPONENTIAL SMOOTHING 60

- ESTIMATION OF SEASONALVARIATION 61

- FITTING THE AUTOREGRESSIVEMODELOF ORDER P (AR(p)) 61

- FORECASTING 62

CHAPTER FOUR

- DATA ANALYSIS 63
- FIRST DIFFERENCE GRAPH OF NUMBER OF TYPHOID FEVER PATIENT TREATED 66

- ESTIMATION OF TREND AND SEASONAL VARIATION USING

MA METHOD 69

- ESTIMATION OF SEASONAL INDICES BY MOVING AVERAGE

USING THE ORIGINAL DATA 70

- DESEASONEI: SERIES 75
- ESTIMATION OF TREND USING LEAST SQUARE METHOD 78

- PREDICTION FOR THE YEAR 2006 USING LEAST

SQUARE METHOD 81

- ESTIMATION OF AUTO COVARIANCE AND AUTOCORRELATION 83
- ESTIMATION OF THE PARTIAL AUTOCORRELATION FUNCTION 89

- USING INFORMATION CRITERIA TO FIT THE BEST ORDER

THAT FITS THE SERIES 93

- FITTING AUTOREGRESSIVE MODEL OF ORDER 3 93
- FORECASTING USING AUTOREGRESSIVE MODEL 93

- CALCULATION OF THE STANDARD ERROR OF LEAST

SQUARE METHOD 95

- CALCULATION OF THE STANDARD ERROR OF THE AUTOREGRESSIVE MODEL OF ORDER 5 i.e. . AR(5) 95

CHAPTER FIVE

- RESULTSAND DISCUSSIONOF FINDING 96
- CONCLUSION 98

- RECOMMENDATIONS 98

REFERENCES 100

LIST OF FIGURES

- Time plot of original data for the number of typhoid fever patients treated for then year 1992 – 2006
- Time plot of first difference for the treated patient with typhoid fever.

- Graph of the seasonal indices for the treated patients with typhoid fever.
- Graph of deseasonalize series with trend using LSM

- Graph of correlogram

LIST OF TABLES

- Quarterly number of typhoid fever patients treated between 1992 – 2006.
- First difference for the number of patients treated with typhoid fever.

iii. | Estimation of trend and seasonal variation using | moving average |

method. | ||

iv. | Seasonal Indices | |

v. | Deseasonalised Series | |

vi. | Estimation of tread using LSM | |

vii. | Prediction for the year 2006. | |

viii. | Estimation of Auto covariance and Autocorrelation |

XI

- INTRODUCTION

Typhoid fever is a systemic infection caused by enteric pathogen called salmonella typhi. The infedion is spread by the faucal-Oral route and is closely associated with poor food -hygiene and inadequate sanitation.

According to David Greenwood et al (2002), in the book medical microbiology, the current incident throughout most of the developed world is about 0.2 causes per 100,000 of the population. Most contact the infection (i.e. typhoid fever) by the consumption of unhygienic food and water. In the World Heath Organization Epidemiological report (2000), about 16 million cases of typhoid fever occur annua11yand globally causing 600,000 deaths. This disease is common in non- industrialized and developing countries.

- DEFINITION OF TIME SERIES

Chatfield (1980) defmes time series as a conecnon of observations made sequentially in time. Examples include the quarterly record of number of typhoid fever patients, the annual production of petroleum, total monthly sales of Departmental store etc. These series are often gathered purposely to identify, detect and study the pattern of variety of variables in relation to reality in order to control a system. In the course of analyzing time series, the different observations reflecting different time periods are taken to be homogenous in it. In simple form, the observations are time independent. It is on this note that the series are often assumed stationary and if not, transformed to achieve stationarity. The reason for stationarity is to remove serial dependency for the time periods. The ultimate goal of time series is to control the system with a view to forecast the likely future pattern of the

phenomenon, achieved through a mathematical model derived from the analysis of the existing data set.

- THE NATURE OF HEALTH TIME SERIES DATA

Keith (1990; 1), “Errors and random perturbations are frequent and important for many of the variables with which time series is concerned, and only on exceptional cases do researchers consider errors of measurement in time itself”. The identification of these random shocks is often difficult for they do not vary over a range of time that could allow easy realization of their impact on the pattern of the series.

- SCOPE OF THE STUDY

For the purpose of this research worl<, “ime series” would be defined as Manordered sequence of values of a variable at equally spaced time intervals. It is observed at discrete time points are equally spaced, either monthly, quarterly, or annually.

Time series analysis accounts for the fact that data points taken over time may have an interval structure (such as autocorrelation, trend or seasonal variation) that should be accounted for. To predict time series, it is necessary to represent the behaviour of the process by mathematical model that can be extrapolated into the future. At this point, an appropriate forecasting technique is developed. The general procedure ls to first obtain the time – plot (graph of the series) and autocorrelation function (ACF) and the partial autocorrelation (PACF) of the series after which the differencing is carried out if necessary.

For the data used in this research work, (number of typhoid fever patients treated from 1992-2006), various transfonnation techniques were used among which are the first difference, seasonal difference, log transformation and removal of seasonality by deseasonalising the series. The simple forecasting methods which according to Gross (1974) are efficient shall be used to predict the future pattern of the series.

- STATEMENT OF THE PROBLEM

This analysis is deliberately designed to investigate influence of the social and health elimination of typhoid disease caused by salmonella typhi. Prescott (2002) and George (1919) in the Microbiology and the Military Surgeon vol. XLV respectively; a very prominent discovery was in the early 1900s in which there were thousands of typhoid fever cases, and many died of the disease. Most of these cases arose when people drank water contaminated with sewage or ate food handled by or prepared by individuals who were shedding the typhoid fever bacterium (salmonella typhi). The most famous carrier of the typhoid bacterium was Mary Manon. Between 1896 and 1906, Mary Mallon worked as a cook in seven homes in New York City. Twenty- eight cases of typhoid fever occurred in these homes while she worked in them. As a result the New York City Health Department had Mary arrested and admitted to an isolation hospital on North Brother Island in New York’s East River. Examination of Mary’s stools showed that she was shedding large numbers of typhoid bacteria though she exhibited no external symptoms of the disease. An article publishedin 1908 in the Journal of the American Medical Association referred to her as “Typhoid Mary”, an epithet by which she is still known today. After being released when she pledged not to cook for others or serve food to them, Mary changed her name and began to work as a

cook again. For five years she managed to avoid capture while continuing to spread typhoid fever. Eventuany, the authority tracked her down. She was held in custody for 23 years until she died in 1938. As a lifetime carrier, Mary Mallon was positively linked with 10 outbreaks of typhoid fever J 53 cases and 3 deaths.

- OBJECTIVE OF THE STUDY

The main objectives of ‘this research work are:

- To identify possible trend in the number of typhoid fever patients in the period of investigation.
- To predict the number of occurrence of typhoid fever based on the past behaviour for the year 2006.

- SOURCE OF DATA

There are two major sources of data, the primary and the secondary sources. The data to be used for this project work (number of typhoid fever patients treated) is from a secondary source. This is a transcription from the Medical Records Department of Lagos State University Teaching Hospital (LASUTH) Ikeja, Lagos State, Nigeria. The data was collected over successive quarterly intervals from 1992 to 2006 (15 years).

- HISTORICAL BACKGROUND OF THE LAGOS STATE UNIVERSITY TEACHING HOSPITAL (LASUTH) 1KEJA, LAGOS STATE

The General Hospital Ikeja was established primarily to provide health care delivery services for the people of Ikeja and its environs. It was initially a cottage Hospital serving a wide and populated area. The foundation stone of Ikeja Genera’ Hospital was ‘aid on the 25th June, 1955 by Honorable S.O. Igbodare, the Minister of Public Health of the Western Region of which Ikeja was a part.

With increase in popu’ation and the need to provide a health care delivery system that is commensurate with the status of the State the Executive Governor of Lagos State, His Excellency, Asiwaju Ahmed Bola Tinubu in April year 2001. This singular act was aimed at providing tertiary health care services to the good people of the State, in addition to the primary and secondary services already in existence. With this status therefore, it became imperative for the hospita’to have its own medical school to produce its own medical personnel. The hospita1 is located at 1/3 Oba Akinjobi Street, Ikeja Lagos State.

Presently, the hospital has the following clinical departments:

- Department of Medicine

- Department of Surgery

- Department of Pediatrics

- Department of Obstetrics and Gynecology

- Department of Anesthesia

- Department of Dentistry

- Department of Radiology

- Department of Physical Medicine and Rehabilitation

- Department of Haematology

- Department of ENT

- Department of Ophtbahnology

- Department of Psychiatry

- Department of Famny Medicine

- Department of Morbid Anatomy

- Department of Nursing

- PROBLEMS ENCOUNTERED DURING DATA COLLECTION

The major problems encountered in this research work in relation to data collection are:

- There were no up- to -date data on previous years of the establishment of the hospital. Then, there was no department for the medical record keeping.
- The time constraint factor. The collection of data from the medical record department of LASUTH, Ikeja undergoes bureaucracy. Thus this delays the data collection process.

- LIMITATION OF THE STUDY

The analysis in this project research work is exposed to some inevitable shortcomings. They are of unavoidable random variations which are beyond human control but which could not distort the reliability of the result of the analysis.

However, this is subjected to a certain amount of variability which Is random in nature and hence the analysis is still under statistical control.

The foremost of the lapses is that the result of this project work can only be valid for the short run forecasting as result of unforeseen Circumstances in the future and the accumulated steady-adjustments to periodic fluctuations in the health services as well as technological advancement which keeps on improving the state of living at a normal rate.

- ASSUMPTION OF THE STUDY

This project is based on the fundamental assumption that an health phenomena and factors which have influenced the structural pattern and behaviour of the series in the past and at present will continue to do so in the same pattern in the future.

- DEFINITION OF TERMS

Xt – Time series

K – Lag value

Ck – Auto covariance of lag k Co – Variance

4>(8)= Characteristic function of AR process 9{B) = Characteristic function of MA process AR (P) = Autoregressive process of order p MA (q) = Moving average of order (q)

ARMA (p, q) = Autoregressive moving average of order p, q

ARlMA (p, d, q) = Autoregressive Integrated Moving Average of order p, d,q

From Bailliere’s (1997), Arnold (1999) and Hornby (1989), the following terms are defined.

Bacterla:-They are kind or germs found in the air, water, and earth, in living and dead bodies and usually in things rotten. Species: – Group of animals or plants within a genus.

Parasite: – Animals or plants that lives on or in another and gets its food from it.

Vector:- This is a secondary host, since it feeds on primary host that ensure transmission of the parasite.

Medical data:-These are data collected on care and prevention of disease and the application of the drugs.

Salmonella typhi: – They are any of the genuses of gram- negative, non- sporing, rod-like bacteria that are parasites of the intestinal track of man and animals. Salmonella typhi: – (s.typhi) are exclusively human pathogens which cause typhoid fever.

Acute: – It is a process that usually begins abruptly and is ever soon.

Disease: -It is any devlatlon from a normal physical state of an organism sufficient to produce current symptoms. It may be chronic or acute.

Chronic:- It is a process that often begins very gradually and then persisted over a long period.

Pathogen: -It is any disease producing agent or micro-organism.

Perforation: – A hole or break in the containing membrane (or wall) of an organ or structure of the body.

Enteric:- Pertaining to the intestine e.g. enteric fever, fever that affects intestine.

Bacillus:- Loosely, the cause of any bacteria infection by a rod-shaped microorganism.

Malaise:- General feeling of illness, without clear signs often particular disease. Constipation:. – Incomplete or infrequent action of the bowels, with consequent filing of the rectum with hard faeces.

Disinfectant: – Substance that clean by destroying germs that cause disease e.g. disinfect a wound, a surgical instrument, and a hospital ward.

Vaccine:- Substance that is injected into the blood-stream and protects the body by making it have a mild fonn of the disease. It is also a suspension of killed organism administered for prevention.

Endemic: – Pertaining to a disease prevalent in a particular locality. Asymptomatic: – Without symptoms.

Typhoid fever: It is a serious bacterial disease that results in fever, weakness, and in severe cases, death.

Paratyphoid fever: -It is a disease that resembles typhoid fever in terms of symptoms, spread, and treatment. It resuHs from infection with salmonella organisms other than salmonella typhi (s.typhi).

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