Stability Analysis of Co-Infection of Malaria-Dengue

Stability Analysis of Co-Infection of Malaria-Dengue

Nisha Sheoran (Department of Mathematics, Gujarat University, Ahmedabad, India) and Moksha H. Satia (Department of Mathematics, Gujarat University, Ahmedabad, India)
Copyright: © 2020 |Pages: 40
DOI: 10.4018/978-1-7998-3741-1.ch007
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Dengue and malaria most commonly occur in tropical and sub-tropical areas. Dengue is a viral infection in a human being caused by a bite of a female aedes mosquito whereas malaria is caused by plasmodium parasite transmitted by a bite of infected mosquito. In this chapter, a mathematical model of co-infection of malaria and dengue is described by deterministic system of non-linear ordinary differential equations. This system considers the force of infection which is applied to dengue susceptible individuals. Moreover, two sub-models, namely malaria-only and dengue-only, are also constructed to study the transmission dynamics. Basic reproduction number is calculated for these models to investigate the existence of the models. The system is proved to be locally and globally stable at its equilibrium points. Stability of these models is also shown through numerical simulation.
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Now-a-days, due to environmental conditions and human behavior, population is prone to many viral infectious diseases. Most commonly occurring diseases in tropical and sub-tropical areas are due to mosquito bites. Some of the vector borne diseases are malaria, dengue, chikungunya, zika fever, filariasis etc. Among them malaria and dengue are most commonly prevailing diseases which leads to death also. Malaria is an infectious parasitic disease transmitted by Anopheles mosquito. There are five parasite species that causes malaria in humans, two of its major threating species are P. falciparum and P. vivax. Some of its symptoms are pain in abdomen, muscles, fatigue, sweating, shivering, vomiting etc. According to world malaria report 2013 published by WHO, malaria is a leading cause of premature death, particularly in children under the age of five, with an estimate of 207 million cases and more than half a million deaths in 2012. Also, death toll reached to one million as of 2018 according to the American mosquito control association. The other tropical disease Dengue which is a viral disease transmitted by the mosquitoes Aedes aegypti and Aedes albopictus, which are found throughout the world. Dengue was first recognised in 1950’s. Around 2.5 billion people, or 40% of the world's population, live in areas where there is a risk of dengue transmission. Some of the dengue symptoms include pain in abdomen and back of the eyes, nausea, skin rashes, vomiting, mild bleeding etc. Symptoms usually appear 4 to 7 days after the mosquito bite and typically last 3 to 10 days. Both the disease starts with some common symptoms for example headache, intense muscle pain, weakness. Which makes it difficult to identify the disease. Therefore, one should go for test of both the diseases.

Mathematical modeling is among the best ways to study dynamics of transmission of many problems. Kermack and McKendrick established basic foundation of mathematical modeling by developing SIR model in 1927. Martcheva in her book “mathematical modeling in epidemiology” described various methods and strategies to solve infectious disease models. The basic Mathematical model SIS of malaria transmission dynamics consisting of two compartments (Ross, 1911) and its modified model by Macdonald, (1957) which together known as Ross-Macdonald model of malaria transmission was developed which laid down the base for constructing future models. A review related to malaria infection due to vectors were discussed containing different models covering every criterion such as age, environment, immunity, socio-economic etc. by Mandal et al., (2011). Various dengue transmission models also exist. One of dengue transmission model considering severe DHF compartment was studied by Nuraini et al., (2007). Global stability of dengue model with the help of saturation and bilinear incidence was discussed by Cai et al., (2009). Rodrigues et al. considered both human and mosquitos’ population and applied control parameter (insecticide) in order to fight against mosquitoes and they have also concluded usage of insecticide to be done in night. Co-infection model has also been studied by some of the researchers. for example, Stability Analysis of Zika – Malaria Co-infection Model with its sub-models zika only, malaria-only for Malaria Endemic Region was studied (Mensah et al.,2018). Mathematical model considering both human population and mosquitos population in dengue-chikungunya co-infection is studied by Aldila and Agustin, 2018. Mathematical analysis for co-infection of HIV-Malaria which showed their co-existence when their reproduction number exceeds unity was contributed by Mukandavire et al., (2009) Salam et al. have reviewed the literature associated with the co-infection of three vector borne diseases malaria-dengue-chikungunya. One of the co-infection malaria-dengue studied by them showed that all possible combination of co-infection of these three diseases were seen in only India and Nigeria.

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