Effective Control Strategies on the Transmission Dynamics of a Vector-Borne Disease
Affiliation(s)
1 BRAC University, Dhaka, Bangladesh. 2 Ahsanullah Universities of Science and Technology, Dhaka, Bangladesh. 3 Department of Mathematics, University of Dhaka, Dhaka, Bangladesh.
1 BRAC University, Dhaka, Bangladesh. 2 Ahsanullah Universities of Science and Technology, Dhaka, Bangladesh. 3 Department of Mathematics, University of Dhaka, Dhaka, Bangladesh.
ABSTRACT
In this paper, we have rigorously analyzed a model to find the effective control strategies on the transmission dynamics of a vector-borne disease. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. The numerical simulations (using MatLab and Maple) of the model reveal that the precautionary measures at the aquatic and adult stage decrease the number of new cases of dengue virus. Numerical simulation indicates that if we take the precautionary measures seriously then it would be more effective than even giving the treatment to the infected individuals.
In this paper, we have rigorously analyzed a model to find the effective control strategies on the transmission dynamics of a vector-borne disease. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. The numerical simulations (using MatLab and Maple) of the model reveal that the precautionary measures at the aquatic and adult stage decrease the number of new cases of dengue virus. Numerical simulation indicates that if we take the precautionary measures seriously then it would be more effective than even giving the treatment to the infected individuals.
Cite this paper
Hossain, S. , Nayeem, J. and Podder, C. (2015) Effective Control Strategies on the Transmission Dynamics of a Vector-Borne Disease. Open Journal of Modelling and Simulation, 3, 111-119. doi: 10.4236/ojmsi.2015.33012.
Hossain, S. , Nayeem, J. and Podder, C. (2015) Effective Control Strategies on the Transmission Dynamics of a Vector-Borne Disease. Open Journal of Modelling and Simulation, 3, 111-119. doi: 10.4236/ojmsi.2015.33012.
References
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http://dx.doi.org/10.1093/bmb/ldq019 [6] Wilder-Smith, A. and Schwartz, E. (2005) Dengue in Travelers. The New England Journal of Medicine, 353, 924-932. http://dx.doi.org/10.1056/NEJMra041927 [7] Varatharaj, A (2010) Encephalitis in the Clinical Spectrum of Dengue Infection. Neurology India, 58, 585-591. http://dx.doi.org/10.4103/0028-3886.68655 [8] Holmes, P. and Guckenheimer, J. (1990) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York Inc., New York. [9] Kautner, I., Robinson, M.J. and Kuhnle, U. (1997) Dengue virus Infection: Epidemiology, Pathogenesis, Clinical Presentation, Diagnosis, and Prevention. The Journal of Pediatrics, 131, 516-524. http://dx.doi.org/10.1016/S0022-3476(97)70054-4 [10] Esteva, L. and Vargas, C. (2003) Coexistence of Different Serotypes of Dengue Virus. Journal of Mathematical Biology, 46, 31-47. http://dx.doi.org/10.1007/s00285-002-0168-4 [11] Esteva, L. and Vargas, C. (1999) A Model for Dengue Disease with Variable Human Population. Journal of Mathematical Biology, 38, 220-240. http://dx.doi.org/10.1007/s002850050147 [12] CDC (2010) Locally Acquired Dengue—Key West, Florida, 2009-2010. Morbidity and Mortality Weekly Report (MMWR), 59, 577-581. [13] Wilder-Smith, A. and Tambyah, P.A. (2007) Severe Dengue Virus Infections in Travelers. The Journal of Infectious Diseases, 195, 1081-1083. http://dx.doi.org/10.1086/512684 [14] CDC (2006) Travel-Associated Dengue—United States, 2005. Morbidity and Mortality Weekly Report (MMWR), 55, 700-702. [15] Koopman, J.S., Prevots, D.R., Mann, M.A.V., Dantes, H.G., Aquino, M.L.Z., et al. (1991) Determinants and Predictors of Dengue Infection in Mexico. American Journal of Epidemiology, 133, 1168-1178. [16] Takahashi, L.T., Maidana, N.A., Ferreira Jr., W.C., Pulino, P. and Yang, H.M. (2005) Mathematical Models for the Aedes aegypti Dispersal Dynamics: Travelling Waves by Wing and Wind. Bulletin of Mathematical Biology, 67, 509- 528. http://dx.doi.org/10.1016/j.bulm.2004.08.005 [17] Feng, Z. and Velasco-Hernandez, X.J. (1997) Competitive Exclusion in a Vector-Host Model for the Dengue Fever. Journal of Mathematical Biology, 35, 523-544.
http://dx.doi.org/10.1007/s002850050064 [18] Struchiner, C.J., Luz, P.M., Codeco, C.T., Coelho, F.C. and Massad, E. (2006) Current Research Issues in Mosquito-Borne Diseases Modelling. Contemporary Mathematics, 410, 349-352. [19] Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. and Massad, E. (2006) Threshold Conditions for a Non-Autonomous Epidemic System Describing the Population Dynamics of Dengue. Bulletin of Mathematical Biology, 68, 2263-2282.
http://dx.doi.org/10.1007/s11538-006-9108-6 [20] Chowell, G., Diaz-Duenas, P., Miller, J.C., Alcazar-Velazco, A., Hyman, J.M., Fenimore, P.W. and Castillo Chavez, C. (2007) Estimation of the Reproduction Number of Dengue Fever from Spatial Epidemic Data. Mathematical Biosciences, 208, 571-589. http://dx.doi.org/10.1016/j.mbs.2006.11.011 [21] Jelinek, T. (2000) Dengue Fever in International Travelers. Clinical Infectious Diseases, 31, 144-147. http://dx.doi.org/10.1086/313889 [22] Tewa, J.J., Dimi, J.L. and Bowang, S. (2007) Lyapunov Functions for a Dengue Disease Transmission Model. Chaos, Solitons & Fractal, 39, 936-941. http://dx.doi.org/10.1016/j.chaos.2007.01.069 [23] Esteva, L. and Vargas, C. (1998) Analysis of a Dengue Disease Transmission Model. Mathematical Biosciences, 150, 131-151. http://dx.doi.org/10.1016/S0025-5564(98)10003-2 [24] Esteva, L. and Vargas, C. (2000) Influence of Vertical and Mechanical Transmission on the Dynamics of Dengue Disease. Mathematical Biosciences, 167, 51-64.
http://dx.doi.org/10.1016/S0025-5564(00)00024-9 [25] Derouich, M. and Boutayeb, A. (2006) Dengue Fever: Mathematical Modelling and Computer Simulation. Applied Mathematics and Computation, 177, 528-544.
http://dx.doi.org/10.1016/j.amc.2005.11.031 [26] Ferguson, N.M., Donnelly, C.A. and Anderson, R.M. (1999) Transmission Dynamics and Epidemiology of Dengue: Insights from Age-Stratied Sero-Prevalence Surveys. Philosophical Transactions of the Royal Society of London B, 354, 757-768. http://dx.doi.org/10.1098/rstb.1999.0428
[1] Garba, S.M., Gumel, A.B. and Abu Bakar, M.R. (2008) Backward Bifurcations in Dengue Transmission Dynamics. Mathematical Biosciencees, 201, 11-25. http://dx.doi.org/10.1016/j.mbs.2008.05.002 [2] Ranjit, S. and Kissoon, N. (2011) Dengue Hemorrhagic Fever and Shock Syndromes. Pediatric Critical Care Medicine, 12, 90-100. http://dx.doi.org/10.1097/PCC.0b013e3181e911a7 [3] WWW.CDC.Gov/Dengue [4] WWW.WHO.int/denguecontrol/faq/en/index6.html [5] Whitehorn, J. and Farrar, J. (2010) Dengue. British Medical Bulletin, 95, 161-173.
http://dx.doi.org/10.1093/bmb/ldq019 [6] Wilder-Smith, A. and Schwartz, E. (2005) Dengue in Travelers. The New England Journal of Medicine, 353, 924-932. http://dx.doi.org/10.1056/NEJMra041927 [7] Varatharaj, A (2010) Encephalitis in the Clinical Spectrum of Dengue Infection. Neurology India, 58, 585-591. http://dx.doi.org/10.4103/0028-3886.68655 [8] Holmes, P. and Guckenheimer, J. (1990) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York Inc., New York. [9] Kautner, I., Robinson, M.J. and Kuhnle, U. (1997) Dengue virus Infection: Epidemiology, Pathogenesis, Clinical Presentation, Diagnosis, and Prevention. The Journal of Pediatrics, 131, 516-524. http://dx.doi.org/10.1016/S0022-3476(97)70054-4 [10] Esteva, L. and Vargas, C. (2003) Coexistence of Different Serotypes of Dengue Virus. Journal of Mathematical Biology, 46, 31-47. http://dx.doi.org/10.1007/s00285-002-0168-4 [11] Esteva, L. and Vargas, C. (1999) A Model for Dengue Disease with Variable Human Population. Journal of Mathematical Biology, 38, 220-240. http://dx.doi.org/10.1007/s002850050147 [12] CDC (2010) Locally Acquired Dengue—Key West, Florida, 2009-2010. Morbidity and Mortality Weekly Report (MMWR), 59, 577-581. [13] Wilder-Smith, A. and Tambyah, P.A. (2007) Severe Dengue Virus Infections in Travelers. The Journal of Infectious Diseases, 195, 1081-1083. http://dx.doi.org/10.1086/512684 [14] CDC (2006) Travel-Associated Dengue—United States, 2005. Morbidity and Mortality Weekly Report (MMWR), 55, 700-702. [15] Koopman, J.S., Prevots, D.R., Mann, M.A.V., Dantes, H.G., Aquino, M.L.Z., et al. (1991) Determinants and Predictors of Dengue Infection in Mexico. American Journal of Epidemiology, 133, 1168-1178. [16] Takahashi, L.T., Maidana, N.A., Ferreira Jr., W.C., Pulino, P. and Yang, H.M. (2005) Mathematical Models for the Aedes aegypti Dispersal Dynamics: Travelling Waves by Wing and Wind. Bulletin of Mathematical Biology, 67, 509- 528. http://dx.doi.org/10.1016/j.bulm.2004.08.005 [17] Feng, Z. and Velasco-Hernandez, X.J. (1997) Competitive Exclusion in a Vector-Host Model for the Dengue Fever. Journal of Mathematical Biology, 35, 523-544.
http://dx.doi.org/10.1007/s002850050064 [18] Struchiner, C.J., Luz, P.M., Codeco, C.T., Coelho, F.C. and Massad, E. (2006) Current Research Issues in Mosquito-Borne Diseases Modelling. Contemporary Mathematics, 410, 349-352. [19] Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. and Massad, E. (2006) Threshold Conditions for a Non-Autonomous Epidemic System Describing the Population Dynamics of Dengue. Bulletin of Mathematical Biology, 68, 2263-2282.
http://dx.doi.org/10.1007/s11538-006-9108-6 [20] Chowell, G., Diaz-Duenas, P., Miller, J.C., Alcazar-Velazco, A., Hyman, J.M., Fenimore, P.W. and Castillo Chavez, C. (2007) Estimation of the Reproduction Number of Dengue Fever from Spatial Epidemic Data. Mathematical Biosciences, 208, 571-589. http://dx.doi.org/10.1016/j.mbs.2006.11.011 [21] Jelinek, T. (2000) Dengue Fever in International Travelers. Clinical Infectious Diseases, 31, 144-147. http://dx.doi.org/10.1086/313889 [22] Tewa, J.J., Dimi, J.L. and Bowang, S. (2007) Lyapunov Functions for a Dengue Disease Transmission Model. Chaos, Solitons & Fractal, 39, 936-941. http://dx.doi.org/10.1016/j.chaos.2007.01.069 [23] Esteva, L. and Vargas, C. (1998) Analysis of a Dengue Disease Transmission Model. Mathematical Biosciences, 150, 131-151. http://dx.doi.org/10.1016/S0025-5564(98)10003-2 [24] Esteva, L. and Vargas, C. (2000) Influence of Vertical and Mechanical Transmission on the Dynamics of Dengue Disease. Mathematical Biosciences, 167, 51-64.
http://dx.doi.org/10.1016/S0025-5564(00)00024-9 [25] Derouich, M. and Boutayeb, A. (2006) Dengue Fever: Mathematical Modelling and Computer Simulation. Applied Mathematics and Computation, 177, 528-544.
http://dx.doi.org/10.1016/j.amc.2005.11.031 [26] Ferguson, N.M., Donnelly, C.A. and Anderson, R.M. (1999) Transmission Dynamics and Epidemiology of Dengue: Insights from Age-Stratied Sero-Prevalence Surveys. Philosophical Transactions of the Royal Society of London B, 354, 757-768. http://dx.doi.org/10.1098/rstb.1999.0428