Mathematical and Electronic Model for The Spread of Malaria in Cameroon
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Malaria occurs mostly in poor tropical and subtropical areas of the world. In many of the countries affected by malaria, it is a leading cause of illness and death. In areas with high transmission, the most vulnerable groups are young children, who have not developed immunity to malaria yet, and pregnant women, whose immunity has been decreased by pregnancy. The costs of malaria to individuals, families, communities, nations are enormous. The model presented in this work is discussed based on two different approaches: In the first part, we have studied a simple SEIR model and have estimated the reproduction number. In the second part of this model, we have considered the SEIR-SEI model of malaria transmission between humans and mosquitoes. We have estimated the reproduction number and discussed the stability of the disease-free and endemic equilibria. These mathematical models are said to be compartmental because the population is partitioned in compartments that represent the states of its members (i.e. either susceptible, exposed, infected, recovered or immune etc) with respect to the disease. Furthermore, knowing that many electrical components or parameters (such as voltage, resistance, capacitance, current etc) have different behaviors in circuits which can be described by differential equations, we have proposed an electronic model for the mathematical model described. The electronic model is just analyzed based on simulations on an electrical circuit of the system of equations given by the mathematical model that describes the dynamics of transmission of malaria.
Malaria is a disease caused by the parasite Plasmodium, which is transmitted by the bite of an infected mosquito. Only the Anopheles genus of the mosquito can transmit Malaria . Our understanding of the malaria parasites begins in 1880 with the discovery of the parasites in the blood of malaria patients by Alphonse Laveran. The discovery that malaria parasites developed in the liver before entering the blood stream was made by Henry Shortt and Cyril Garnham in 1948 and the final stage in the life cycle, the presence of dormant stages in the liver, was conclusively demonstrated in 1982 by Wojciech Krotoski .
Malaria is an ancient disease having a huge social, economic, and health burden. It is predominantly present in the tropical countries. Even though the disease has been investigated for hundreds of years, it still remains a major public health problem with 91 countries. According to the latest World malaria reports, there were 241 million cases of malaria in 2020 compared to 227million cases in 2019. The estimated number of malaria deaths stood at 627,000 in 2020 – an increase of 69000 deaths over the previous year. While about two thirds of these deaths (47 000) were due to disruptions during the COVID-19 pandemic, the remaining one third of deaths (22000) reflect a recent change in WHO’s methodology for calculating malaria mortality (irrespective of COVID-19 disruptions). The new cause-of-death methodology was applied to 32 countries in sub-Saharan Africa that shoulder about 93% of all malaria deaths globally. Applying the methodology revealed that malaria has taken a considerably higher toll on African children every year since 2000 than previously thought. The WHO African Region continues to carry a disproportionately high share of the global malaria burden. In 2020 the Region was home to 95% of all malaria cases and 96% of deaths. Children under 5 years of age accounted for about 80% of all malaria deaths in the Region. Four African countries accounted for just over half of all malaria deaths worldwide: Nigeria (31.9%), the Democratic Republic of the Congo (13.2%), United Republic of Tanzania (4.1%) and Mozambique (3.8%) .
It is observed from research that across Africa, millions of people still lack access to the tools they need to prevent and treat the disease. Malaria has for many years been considered as a global issue, and many epidemiologists and other scientists invest their effort in learning the dynamics of malaria and to control its transmission. From interactions with those scientists, mathematicians have developed a significant and effective tool, called mathematical models of malaria, giving an insight into the interaction between the host and vector population, the dynamics of malaria, how to control malaria transmission, and eventually how to eradicate it. A huge set of epidemiology models have been formulated mathematically, analysed and applied to many infectious diseases.
In 1999, Ngwa GA, Shu WS developed and analysed an SEIRS model to study the dynamics and transmission of malaria, involving variable human and mosquito populations. According to their results, there is a threshold parameter Ro and the disease can persist if and only if Ro > 1 and the Disease-Free Equilibrium (DFE) always exists and is locally stable if Ro < 1, and unstable if Ro > 1. Their model was also globally stable when Ro ≤ 1. They confirmed their results with numerical simulations. Their model provides a frame work for studying control strategies for the containment of malaria .
Another model which is related to this work is that of Olaniyi and Obabiyi, they used a system of seven-dimensional ODE’S to modeling the transmission of plasmodium falciparum malaria between humans and mosquitoes with non-linear forces of infection in form of saturated incidence rates, these incidence rates produce antibodies in response to the presence of parasite causing malaria in both human and mosquito populations. They investigated the stability analysis of (DFE) and according to their results, (DFE) was asymptotically stable when Ro < 1 and unstable when Ro > 1. They also determined the existence of the unique Endemic Equlibrium (EE) under certain conditions, and their numerical simulation confirms the analytical result .
Nita and Gupta, modelled the basic of SEIR model and applied it to vector borne disease (malaria). They carried out the sensitivity analysis of the model using data from India. According to their results, the sensitivity analysis was very important, and it is the most sensitive aspect to be taken care of in their model .
Jia Li , developed an SEIR malaria model with stage-structured mosquitoes. They included metamorphic stages in the mosquito population and a simple stage mosquito population is introduced, where the mosquito population is divided into two classes namely, the aquatic stage in one class and all adults in the other class. According to their results the different dynamical behaviour of the models in their study, compared to other the bahaviour of most classical epidemiological models, and the possible occurrence of backward bifurcation make control of malaria more difficult.
The above are few of the models developed by scientists to give an insight into the interaction between the host and vector population, the dynamics of malaria, how to control malaria transmission, and eventually how to eradicate it.
In this work, inspired by the work of Isaac Kwasi Adu, Kumasi Technical University, and Mojeeb Osman, International University of Africa , we try to address this by first introducing an SEIR model, and then apply it to malaria transmission between mosquitoes and human. We extend the model in by introducing exposed class for humans and mosquitoes. Our main objective of this study is to investigate the stability analysis and also to study the important parameters in the transmission of malaria and try to develop effective ways for controlling the disease. The mathematical models developed in this work are supposed to be as simple so that medical authorities can use it to compute different scenarios for the future concerning the progression of the disease. Also, governmental authorities will also be able to use it to grade the measures they put in place to limit the spread of the virus, and eventually know if the measures implemented should be revised or not so as to achieve desired results.
Mathematical theories and models are used to analyze both data and new ideas in epidemiology . The conventional scientific approach is to observe a phenomenon, generate a hypothesis and design experiments to test the hypothesis. However, experiments in epidemiology are difficult to design, with serious ethical issues. A mathematical model, on the other hand, is a description of a phenomenon or situation based on a hypothesis. The general process involves certain assumptions on disease propagation, formulation of the assumption in mathematical terms and translation into a mathematical problem. The mathematical problem then becomes the model for the epidemic. The foundation of mathematical epidemiology was laid by the contribution of several biologists and physicians such as P.D. Enko, W.H. Hammer, Sir R.A. Ross, A.G. McKendrick and W.O. Kermack . However, the first mathematical model in epidemiology was developed by Bernouli to study the variolation against small pox in increasing life expectancy.
This work equally proposes an electronic model for the mathematical model developed. This electronic model is actually analyzed based on simulations on electrical circuits of the equations describing the mathematical model. Very few works have been carried out for the development of such electronic models, making the field for the improvement of this electronic model of the evolution of endemics (malaria in this case) large. Most often, simulation of mathematical models of the evolution of endemics are computer-based than electronic-based.
Now concerning how this work will be organized, after the present introduction, we present a general concept on endemics in chapter 1, where also the motivation and the problem statement of the work shall be clearly said. Then in chapter 2, we will look at the development of the mathematical model, and in chapter 3 we will look at the development of the electronic model.