Application of Mathematical Modeling: A Mathematical Model for Dengue Disease in Bangladesh
Автор: Nazrul Islam, J.R.M. Borhan, Rayhan Prodhan
Журнал: International Journal of Mathematical Sciences and Computing @ijmsc
Статья в выпуске: 1 vol.10, 2024 года.
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A virus spread by mosquitoes called dengue fever affects millions of people each year and is a serious threat to world health. More than 140 nations are affected by the illness of dengue fever. Therefore, in this paper, a Susceptible-Infectious-Recovered (SIR) mathematical model for the host (human) and vector (dengue mosquitoes) has been presented to describe the transmission of dengue in Bangladesh. In the model the vector are related with two compartments that are susceptible and infective and host are related with three compartments that are susceptible, infective, and recovered. By these five compartments, five connected nonlinear ordinary differential equations (ODEs) are produced. As a result of non dimensionalization, a system of three nonlinear ODEs has been generated. The reproductive number and equilibrium points have been estimated for different cases. In order to compute the infection rate, data for infected human populations have been gathered from multiple health institutes in Bangladesh. MATLAB has been utilized to construct numerical simulations of different compartments in order to examine the impact of critical parameters on the disease’s propagation and to bolster the analytical findings. The simulated outcomes for susceptible, infected, and eliminated in graphical formats have been displayed. The paper’s main goal is to emphasize the uniqueness of computational analysis of the SIR mathematical model for the dengue fever.
Dengue Disease, Endemic Equilibrium, Mathematical Model, Reproduction Number, Numerical Simulations
Короткий адрес: https://sciup.org/15019072
IDR: 15019072 | DOI: 10.5815/ijmsc.2024.01.03
Список литературы Application of Mathematical Modeling: A Mathematical Model for Dengue Disease in Bangladesh
- Kandhway, K. and Kuri, J., How to run a campaign: optimal control of SIS and SIR information epidemics, Applied Mathematics and Computation, 231, 79-92, 2014. https://doi.org/10.1016/j.amc.2013.12.164.
- Rodrigues, H. S., Application of SIR epidemiological model: new trends, International journal of applied mathematics and informatics, 10, 92-97, 2016. https://arxiv.org/pdf/1611.02565.pdf.
- Ehrhardt, M., Gasper, J. and Kilianova, S., SIR-based mathematical modeling of infectious diseases with vaccination, Journal of Computational Science, 37, 101027, 2019. https://doi.org/10.1016/j.jocs.2019.101027.
- Zaman, G., Kang, Y. H. and Jung, H., Stability analysis and optimal vaccination of an SIR epidemic model, Science Direct, 93, 240–249, 2008. https://doi.org/10.1016/j.biosystems.2008.05.004.
- Barro, M., Guiro, A. and Ouedraogo, D., Optimal control of a SIR epidemic model with general incidence function and a time delays, CUBO A Mathematical Journal, 20(02), 53–66, 2018. http://dx.doi.org/10.4067/S0719-06462018000200053.
- Kimaro, M. A., Massawe, E. S. and Makinde, O. D., Modelling the optimal control of transmission dynamics of mycobacterium ulceran infection, Open Journal of Epidemiology, 05(04), 229-243, 2015. https://10.4236/ojepi.2015.54027.
- Javeed, S., Ahmed, A., Khan, M. S. and Javed, M. A., Stability analysis and solutions of dynamical models for dengue, Journal of Mathematics, 50(2), 45-67, 2018.
- Aldila, D., Gotz, T. and Soewono, E., An optimal control problem arising from a dengue disease transmission model, Mathematical Biosciences, 242(1), 9-16, 2013. https://doi.org/10.1016/j.mbs.2012.11.014.
- Chiyaka, C., Garira, W. and Dube, S., Transmission model of endemic human malaria in a partially immune population, Mathematical and Computer Modelling, 46(5–6), 806-822, 2007. https://doi.org/10.1016/j.mcm.2006.12.010.
- Gaff, H. and Schaefer, E., Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6, 469-492, 2009.
- Hattaf, K. and Yousf, N., Optimal control of a delayed HIV infection model with immune response using an efficient numerical method, International Scholarly Research Network, 1-7, 2012. https://doi.org/10.5402/2012/215124.
- Kaddar, A., On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate, Electronic Journal of Differential Equations, 2009(133), 1-7, 2009.
- Anggriani, N., Supriatna, A. K. and Soewono, E., The existence and stability analysis of the euilibria in dengue disease infection model, Journal of Physics: Conference Series, 622, 1-10, 2015.
- Zhou, Y., Liu and H., Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38(3–4), 299-308, 2003.
- Zhang, F., Li, Z. and Zhang, F., Global stability of an SIR epidemic model with constant infectious period, Applied Mathematics and Computation, 199(1), 285-291, 2008.http://dx.doi.org/10.1016/j.amc.2007.09.053.
- Feng, Z. and Velásco-Hernández, J. X., Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (5) 523-544, 1997.
- Ji, W., Zou, S., Liu, J., Sun, Q. and Xia, L., Dynamic of non-autonomous vector infectious disease model with cross infection, American Journal of Computational Mathematics, 10(04), 591-602, 2020.
- [Kyrychkoa, Y. N. and Blyuss, K. B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Analysis: Real World Applications, 6, 495 – 507, 2004.
- Syafruddin and Noorani, M. S. M., Lyapunov function of SIR and SEIR model for transmission of dengue fever disease, Int. J. Simulation and Process Modelling, 8(2-3), 177–184, 2013. https://doi.org/10.1504/IJSPM.2013.057544.
- Mohajan, H. K., Mathematical analysis of SIR model for COVID-19 transmission, Journal of Innovations in Medical Research, 1(2), 1-35, 2022. https://doi.org/10.56397/JIMR/2022.09.01.
- Bernardi, F. and Aminian, M., Epidemiology and the SIR model: historical context to modern applications, CODEE Journal, 14(4), 1-21, 2021.
- Bhattacharya, P., Paul, S. and Biswas, P., Mathematical modeling of treatment SIR model with respect to variable contact rate, International Proceedings of Economics Development and Research, 83, 34-41, 2015.
- Sanchez, D. A., Ordinary differential equations and stability theory an introduction, The Dover Publications, 1979.
- Acemoglu, D., Chernozhukov, V., Werning, I. and Whinston, M. D., A multi-rRisk SIR model with optimally targeted lockdown, National Bureau of Economic Research, 27102, 1-38, 2020.
- Heesterbeek, J. A. and Roberts, M. G., The type-reproduction number T in models for infectious disease control, Mathematical Biosciences, 206(1), 3-10, 2007.
- Moghadas, S. M. and Gumel, A. B., Global stability of a two-stage epidemic model with generalized non-linear incidence, Mathematics and Computers in Simulation, 60(1–2), 107-118, 2002.
- Diaz, J. P., Computational analysis of the sir mathematical model for the dengue fever, Theses and Dissertations – UTRGV, 29, 2015.
- Khalid, M., M. Sultana, M. and Khan, F. S., Numerical solution of SIR model of dengue fever, International Journal of Computer Applications, 118(21), 1-4, 2015.
- Chakraborty, A. K., Haque, M. A. and Islam, M. A., Mathematical modelling and analysis of dengue transmission in Bangladesh with saturated incidence rate and constant treatment function, Communication in Bio-mathematical Sciences, 3(2) ,101-113, 2021.https://doi.org/10.5614/cbms.2020.3.2.
- Syafruddin and Noorani, M. S. M., A SIR model for spread of dengue fever disease (simulation for south Sulawesi, Indonesia and Selangor, Malaysia), World Journal of Modeling and Simulation, 9(2), 96-105, 2013.
- Dengue out-breaks in Bangladesh, 2023. https://en.prothomalo.com/bangladesh/19ugplpds8.
- Institute of Epidemiology Disease Control and Research (IEDCR), B.E newsletter on dengue, December 2023. https://www.iedcr. gov.bd/index.php/dengue.
