A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate

ABSTRACT

A model of viral infection of monocytes population by dengue virus is formulated in a system of four ordinary differenttial equations. The model takes into account the immune response and the incidence rate of susceptible and free virus particle as Beddington-DeAngelis functional response. By constructing a block, the global stability of the unin-fected steady state is investigated. This steady state always exists. If this is the only steady state, then it is globally asymptotically stable. If any infected steady state exists, then uninfected steady state is unstable and one of the infected steady states is locally asymptotically stable. These different cases depend on the values of the basic reproduction ratio and the other parameters.

A model of viral infection of monocytes population by dengue virus is formulated in a system of four ordinary differenttial equations. The model takes into account the immune response and the incidence rate of susceptible and free virus particle as Beddington-DeAngelis functional response. By constructing a block, the global stability of the unin-fected steady state is investigated. This steady state always exists. If this is the only steady state, then it is globally asymptotically stable. If any infected steady state exists, then uninfected steady state is unstable and one of the infected steady states is locally asymptotically stable. These different cases depend on the values of the basic reproduction ratio and the other parameters.

Cite this paper

H. Ansari and M. Hesaaraki, "A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate,"*Applied Mathematics*, Vol. 3 No. 2, 2012, pp. 177-184. doi: 10.4236/am.2012.32028.

H. Ansari and M. Hesaaraki, "A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate,"

References

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[5] M. A. Nowak and R. M. May, “Virus Dynamics: Mathematical Principles of Immunology and Viroloy,” Oxford University Press, Oxford, 2000.

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[7] C. Castillo-Chavez, Z. Feng and W. Huang, “On the Computation of R0 and Its Role on Global Stability,” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002, pp. 229-250. doi:10.1007/978-1-4613-0065-6

[8] L. Esteva and C. Vargas, “Analysis of a Dengue Disease Transmission Model,” Mathematical Biosciences, Vol. 150, No. 2, 1998, pp. 131-151. doi:10.1016/S0025-5564(98)10003-2

[9] L. Esteva and C. Vargas, “A Model for Dengue Disease with Variable Human Population,” Journal of Mathematical Biology, Vol. 38, No. 3, 1999, pp. 220-240. doi:10.1007/s002850050147

[10] L. Esteva and C. Vargas, “Coexistence of Different Serotypes of Dengue Virus,” Journal of Mathematical Biology, Vol. 46, No. 1, 2003, pp. 31-47. doi:10.1007/s00285-002-0168-4

[11] Z. Feng and J. X. Velasco-Hernandez, “Competitive Exclusion in a Vector-Host Model for the Dengue Fever,” Journal of Mathematical Biology, Vol. 35, No. 5, 1997, pp. 523-544. doi:10.1007/s002850050064

[1] WHO, “Tropical Disease Research, Making Health Research Work for Poor People,” Progress 2003-2004, World Health Organization, Geneva, 2005.

[2] D. J. Gubler, “Dengue and Dengue Hemorrhagic Fever,” Clinical Microbiology Reviews, Vol. 11, No. 3, 1998, pp. 480-496.

[3] E. A. Henchal and J. R. Putnak, “The Dengue Viruses,” Clinical Microbiology Reviews, Vol. 3, No. 4, 1990, pp. 376-396.

[4] N. Nuraini, H. Tasman, E. Soewono and K. A. Sidarto, “A With-In Host Dengue Infection Model with Immune Response,” Mathematical and Computer Modelling, Vol. 49, No. 5-6, 2009, pp. 1148-1155. doi:10.1016/j.mcm.2008.06.016

[5] M. A. Nowak and R. M. May, “Virus Dynamics: Mathematical Principles of Immunology and Viroloy,” Oxford University Press, Oxford, 2000.

[6] N. Bellomo, N. K. Li and P. Maini, “On the Foundation of Cancer Modeling: Selected Topics, Speculations and Perspectives,” Mathematical Models and Methods in Applied Sciences, Vol. 18 No. 4, 2008, pp. 593-646. doi:10.1142/S0218202508002796

[7] C. Castillo-Chavez, Z. Feng and W. Huang, “On the Computation of R0 and Its Role on Global Stability,” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002, pp. 229-250. doi:10.1007/978-1-4613-0065-6

[8] L. Esteva and C. Vargas, “Analysis of a Dengue Disease Transmission Model,” Mathematical Biosciences, Vol. 150, No. 2, 1998, pp. 131-151. doi:10.1016/S0025-5564(98)10003-2

[9] L. Esteva and C. Vargas, “A Model for Dengue Disease with Variable Human Population,” Journal of Mathematical Biology, Vol. 38, No. 3, 1999, pp. 220-240. doi:10.1007/s002850050147

[10] L. Esteva and C. Vargas, “Coexistence of Different Serotypes of Dengue Virus,” Journal of Mathematical Biology, Vol. 46, No. 1, 2003, pp. 31-47. doi:10.1007/s00285-002-0168-4

[11] Z. Feng and J. X. Velasco-Hernandez, “Competitive Exclusion in a Vector-Host Model for the Dengue Fever,” Journal of Mathematical Biology, Vol. 35, No. 5, 1997, pp. 523-544. doi:10.1007/s002850050064