Mathematical Model for Dengue Disease with Maternal Antibodies
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.
[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.
[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.
@article{"International Journal of Medical, Medicine and Health Sciences:61577", author = "Rujira Kongnuy and Puntani Pongsumpun and I-Ming Tang", title = "Mathematical Model for Dengue Disease with Maternal Antibodies", 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.", volume = "5", number = "4", pages = "170-10", }
