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Year 2016, Volume: 4 Issue: 2, 58 - 70, 30.10.2016
https://doi.org/10.36753/mathenot.421458

Abstract

References

  • [1] Bailey, N., The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975.
  • [2] Bhatt, S., Gething, P.W., Brady, O.J., Messina, J.P., Farlow, A.W., Moyes, C.L. et al., The global distribution and burden of dengue. Nature, 496 (2013), 504-507.
  • [3] Brady, O.J., Gething, P.W., Bhatt, S., Messina, J.P., Brownstein, J.S., Hoen, A.G. et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus. PLOS Negl Trop Dis 6 (20120), no. 8.
  • [4] Bronson, R., Schaum’s Outline of Differential Equations, 4th Edition, McGraw-Hill Education, New York, 2014.
  • [5] Butcher, J.C., Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, New York, 2008.
  • [6] Cyganowski, S., Kloeden, P. and Ombach, J., From Elementary Probability to Stochastic Differential Equations with MAPLE , Springer-Verlag, New York, 2001.
  • [7] Dietz, K., Transmission and control of arbovirus diseases. In D. Ludwig and K. L. Cooke, editors, Epidemiology, 104–121. SIAM, 1975.
  • [8] Esteva, L. and Vargas, C., Analysis of a dengue disease transmission model. Mathematical Biosciences, 150 (1998), 131-151.
  • [9] Feller W., An Introduction to Probability Theory and Its Applications, Volume I, 3rd Edition John Wiley & Sons, Inc., New York, 1968.
  • [10] Feller W., An Introduction to Probability Theory and Its Application, Volume II, John Wiley & Sons, Inc., New York, 1971.
  • [11] Imran, M., Hassan, M., Dur-E-Ahmad, M. and Khan, A., A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage. Journal of Biological Dynamics, 7 (2013), no. 1, 276-301.
  • [12] Kermack, W.O. and McKendrick, A.G., A Contribution the Mathematical Theory of Epidemics. Proceedings of The Royal Society A, 115 (1927), no. 772, 700-721.
  • [13] Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Second Edition, SpringerVerlag, New York, 1995.
  • [14] Kolmogorov A. N., Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1956.
  • [15] Lahrouz, A., Omari, L., Kiouach, D. and Belmaati, A., Deterministic and Stochastic Stability of a Mathematical Model of Smoking. Statistics and Probability Letters, 81 (2011), 1276-1284.
  • [16] Martcheva, M., An Introduction to Mathematical Epidemiology, Springer Science+Business Media, New York, 2015.
  • [17] Merdan M., Bekiryazici Z. and Kesemen T., Stochastic and Deterministic Stability of Models for Hepatitis C, 7th International Conference on Mathematical Analysis, Differential Equations and their Applications, September 2015, Baku, Abstracts Book, p. 114.
  • [18] Merdan, M., and Khaniyev, T., On the Behaviour of Solutions under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model. International Journal of Biotechnology and Biochemistry, 4 (2008), no. 1, 75-100.
  • [19] Pang, L., Zhao, Z., Liu, S., Zhang, X., A Mathematical Model Approach for Tobacco Control in China. Applied Mathematics and Computation, 259 (2015), 497-509.
  • [20] Phaijoo, G.R. and Gurung, D.B., Mathematical Study of Biting Rates of Mosquitoes in Transmission of Dengue Disease, Journal of Science. Engineering and Technology, 11 (2015), no. 2, 25-33.
  • [21] Shiryayev A. N., Gradute Texts in Mathematics: Probability, Spring Science+ Business Media, LLC, New York, 1984.
  • [22] Soong, T.T., Random Differential Equations in Science and Engineering, Academic Press Inc., New York, 1973.
  • [23] Tan, W. and Wu, H., Stochastic Modeling of the Dynamics of CD4+ T-Cell Infection by HIV ad Some Monte Carlo Studies. Mathematical Biosciences, 147 (1998), 173-205.
  • [24] Yaacob, Y., Analysis of a Dengue Disease Transmission Model without Immunity. Matematika, 23 (2007), no. 2, 75-81.
  • [25] World Health Org., Dengue and severe dengue, http://www.who.int/mediacentre/factsheets/fs117/en/, 23 March 2016.

Mathematical Modeling of Dengue Disease under Random Effects

Year 2016, Volume: 4 Issue: 2, 58 - 70, 30.10.2016
https://doi.org/10.36753/mathenot.421458

Abstract

In this study, the deterministic mathematical model of Dengue disease is examined under Laplacian
random effects. Random variables with Laplace distribution are used for randomizing the deterministic
parameters. Simulations of the numerical results of the equation system are made with Monte-Carlo
methods and the results are used for commenting on the disease. Comments are made on the random
behavior of the components of the model after the calculation of their numerical characteristics like the
expected value, variance, standard deviation, confidence interval and moments along with the coefficients
of skewness and kurtosis from the results of the simulations. Results from the deterministic model are
compared with the results from the random model to point out the possible contribution of random
modeling to mathematical analysis studies on the disease. 

References

  • [1] Bailey, N., The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975.
  • [2] Bhatt, S., Gething, P.W., Brady, O.J., Messina, J.P., Farlow, A.W., Moyes, C.L. et al., The global distribution and burden of dengue. Nature, 496 (2013), 504-507.
  • [3] Brady, O.J., Gething, P.W., Bhatt, S., Messina, J.P., Brownstein, J.S., Hoen, A.G. et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus. PLOS Negl Trop Dis 6 (20120), no. 8.
  • [4] Bronson, R., Schaum’s Outline of Differential Equations, 4th Edition, McGraw-Hill Education, New York, 2014.
  • [5] Butcher, J.C., Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, New York, 2008.
  • [6] Cyganowski, S., Kloeden, P. and Ombach, J., From Elementary Probability to Stochastic Differential Equations with MAPLE , Springer-Verlag, New York, 2001.
  • [7] Dietz, K., Transmission and control of arbovirus diseases. In D. Ludwig and K. L. Cooke, editors, Epidemiology, 104–121. SIAM, 1975.
  • [8] Esteva, L. and Vargas, C., Analysis of a dengue disease transmission model. Mathematical Biosciences, 150 (1998), 131-151.
  • [9] Feller W., An Introduction to Probability Theory and Its Applications, Volume I, 3rd Edition John Wiley & Sons, Inc., New York, 1968.
  • [10] Feller W., An Introduction to Probability Theory and Its Application, Volume II, John Wiley & Sons, Inc., New York, 1971.
  • [11] Imran, M., Hassan, M., Dur-E-Ahmad, M. and Khan, A., A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage. Journal of Biological Dynamics, 7 (2013), no. 1, 276-301.
  • [12] Kermack, W.O. and McKendrick, A.G., A Contribution the Mathematical Theory of Epidemics. Proceedings of The Royal Society A, 115 (1927), no. 772, 700-721.
  • [13] Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Second Edition, SpringerVerlag, New York, 1995.
  • [14] Kolmogorov A. N., Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1956.
  • [15] Lahrouz, A., Omari, L., Kiouach, D. and Belmaati, A., Deterministic and Stochastic Stability of a Mathematical Model of Smoking. Statistics and Probability Letters, 81 (2011), 1276-1284.
  • [16] Martcheva, M., An Introduction to Mathematical Epidemiology, Springer Science+Business Media, New York, 2015.
  • [17] Merdan M., Bekiryazici Z. and Kesemen T., Stochastic and Deterministic Stability of Models for Hepatitis C, 7th International Conference on Mathematical Analysis, Differential Equations and their Applications, September 2015, Baku, Abstracts Book, p. 114.
  • [18] Merdan, M., and Khaniyev, T., On the Behaviour of Solutions under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model. International Journal of Biotechnology and Biochemistry, 4 (2008), no. 1, 75-100.
  • [19] Pang, L., Zhao, Z., Liu, S., Zhang, X., A Mathematical Model Approach for Tobacco Control in China. Applied Mathematics and Computation, 259 (2015), 497-509.
  • [20] Phaijoo, G.R. and Gurung, D.B., Mathematical Study of Biting Rates of Mosquitoes in Transmission of Dengue Disease, Journal of Science. Engineering and Technology, 11 (2015), no. 2, 25-33.
  • [21] Shiryayev A. N., Gradute Texts in Mathematics: Probability, Spring Science+ Business Media, LLC, New York, 1984.
  • [22] Soong, T.T., Random Differential Equations in Science and Engineering, Academic Press Inc., New York, 1973.
  • [23] Tan, W. and Wu, H., Stochastic Modeling of the Dynamics of CD4+ T-Cell Infection by HIV ad Some Monte Carlo Studies. Mathematical Biosciences, 147 (1998), 173-205.
  • [24] Yaacob, Y., Analysis of a Dengue Disease Transmission Model without Immunity. Matematika, 23 (2007), no. 2, 75-81.
  • [25] World Health Org., Dengue and severe dengue, http://www.who.int/mediacentre/factsheets/fs117/en/, 23 March 2016.
There are 25 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Zafer Bekiryazici

Mehmet Merdan

Tülay Kesemen

Mohammed Najmuldeen

Publication Date October 30, 2016
Submission Date June 24, 2016
Published in Issue Year 2016 Volume: 4 Issue: 2

Cite

APA Bekiryazici, Z., Merdan, M., Kesemen, T., Najmuldeen, M. (2016). Mathematical Modeling of Dengue Disease under Random Effects. Mathematical Sciences and Applications E-Notes, 4(2), 58-70. https://doi.org/10.36753/mathenot.421458
AMA Bekiryazici Z, Merdan M, Kesemen T, Najmuldeen M. Mathematical Modeling of Dengue Disease under Random Effects. Math. Sci. Appl. E-Notes. October 2016;4(2):58-70. doi:10.36753/mathenot.421458
Chicago Bekiryazici, Zafer, Mehmet Merdan, Tülay Kesemen, and Mohammed Najmuldeen. “Mathematical Modeling of Dengue Disease under Random Effects”. Mathematical Sciences and Applications E-Notes 4, no. 2 (October 2016): 58-70. https://doi.org/10.36753/mathenot.421458.
EndNote Bekiryazici Z, Merdan M, Kesemen T, Najmuldeen M (October 1, 2016) Mathematical Modeling of Dengue Disease under Random Effects. Mathematical Sciences and Applications E-Notes 4 2 58–70.
IEEE Z. Bekiryazici, M. Merdan, T. Kesemen, and M. Najmuldeen, “Mathematical Modeling of Dengue Disease under Random Effects”, Math. Sci. Appl. E-Notes, vol. 4, no. 2, pp. 58–70, 2016, doi: 10.36753/mathenot.421458.
ISNAD Bekiryazici, Zafer et al. “Mathematical Modeling of Dengue Disease under Random Effects”. Mathematical Sciences and Applications E-Notes 4/2 (October 2016), 58-70. https://doi.org/10.36753/mathenot.421458.
JAMA Bekiryazici Z, Merdan M, Kesemen T, Najmuldeen M. Mathematical Modeling of Dengue Disease under Random Effects. Math. Sci. Appl. E-Notes. 2016;4:58–70.
MLA Bekiryazici, Zafer et al. “Mathematical Modeling of Dengue Disease under Random Effects”. Mathematical Sciences and Applications E-Notes, vol. 4, no. 2, 2016, pp. 58-70, doi:10.36753/mathenot.421458.
Vancouver Bekiryazici Z, Merdan M, Kesemen T, Najmuldeen M. Mathematical Modeling of Dengue Disease under Random Effects. Math. Sci. Appl. E-Notes. 2016;4(2):58-70.

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