Improved Variational Iteration Solutions to the SIR Model of Dengue Fever Disease for the Case of South Sulawesi
DOI:
https://doi.org/10.5614/j.math.fund.sci.2020.52.3.4Keywords:
approximation methods, dengue fever, SIR model, susceptible-infected- recovered, variational iterationsAbstract
The susceptible-infected-recovered (SIR) model of the spread of dengue fever for the case of South Sulawesi is considered. Rangkuti's variational iteration method (RVIM) is recalled. This paper makes four contributions. The first one is to provide a successive approximation method (SAM) for solving the considered model. The second one is to show that SAM and RVIM are identical. Thirdly, a modification of RVIM is proposed. Fourthly, it is shown that the modification leads to an improvement of the accuracy of the method. Both RVIM and the improved version are quite accurate for short time periods. However, the improved version is more accurate and is able to provide more realistic explicit solutions to the model.References
Arquam, M., Singh, A. & Cherifi, H., Impact of Seasonal Conditions on Vector-Borne Epidemiological Dynamics, IEEE Access, 8, Art. 9096282, pp. 94510-94525, 2020.
Iggidr, A. & Souza, M.O., State Estimators for Some Epidemiological Systems, Journal of Mathematical Biology, 78(1-2), pp. 225-256, 2019.
Makinde, O.D., Adomian Decomposition Approach to a SIR Epidemic Model with Constant Vaccination Strategy, Applied Mathematics and Computation, 184(2), pp. 842-848, 2007.
Makinde, O.D., On Non-perturbative Approach to Transmission Dynamics of Infectious Diseases with Waning Immunity, International Journal of Nonlinear Sciences and Numerical Simulation, 10(4), pp. 451-458, 2009.
Makinde, O.D. & Okosun, K.O., Impact of Chemotherapy on Optimal Control of Malaria Disease with Infected Immigrants, Biosystems, 104(1), pp. 32-41, 2011.
Massawe, L.N., Massawe, E.S. & Makinde, O.D., Dengue in Tanzania - Vector Control and Vaccination, American Journal of Computational and Applied Mathematics, 5(2), pp. 42-65, 2015.
Okosun, K.O. & Makinde, O.D., A Co-infection Model of Malaria and Cholera Diseases with Optimal Control, Mathematical Biosciences, 258, pp. 19-32, 2014.
Wagner, C.E., Hooshyar, M., Baker, R.E., Yang, W., Arinaminpathy, N., Vecchi, G., Metcalf, C.J.E., Porporato, A. & Grenfell, B.T., Climatological, Virological and Sociological Drivers of Current and Projected Dengue Fever Outbreak Dynamics in Sri Lanka, Journal of the Royal Society, Interface, 17(167), Art. 20200075, pp. 1-15, 2020.
Windarto, Khan, M.A. & Fatmawati, Parameter Estimation and Fractional Derivatives of Dengue Transmission Model, AIMS Mathematics, 5(3), pp. 2758-2779, 2020.
Zhu, M., Lin, Z. & Zhang, L., Spatial-Temporal Risk Index and Transmission of a Nonlocal Dengue Model, Nonlinear Analysis: Real World Applications, 53, Art. 103076, pp. 1-22, 2020.
Cahyono, E.S., Faruk, A. & Suprihatin, B., Analysis of Dengue Fever Disease Transmission Using Suspected-Infected-Recovered (SIR) Model, Journal of Physics: Conference Series, 1282(1), Art. 012013, pp. 1-7, 2019.
Hamdan, N.I. & Kilicman, A., A Fractional Order SIR Epidemic Model for Dengue Transmission, Chaos, Solitons and Fractals, 114, pp. 55-62, 2018.
Nur, W., Rachman, H., Abdal, N.M., Abdy, M. & Side, S., SIR Model Analysis for Transmission of Dengue Fever Disease with Climate Factors Using Lyapunov Function, Journal of Physics: Conference Series, 1028(1), Art. 012117, pp. 1-7, 2018.
Silva, T. & Montalvao, J., Inversion of the SIR-SI System for Estimation of Human-Vector Contact Rate and Prediction of Dengue Cases, IEEE Latin America Transactions, 17(9), Art. 8931142, pp. 1482-1490, 2019.
Fakhruddin, M., Nuraini, N. & Indratno, S.W., Mathematical Model of Dengue Transmission Based on Daily Data in Bandung, AIP Conference Proceedings, 2084(1), Art. 020013, pp. 1-8, 2019.
Ramadhan, N.R., Side, S., Sidjara, S., Irwan & Sanusi, W., Numerical Solution of SIRS Model for Transmission of Dengue Fever Using Homotopy Perturbation Method in Makassar, AIP Conference Proceedings, 2192(1), Art. 060015, pp. 1-8, 2019.
Rangkuti, Y.M., Side, S. & Noorani, M.S.M., Numerical Analytic Solution of SIR Model of Dengue Fever Disease in South Sulawesi Using Homotopy Perturbation Method and Variational Iteration Method, Journal of Mathematical and Fundamental Sciences, 46(1), pp. 91-105, 2014.
Binder, M. & Pilyugin, S.S., Stability Analysis of a Deterministic Model of Zika/Dengue Co-Circulation, International Journal of Biomathematics, 12(4), Art. 1950045, pp. 1-39, 2019.
Hamdan, N.I. & Kilicman, A., Basic Epidemic Model of Dengue Transmission Using the Fractional Order Differential Equations, Malaysian Journal of Science, 38, pp. 1-18, 2019.
Lamwong, J. & Pongsumpun, P., Mathematical Model for 4 Serotypes of Dengue Virus with Vaccination, Proceedings of the 2nd European Conference on Electrical Engineering and Computer Science (EECS 2018), Art. 8910025, pp. 152-159, 2018.
Perera, S.S.N., Insurance Model to Estimate the Financial Risk due to Direct Medical Cost on Dengue Outbreaks, Studies in Systems, Decision and Control, 200, pp. 629-663, 2019.
Side, S. & Noorani, S.M., A SIR Model for Spread of Dengue Fever Disease (Simulation for South Sulawesi, Indonesia and Selangor, Malaysia), World Journal of Modelling and Simulation, 9(2), pp. 96-105, 2013.
Esteva, L. & Vargas, C., Analysis of a Dengue Disease Transmission Model, Mathematical Biosciences, 150(2), pp. 131-151, 1998.
Yaacob, Y., Analysis of a Dengue Disease Transmission Model without Immunity, Matematika, 23(2), pp. 75-81, 2007.
Agarwal, R.P. & O'Regan, D., An Introduction to Ordinary Differential Equations, Springer, New York, 2008.
Mungkasi, S., Variational Iteration and Successive Approximation Methods for a SIR Epidemic Model with Constant Vaccination Strategy, Applied Mathematical Modelling, 90, pp. 1-10, 2021.
He, J-H., Variational Iteration Method - A Kind of Non-Linear Analytical Technique: Some Examples, International Journal of Non-Linear Mechanics, 34(4), pp. 699-708, 1999.
He, J-H., Variational Iteration Method for Autonomous Ordinary Differential Systems, Applied Mathematics and Computation, 114(2-3), pp. 115-123, 2000.
