A General Model for Hepatitis B Disease with Age-Dependent Susceptibility and Transmission Probabilities

Danga Duplex Elvis Houpa, Tagne Eric Miamdjo, Tchaptchie Yannick Kouakep

Department of Mathematics and Computer Science, Ngaoundéré University, Ngaoundere, Cameroon.

DOI: 10.4236/am.2014.54068                

Abstract

A SEI model for hepatitis B is constructed where the susceptibility and other crucial transmission probabilities depend on the chronological age and the basic reproduction rate R₀ is derived. Under suitable (biological and mathematical) assumptions in a closed population, results of Houpa D. D. E. et al. [1] are extended from constant case of p and q to age-dependent case: the disease-free equilibrium is globally asymptotically stable (GAS) if R₀ < 1. On the other hand, R₀ > 1 induces that endemic equilibrium is GAS and the system is uniformly persistent.

Keywords

A General Model for Hepatitis B Disease with Age-Dependent Susceptibility and Transmission Probabilities

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Houpa, D.D.E., Miamdjo Tagne, E. and Kouakep, T.Y. (2014) A Model for Hepatitis B Disease with Age-Dependent Susceptibility. JMCS, 4, accepted. http://scik.org/index.php/jmcs/article/view/1431
[2]	Zuckerman, A.J. (1976) The A, B, C Viruses. Nature, 259, 363-364.
[3]	Fall, A.A., Gauthier, S. and Abderrahman, I. (2010) Modélisation de la Transmission Verticale de l’Hépatite B. CARI 2010 Report.
[4]	WHO (2013) Centre des Médias: Hépatite B. http://www.who.int/mediacentre/factsheets/fs204/en/index.html
[5]	Pasquini, P. and Cvjetanovic, B. (1988) Mathematical Models of Hepatitis B Infection. Annali dell’Istituto Superiore di Sanità, 24, 245-250.
[6]	Bonzi, B., Fall, A.A., Iggidr, A. and Sallet, G. (2011) Stability of Differential Susceptibility and Infectivity Epidemic Models. Journal of Mathematical Biology, 62, 39-64. http://dx.doi.org/10.1007/s00285-010-0327-y
[7]	Inaba, H. (1990) Threshold and Stability Results for an Age-Structured Epidemic Model. Journal of Mathematical Biology, 28, 411-434. http://dx.doi.org/10.1007/BF00178326
[8]	Nokes, D.J., Hall, A.J., Edmunds, W.J., Medley, G.F. and Whittle, H.C. (1993) The Influence of Age on the Development of the Hepatitis B carrier State. Proceedings of the Royal Society B: Biological Sciences, 253, 197-201. http://dx.doi.org/10.1098/rspb.1993.0102
[9]	Zou, L., Ruan, S. and Zhang, W. (2010) An Age-Structured Model for Transmission Dynamics of Hepatitis B. SIAM, 70, 3121-3139.
[10]	Kouakep, T.Y., Ducrot, A. and Houpa, D.D.E. (2013) A Model for Hepatitis B with Chronological and Infection Ages. Applied Mathematical Sciences, 7, 5977-5993.
[11]	Melnik, A.V. and Korobeinikov, A. (2013) Lyapunov Functions and Global Stability for SIR and SEIR Models with Age-Dependent Susceptibility. Mathematical Biosciences and Engineering, 10, 369-378. http://dx.doi.org/10.3934/mbe.2013.10.369
[12]	WHO (1996) Hepatitis B and Breastfeeding. http://www.who.int/maternal_child_adolescent/documents/pdfs/hepatitis_b_and_breastfeeding.pdf
[13]	Magal, P., McCluskey, C.C. and Webb, G.F. (2010) Liapunov Functional and Global Asymptotic Stability for an Infection-Age Model. Applicable Analysis, 89, 1109-1140. http://dx.doi.org/10.1080/00036810903208122
[14]	Magal, P. (2001) Compact Attractors for Time-Periodic Age Structured Population Models. Electronic Journal of Differential Equations, 2001, 1-35.
[15]	Magal, P. and Ruan, S. (2009) On Semilinear Cauchy Problems with Non-Dense Domain. Advances in Differential Equations, 14, 1041-1084.
[16]	Thieme, H.R. (1997) Quasi-Compact Semigroups via Bounded Pertubation. In: Arino, O., Axelrod, D. and Kimmel, M., Eds., Advances in Mathematical Population Dynamics-Molecules, Cells and Man, World Scientific Publishing, River Edge, 691-711.
[17]	Djidjou, D.R. and Ducrot, A. (2013) An Age-Structured Within-Host Model for Multistrain Malaria Infections. SIAM Journal on Applied Mathematics, 73, 572-593. http://dx.doi.org/10.1137/120890351
[18]	Sell, G.R. and You, Y. (2002) Dynamics of Evolutionary Equations. Springer, New York. http://dx.doi.org/10.1007/978-1-4757-5037-9
[19]	Hale, J.K. (1986) Asymptotic Behavior and Dynamics in Infinite Dimensions, in Nonlinear Differential Equations. Hale, J.K. and Martinez-Amores, P., Eds., Pitman, Marshfield.
[20]	Hale, J.K. (1988) Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs 25, American Mathematical Society, Providence.
[21]	Hale, J.K. and Waltman, P. (1989) Persistence in Infinite-Dimensional Systems. SIAM Journal on Mathematical Analysis, 20, 388-395. http://dx.doi.org/10.1137/0520025
[22]	LaSalle, J.P. (1976) The Stability of Dynamical Systems. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970432
[23]	Sall Diallo, A., Sarr, M., Fall, Y., Diagne, C. and Kane, M.O. (2004) Hepatitis B Infection in Infantile Population of Senegal. Dakar Medical, 49, 136-142.
[24]	Magal, P. (2009) Perturbation of a Globally Stable Steady State and Uniform Persistence. Journal of Dynamics and Differential Equations, 21, 1-20. http://dx.doi.org/10.1007/s10884-008-9127-0
[25]	Ducrot, A., Houpa, D.D.E. and Kouakep, T.Y. (2013) An Age-Structured Model with Differential Infectiousness: Application to Hepatitis B Virus Transmission.