ABSTRACT
Mathematical model for malaria transmission was formulated using ordinary differential equation. The equations were modeled compartmentally. Basic reproduction number was calculated using the spectral radius of the new generation matrix. The disease free equilibrium is stable for R0 < 1 and unstable otherwise. Also R0 = 1 is the perfect bifurcation of the model, since it is found that at R0 = 1 the endemic and disease free equilibrium coincides. We also showed that interaction between mosquitoes and recovered humans is negligible ie R0 < < 1 the average number of infection by an infectious human reduces and so implies that adequate treatment and control reduces malaria transmission.
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
Mathematical models are representations of physical phenomena and concepts using mathematical language. The system been modeled will now be represented in a mathematical equation which provides detailed description of the system’s operational method(s).
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family such as the integral curves of a family of vector fields, and the solutions of a family of ordinary differential equation. Bifurcation is most commonly applied in the study of dynamical systems. A bifurcation occurs when a small change in the parameter values of a system causes a sudden qualitative or topological change in the system. It occurs in both continuous systems ( ODEs, DDEs, PDEs) and discrete systems ( described by maps).
Malaria is an infectious disease caused by a parasite that live part of its life in humans and part in mosquito. It is responsible for the death of many people worldwide especially in Africa and children are the greatest victim. The greatest havoc caused by malaria worldwide was during the World War II when it was responsible for more deaths than the deaths by gun.
Malaria is primarily caused by parasite of genus plasmodium.
Plasmodium is a single-celled parasite and there are about hundred species of plasmodium. However, plasmodium causing malaria in humans earlier on was believed to be four viz plasmodium falciparum, plasmodium vivax, plasmodium malariae and plasmodium ovale. The recent work of an Indian Doctor, Dr. Knowlesi has disproved this as he discovered another plasmodium parasite that causes malaria in human beings, it goes by the name plasmodium knowlesi.
Malaria is caused when an infected anopheles mosquito bites a human being, the plasmodium parasite in form of sporozite is injected into the human system via saliva from the mosquito. The developmental process of the sporozite through schizont, to merozoite is what causes malaria in humans.
Initially the treatment of malaria was not known owing to the fact that the actual cause of it was unknown. Historically, malaria was thought of as being the result of inhaling bad air. Malaria has its origin from Italy “Mal aria” or bad air.
A French scientist, Alphonse Laveran, discovered that the real cause of malaria is the singled-celled plasmodium parasite and years later, scientist discovered that anopheles mosquitoes are the vectors responsible for transmitting malaria.
It was not until these were known that serious research into the treatment of malaria commenced. The first of it was the quinine treatment from Spain, followed by the chloroquine. Chloroquine treatment was very successful but sooner than later, malaria started developing resistance towards these treatments and this led to aggressive research into drugs for the treatment of malaria. Today, there are many effective drugs for the treatment of malaria as there are for body pain.
It should be observed here that just as there are many drugs for the treatment of malaria, malaria have also developed resistance to many, and this led to stronger quest and desire to eliminate malaria. Since the vector carrying plasmodium, anopheles mosquito are known, it was considered that another way of fighting malaria is through the control of mosquitoes. World Health Organization recommended mosquito treated nets, neat environments, clearing of stagnant water and bushes. These help to reduce the mosquito population since they are deprived of their breeding places.
In this work, we intend to design a mathematical model that will x-ray the transmission dynamics of malaria from female anopheles mosquitoes to human under treatment control.
1.2 Objectives of the Study
The objectives of our study include but not restricted to the following:-
- Carry out a detailed research on malaria transmission
- Study the metamorphosis of plasmodium parasite
- Construct a mathematical model for the transmission of malaria under treatment and control.
- Provide useful and analytical contributions that will help in exterminating malaria in our society.
- Pave way for further studies into the transmission dynamics of malaria.
1.3 Scope/Limitations
This study is on the transmission of malaria under treatment and control, the mathematical models was designed using compartments and various parameters. Ordinary differential equation was used. We relied on literature for the various values assigned to various parameters.
Our greatest constraint is lack of experimental materials for the research. Financial constraints also affected our deeper analysis. It was also impossible for us to determine the gonotrophic cycle of the mosquito and its anthropophilic rate.
1.4 Significance of the Study
There is a popular saying in football that the best form of defense is attack. With the knowledge of malaria and its transmission mechanism, the cheaper and best form of treatment is by attacking the vector host, mosquito, by adherence to the various controls and prevention techniques enumerated.
This study will add to the existing literatures and pave way for further researches into transmission and eradication of malaria in our society.
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