BASIC EPIDEMIC MODEL OF DENGUE TRANSMISSION USING THE FRACTIONAL ORDER DIFFERENTIAL EQUATIONS
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Published:
Feb 28, 2019
Keywords:
Dengue Epidemic Fractional order Stability Reproduction number
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Abstract
Dengue is normally emerging in tropical and subtropical countries and now has become a serious health problem. In Malaysia, dengue is considered endemic for the past few years. A reliable mathematical model of dengue epidemic is crucial to provide some means of interventions in controlling the spread of the disease. Many mathematical models have been proposed and analyzed in the literature, but very little of them used fractional order derivative in analyzing the dengue transmission. In this paper, a study on a basic fractional order epidemic model of dengue transmission is conducted using the SIR-SI model, including the aquatic phase of the vector. The population size of the human is assumed to be constant. The threshold quantity R0 is attained by the next generation matrix method. The preliminary result of the study is presented. It has shown that the disease-free equilibrium is locally asymptotically stable when R0 < 1, and unstable when R0 > 1. In other words, the dengue disease is eliminated if R0 < 1, and it approaches a positive endemic equilibrium if R0 > 1. Finally, some numerical results are presented based on the real data in Malaysia in 2016.
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Hamdan, N. ‘Izzati, & Kilicman, A. (2019). BASIC EPIDEMIC MODEL OF DENGUE TRANSMISSION USING THE FRACTIONAL ORDER DIFFERENTIAL EQUATIONS. Malaysian Journal of Science, 38(Sp 1), 1–18. https://doi.org/10.22452/mjs.sp2019no1.1
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References
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(n.d.). Retrieved February 2018, from From the Desk of the Director-General of Health Malaysia: https://kpkesihatan.com/2016/01/28/terkini-situasi-denggi-2016/.
Agarwal, R., Ntouyas, S., Ahmad, B., & Alhothuali, M. (2013). Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Advances in Difference Equation, 128.
Bailey, N. (1975). The Mathematical Theory of Infectious Disease and Its Applications. London: Griffin.
Demirci, E., Unal, A., & Ozalp, N. (2011). A fractional order SEIR model with density dependent death rate. Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics, 40(2), 287-295.
Dengue Situation Updates 2016. (n.d.). Retrieved March 2018, from World Health Organization Western Pacific Region: http://iris.wpro.who.int/handle/10665.1/14182.
Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer.
Diethelm, K. (2013). A fractional calculus based model for he simulation of an outbreak of dengue fever. Nonlinear Dynamics, 71(4), 613-619.
Diethelm, K., & Freed, A. (1999). The FracPECE Subroutine for the numerical solution of differential equation of fractional order. Orschung und Wissenschaftliches Rechnen, (pp. 57-71).
Diethelm, K., & Freed, A. D. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 260(2), 229-248.
Esteva, L., & Vargas, C. (1998). Analysis of dengue transmission model. Mathematical Biosciences, 15(2), 131-151.
Esteva, L., & Yang, H. (2015). Assessing the effects of temperature and dengue virus load on dengue transmission. Journal of Biological Systems, 23(4).
Garrappa, R. (2015). Trapezoidal methods for fractional equations: Theoretical and computational aspects. Mathematics and Computers in Simulation, 11, 96-112.
Hamdan, N., & Kilicman, A. (2018). A fractional order SIR epidemic model for dengue transmission. Chaos, Solitons and Fractals, 114, 55-62.
Hamdan, N., & Kilicman, A. (2019). Analysis of the fractional order dengue transmission model: a case study in Malaysia. Advances in Difference Equations, 2019, 1-13.
Kermack, W., & McKendrick, A. (1996). A contribution to the mathematical theory of epidemics. Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2, pp. 963-968.
Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. Proceedings of The Computational Engineering in Systems Application, 2, pp. 963-968.
McCall, P. J., & Kelly, D. W. (2002). Learning and memory in disease vectors. Trends. Parasitols, 18(10), 429-433.
Rodrigues, H. S., Monteiro, M. T., Torres, D. F., & Zinober, A. (2012). Dengue disease, basic reproduction number and control. International Journal of Computer Mathematics, 89(3), 334-346.
Sardar, T., Rana, S., & Chattopadhyay, J. (2014). A mathematical model of dengue transmission with memory. Commun Nonlinear Sci Numer Simulat, 22, 511-525.
Sardar, T., Rana, S., Bhattacharya, S., Al-Khaled, K., & Chattopadhyay, J. (2015). A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Mathematical Biosciences, 263, 18-36.
Syafruddin, S., & Noorani, M. (2012). SEIR model for transmission of dengue fever in Selangor Malaysia. International Journal of Modern Physics: Conference Series, 9, 380-389.
(n.d.). Retrieved February 2018, from From the Desk of the Director-General of Health Malaysia: https://kpkesihatan.com/2016/01/28/terkini-situasi-denggi-2016/.
Agarwal, R., Ntouyas, S., Ahmad, B., & Alhothuali, M. (2013). Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Advances in Difference Equation, 128.
Bailey, N. (1975). The Mathematical Theory of Infectious Disease and Its Applications. London: Griffin.
Demirci, E., Unal, A., & Ozalp, N. (2011). A fractional order SEIR model with density dependent death rate. Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics, 40(2), 287-295.
Dengue Situation Updates 2016. (n.d.). Retrieved March 2018, from World Health Organization Western Pacific Region: http://iris.wpro.who.int/handle/10665.1/14182.
Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer.
Diethelm, K. (2013). A fractional calculus based model for he simulation of an outbreak of dengue fever. Nonlinear Dynamics, 71(4), 613-619.
Diethelm, K., & Freed, A. (1999). The FracPECE Subroutine for the numerical solution of differential equation of fractional order. Orschung und Wissenschaftliches Rechnen, (pp. 57-71).
Diethelm, K., & Freed, A. D. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 260(2), 229-248.
Esteva, L., & Vargas, C. (1998). Analysis of dengue transmission model. Mathematical Biosciences, 15(2), 131-151.
Esteva, L., & Yang, H. (2015). Assessing the effects of temperature and dengue virus load on dengue transmission. Journal of Biological Systems, 23(4).
Garrappa, R. (2015). Trapezoidal methods for fractional equations: Theoretical and computational aspects. Mathematics and Computers in Simulation, 11, 96-112.
Hamdan, N., & Kilicman, A. (2018). A fractional order SIR epidemic model for dengue transmission. Chaos, Solitons and Fractals, 114, 55-62.
Hamdan, N., & Kilicman, A. (2019). Analysis of the fractional order dengue transmission model: a case study in Malaysia. Advances in Difference Equations, 2019, 1-13.
Kermack, W., & McKendrick, A. (1996). A contribution to the mathematical theory of epidemics. Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2, pp. 963-968.
Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. Proceedings of The Computational Engineering in Systems Application, 2, pp. 963-968.
McCall, P. J., & Kelly, D. W. (2002). Learning and memory in disease vectors. Trends. Parasitols, 18(10), 429-433.
Rodrigues, H. S., Monteiro, M. T., Torres, D. F., & Zinober, A. (2012). Dengue disease, basic reproduction number and control. International Journal of Computer Mathematics, 89(3), 334-346.
Sardar, T., Rana, S., & Chattopadhyay, J. (2014). A mathematical model of dengue transmission with memory. Commun Nonlinear Sci Numer Simulat, 22, 511-525.
Sardar, T., Rana, S., Bhattacharya, S., Al-Khaled, K., & Chattopadhyay, J. (2015). A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Mathematical Biosciences, 263, 18-36.
Syafruddin, S., & Noorani, M. (2012). SEIR model for transmission of dengue fever in Selangor Malaysia. International Journal of Modern Physics: Conference Series, 9, 380-389.