Malaria and cholera co-dynamic model analysis furnished with fractional-order differential equations

Year 2023, Volume: 3 Issue: 1, 33 - 57, 31.03.2023

Livinus L. Iwa Ugochukwu K. Nwajeri Anne O. Atede Augustine B. Panle Kenneth U. Egeonu

https://doi.org/10.53391/mmnsa.1273982

Cited By: 3

Abstract

This paper presents malaria and cholera co-dynamics under Caputo-Fabrizio derivative of order $\alpha\in(0,1)$ varied with some notable parameters in the fractional system. The fractional order system comprises ten compartments divided into human and vector classes. The human population is exposed to obnoxious diseases such as malaria and cholera which can lead to an untimely death if proper care is not taken. As a result, we present the qualitative analysis of the fractional order system where the existence and uniqueness of the solution using the well-known Banach and Schauder fixed point theorems. The numerical solution of the system is achieved through the famous iterative Atangana-Baleanu fractional order Adams-Bashforth scheme. The numerical algorithm obtained from the scheme is used for graphic simulation for different fractional orders $\alpha\in (0,1)$. The figures produced using various fractional orders show total convergence and stability as time increases. It is also evident that stability and convergence are achieved as the fractional orders tend to 1. The actual behavior of the fractional co-dynamical system of the diseases is established also in the numerical simulation.

Keywords

Malaria disease, cholera disease, fractional derivative, stability, co-infection

References

• Birhanie, M., Tessema, B., Ferede, G., Endris, M., & Enawgaw, B. Malaria, typhoid fever, and their coinfection among febrile patients at a rural health center in Northwest Ethiopia: a cross-sectional study. Advances in Medicine, 531074, (2014).
• World malaria report 2019. https://www.who.int/malaria/publications/world-malariareport-2019/en/.(2019), Meeting Report, Access date: 25th November 2022.
• World Health Organization, www.who.int/news-room, Access date: 30th March 2022.
• Centres for Disease Control and Prevention (CDC). Cholera - Vibrio cholera infection, (2020). https://www.cdc.gov/cholera/general/index.html. Access date: 12th August 2022.
• Ross, S. The Prevention of Malaria Dutton: New York, NY, USA, (1911).
• Okosun, K.O. & Makinde O.D. A co-infection model of malaria and cholera diseases with optimal control. Mathematical Biosciences, 258, 19-32, (2014).
• Egeonu, K.U., Omame, A., & Inyama, S.C. A co-infection model for two-strain malaria and cholera with optimal control. International Journal of Dynamics and Control, 9, 1612–1632, (2021).
• Mandal, S., Sarkar, R.R., & Sinha, S. Mathematical models of malaria-a review. Malaria Journal, 10, 202, (2011).
• Oke, S.I., Ojo, M.M., Adeniyi, M.O., & Matadi, M.B. Mathematical modeling of malaria disease with control strategy. Communication in Mathematical Biology and Neuroscience, (2020).
• Osman, M.A.E., Adu, I.K., Simple mathematical model for malaria transmission. Journal of Advances in Mathematics and Computer Science, 25(6), 1-24, (2017).
• Tilahun, G.T., Woldegerima, W.A., & Wondifraw, A. Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission. Advances in Difference Equation, 2020, (2020).
• Hntsa, K.H., & Kahsay, B.N. Analysis of cholera epidemic controlling using mathematical modeling. International Journal of Mathematics and Mathematical Sciences, 2020, 1-13, (2020).
• Nwajeri, U.K., Panle, A.B., Omame, A., Obi M.C., & Onyenegecha, C.P. On the fractional order model for HPV and Syphilis using non-singular kernel. Results in Physics, 37, 105463, (2022).
• Omame, A., Isah, M.E., Abbas, M., Abdel-Aty, A.H, & Onyenegecha, C.P. A fractional order model for Dual Variants of COVID-19 and HIV co-infection via Atangana-Baleanu derivative. Alexandria Engineering Journal, 61(12), 9715-9731, (2022).
• Nwajeri, U.K., Omame, A., & Onyenegecha, C.P. Analysis of a fractional order model for HPV and CT co-infection. Results in Physics, 28, 104643, (2021).
• Ogunrinde, R.B., Nwajeri, U.K., Fadugba, S.E., Ogunrinde, R.R., & Oshinubi, K.I. Dynamic model of COVID-19 and citizens reaction using fractional derivative. Alexandria Engineering Journal, 60(2), 2001-2012, (2021).
• Ahmed, I., Baba, I.A., Yusuf, A., Kumam, P., & Kumam, W. Analysis of Caputo fractional-order model for COVID-19 with lockdown. Advances in Difference Equations, 394, (2020).
• Almeida, R., Cruz, A.M.C.B., Martins, N., & Monteiro, M.T.T. An epidemiological MSEIR model described by the Caputo fractional derivative. International Journal of Dynamics and Control, 7, 776-784, (2019).
• Karaji, P.T., & Nyamoradi, N. Analysis of a fractional SIR model with general incidence function. Applied Mathematics Letters, 108, 106499, (2020).
• Lin, W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications, 332(1), 709-726, (2007).
• Tuan, N.H., Mohammadi, H., & Rezapour, S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos, Solitons & Fractals, 140, 110107, (2020).
• Alrabaiah, H., Ur-Rahman, M., Mahariq, I., Bushnaq, S., & Arfan, M. Fractional order analysis of HBV and HCV co-infection under ABC derivative. Fractals, 30(01), 2240036, (2022).
• Wei-Yun, S., Yu-Ming, C., Ur-Rahman, M., Mahariq, I., & Zeb, A. Mathematical analysis of HBV and HCV co-infection model nonsingular fractional order derivative. Results in Physics, 28, 104582, (2021).
• Arafa, A.A.M., Rida, S.Z, & Khalil, M. A fractional-order model of HIV infection with drug therapy effect. Journal of the Egyptian Mathematical Society, 22(3), 538-543, (2014).
• Baleanu, D., Jajarmi, A., Sajjadi, S.S., & Mozyrska, D. A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(8), 083127, (2019).
• Liu, X., Arfan, M., Ur Rahman, M., & Fatima, B. Analysis of SIQR type mathematical model under Atangana-Baleanu fractional differential operator. Computer Methods in Biomechanics and Biomedical Engineering, 26(1), 98-112, (2022).
• Losada, J., & Nieto, J.J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Application, 1(2), 87-92, (2015).
• Özköse, F., Şenel, M.T., & Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).
• Omame, A., Abbas, M., & Onyenegecha, C.P. A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana-Baleanu derivative. Chaos Solitons & Fractals, 153, 111486, (2021).
• Uçar, E., Uçar, S., Evirgen, F., & Özdemir, N. A fractional SAIDR model in the frame of Atangana-Baleanu derivative. Fractal and Fractional, (2021).
• Uçar, S. Analysis of a basic SEIRA model with Atangana-Baleanu derivative. AIMS Mathematics, (2020).
• Omame, A., Abbas, M., & Abdel-Aty, A.H. Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives. Chaos Solitons & Fractals, 162, 112427, (2022).
• Blayneh, K.W., Cao, Y., & Kwon, H.D. Optimal control of vector-borne disease: treatment and prevention. Discrete and Continuous Dynamical Systems B, 11(3), 587-611, (2009).
• Ishikawa, H., Ishii, A., Nagai, N., Ohmae, H., Harada, M., Suguri, S., & Leafasia, J. A mathematical model for the transmission of Plasmodium vivax malaria. Parasitology International, 52(1), 81-93, (2003).
• Aron, J.L., & May, R.M. The population dynamics of malaria. In: Anderson RM(ed) Population dynamics of infectious diseases (pp. 139-179). London: Chapman and Hall, (1982).
• Smith, R.J., & Hove-Musekwa, S.D. Determining effective spraying periods to control malaria via indoor residual spraying in Sub-Saharan Africa. Journal of Applied Mathematics and Decision Sciences, 745463, (2008).
• Buonomo, B. Analysis of a malaria model with mosquito host choice and bed-net control. International Journal of Biomathematics, 8(6), 1550077, (2015).
• Nielan, R.L.M., Schaefer, E., Gaff, H., Fister, K.R., & Lenhart, S. Modeling optimal control intervention strategies for cholera. Bulletin of Mathematical Biology, 72, 2004-2018, (2010).
• Owolabi, K.M., & Atangana, A. Numerical Methods for Fractional Differentiation. Singapore: Springer Nature, (2019).
• Thabet, S.T.M., Abdo, M.S., & Shah, K. Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo–Fabrizio derivative. Advances in Differential Equations, 184, (2021).
• Van den Driessche, P., & Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1-2), 29-48, (2002).
• Toufik, M., & Atangana, A. New numerical approximation of fractional derivative with nonlocal and non-singular kernel: application to chaotic models. The European Physical Journal Plus, 132, 444, (2017).