## Analysis of malaria dynamics using its fractional order mathematical model.(English)Zbl 07452316

Summary: In this paper, we have studied dynamics of fractional order mathematical model of malaria transmission for two groups of human population say semi-immune and non-immune along with growing stages of mosquito vector. The present fractional order mathematical model is the extension of integer order mathematical model proposed by Ousmane Koutou et al. For this study, Atangana-Baleanu fractional order derivative in Caputo sense has been implemented. In the view of memory effect of fractional derivative, this model has been found more realistic than integer order model of malaria and helps to understand dynamical behaviour of malaria epidemic in depth. We have analysed the proposed model for two precisely defined set of parameters and initial value conditions. The uniqueness and existence of present model has been proved by Lipschitz conditions and fixed point theorem. Generalised Euler method is used to analyse numerical results. It is observed that this model is more dynamic as we have considered all classes of human population and mosquito vector to analyse the dynamics of malaria.

### MSC:

 34A34 Nonlinear ordinary differential equations and systems 34B60 Applications of boundary value problems involving ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 92B05 General biology and biomathematics
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### References:

  S.G. Samko et al., Fractional Integrals and Derivatives Theory and Applications, Gorden and Breach, New York, 1993.  I. Podlubny, Fractional Differential Equation, Academic Press, New York, 1999. · Zbl 0924.34008  R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001. · Zbl 0998.26002  O. Marom et al., A comparison of numerical solutions of fractional diffusion models in finance, Non linear Analysis: Real World Applications 10 (2009), 3435-3442. · Zbl 1180.91308  Peter Amoako-Yirenkyi1 et al., A new construction of a fractional derivative mask for image edge analysis based on Riemann-Liouville fractional derivative, Advances in Difference Equations 2016 (2016), 1-23. · Zbl 1419.34009  V.K. Srivastava et al., Two dimensional time fractional-order biological population model and its analytical solution, Egyptian Journal of Basic and Applied Sciences 1 (2014), 71-76.  H. Sheng et al., Fractional Processes and Fractional order Signal Processing: Techniques and Applications, Signals and Communication Technology, Springer, New York, USA, 2012.  D. Baleanu et al., Fractional Dynamics and Control, Springer, New York Dordrecht, London, 2012.  T.J. Anastasio et al., The fractional order dynamics of brain stem vestibulo-oculomotor neirons, Biological Cybernetics 72 (1994), 69-79.  N. OZalp et al., A fractional order SEIR model with vertical transmission, Mathematical and Computer Modelling 54 (2011), 1-6. · Zbl 1225.34011  N.H. Sweilam et al., Optimal control for fractional tuberculosis infection model including the impact of diabetes and resistant strains, Journal of Advanced Research 17 (2019), 125-137.  Carla M.A. Pinto et al., Fractional model for malaria transmission under control strategies, Computers and Mathematics with Applications 66 (2013), 908-916.  D.D. Pawar et al., Numerical solution of fractional order mathematical model of drug resistant tuberculosis with two line treatment, Journal of Mathematics and Computational Science 10 (2019), 262-276.  Kumar Devendra et al., A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Advances in Difference Equations 2019 (2019), 1-19. · Zbl 1458.65118  Ousmane Koutou et al., Mathematical modelling of malaria transmission global dynamics: taking into account the immature stages of vectors, Advances in Difference Equations 2018 (2018), 1-34. · Zbl 1445.34030  I. Khan et al., New idea of Atangana and Baleanu fractional derivatives to human blood flow in nano fluids, Chaos An Interdisciplinary Journal of Nonlinear Science 29 (2019), 1-10.  M.A. Ullah Khan et al., A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos, Solitons and Fractals 116 (2018), 227-238. · Zbl 1442.92150  S. Ucar et al, Analysis of a basic SEIRA model with Atangana-Baleanu derivative, AIMS Mathematics 5 (2020), 1411-1424.  Bakary Traore et al., A mathematical model of malaria transmission with structured vector and seasonality, Journal of Applied Mathematics 2017, (2017), 1-15. · Zbl 1437.92081
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