MATHEMATICAL MODEL OF MALARIA TRANSMISSION RELATING TO AGE GROUP. A CASE STUDY OF HAJIYA GAMBO SAWABA GENERAL HOSPITAL ZARIA

This project focuses on the mathematical modelling of malaria transmission using the susceptible, infected and recovered (S I R) model to describe the occurrence of malaria transmission in different age groups using Euler’s formula and MATLAB to solve and obtain graph for the numerical solution of the model. The study was conducted using the data obtain from Hajiya Gambo Sawaba General Hospital, Zaria, Kaduna state. The result of the analysis shows that there is positive occurrence of malaria parasite transmission cause by mosquitoes among both the age groups below five years and above five years, where the high occurrence of the infected population is in the age group above five during the period of the study.    

 

Malaria is a mosquito-borne infectious disease that affects humans and other animals. Malaria causes symptoms that typically include fever, fatigue, vomiting, and headaches. In severe cases, it can cause jaundice, seizures, coma, or death. Symptoms usually begin ten to fifteen days after being bitten by an infected mosquito. If not properly treated, people may have recurrences of the disease months later. In those who have recently survived an infection, reinfection usually causes milder symptoms. This partial resistance disappears over months to years if the person has no continuing exposure to malaria.

Malaria is a serious and sometimes fatal disease caused by a parasite that commonly infects a certain type of mosquito which feeds on humans. People who get malaria are typically very sick with high fevers, shaking chills, and flu-like illness. Four kinds of malaria parasites infect humans: Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, and Plasmodium malariae. In addition, Plasmodium knowlesi, a type of malaria that naturally infects macaques in Southeast Asia, also infects humans, causing malaria that is transmitted from animal to human (“zoonotic” malaria). Plasmodium falciparum is the type of malaria that is most likely to result in severe infections and if not promptly treated, may lead to death. Although malaria can be a deadly disease, illness and death from malaria can usually be prevented.

Malaria is a mosquito-borne disease caused by a parasite. People with malaria often experience fever, chills, and flu-like illness. Left untreated, they may develop severe complications and die. In 2020 an estimated 241 million cases of malaria occurred worldwide and 627,000 people died, mostly children in sub-Saharan Africa. About 2,000 cases of malaria are diagnosed in the United States each year. The vast majority of cases in the United States are in travelers and immigrants returning from countries where malaria transmission occurs, many from sub-Saharan Africa and South Asia.

The malaria parasite is transmitted to human by mosquitoes belonging to the genus Anopheles. Malaria parasite can also be transmitted through blood transfusion, organ transplant, or the shared use of needles or syringes (CDC, 2014), and from mother to child (congenital malaria). The nature of morbidity can also be affected by the stability of transmission. As transmission intensity decreases, the cumulative risk for experiencing a severe disease episode during childhood increases (Snow et al., 1998). Severe malaria becomes less likely as children grow older, but when severe malaria does occur, 8-15years old children (60.6%) are more likely to develop life-threatening cerebral manifestations than those who are 4-7 years old (28.2%) or younger (11.3%) (Imbertet al., 1997). Thus, cerebral malaria is more likely to develop from malaria infections in epidemic-prone regions, which may in part account for the high case fatality rates noted during epidemics. Many biological and environmental factors shape the character of malaria in a given location. Nearly all the people who live in endemic areas are exposed to infection repeatedly. Those who survive malaria in childhood gradually build up some immunity. They may carry the infection, serving as reservoirs for transmission by mosquitoes without developing severe disease (Sullivan et al., 1999). In other areas, where the infection rate is low, people do not develop immunity because they rarely are exposed to the disease. This makes them more susceptible to the ravages of an epidemic. 

The symptoms of malaria include recurring attacks of chills, followed by fever and then sweating. National Institute of Allergy and Infectious Disease (2009). Other symptoms that may be presents include headache, muscle aches, joints pain, tiredness, nausea, vomiting, diarrhea, anemia, and jaundice. Enlargement of the spleen, thrombocytopenia, hypoglycemia, pulmonary or renal dysfunction and neurologic changes can also occur. If untreated Plasmodium falciparum can lead to kidney failure, seizures, mental confusions, coma and death.

Symptoms first appear about 10 to 15 days after infection, when red blood cells are infected and burst at the same time, these attacks recure at regular intervals. Attacks occur every 2 days for Plasmodium vivax and Plasmodium ovale infections and every 3 days for Plasmodium malariae. In the case of Plasmodium vivax, the patient may note experience any symptoms between attacks and the attacks subside in a few weeks even without treatment. In the case of Plasmodium falciparum, the patient will probably feel miserable between attacks and may die without treatment.

In countries where malaria is endemic, control is aimed at reducing the health impact on a population. Complete elimination of the disease is not currently a realistic goal in these countries. The method of control includes diagnosis and treatment of infected patients, infection prevention through vector control and prevention through the use of antimalarial medications especially for pregnant women and infants. Vector control includes the use of insecticides, treated bed nets and spraying insecticide in houses as well as destroying larval breeding sites. Antimalarial drug cannot prevent infection through mosquito bites, but they destroy the parasite in the human’s blood stream, preventing the disease from developing. Treatments for infected patients include oral or intravenous administration of antimalarial drugs such as, chloroqiune, sulfadoxine-pyrimethamine, mefloquine, atovaquoune-proguanil, quinime doxycycline or artemisiderivatives. Once treatment is begun most patients improve within 24 to 48 hours. However, if the patient is infected with Plasmodium falciparum, the fever can persist for 5 days. The medication primaquine acts against the domant parasite liver forms and prevents relapses of the disease.

Prevention of malaria can be aimed at preventing infection avoiding mosquito bites at preventing the disease by using antimalarial drugs. Travelers to endemic areas can take antimalarial drugs as a preventative measure and while in the endemic area’s travelers should wear insect repellent with DEET, cover up their skin with clothing and sleep under mosquito netting.

Currently there is no approved vaccine for malaria due to the complex life cycle of the parasite, its antigens are constantly changing. This makes creating a vaccine against the varying antigens is a difficult task. Scientists do not yet completely understand the complex immune response that protect humans against malaria. Currently many scientists are working on developing a malaria vaccine, and because other methods of fighting the disease have not succeeded in eliminating it, the search for a vaccine is considered one of the most important research project in public health.

This research is mainly on Zaria, to study malaria transmission dynamics among the selected age groups classified as below five (<5) and above five (>5). A mathematical model is used to describe the effect of malaria on the different age groups in Zaria. It does not consider any other place. Due to time constrain, data collected for the research is limited to a hospital in Zaria local government, Kaduna State.

So far Zaria is not having any model which I know of, that the people are using to study malaria transmission dynamic, and it is because of that we are now trying to develop a mathematical model which will help the people to understand and investigate malaria disease in the set of age groups.

The aim of this study is to develop a mathematical model of malaria transmission dynamic for Zaria, to help in investigating and explaining the prevalence of malaria and to make prediction about malaria disease.

The above aim can be achieved through the following objectives

Charles Louis Alphonse Laveran (1845-1922) was a French army doctor during the Franco-Prussian War. He later authored a treatise on military medicine. In it he challenged the traditional wisdom regarding malaria's ecology—namely, that the disease was restricted to low-lying humid plains. Laveran noted that malaria also could occur in temperate zones, and that not all tropical areas were plagued by the disease. Although malaria had been linked with swamps ever since the condition known as Roman fever inspired the name mal'aria (“bad air”), Laveran knew from contemporary scientific articles that many diseases previously ascribed to miasmas, or evil vapors, were in fact caused by microbes. Thus, he predicted: “Swamp fevers are due to a germ” (Jarcho, 1984).

After transferring to a new post on Algeria's North African coast, Laveran investigated his theory. On October 20, 1880, while looking through a crude microscope at the blood of a febrile soldier, he saw crescent-shaped bodies that were nearly transparent except for one small dot of pigment. In preceding decades the brownish-black pigment hemozoin (now known to be the product of hemoglobin digestion by the malaria parasite) had been found in cadaveric spleens and blood of malaria victims by several investigators including Meckel, Virchow, and Frerichs. Laveran subsequently examined blood specimens from 192 malaria patients and saw pigment-containing crescents in 148 sufferers (Laveran, 1978). He ultimately recognized four distinct forms in human blood that would prove to be the malaria parasite in different stages of its life cycle: the female and male gametocyte, schizont and trophozoite stages.

Dynamical System is a set of relationships among two or more measurable quantities, in which a fixed rule describes how the quantities evolve over time in response to their own values.

A mathematical model is a description of a system using mathematical concepts and language to help explain then system and to study the effect of different components and to make prediction about behavior.

Its objective is to ensure that a model matches the system modeled so that the conclusion drawn about the system model is a reasonable conclusion about the real system as well. A model is valid if its base faithfully reproduces the behavior of the system.

Model is an abstract concept of an ideal system. It captures the essence of the detail.

Susceptible: These are people that are not infected with the disease yet. However, they are not immune to it either, and so they can become infected with the disease in the future.

Infected Or Infectious: These are people that are infected with the disease and can transmit the disease to susceptible people.

Recovered: These are people who have recovered from the disease and are immune.