Numerical study to predict the causes and effects of Dengue fever by transformations of Euler and Runge – Kutta methods in Susceptible – Infectious – Recovered modeling

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ABSTRACT

The infectious disease can be controlled and prevented by understanding the causes and effects of it. We consider the vector borne diseases; Dengue caused by mosquitoes mainly the Aedes Aegyptic mosquito. We apply Susceptible – Infectious – Recovered modeling to study the dynamics of the spread of infectious disease and adapt mathematical model which consist of mathematical concepts and description to represent the spread of disease. It is not easy to solve three basic equations of SIR model. Therefore, we apply Euler and Fourth order Runge- Kutta methods. These two strategies are practically well suited and very efficient. Therefore we use to solve the initial value problem of ordinary differential equations. We also compare the result of Euler and Runge- Kutta methods and discuss their performances .In this work, we consider the population in the districts where there is vast effect of Dengue fever.

Key Words

Infectious disease- Susceptible-Infectious-Recovered – Euler method-Runge- Kutta method

INTRODUCTION

Infectious diseases are disorders by biological agents like bacterium, virus, protozoan, parasite or fungus. Some infectious diseases are caused by human beings from person to person and by animals. Insects and animals transmit the diseases through bites. The diseases even spread through contaminated food, water and other micro organisms in the environment.

The symptoms and causes may vary depending upon the diseases. If the diseases are diagnosed in the initial stage, they can be cured easily and the spread of the diseases can be controlled. Some of the major infectious diseases in India are

Tuberculosis

Malaria

Typhoid

Cholera

Dengue fever

Hepatitis

Yellow fever

Chikungunya

Jaundice

Flu

Influenza

The infectious disease taken for studies is dengue fever. Many children are being affected by the disease.

Dengue fever is caused by Mosquito namely the Aedes Aegyptic (Mosquito). Aedes Aegyptic bites during the daytime. The bites of even single Aedes Aegyptic can cause dengue fever. Dengue was first discovered in 1946. Later in 1963-1964, the spread of dengue fever was recorded on the eastern coast of India. It has spread north and south simultaneously. It starts with mild fever and extends to severe hemorrhagic diseases that end in death. The other names of dengue were “break bone fever” or “dandy fever”.

The major symptoms of the diseases are sudden fever and headache followed by skin rashes, severe pains in the joints, muscles and bones. In severe cases, it may lead to excessive bleeding known as hemorrhaging due to reduction in count of blood Platelets accompanied by blood plasma leakage and low blood pressure which leads to death.

Dengvaxia is the first dengue vaccine invented by Sanofi Pasteur in Mexicoin December 2015.The vaccine is given to the infected person in three doses series from nine to forty five years of age living in endemic areas.

The highest death rate is recorded in Tamil Nadu. National Vector Borne Control programme (NVBDCP) been reported nearly forty deaths have been reported in Tamil Nadu from 14,465 dengue cases in India. Unfortunately, Chennai has higher consistence of dengue cases, while Vellore and Kanchipuram takes the next place.

In this research, we analyze the spread and an effect of dengue fever .It is essential to know how to take preventive measure, control the spread of diseases. Therefore we decided to study the spread and control of diseases by mathematical model.

Mathematical models play an important role of understanding, comparing, planning, implementation, evaluating and optimizing various detection, prevention therapy and control programs of infectious diseases, dengue. The main objective is to apply mathematical modeling to examine and to analyze the viral dynamics of the dengue fever. We use Susceptible-Infectious -Recovered (SIR) model to study the dengue epidemics in 2017.

OBJECTIVES

To apply SIR model to predict the sudden occurrence of dengue.

To determine the effect of the initial number of infectives’ of the population.

To compare the extracted data with real data and fit with SIR model.

SCOPE

In this study, we will investigate the outbreaks of Dengue fever by applying mathematical model. The data is collected from the three districts. The calculation is done by using tools of MATLAB software.

METHODOLOGY

Introduction to epidemic modeling is usually made through one of the first epidemic models proposed by Kermack and Mc Kendrick in 1927, a model known as the Susceptible-Infected -Recovered (SIR) epidemic model to predict the outbreak of the diseases. Susceptible – Infectious – Recovered model is used to study the spread of the epidemics through the interaction of the three compartments (variables). The SIR model is known as compartmental model where the population is divided into three distinct classes namely

Susceptible, S – People who can susceptible to the disease.

Infective, I – People who can have the disease and can transmit the disease

Recovered/Removed, R – Those who have either had the disease or recovered, immuned or isolated until recovered.

The progress of individuals can be expressed systematically by

Let us assume the population as fixed with N, number of people where there are no birth and death by natural cause. The assumption is made for sixty days with no birth and death cases.

Therefore N = S+ I + R

Since the population is fixed, we can split the population into three compartments. These variables changes with respect to the time duration.

Initially let us assume the value of t = time is zero.

The model uses two parameters namely

?– The infection rate of the disease

? – The rate of recovery from the disease where ?, ? ; 0.

Based on the above assumption, the model uses three ordinary differential equations

ds/dt = -?IS

ds/dt refers to the rate of change of the number of pupil susceptible to the disease at time t. The value of ds/dt decreases proportionally to I so that the person get infected,

The only possible way of leaving the susceptible group is getting infected with the diseases itself. Therefore the general count of number of person who is susceptible is determined by the count of person who is already susceptible. Every individual has equal chance of becoming infected with the disease.

dI/dt = ?IS – ?I

dI/dt refers to the rate of number of pupil infected by the diseases. This is not an independent variable on the count of the person susceptible and the count of the person infected and also the rate of infection of the diseases between the two compartments. When the count of infected person increases, the population of S decreases. Therefore dI/dt is inversely proportional to ds/dt .

dR/dt = ?I refers to the rate of change of the number of pupil recovered over time. The rate of the count of pupil recovering is dependent upon the count of pupil infected as in order to become either recovered from the disease or removed. Therefore the increases in rate of change of recovered/removed person are proportional to the rate of the disease being infected.

To solve easily the three basic equations of SIR model with a certain formula solution, we use Euler and fourth- order Runge-Kutta methods (RK4).We discuss the numerical comparisons between Euler method and Runge-Kutta methods.

The primary focus is to apply SIR model to predict the outbreaks of vector borne diseases. The secondary objective is to determine the effect of the initial number of infective of the population, compare and fit with model. The study of the outbreak of various vector borne diseases and the comparison of their spread using SIR model will lead the development of this research.

PARAMETERIZATION OF THE MODEL

To calculate ? – infection rate and ?– recovery/ removed rate, we define two more parameters namely

P = Period of disease for those recovered.

M = Mortality rate for those who die per day

The rate at which the diseases is spread is given by

? = 1/P

The infection rate of the diseases

?= M/S

We proceed the research by transforming the Runge – Kutta equations and Euler equations for SIR modeling.

CHAPTER SCHEMER

Chapter – 1

Introduction

Objectives

Scope

Limitation

Chapter – 2

Review of Literature

Chapter-3

Theoretical frame work of the study

Chapter – 4

Calculation

Analysis

Chapter – 5

Findings

Suggestion

Conclusion

WORK PLANNER

Submission – March 2019

Second draft – February 2019

First draft – January 2019

Conclusion and Suggestion – December 2018

Data calculation – November 2018

Review of literature – October 2018

MATLAB Software class and Abstract – September 2018

REFERENCE

Mathematical Model of the spread and control of Ebola Virus Disease. – Durojaye M.O,Ajie I.J.

www.who.int/dengue control/mosquito/en/

Cause and symptoms of Dengue. www.godrejhit.com/hit-blogs/dengue/cause-and -symptoms-of-dengue

https://www.medicinenet.com/dengue-fever/article/article.htm

www.who.int/we/2016/wer9130.pdf/ua=1 weekly epidemiological record releve epidemiologique heb domadaire.

Highest number of dengue deaths in Tamil Nadu. Deccan Chronicle / Shweta Tripathi – October 27 ,2017

American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers.Numerical Study of Kermack-Mckendrik SIR Model to Predict the Outbreak of Ebola Virus Diseases Using Euler and Fourth Order Runge-Kutta Methods Md. Tareque Hossaina , Md. Musa Miahb*, Md. Babul Hossainc a Department of Textile Engineering, City University, Birulia, Savar, Dhaka, Bangladesh b,cDepartment of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh a Email: [email protected], b Email: [email protected], c Email: [email protected]

https://timesofindia.indiatimes.com/city/chennai/in-tamil-nadu-its-a-war-on-dengue/articleshow/59583979.cms

https://www.livemint.com/Science/LHBiyAUyUMf1jr2UoXnJGJ/Dengue-deaths-on-the-rise-in-Tamil-Nadu.html

https://www.deccanchronicle.com/nation/current-affairs/271017/highest-number-of-dengue-deaths-in-tamil-nadu.html

https://www.researchgate.net/figure/DengueDHF-affected-districts-in-Tamil-Nadu-India-The-present-study-was-carried-out-in_fig1_12268880

https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology

https://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-the-differential-equation-model