# Dynamics of cholera disease by using two recent fractional numerical methods

## Authors

• Pushpendra Kumar Department of Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, Punjab-151001, India
• Vedat Suat Erturk Department of Mathematics, Ondokuz Mayis University, Atakum-55200, Samsun, Turkey

## Keywords:

Cholera disease, mathematical model, generalized Liouville-Caputo fractional derivative, numerical methods, graphical simulations

## Abstract

In this paper, we simulate an epidemic model of cholera disease in the sense of generalized Liouville-Caputo fractional derivative. We provide the results related to the existence of a unique solution by using some well-known theorems. Numerical solutions of the given model are derived by using two different numerical methods along with their importance. A number of graphs are plotted to understand the given cholera disease dynamics. The main motivation to do this research is to understand the given disease dynamics as well as the efficiency of both methods which are very recent to the literature.

## References

Piarroux, R., Barrais, R., Faucher, B., Haus, R., Piarroux, M., Gaudart, J., Magloire, R. & Raoult, D. Understanding the cholera epidemic, Haiti. Emerging infectious diseases, 17(7), 1161, (2011).

Tappero, J.W. & Tauxe, R.V. Lessons learned during public health response to cholera epidemic in Haiti and the Dominican Republic. Emerging infectious diseases, 17(11), 2087, (2011).

Lemos-Paião, A.P., Silva, C.J. & Torres, D.F. An epidemic model for cholera with optimal control treatment. Journal of Computational and Applied Mathematics, 318, 168-180, (2017).

Eustace, K.A., Osman, S. & Wainaina, M. Mathematical modelling and analysis of the dynamics of cholera. Global Journal of Pure and Applied Mathematics, 14(9), 1259-1275, (2018).

Sun, G.Q., Xie, J.H., Huang, S.H., Jin, Z., Li, M.T. & Liu, L. Transmission dynamics of cholera: Mathematical modeling and control strategies. Communications in Nonlinear Science and Numerical Simulation, 45, 235-244, (2017).

Njagarah, J.B.H. & Tabi, C.B. Spatial synchrony in fractional order metapopulation cholera transmission. Chaos, Solitons & Fractals, 117, 37-49, (2018).

Kilbas, A., Srivastava, H.M., and Trujillo, J.J. Theory and Applications of Fractional Differential Equations. Elsevier Science, (2006).

Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Elsevier, (1998).

Kumar, P. & Erturk, V.S. The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative. Mathematical Methods in the Applied Sciences, (2020).

Kumar, P., Erturk, V.S., Abboubakar, H. & Nisar, K.S. Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives. Alexandria Engineering Journal, 60(3), 3189-3204, (2021).

Gao, W., Veeresha, P., Baskonus, H.M., Prakasha, D.G. & Kumar, P. A New Study of Unreported Cases of 2019-nCOV Epidemic Outbreaks. Chaos, Solitons & Fractals,138, 109929, (2020).

Nabi, K.N., Abboubakar, H. & Kumar, P. Forecasting of COVID-19 pandemic: From integer derivatives to fractional derivatives. Chaos, Solitons & Fractals, 141, 110283, (2020).

Nabi, K.N., Kumar, P. & Erturk, V.S. Projections and fractional dynamics of COVID-19 with optimal control strategies. Chaos, Solitons & Fractals, 145, 110689, (2021).

Kumar, P., Erturk, V.S., & Murillo-Arcila, M. A new fractional mathematical modelling of COVID-19 with the availability of vaccine. Results in Physics, 24, 104213, (2021).

Naik, P.A., Yavuz, M., Qureshi, S., Zu, J. & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).

Özköse, F., & Yavuz, M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in biology and medicine, 105044, (2022).

Ikram, R., Khan, A., Zahri, M., Saeed, A., Yavuz, M. & Kumam, P. Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay. Computers in Biology and Medicine, 141, 105115, (2022).

Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A. & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(2), 56-66, (2021).

Kumar, P., Erturk, V.S., Yusuf, A. & Kumar, S. Fractional time-delay mathematical modeling of Oncolytic Virotherapy. Chaos, Solitons & Fractals, 150, 111123, (2021).

Abboubakar, H., Kumar, P., Erturk, V.S. & Kumar, A. A mathematical study of a Tuberculosis model with fractional derivatives. International Journal of Modeling, Simulation, and Scientific Computing, 2150037, (2021).

Abboubakar, H., Kumar, P., Rangaig, N.A. & Kumar, S. A Malaria Model with Caputo-Fabrizio and Atangana-Baleanu Derivatives. International Journal of Modeling, Simulation, and Scientific Computing, 12(02), 2150013, (2020).

Kumar, P., Erturk, V.S., Yusuf, A. & Sulaiman, T.A. Lassa hemorrhagic fever model using new generalized Caputo-type fractional derivative operator. International Journal of Modeling, Simulation, and Scientific Computing, 12(06), 2150055, (2021).

Kumar, P., Erturk, V.S., Yusuf, A., Nisar, K.S. & Abdelwahab, S.F. A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives. Results in Physics, 25, 104281, (2021).

Kumar, P., Erturk, V.S. & Murillo-Arcila, M. A complex fractional mathematical modeling for the love story of Layla and Majnun. Chaos, Solitons & Fractals, 150, 111091, (2021).

Kumar, P. & Erturk, V.S. Environmental persistence influences infection dynamics for a butterfly pathogen via new generalised Caputo type fractional derivative. Chaos, Solitons & Fractals, 144, 110672, (2021).

Kumar, P., Erturk, V.S., Banerjee, R., Yavuz, M. & Govindaraj, V. Fractional modeling of plankton-oxygen dynamics under climate change by the application of a recent numerical algorithm. Physica Scripta, 96(12), 124044, (2021).

Kumar, P., Erturk, V.S. & Nisar, K.S. Fractional dynamics of huanglongbing transmission within a citrus tree. Mathematical Methods in the Applied Sciences, (2021).

Kumar, P., Erturk, V.S. & Almusawa, H. Mathematical structure of mosaic disease using microbial biostimulants via Caputo and Atangana–Baleanu derivatives. Results in Physics, 24, 104186, (2021).

Joshi, H. & Jha, B.K. Fractional-order mathematical model for calcium distribution in nerve cells. Computational and Applied Mathematics, 39(2), 1-22, (2020).

Joshi, H. & Jha, B.K. On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory. The European Physical Journal Plus, 136(6), 1-15, (2021).

Joshi, H. & Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(2), 84-94, (2021).

Joshi, H. & Jha, B.K. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects. International Journal of Nonlinear Sciences and Numerical Simulation, (2021).

Veeresha, P. A numerical approach to the coupled atmospheric ocean model using a fractional operator. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 1-10, (2021).

Baishya, C. & Veeresha, P. Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel. Proceedings of the Royal Society A, 477(2253), 20210438, (2021).

Veeresha, P. & Baleanu, D. A unifying computational framework for fractional Gross–Pitaevskii equations. Physica Scripta, 96(12), 125010, (2021).

Akinyemi, L., Nisar, K.S., Saleel, C.A., Rezazadeh, H., Veeresha, P., Khater, M.M. & Inc, M. Novel approach to the analysis of fifth-order weakly nonlocal fractional Schrödinger equation with Caputo derivative. Results in Physics, 31, 104958, (2021).

Okposo, N.I., Veeresha, P. & Okposo, E.N. Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons. Chinese Journal of Physics, (2021).

Odibat, Z. and Baleanu, D. Numerical simulation of initial value problems with generalized caputo-type fractional derivatives. Applied Numerical Mathematics, 156, 94-105, (2020).

Erturk, V.S. & Kumar, P. Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives. Chaos, Solitons & Fractals, 139, 110280, (2020).

Kumar, P. & Erturk, V.S. A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives. Mathematical Methods in the Applied Sciences, 1-14, (2021).

Odibat, Z., Erturk, V.S., Kumar, P. & Govindaraj, V. Dynamics of generalized Caputo type delay fractional differential equations using a modified Predictor-Corrector scheme. Physica Scripta, 96(12), 125213, (2021).

Kumar, P., Erturk, V.S. & Kumar, A. A new technique to solve generalized Caputo type fractional differential equations with the example of computer virus model. Journal of Mathematical Extension, 15, (2021).

## Published

2021-12-21
CITATION METRICS
DOI: 10.53391/mmnsa.2021.01.010

## How to Cite

Kumar, P., & Erturk, V. S. (2021). Dynamics of cholera disease by using two recent fractional numerical methods. Mathematical Modelling and Numerical Simulation With Applications, 1(2), 102–111. https://doi.org/10.53391/mmnsa.2021.01.010

## Section

Research Articles  