OPTIMAL CONTROL OF VECTOR-BORNE DISEASE WITH DIRECT TRANSMISSION
DOI:
https://doi.org/10.11113/jt.v76.5822Keywords:
Epidemic model, vector-borne disease, direct transmission, optimal controlAbstract
This paper introduces the usage of three controls as a way to reduce the occurrence of vector-borne disease. The governing equation of the dynamical system used in this paper describes both direct and indirect transmission mode of vector-borne disease. This means that the disease can be transmitted in two different ways. First, it can be transmitted through mosquito bites and the other is through human blood transfusion. The three controls that are incorporated in the dynamical system include a measurement of basic practice for blood donation procedure, self-prevention effort and vector control strategy by health authority. The optimality system of the three controls is characterized using optimal control theory and the existence and uniqueness of the optimal control are established. Then, the effect of the incorporation of the three controls is investigated by performing numerical simulation.Â
References
Feng, Z. and J. X. Velasco-Hernández. 1997. Competitive Exclusion in a Vector-host Model for the Dengue Fever. Journal of Mathematical Biology. 35: 523-544.
Esteva, L. and C. Vargas. 1998. Analysis of a Dengue Disease Transmission Model. Mathematical Biosciences. 150: 131-151.
Ferguson, N., R. Anderson, and S. Gupta. 1999. The Effect of Antibody-dependent Enhancement on the Transmission Dynamics and Persistence of Multiple-strain Pathogens. Proceedings of the National Academy of Sciences of the United States of America. 96: 790-794.
Ngwa, G.A. and W. S. Shu. 2000. A Mathematical Model for Endemic Malaria with Variable Human and Mosquito Populations. Mathematical and Computer Modelling. 32: 747-763.
Esteva, L. and M. Matias. 2001. A Model for Vector Transmitted Diseases with Saturation Incidence. Journal of Biological Systems. 9: 235-245.
Kelly, S., T. N. Le, J. A. Brown, E.W. Lawaczek, M. Kuehnert, I. B. Rabe, J. E. Staples and M. Fischer. 2013. Fatal West Nile Virus Infection after Probable Transfusion-Associated Transmission-Colorado 2012. Morbidity and Mortality Weekly Report. 62(31): 622-624.
Kitchen, A. D. and P. L. Chiodini. 2006. Malaria and Blood Transfusion. Vox Sanguinis. 90: 77-84.
Tambyah, P. A., E. S. C. Koay, M. L. M. Poon, R. V. T. P. Lin and B. K. C. Ong. 2008. Dengue Hemorrhagic Fever Transmitted by Blood Transfusion. N Engl J Med. 359: 1526-1527.
Punzel., M., G. Korukluoğlu, D. Y. Caglayik, D. Menemenlioglu, S. C. Bozdag, E. Tekgündüz, F. Altuntaş, R. de Mendonca Campos, B. Burde, S. Günther, D. Tappe, D. Cadar and J. Schmidt-Chanasit. 2014. Dengue Virus Transmission by Blood Stem Cell Donor after Travel to Sri Lanka, 2013. Emerging Infectious Diseases. 20(8): 1366-1369.
Stramer, S. L., J. M. Linnen, J. M. Carrick, G. A. Foster, D. E. Krysztof, S. Zou, R. Y. Dodd, L. M. Tirado-Marrero, E. Hunsperger, G. A. Santiago, J. L. Muñoz-Jordan and K.M. Tomashek. 2012. Dengue Viremia in Blood Donors Identified by RNA and Detection of Dengue Transfusion Transmission during the 2007 Dengue Outbreak in Puerto Rico. Transfusion. 52: 1657-1666.
Harif, N. F., Z. S. A. Kader, S. R. Joshi and N. M. Yusoff. 2014. Seropositive Status of Dengue Virus Infection among Blood Donors in North Malaysia. Asian J. Transfus. Sci. 8(1): 64.
Wei, H.-M., X.-Z. Li and M. Martcheva. 2008. An Epidemic Model of Vector-borne Disease with Direct Transmission and Time Delay. Journal of Mathematical Analysis and Applications. 342: 895-908.
Cai, L. and X. Li. 2010. Analysis of a Simple Vector-host Epidemic Model with Direct Transmission. Discrete Dynamics in Nature and Society. 2010: 1-12.
Lashari, A. A. and G. Zaman. 2011. Global Dynamics of Vector-borne with Horizontal Transmission in Host Population. Computers and Mathematics with Applications. 61: 745-754.
Cai, L.-M., X. Z. Li and Z. Li. 2013. Dynamical Behavior of an Epidemic Model for a Vector-borne Disease with Direct Transmission. Chaos, Solitons & Fractals. 46: 54-64.
Brauer, F. and F. Castillo – Chavez. 2012. Mathematical Models in Population Biology and Epidemiology. Second Edition. New York: Springer Science+Business Media.
Lashari, A. A. and G. Zaman. 2012. Optimal Control of a Vector Borne Disease with Horizontal Transmission. Nonlinear Analysis: Real World Applications. 13: 203-212.
Lashari, A.A., K. Hattaf, G. Zaman and X.-Z. Li. 2013. Backward Bifurcation and Optimal Control of a Vector Borne Disease. Applied Mathematics & Information Sciences. 7(1): 301-309.
Blayneh, K. W., A. B. Gumel, S. Lenhart and T. Clayton. 2010. Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus. Bulletin of Mathematical Biology. 72: 1006-1028.
Fleming, W. H. and R. W. Rishel. 1975. Deterministic and Stochastic Optimal Control. United States of America: Springer-Verlag New York Inc.
Lukes, D. L. 1982. Differential Equations: Classical to Controlled, Mathematics in Science and Engineering. New York: Academic Press.
Pontryagin, L. S., V. G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko. 1964. The Mathematical Theory of Optimal Processes. New York: Pergamon Press Ltd.
Lenhart, S. and J. T. Workman. 2007. Optimal Control Applied to Biological Models. Taylor & Francis Group, LLC.
Kar, T. K. and S. Jana. 2013. Application of Three Controls Optimally in a Vector-borne Disease - a Mathematical Study. Communications in Nonlinear Science and Numerical Simulation. 18: 2868-2884.
Coutinho, F. A. B., M. N. Burattini, L.F. Lopez and E. Massad. 2006. Threshold Conditions for a Non-autonomous Epidemic System Describing the Population Dynamics of Dengue. Bulletin of Mathematical Biology. 68: 2263-2282.
Pongsumpun, P., D. G. Lopez, C. Favier, L. Torres, J. Llosa and M. A. Dubois. 2008. Dynamics of Dengue Epidemics in Urban Contexts. Tropical Medicine and International Health. 13(9): 1180-1187.
Esteva, L. and C. Vargas. 1999. A Model for Dengue Disease with Variable Human Population. Journal of Mathematical Biology. 38: 220-240.
Cai, L., S. Guo, X. Li and Ghosh, M. 2009. Global Dynamics of a Dengue Epidemic Mathematical Model. Chaos, Solitons and Fractals. 42: 2297-2304.
Downloads
Published
Issue
Section
License
Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.
