Mathematical Model for Dengue Disease with Maternal Antibodies

Authors: Rujira Kongnuy, Puntani Pongsumpun, I-Ming Tang

Abstract:

Mathematical models can be used to describe the dynamics of the spread of infectious disease between susceptibles and infectious populations. Dengue fever is a re-emerging disease in the tropical and subtropical regions of the world. Its incidence has increased fourfold since 1970 and outbreaks are now reported quite frequently from many parts of the world. In dengue endemic regions, more cases of dengue infection in pregnancy and infancy are being found due to the increasing incidence. It has been reported that dengue infection was vertically transmitted to the infants. Primary dengue infection is associated with mild to high fever, headache, muscle pain and skin rash. Immune response includes IgM antibodies produced by the 5th day of symptoms and persist for 30-60 days. IgG antibodies appear on the 14th day and persist for life. Secondary infections often result in high fever and in many cases with hemorrhagic events and circulatory failure. In the present paper, a mathematical model is proposed to simulate the succession of dengue disease transmission in pregnancy and infancy. Stability analysis of the equilibrium points is carried out and a simulation is given for the different sets of parameter. Moreover, the bifurcation diagrams of our model are discussed. The controlling of this disease in infant cases is introduced in the term of the threshold condition.

Keywords: Dengue infection, equilibrium states, maternalantibodies, pregnancy and infancy.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079412

References:

[1] E. A. Henchal, and J.R. Putnak, "The dengue viruses," Clin Microbiol Rev., vol. 3, pp. 376-396, 1990.
[2] World Health Organization , Dengue Haemorrhagic fever:Diagnosis treatment and control., Geneva, 1997.
[3] K. B. Platt, K. J. Linthicum, K. S. Myint, B. L. Innis, K. Lerdthusnee, and D. W.Vaughn, "Impact of dengue virus infection on feeding behavior of Aedes aegypti," Am J Trop Med Hyg., vol. 57, pp. 119-125, 1997.
[4] P. K. Russell, S. Udomsakdi, and S. B. Halstead, "Antibody response in dengue and dengue hemorrhagic fever," Jpn J Med Sci Biol., vol. 20(suppl.), pp. 103-108, 1967.
[5] R. M. Scott, and P. K. Russell, "Complement fixation blocking activity of antidengue IgM antibody," J Immunol., vol. 109, pp. 875-881, 1972.
[6] OHP. Pepper, "A note of David Bylon and dengue," Annals of Int Med History., vol. 3, pp. 363-368, 1941.
[7] N. Bhamarapravate, V. Boonpucknavig, S. Boonpucknavig, and B.Petchclai, "Immune complexes in dengue hemorrhagic fever," J Med Assoc Thailand., vol. 61(suppl 3), pp. 62-66, 1978.
[8] S. Nimmannitya, "Clinical spectrum and management of dengue hemorrhagic fever," Southeast Asian J Trop Med Publ Health., vol. 18, pp. 392-397, 1987.
[9] S. B. Halstead, "Dengue:hematologic aspects," Seminars in Hematology., vol. 19, pp. 116-131, 1982.
[10] Division of Epidemiology, Ministry of Public Health, Thailand, Annual Epidemiological Survillance Report., 1997-2007.
[11] S. B. Halstead, S. Nimmannitya, and S. N. Cohen, "Observations related to pathogenesis of dengue hemorrhagic fever, IV.Relation of disease severity to antibody response and virus recovered," Yale J Biol Med., vol. 42, pp. 311-328, 1970.
[12] S. C. Kliks, S. Nimmanitya, A. Nisalak, and D.S. Burke, "Evidence that maternal dengue antibodies are important in the development of dengue hemorrhagic fever in infants," Am J Trop Med Hyg., vol. 38, pp. 411-419, 1988.
[13] D. J. Gubler, "The global pandemic of dengue/dengue hemorrhagic fever:current status and prospects for the future," Ann Acad Med Singapore., vol. 27, pp. 227-234, 1998.
[14] S. B. Halstead, "Observations related to pathogenesis of dengue hemorrhagic fever, IV.Relation of disease severity to antibody response and virus recovered," Yale J Biol Med., vol. 42, pp. 350-362, 1970.
[15] D. J. Gubler, "Dengue and dengue hemorrhagic fever," Clin Microbiol Rev., vol. 11, pp. 480-496, 1998.
[16] A. L. Rothman, and F. A. Ennis, "Immunopathogenesis of dengue hemorrhagic fever," Virology., vol. 257, pp. 1-6, 1999.
[17] L. Kabilan, S. Balasubramanian, S. M. Keshava, V. Thenmozhi, G. Sehar, S. C. Tewari, N. Arunachalam, R. Rajendran, and K. Satyanarayana, "Dengue disease spectrum among infants in the 2001 dengue epidemic in Chennai, Tamil Nadu, India," J Clin Microbiol., vol. 41, pp. 3919-3921, 2003.
[18] H. W. Hethcote, Three basic epidemiological models in Applied Mathematical Ecology., Springer-Verlag, Berlin, pp. 119-144, 1989.
[19] H. W. Hethcote, and J. W. Van Ark, Modeling HIV Transmission and AIDS in the United States., Lecture Notes in Biomath, Springer-Verlag, Berlin, 1992.
[20] L. Esteva, and C. Vargas, "Analysis of a dengue disease transmission model," Math Biosci., vol. 150, pp. 131-151, 1998.
[21] R. Kongnuy, P. Pongsumpun, and I-Ming Tang, "Analysis of a Mathematical Model for Dengue Disease in Pregnant Cases," Int J Biomed Sci., vol. 3, pp. 192-199, 2008.
[22] R. Ross , The Prevention of Malaria, Second Edition, Murray, London.
[23] M. Robert, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 1973.
[24] P. Pongsumpun, and I-Ming Tang, "Mathematical model for the transmission of Plasmodium Vivax Malaria," Int J math models and methods in applied sci., vol. 3, pp. 117-121, 2007.
[25] P. Pongsumpun , and R. Kongnuy, "Model for the transmission of dengue disease in pregnant and non-pregnant patients," Int J math models and methods in applied sci., vol. 3, pp. 127-132, 2007.
[26] F. C. Coelho, C. T. Codeco, and C. J. Struchiner, "Complete treatment of uncertainties in a model for dengue R0 estimation," Cad Saude Publica., vol. 24, pp. 853-861, 2008.
[27] P. Pongsumpun, and I-Ming Tang, "Transmission Model for Plasmodium Vivax Malaria:Conditions for Bifurcation," Int J Biomed Sci., vol. 3, pp. 161-168, 2008.
[28] P. Pongsumpun, and I-Ming Tang, "Mathematical Model for the Transmission for P. Falciparum and P. Vivax Malaria along the Thai- Myanmar Border," Int J Biomed Sci., vol. 3, pp. 200-207, 2008.
[29] P. Pongsumpun, and I-Ming Tang, "Effect of the Seasonal Variation in the Extrinsic Incubation Period on the Long Term Behavior of the Dengue Hemorrhagic Fever Epidemic," Int J Biomed Sci., vol. 3, pp. 208-214, 2008.