Effect of Time Delay on the Transmission of Dengue Fever
The effect of a time delay on the transmission on
dengue fever is studied. The time delay is due to the presence of an
incubation period for the dengue virus to develop in the mosquito
before the mosquito becomes infectious. The conditions for the
existence of a Hopf bifurcation to limit cycle behavior are
established. The conditions are different from the usual one and they
are based on whether a particular third degree polynomial has
positive real roots. A theorem for determining whether for a given
set of parameter values, a critical delay time exist is given. It is
found that for a set of realistic values of the parameters in the model,
a Hopf bifurcation can not occur. For a set of unrealistic values of
some of the parameters, it is shown that a Hopf bifurcation can occur.
Numerical solutions using this last set show the trajectory of two of
the variables making a transition from a spiraling orbit to a limit
cycle orbit.
[1] D.J. Gubler DJ, Dengue and Dengue Hemorrhagic Fever, Clin.
Mirobiol. Rev. 11 (1998) 480.
[2] World Health Organization, Dengue hemorrhagic fever: diagnosis,
treatment, prevention and control, 2nd Ed. (1997) WHO..
[3] A. Martin and S. Ruan, Predator-prey models with delay and prey
harvesting. J. Math. Biol. 43 (2001) 247.
[4] Q.L.A. Khan, D. Greenhalgh, Hopf bifurcation in epidemic models
with a time delay in vaccination. IMA J. Math. Appl. Med. Bio. 16
(1998) 113.
[5] Y. Xiao, L. Chen, Modeling and analysis of a predator-prey model
with disease in prey. Math. Biosci. 171 (2001) 59.
[6] S. Ruan, J. Wei, On the zeros of a third degree exponential
polynomial with application to a delay model for control of
testosterone secretion. IMA J. Math. Appl. Med. Biol.18 (2001) 41.
[7] J. Tam, Delay effect in a model for virus replication.
IMA J. Math. Appl. Med. Biol. 16 (1999) 29.
[8] J.E. Marsden, M. McCracken, The Hopf Bifurcation
and Its Application, (Springer-Verlag, Berlin(1976)).
[9] L. Esteva, C. Vargas C, Analysis of a dengue disease transmission
model. Math. BioSci. 150 (1998) 131.
[10] C.A. Marques, O.P. Forattini, E. Massad,. The basic reproduction
number for dengue fever in Sao Paulo state, Brazil: 1990-1991
epidemic. Trans. Roy. Soc. Trop. Med Hyg. 88 (1994), 58.
[11] G Caughley, The elephant problem-an alternative hypothesis. East
Aft. Wildl. J. 14 (1976) 265.
[12] K.J. Duffy, B.R. Page, J.H. Swart, V.B. Bajic, Realistic parameter
assessment for a well known elephant-tree ecosystem model reveals
that limit cycles are unlikely. Ecol. Mod. 121 (1999) 115.
[13] SI Hay, MF Myers, DS Burke, DW Vaughn, T Endy, N Ananda, GD
Shanks, RW Snow, DJ Rogers, Etiology of interepidemic periods of
mosquito-born disease. PNAS 97 (2000) 9335.
[14] S. Dowell, Seasonal variation in host susceptibility and cycles of
certain infectious diseases. Emer. Inf. Dis. 7 (2001) 369.
[15] Y. Kuang, in: Delay differential equations with appli-cation to
population dynamics, (Academic Press, New York, 1993) page 66.
[1] D.J. Gubler DJ, Dengue and Dengue Hemorrhagic Fever, Clin.
Mirobiol. Rev. 11 (1998) 480.
[2] World Health Organization, Dengue hemorrhagic fever: diagnosis,
treatment, prevention and control, 2nd Ed. (1997) WHO..
[3] A. Martin and S. Ruan, Predator-prey models with delay and prey
harvesting. J. Math. Biol. 43 (2001) 247.
[4] Q.L.A. Khan, D. Greenhalgh, Hopf bifurcation in epidemic models
with a time delay in vaccination. IMA J. Math. Appl. Med. Bio. 16
(1998) 113.
[5] Y. Xiao, L. Chen, Modeling and analysis of a predator-prey model
with disease in prey. Math. Biosci. 171 (2001) 59.
[6] S. Ruan, J. Wei, On the zeros of a third degree exponential
polynomial with application to a delay model for control of
testosterone secretion. IMA J. Math. Appl. Med. Biol.18 (2001) 41.
[7] J. Tam, Delay effect in a model for virus replication.
IMA J. Math. Appl. Med. Biol. 16 (1999) 29.
[8] J.E. Marsden, M. McCracken, The Hopf Bifurcation
and Its Application, (Springer-Verlag, Berlin(1976)).
[9] L. Esteva, C. Vargas C, Analysis of a dengue disease transmission
model. Math. BioSci. 150 (1998) 131.
[10] C.A. Marques, O.P. Forattini, E. Massad,. The basic reproduction
number for dengue fever in Sao Paulo state, Brazil: 1990-1991
epidemic. Trans. Roy. Soc. Trop. Med Hyg. 88 (1994), 58.
[11] G Caughley, The elephant problem-an alternative hypothesis. East
Aft. Wildl. J. 14 (1976) 265.
[12] K.J. Duffy, B.R. Page, J.H. Swart, V.B. Bajic, Realistic parameter
assessment for a well known elephant-tree ecosystem model reveals
that limit cycles are unlikely. Ecol. Mod. 121 (1999) 115.
[13] SI Hay, MF Myers, DS Burke, DW Vaughn, T Endy, N Ananda, GD
Shanks, RW Snow, DJ Rogers, Etiology of interepidemic periods of
mosquito-born disease. PNAS 97 (2000) 9335.
[14] S. Dowell, Seasonal variation in host susceptibility and cycles of
certain infectious diseases. Emer. Inf. Dis. 7 (2001) 369.
[15] Y. Kuang, in: Delay differential equations with appli-cation to
population dynamics, (Academic Press, New York, 1993) page 66.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61071", author = "K. Patanarapelert and I.M. Tang", title = "Effect of Time Delay on the Transmission of Dengue Fever", abstract = "The effect of a time delay on the transmission on
dengue fever is studied. The time delay is due to the presence of an
incubation period for the dengue virus to develop in the mosquito
before the mosquito becomes infectious. The conditions for the
existence of a Hopf bifurcation to limit cycle behavior are
established. The conditions are different from the usual one and they
are based on whether a particular third degree polynomial has
positive real roots. A theorem for determining whether for a given
set of parameter values, a critical delay time exist is given. It is
found that for a set of realistic values of the parameters in the model,
a Hopf bifurcation can not occur. For a set of unrealistic values of
some of the parameters, it is shown that a Hopf bifurcation can occur.
Numerical solutions using this last set show the trajectory of two of
the variables making a transition from a spiraling orbit to a limit
cycle orbit.", keywords = "Dengue fever transmission, time delay, Hopfbifurcation, limit cycle behavior", volume = "1", number = "10", pages = "493-9", }