# Mathematical Modelling and Analysis of Dengue Transmission in Bangladesh with Saturated Incidence Rate and Constant Treatment Function

## Authors

• Amit Kumar Chakraborty Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114
• M. A. Haque Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114
• M. A. Islam Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114

## Keywords:

Dengue Disease, Endemic Equilibrium, Global Stability, Lyapunov Function, Sensitivity Analysis.

## Abstract

Dengue is one of the major health problems in Bangladesh and many people are died in recent years due to the severity of this disease. Therefore, in this paper, a SIRS model for the human and SI model for vector population with saturated incidence rate and constant treatment function has been presented to describe the transmission of dengue. The equilibrium points and the basic reproduction number have been computed. The conditions which lead the disease free equilibrium and the endemic equilibrium have been determined. The local stability for the equilibrium points has been established based on the eigenvalues of the Jacobian matrix and the global stability has been analyzed by using the Lyapunov function theory. It is found that the stability of equilibrium points can be controlled by the reproduction number. In order to calculate the infection rate, data for infected human populations have been collected from several health institutions of Bangladesh. Numerical simulations of various compartments have been generated using MATLAB to investigate the influence of the key parameters for the transmission of the disease and to support the analytical results. The effect of treatment function over the infected compartment has been illustrated. The sensitivity of the reproduction number concerning the parameters of the model has been analyzed. Finally, the most sensitive parameter that has the highest effect over reproduction number has been identified.

## Author Biography

### Amit Kumar Chakraborty, Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114

Assistant Professor,Â Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114, Bangladesh

## References

Gupta, N., Srivastava, S., Jain, A. and Chaturvedi, U., Dengue in india, The Indian journal of medical research, 136(3), 373–390, 2012.

Laura, L., Supriatna, A. K., Khumaeroh, M. S. and Anggriani, N., Biological and mechanical transmission models of dengue fever, Communication in Biomathematical Sciences, 2(1), 12–22, 2019.

Esteva, L. and Vargas, C., Coexistence of different serotypes of dengue virus, Journal of mathematical biology, 46(1), 31–47, 2003.

Side, S. and Noorani, 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), 96–105, 2013.

Gbadamosi, B., Ojo, M., Oke, S. and Matadi, M., Qualitative analysis of a dengue fever model, Mathematical and Computational Applications, 23(3), 33, 2018.

Organization, W. H. and Others, Dengue and severe dengue, Tech. Rep., Regional Office for the Eastern Mediterranean, 2014.

Chanprasopchai, P., Tang, I. and Pongsumpun, P., SIR model for dengue disease with effect of dengue vaccination, Computational and mathematical methods in medicine, Hindawi, 2018, 2018.

Nur, W., Rachman, H., Abdal, N., Abdy, M. and Side, S., SIR model analysis for transmission of dengue fever disease with climate factors using lyapunov function, In Journal of Physics: Conference Series, 1028(1), 012117, IOP Publishing, 2018.

Derouich, M., Boutayeb, A. and Twizell, E., A model of dengue fever, BioMedical Engineering OnLine, 2(1), 4, 2003.

Lopez, L., CESMAG, I. U., Loaiza, A. and Tost, G., A mathematical model for transmission of dengue, Applied Mathematical Sciences, 10(7), 345–355, 2016.

Guo, S., Li, X. and Ghosh, M., Analysis of a dengue disease model with nonlinear incidence, Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, 2013.

Esteva, L. and Vargas, C., Analysis of a dengue disease transmission model, Mathematical biosciences, 150(2), 131–151, 1998.

Rodrigues, H., Monteiro, M. and Torres, D., Sensitivity analysis in a dengue epidemiological model, In Conference Papers in Science, Hindawi, vol. 2013, 2013.

Puspita, J. W., Fakhruddin, M., Fahlena, H., Rohim, F. and Sutimin, S., On the reproduction ratio of dengue incidence in Semarang, Indonesia 2015-2018, Communication in Biomathematical Sciences, 2(2), 118–126, 2019.

Baez-Hernandez, N., Casas-Martinez, M., Danis-Lozano, R. and Velasco-Hernandez, J., A mathematical model for dengue and chikungunya in mexico, BioRxiv 122556, 2017.

Manore, C., Hickmann, K., Xu, S., Wearing, H. and Hyman, J., Comparing dengue and chikungunya emergence and endemic transmission in a. aegypti and a. albopictus, Journal of theoretical biology, 356, 174–191 2014.

Carvalho, S., da Silva, S. and da Cunha Charret, I., Mathematical modeling of dengue epidemic: control methods and vaccination strategies,Theory in Biosciences, 138(2), 223–239 2019.

Hossain, S., Nayeem, J., Podder, C. and Others, Effective control strategies on the transmission dynamics of a vector-borne disease, Open Journal of Modelling and Simulation, 3(3), 111 2015.

Braselton, J. and Bakach, I., A survey of mathematical models of dengue fever, Journal of Computer Science Systems Biology, 8(5), 255, 2015.

Feng, Z. and Velasco-Hern andez, J,. Competitive exclusion in a vector-host model for the dengue fever, Journal of mathematical biology, 35(5), 523–544, 1997.

Aguiar, M. and Stollenwerk, N., Mathematical models of dengue fever epidemiology: multi-strain dynamics, immunological aspects associated to disease severity and vaccines, Communication in Biomathematical Sciences, 1(1), 1–12 2017.

Hethcote, H., The mathematics of infectious diseases, SIAM review, 42(4), 599–653 2000.

Ozair, M., Lashari, A., Jung, I. and Okosun, K., Stability analysis and optimal control of a vector-borne disease with non linear incidence, Discrete Dynamics in Nature and Society, 2012.

Zhang, X. and Liu., X., Backward bifurcation of an epidemic model with saturated treatment function, Journal of mathematical analysis and applications, 348(1), 433–443, 2008.

Xiao, D. and Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical biosciences, 208(2), 419–429, 2007.

Wang, W., Epidemic models with nonlinear infection forces, Mathematical Biosciences Engineering, 3(1), 267, 2006.

Ruan, S. and Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188(1), 135–163, 2003.

Liu, W., Hethcote, H. and Levin, S, Dynamical behavior of epidemiological models with nonlinear incidence rates, Journal of mathematical biology, 25(4), 359–380, 1987.

Cai, L. and Li, X., Global analysis of a vector-host epidemic model with nonlinear incidences, Applied Mathematics and Computation, 217(7), 3531–3541, 2010.

Capasso, V. and Serio, G., A generalization of the kermack-mckendrick deterministic epidemic model, Mathematical Biosciences, 42(1-2), 43–61, 1978.

Olaniyi, S. and Obabiyi, O., Qualitative analysis of malaria dynamics with nonlinear incidence function, Applied Mathematical Sciences, 8(78), 3889–3904, 2014.

Zaman, G., Lashari, A., Chohan, M. and Al Buraimi, O., Dynamical features of dengue disease with saturating incidence rate, International Journal of Pure and Applied Mathematics, 76(3), 383–402, 2012.

Hu, Z., Ma, W. and Ruan, S., Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Mathematical biosciences, 238(1), 12–20, 2012.

Wang, W. and Ruan, S., Bifurcations in an epidemic model with constant removal rate of the infectives, Journal of Mathematical Analysis and Applications, 291(2), 775–793, 2004.

Wang, W., Backward bifurcation of an epidemic model with treatment., Mathematical biosciences, 201(1-2), 58–71, 2006.

Jansen, W., Lakshmikantham, V., Leela, S. and Martynyuk, A., Stability analysis of nonlinear systems, Astronomische Nachrichten, 316(1), 67–67, 1995.

Cai, L. and Li, X., Analysis of a SEIV epidemic model with a nonlinear incidence rate, Applied Mathematical Modelling, 33(7), 2919–2926, 2009.

Van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180(1-2), 29–48, 2002.

Roussel, M., Stability analysis for ODEs, Nonlinear Dynamics, lecture notes, University Hall, Canada, 2005.

LaSalle, J., The stability of dynamical systems, SIAM, vol. 25, 1976.

Institute of Epidemiology Disease Control and Research (IEDCR), B.E newsletter on dengue, December 2019. https://www.iedcr.gov.bd/index.php/dengue/

Chitnis, N., Hyman, J. and Cushing, J., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of mathematical biology, 70(5), 1272, 2008.

Tay, C. J., Dynamical behavior of secondary dengue infection model, Communication in Biomathematical Sciences, 2(1), 1–11, 2019.