International Journal of Research and Innovation in Applied Science (IJRIAS) |Volume VIII, Issue I, January 2023|ISSN 2454-6194

 A Mathematical Measure to Fight Against Malaria and Exterminate Anopheles Mosquitoes

Atanyi Yusuf Emmanuel¹, H.K Oduwole², Utalor Kate Ifeoma^3
1Department of Mathematics, Federal University of Lafia, PMB 146, Lafia, Nigeria
²Department of Mathematics, Nasarawa State University, Keffi,PMB 1022, Keffi, Nigeria
³Mathematical Centre, Abuja, Nigeria
*Correspondence
Received: 13 January 2023; Received: 27 January 2023; Accepted: 04 February 2023; Published: 28 February 2023

Modelling the effects of three natural predators on the aquatic and adult anopheles’ mosquitoes in the control of malaria transmission was aimed at eradicating anopheles’ larva, pupa and adult anopheles’ mosquito by introduction of natural predators “copepods, tadpoles and purple martins” (organism that eat up mosquito at larva, pupa, and adult stages), so that there should not be anopheles’ adult mosquito for malaria transmission in our society. This new proposed model is a control flow diagram of predator-prey interaction model in mosquito life-cycle that considers an open population of mosquito and predators. The population is sub-divided based on mosquito life-cycle and natural predators. Under a mosquito life-cycle, the population is divided into four compartments, Egg compartment E(t), Larva compartment L(t), Pupa compartment P(t), and Adult compartment A(t), and natural predators, it is divided into three compartments, namely; Copepods C_P(t), Tadpole T_P (t) and Purple martins P_M(t). These models provide understanding for control of malaria in our environments, especially when the models are based on the ecology of the vector population and sound understanding of variables and parameters relevant for transmission. The model equations were derived using the model variables and parameters. The stability analysis of the free equilibrium states were analyzed using equilibrium point, elimination, substitution methods, idea of Beltrami’s and Diekmann’s conditions. From the stability analysis of steady state, we observed that the model free equilibrium state is stable, this implies that the equilibrium point or steady state is stable and the stability of the model(3.13.1) – (3.13.8) means, there will not be anopheles adult mosquito in our society for malaria transmission and from the idea of Beltrami’s and Diekmann’s conditions we observed that the Determinant of the Jacobian matrix is greater than zero(Det⁡〖{j}〗>0),Trace of the Jacobian matrix is less than zero(Tr{j}<0) and the basic reproduction number is less than one ( R₀<1) which implies that the model disease free equilibrium state is stable. Hence the number of larva that transform to pupa is almost zero and the number of pupa that develop to adult is minimal and number of adult that escape to vector stage are inconsequential, that means the life-cycle could be broken at the larva, pupa, and adult stages with the introduction of natural predators, with the natural implication there will not be anopheles adult mosquito for malaria transmission and we also use maple for symbolical and numerical solution and presented the results graphically. The contribution of this research work to knowledge is to bring out the control flow diagram of prey-predator interaction, mathematical models, Identify the ability to control and eradicate malaria through stability analysis and numerical experiments showing the effect of the introduction of three natural predators on the larva, pupa and adult stages of the adult Anopheles mosquito( biological inoffensive method) which will contribute to the eradication of adult anopheles’ mosquito, which will also lead to the elimination of malaria in our society.

I. Introduction

1. 1 Background to the Study

The Anopheles vector system in Nigeria and in sub-Saharan Africa is probably the strongest that exists for human Plasmodium. Contact with human vectors, particularly An. gambiae s.l., shows remarkable stability and flexibility, resulting in extremely high vaccination rates under different seasonal and geographic ecological conditions (Mokuolu et al., 2018). Malaria remains a leading cause of death and disease in most tropical regions of the world, where it is endemic in 106 countries. In 2010, out of a total of 216 million cases of malaria, around 81% occurred in Africa and 13% in Southeast Asia1. The majority (91%) of the estimated 665,000 malaria deaths occur in Africa and primarily affect children under the age of five (86%). In America in 2010 there were more than 670,000 confirmed cases of malaria with 133 deaths from malaria. The transmission is active in 21 countries and puts approximately 20% of the US population at risk. Malaria severely limits economic development and is a cause of poverty in most countries where the disease is endemic. Malaria remains an ongoing problem in sub-Saharan Africa, and while great strides have been made over the past 15 years, millions of people are still at risk of contracting the parasite (Patouillard et al., 2017).

Africa offers a stable and ecologically diverse ecosystem and hosts the world’s highest vectors of malaria (Bernard et al., 2020) and is expected to remain so in the future. Climate change (Adigun et al., 2015). The main vectors of Anopheles malaria in sub-Saharan Africa are Anopheles funestus s.s. and three members of the Anopheles gambiae complex: An. Gambiae s.s., Anopheles coluzzii and Anopheles arabiensis (Molinaro et al., 2015), which play a role in the transmission of malaria in their distribution area, e.g. the groups Anopheles moucheti and Anopheles nili (Rajeswari, 2017) and another of secondary or random vectors (Antonio-nkondjio et al., 2006). Considering that the genus Anopheles includes more than 500 species worldwide, of which only a few are considered important species for the transmission of malaria (Garcia Guerra et al., 2014).
The morphological identification of species is crucial for allocating scarce resources solely to the fight against malaria vectors. Species groups and species complexes are common within the genus Anopheles (Harbach & Besansky, 2014) and this complicates vector control because not all species in a complex share similar behaviors or similar roles in transmission malaria disease (Velickovic & Leicht, 2002).
Mosquitoes of the family Culicidae are considered a nuisance and a major public health problem because their females feed on human blood and therefore transmit extremely harmful diseases such as malaria, yellow fever and filariasis (Tsoka-Gwegweni & Okafor, 2014). They are estimated to transmit diseases to more than 700 million people each year and are responsible for the death of around 1 in 17 people (“Malaria Policy Advisory Committee to the WHO: Conclusions and Recommendations of Eighth Biannual Meeting (September 2015),” 2016). Effective transmission of mosquito-borne diseases requires successful contact between female mosquitoes and their hosts (Vanelle et al., 2012a). Among Anopheles, members of the genus Anopheles are best known for their role in the global transmission of malaria and filariasis (“Malaria Policy Advisory Committee to the WHO: Conclusions and Recommendations of Fifth Biannual Meeting (March 2014),” 2014). Among these diseases, malaria, caused by the Plasmodium parasite, is one of the deadliest diseases in the world (“Malaria Policy Advisory Committee to the WHO: Conclusions and Recommendations of Sixth Biannual Meeting (September 2014),” 2015). (“Malaria Vaccine: WHO Position Paper, January 2016 – Recommendations,” 2018) reported approximately 207 million cases of malaria in 2012, of which 200 million (80.0%) were on the affected continent. Patterns of disease spread, transmission, and intensity depend on the degree of urbanization and distance from vector breeding sites (MCNAMARA, 2005). The endemicity of malaria in each region is determined, among other things, by native Anopheles mosquitoes, their abundance, diet, resting behavior and Plasmodium infectivity (Atta & Reeder, 2014). The Federal Ministry of Health in Abuja reported that at least 50.0% of Nigerians suffer from some form of malaria, making it the most significant health problem in Nigeria (UM & AN, 2016). The high transmission rate and prevalence of malaria is the result of the various mosquito breeding sites, including convenient water reservoirs such as cans, old tires, tree holes, cisterns, open pools, drains,

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