MATHEMATICAL MODELLING OF CAUSES AND CONTROL OF MALARIA – MATHEMATICS PROJECT TOPICS AND MATERIALS
Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female Anopheles mosquitoes. A mathematical model describes the dynamics of malaria and human population compartments in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of the study is to understand the important parameters in the transmission and spread of endemic malaria disease, and try to find appropriate solutions and strategies for its prevention and control by applying mathematical modelling. The malaria model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs)with 4 variables for humans and 3 variables for mosquitoes. Qualitative analysis of the model applies dimensional analysis, scaling, and perturbation techniques in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability. In that case, the endemic state has a unique equilibrium, re-invasion is always possible, and the disease persists within the human population. These simulations show the behavior of the populations in time and the stability of disease-free and endemic equilibrium points. Numerical simulation of the model suggests that the most effective strategy for controlling or eradicating malaria is to combine the use of insecticide-treated bed nets, indoor residual spraying and chemotherapy, but the best strategy is to reduce the biting rate of the female anopheles mosquito through the use of insecticide-treated bed nets and indoor residual spraying since the malaria parasite has developed resistance to some of the antimalarial drugs.
TABLE OF CONTENTS
1.1Background Of The Study
1.2 Aims and Objectives
2.1 Introduction of Exposed Class In Mosquito Population
2.2 Age And Exposed Class In Human Population
2.3 Migration And Visitation
2.4 Social And Economic Factors
2.5 Varying Population Size
2.6 Other Immunity Models
2.7 Host-Pathogen Variability And Resistant Strain Models
2.8 Environmental Factors
2.9 The Effect Of Sickle- Cell Gene On Malaria
2.10 Malaria Control
2.11 Vector Control and Protection Against Mosquito Bites
2.12 Case Management
2.13 Prophylactic Drugs
2.15 Stochastic Models
3.1 Formulation Of The Model
3.2 Analysis Of The Model
3.3 Equations Of The Model
3.4 Existence Of Equilibrium Points Without Disease
3.5 The Endemic Equilibrium Point
4.1 Estimation Of Parameters
4. 2 Population Data For Mosquitoes
4.3 Equations Of The Model
4.4 Disease-Free Equilibrium Points
4.5 The Endemic Equilibrium Point
5.1 Summary And Conclusion
1.1 Background Of The Study
Malaria is a life-threatening disease caused by a protozoan parasite called Plasmodium, which lives part of its life in humans and part in Anopheles mosquitoes. The development of malaria parasites in a human host commences in the liver cells where the malaria parasites undergo asexual multiplication to produce merozoites that are eventually released into the blood stream to invade red blood cells. The infected red blood cells burst after 2–3 days to release merozoites and gametocytes into the blood stream. This is associated with the clinical symptoms of the disease. Anopheles mosquitoes become infected when they feed and ingest human blood that contains mature gametocytes. The gametocytes develop into male and female gametes that fertilize to become zygotes in the mid-gut wall of the mosquito. The zygote elongates to become
ookinete and penetrates the mid-gut epithelium that later develop and ultimately produce sporozoites which become infective when they migrate to the salivary glands [Tumwiine et al, (2007)]. The disease is endemic in tropical and subtropical regions, including Africa, Asia, Latin America, the Middle East and some parts of Europe. According to the WHO report 2010, it is estimated that the number of cases of malaria rose from 233 million in 2000 to 244 million in
2005 but decreased to 225 million in 2009 and the number of deaths due to malaria is estimated to have decreased from 985 000 in 2000 to 781 000 in 2009. Most of the malaria related deaths occur mostly in sub-Saharan Africa and in children less than five years. There are more than 100 different species of Plasmodium parasites, but only four species of parasites can cause infections in humans, namely Plasmodium falciparum, Plasmodium vivax, Plasmodium malariae and Plasmodium ovale [Understanding Malaria, Fighting an Ancient Scourge, February 2007, www.niaid.nih.gov]. The following three species are found in Nigeria: Plasmodium falciparum, Plasmodium malariae and Plasmodium ovale. Plasmodium falciparum is responsible for most of the deaths and morbidity associated with malaria in Nigeria, accounting for about (90- 98) % of malaria cases. Only infected female Anopheles mosquitoes can transmit malaria and they must have been infected through a previous blood meal taken on an infected person. When a mosquito bites an infected person, a small amount of blood is taken in which contains microscopic malaria parasites. About 1 week later, when the mosquito takes its next blood meal, these parasites mix with the mosquito’s saliva and are injected into the person being bitten. There are three species that transmit the disease in Nigeria: Anopheles gambiae, Anopheles arabiensis and Anopheles funestus. Because the malaria parasite is found in red blood cells of an infected person, malaria can also betransmitted through blood transfusion, organ transplant, or the shared use of needles or syringes contaminated with blood. Malaria may also be transmitted from a mother to her unborn infant before or during delivery (“congenital” malaria) [Malaria.com, 2011)]. The early people attributed the malaria fevers to evil spirits, angered deities, demons, or the black magic of sorcerers. The ancient Chinese believed the frightening symptoms and signs to be the work of three demons, one with a hammer, one with a bucket of cold water, and a third with a stove. The ancient Romans worshiped a fever goddess, three demons rolled into one. Babylonian cuneiform script attributes malaria to a god, pictured as a mosquito-like insect. In 800 BCE the Indian sage Dhanvantari wrote that bites of mosquitoes could causes diseases, fever, shivering etc. In 1696 Morton presented the first detailed description of the clinical picture of malaria and its treatment with cinchona. Fransesco Torti, professor of medicine at Modena, accurately described the intricate course of the disease that was curable by the cinchona in 1712. One American physician, James K. Mitchell, wrote that malaria was due to certain spores present in marshy regions. This possible relationship was so firmly established that it gave the two most frequently used names to the disease malaria, which later shortened to one word ‘malaria’. The term malaria is derived from the Italian words mala “bad” and aria “air” which was used by the Italians to describe the cause of intermittent fevers associated with exposure to marsh air or miasma. Up to that point the various intermittent fevers had been called jungle fever, marsh fever, paludal fever, or swamp fever. In 1884, Russian physiologist, Basil Danielewsky identified parasites of malaria in the blood of wild birds and in the same year, Marchiafava and Celli demonstrated active amoeboid ring in unstained blood and named it Plasmodium. The name chosen for the parasite by them turned out to be an incorrect one, since the organism is not actually a plasmodium. But the name stuck despite years of haggling. In August 20, 1897, Ronald Ross demonstrated oocysts in the gut of anopheline mosquito at Secunderabad, India, proving that mosquito was the vector for malaria. In September 1898, Italian physician Giovanni Battista Grassi was able to report that this insect, Anopheles claviger, was the carrier of human malaria. In 1973 human protection from malaria by vaccination was first reported. For about 20 years, progress occurred mainly in experimental models rather than in human vaccine trials. In 1987, Dr. Manuel Elkin Patarroyo, a Colombian biochemist, developed the first synthetic Spf66 vaccine against P. falciparum parasite. But phase III trials showed that lacked efficacy. During the past 5 years, many candidate vaccine approaches have been tested in clinical trials. The genome sequences of Anopheles gambiae and Plasmodium falciparum were published in 2002 and those of P. vivax and P. knowlesi in 2008.
In Nigeria traditional herbalists were using medicinal plants to treat malaria before the introduction of orthodox medicine. Some of the plants species commonly used are Neem tree and its leaves, pawpaw leaves, etc. In 1950s, Nigeria with the support from WHO, added indoor residual spraying using DDT as one of the measures to control mosquitoes.
According to [Martcheva and Hoppensteadt (2010)], WHO, the World Bank and several charitable organizations launched in 1998 the Roll Back Malaria Partnership (RBM), a global initiative that coordinates actions against malaria. The mission of the RBM Partnership has more recently been outlined in its Global Malaria Action Plan. Some of the major goals of the Partnership are (1) Reduce global malaria cases from 2000 levels by 50% in 2010 and by 75% in 2015; (2) Reduce global malaria deaths from 2000 levels by 50% in 2010, and to near zero by 2015; (3) Eliminate malaria in 8-10 countries by 2015; and eventually (4) Achieve eradication of malaria world- wide. Since 1998, Nigeria has committed itself to the RBM Initiative. The country drew up a
‘Medium Term Strategic Plan for Malaria Control in Nigeria’ (1998-2002), which sought to improve the coverage of malaria control activities by adopting an inter-sectoral approach involving and promoting partnership with the private sector and the community. It has also committed itself to the Abuja Declaration on Roll Back Malaria in Africa, which similarly seeks to achieve specific targets on malaria prevention and control. The P. falciparum parasite has developed resistance to commonly used anti-malarials such as Chloroquine which poses a serious challenge to the mono therapies. In this regard, in 2002 using Artemisinin based combination therapies (ACTs) following WHO recommendations for all countries experiencing resistance to mono-therapies in the treatment of falciparum malaria. Artesunate-Amodiaquine was selected as the first line drug for the treatment of uncomplicated malaria, but due to the challenges such as adverse drug reactions, lack of other treatment options and safety concerns faced by the Health sector, two additional ACTs namely; Artemether-Lumefantrine and Dihydroartemisinin/Piperaquine were also added. Quinine or Intramuscular Artemether is the drugs of choice for treating complicated malaria. Pregnant women with severe malaria are put on parenteral Quinine (Intravenous or Intramuscular in all trimesters) until the patient can take oral preparations. According to UNICEF Nigeria Fact Sheet July 2007 report, as part of the measures to prevent Malaria, Nigeria health Service (GHS) in cooperation with local government authorities and UNICEF has distributed Insecticide Treated Nets (ITNs) to over 20 per cent of children below 5 and pregnant women through community bed nets sales agents, antenatal clinics and child immunization clinics in.
Malaria has a huge social, economic and health burden on the world, particularly in the tropical countries. According to World Health Organization (WHO) report 2010, the amount estimated for malaria control in 2010 was more than US$6 billion, but received US$1.8 billion from the international sources, which represents less than 30% of the amount aiming at. It is therefore means that, the government of malaria endemic country will receive less financial support from WHO and rest of the budget for malaria will come from country’s Gross Domestic Product ( GDP) , which will negatively affect the country’s economic growth.
1.2 Aims And Objectives
The aim is to formulate Mathematical model for the spread and control of Malaria
The objectives are:
• To formulate Mathematical model for the spread and control of Malaria
• To establish the existence of Equilibrium Point and Endemic equilibrium point
• To carry out the stability analysis of the disease free equilibrium state.
• To draw conclusion on the nature of the disease so as to help policy makers know the most appropriate control measures to be used.