Mathematical Modelling Of Malaria Disease in Busia County, Kenya
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Date
2024-02
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Kenyatta University
Abstract
Background: Millions of people throughout the world die every year from malaria, an illness spread by the bite
of an infected female Anopheles mosquito. Busia, a county in Kenya, has been recorded to have the highest
prevalent cases of 37% in Kenya. However, Busia has often been ignored in the mathematical modelling of
malaria in Kenya. The SEIR model is widely used in mathematical simulations of malaria transmission. However,
the paradigm is no longer relevant to malaria cases since asymptomatic Plasmodium parasites persist in the
systems of persons who have recovered from malaria. In this study, the human subpopulation carrying the
plasmodium parasites but are not suffering from malaria are included in the mathematical model. Therefore, this
study presents SIRSp model to study the trend of malaria disease in BusiaCounty,Kenya. The mathematical model
is analysed in the human population by assuming that the disease's infection rate is constant and is based not
only on the number of people who are infected but also on the number of people who are susceptible to the disease.
The reproduction number in human and in mosquitoes are obtained and the equilibrium point shows that the
disease-free equilibrium point is always stable. This suggests the possibility of eradicating malaria in Busia
County. From the numerical simulations, it is found out that the infected humans increase with the force of
infection. Increase in the rate of recovery from malaria reduces the number of infected humans and the infectious
mosquito subpopulation but increases the susceptible human subpopulation.
Methodology:
The stability of the system is established and it shows that the system is always stable when the subpopulations
start in the neighbourhood of the disease-free equilibrium point. The numerical solution of the system is sought
using an adaptive step-size Runge-Kutta-Fehlberg (RKF45) method. The parameter estimation was carried out
by using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and the optimal parameter values are obtained
as;𝛬ℎ = 1, 𝛾 = 1, 𝛼 = 1, 𝑟 = 0.39195362, 𝛬𝑚 = 1, 𝛽 = 1, 𝜇𝑚 = 0, 𝑝 = 0.72426444, 𝑞 = 0.23809668.
Optimisation of the parameters is done by comparing the numerical results with the real-world data. The optimal
parameter values are obtained as;𝛬ℎ = 1, 𝛾 = 1, 𝛼 = 1, 𝑟 = 0.39195362, 𝛬𝑚 = 1, 𝛽 = 1, 𝜇𝑚 = 0.To show the
fitting of the optimal values against the real-world data, we plotted the graphs.
Results and Conclusions: In this model the parameters are optimised and predicted the rate of human infection,
the rate at which mosquitoes get infected and the rate at which human beings recover. The results shows that an
increase in the rat of recovery of an infected human reduces the infection in the human population The optimised
model for the infected human subpopulation agrees well with the real-world data as time proceeds
Description
Research Article