MATHEMATICAL MODELING OF MALARIA: A SUSCEPTIBLE, INFECTED, RECOVERED MODEL (SIR MODEL)

 

Malaria is a mosquito-borne infectious disease that affects humans and other animals.  Malaria causes symptoms that typically include fever, tiredness, vomiting, and headaches. In severe cases it can cause yellow skin, 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. It is caused by single-celled microorganisms of the Plasmodium group. The disease is most commonly spread by an infected female Anopheles mosquito. The mosquito bite introduces the parasites from the mosquito's saliva into a person's blood. The parasites travel to the liver where they mature and reproduce. Five species of Plasmodium can infect and be spread by humans. Most deaths are caused by P. falciparum because P. vivax, P. ovale, and P. malariaegenerally cause a milder form of malaria. Malaria is typically diagnosed by the microscopic examination of blood using blood films, or with antigen-basedrapid diagnostic tests. The risk of disease can be reduced by preventing mosquito bites through the use of mosquito nets and insect repellents, or with mosquito control measures such as spraying insecticides and draining standing water. The disease is widespread in the tropical and subtropical regions that exist in a broad band around the equator. This includes much of Sub-Saharan Africa, Asia, and Latin America. In 2016, there were 216 million cases of malaria worldwide resulting in an estimated 445,000 to 731,000 deaths. Approximately 90% of both cases and deaths occurred in Africa. Rates of disease have decreased from 2000 to 2015 by 37%, during which there were 198 million cases. Malaria is commonly associated with poverty and has a major negative effect on economic development. In Africa, it is estimated to result in losses of US$12 billion a year due to increased healthcare costs, lost ability to work, and negative effects on tourism. Malaria is currently affecting more people in the World than any other disease. It is currently endemic in over 100 countries and is one of the 10 most prevalent and deadly diseases in the world. The disease is caused by tropical parasite that kills people more than any other communicable disease except tuberculosis. Between300 to 500 million clinical cases occur every year with over 1.2 to 2.7 million deaths, of which 90% occur in sub-Saharan Africa. Malaria menace has become an economic burden in tropical Africa. According to the report of the American Association for the Advancement of Science (AAAS) Washington, D.C 1991 on Malaria and Development in Africa, pregnant women and children under the age of five are at high risk of Malaria morbidity and mortality. The World Health Organization (1994)stated that some 90% of the World’s Malaria occurs in Africa because the World wide eradication programme of 1960s which successfully remove Malaria from North America and Europe, exclude sub-Saharan African altogether, due to the lack of technological capability in individual countries and because Malaria was so huge that eradication was considered not feasible. It is also reported that 255 children in Africa die every 2.5Hours, while about 2173 children under the age of 5 die daily in the continent from malaria. Indeed the African region lies in areas where the population is at risk of getting malaria since 74% of the population live in highly endemic areas where malaria transmission is intense. It is responsible for about 20-30% infant mortality, 10% of hospitals admissions, and 20-30% outpatient case in Africa. In Nigeria alone, 60 million people experience Malaria attack at least twice in a year, with no less than 80% of the population exposed to the disease. Scott (2000) ascribed 90% of health problem caused by Malaria to environmental conditions. To corroborate this, Paul, (1997) emphasized the role of temperature on the range, development, timing and intensity of Malaria outbreak. He described mosquito as hot weather insects that have fixed thresholds for survival. For instance, Anopheles mosquito and Falciparum malaria transmission are sustained only where the winter temperature is kept above 160 C.It was observed that P. falciparum transmission was limited by low temperature in areas of high altitude. In Kano metropolis, a detailed study of 278 households made up of 3071 individuals that inhabit around ten non-water outlet ponds from various segments of the metropolis revealed that Malaria is the most common sickness among them. On the average about two members of a household suffered from malaria fever monthly, with females and children having high frequencies of and vulnerable to malaria attack .

Malaria is a mosquito-borne infectious disease that affects humans and other animals. It is characterized by symptoms such as fever, fatigue, vomiting, and headaches. In severe cases, it can lead to yellow skin, seizures, coma, or even death. Typically, symptoms manifest around ten to fifteen days after an individual has been bitten by an infected mosquito. Without appropriate treatment, individuals may experience recurring bouts of the disease several months later.

The SIR (Susceptible-Infectious-Recovered) model, originally proposed by W.O. Kermack and A.G. McKendrick in 1927, has been instrumental in studying the dynamics of malaria transmission. 

The primary aim and objective of this research is to mathematically model malaria transmission using the SIR (Susceptible-Infectious-Recovered) model and is achieved through the following:

This study will utilize the SIR model to mathematically model malaria. It will focus on describing the dynamics among susceptible, infectious, and recovered individuals affected by malaria. This research is confined to the application of the SIR model exclusively for modeling the malaria parasite. 

Anopheles Mosquito: Loose terminology for species in the anopheles genus of mosquitoes, some of  which may transmit various parasites, plasmodium, that are the cause of malaria.

Model: Simplified description, especially a mathematical one, of a system or process, to assist calculations and predictions

You can create various kinds of models for a system, but useful ones have several important properties. Here is a very brief summary of what a good model should look like: A good model is simple, valid, and robust.

Simplicity of a model is really the key essence of what modeling is all about. The main reason why we want to build a model is that we want to have a shorter, simpler description of reality. As the famous principle of Occam’s razor says, if you have two models with equal predictive power, you should choose the simpler one. This is not a theorem or any logically proven fact, but it is a commonly accepted practice in science. Parsimony is good because it is economical (e.g., we can store more models within the limited capacity of our brain if they are simpler) and also insightful (e.g., we may find useful patterns or applications in the models if they are simple). If you can eliminate any parameters, variables, or assumptions from your model without losing its characteristic behavior, you 

should.

Validity of a model is how closely the model’s prediction agrees with the observed reality. This is of utmost importance from a practical viewpoint. If your model’s prediction doesn’t reasonably match the observation, the model is not representing reality and is probably useless. It is also very important to check the validity of not only the predictions of the model but also the assumptions it uses, i.e., whether each of the assumptions used in your model makes sense at its face value, in view of the existing knowledge as well as our common sense. Sometimes this “face validity” is more important in complex systems modeling, because there are many situations where we simply can’t conduct a quantitative comparison between the model prediction and the observational data. Even if this is the case, you should at least check the face validity of your model assumptions based on your understanding about the system and or the phenomena. Note that there is often a trade-off between trying to achieve simplicity and validity of a model. If you increase the model complexity, you may be able to achieve a better fit to the observed data, but the model’s simplicity is lost and you also have the risk of over fitting that is, the model prediction may become adjusted too closely to a specific observation at the cost of generalizability to other cases. You need to strike the right balance between those two criteria.

Robustness of a model is how insensitive the model’s prediction is to minor variations of model assumptions and or parameter settings. This is important because there are always errors when we create assumptions about, or measure parameter values from the real world. If the prediction made by your model is sensitive to their minor variations, then the conclusion derived from it is probably not reliable. But if your model is robust, the conclusion will hold under minor variations of model assumptions and parameters, therefore it will more likely apply to reality, and we can put more trust in it.

When real world problem can be described in mathematical language, a mathematical model is developed. A model is to serve as the abstract mathematical construct related to part of reality and created for a particular purpose. Mathematical modeling involves imaginative mathematical skill. For this reason, a clear set of rules for designing a mathematical model is scarce. There are a number of approaches for scientific model. A way of classifying various kinds of modeling approaches is to put them into the following two major families:

In this approach, researchers try to specify the actual state of a system at a given time point (or at multiple time points) in a descriptive manner. Taking a picture, creating a miniature (this is literally a “model” in the usual sense of the word), and writing a biography of someone, all belong to this family of modeling effort. This can also be done using quantitative methods (e.g., equations, statistics, computational algorithms), such as regression analysis and pattern recognition. They all try to capture “what the system looks like.”

In this approach, researchers try to come up with dynamical rules that can explain the observed behavior of a system. This allows researchers to make predictions of its possible (e.g., future) states. Dynamical equations, theories, and first principles, which describe how the system will change and evolve over time, all belong to this family of modeling effort. This is usually done using quantitative methods, but it can also be achieved at conceptual levels as well (e.g., Charles Darwin’s evolutionary theory). They all try to capture “how the system will behave.”

Both modeling approaches are equally important in science and engineering. For example, observation of planetary movement using telescopes in the early 17th century generated a lot of descriptive information about how they actually moved. This information was already a model of nature because it constituted a simplified representation of reality. In the meantime, Newton derived the law of motion to make sense out of observational information, which was a rule-based modeling approach that allowed people to make predictions about how the planets would or could move in the future or in a hypothetical scenario. In other words, descriptive modeling is a process in which descriptions of a system are produced and accumulated, while rule-based modeling is a process in which underlying dynamical explanations are built for those descriptions. These two approaches take turns and form a single cycle of the scientific modeling effort. The rule-based modeling plays a particularly important role in complex systems science. More specifically, developing a rule-based model at microscopic scales and studying its macroscopic behaviors through computer simulation and or mathematical analysis is almost a necessity to understand emergence and self-organization of complex systems. A typical cycle of rule-based modeling effort goes through the following steps 

This seems okay, and it doesn’t contain the logical problem of “proving a hypothesis” However, there is still one particular step that is fundamentally difficult .The second step (“Reflect on possible rules that might cause the system’s characteristics seen in the observation.”) is particularly challenging to modelers. This is because this step is so deeply interwoven with the modeler’s knowledge, experience, and everyday cognitive processes. It is based on who you are, what you know, and how you see the world. it is, ultimately, a personal thinking process, which is very difficult to teach or to learn in a structured way. Coming up with a model is inherently a personal process, which depends on your own knowledge, experience, and worldview. There is no single algorithm or procedure one can follow to develop a good model. The modeling process is a full-scale interaction between the external world and your whole, intellectual self. To become a good modeler, you will need to gain diverse knowledge and experience and develop rich worldviews. This is why it would be very difficult to be taught.

The clinical manifestation of malaria is varied. The classical description of an individual from shaking chills through intense fever to drenching sweats in characteristics but not universal. In areas where malaria is common, infected individuals may have symptoms that mimic other disease making a correct diagnosis is especially difficult in children who may have blood levels of parasite but relatively mild symptoms. In areas where health workers are unfamiliar with malaria or with patient in whom the range of symptoms may not point clearly to malaria, misdiagnosis is a serious problem. Severe and complicated malaria generally due to infection of plasmodium falciparum is a medical emergency in the absence of prompt intervention; the patient’s condition can deteriorate rapidly, often ending in death. About 80% of deaths from the disease result from cerebral malaria, a state of altered consciousness, hypoglycemia, severe anemia, pulmonary edema and shock may also play a role in fatal malaria.