Modelling and Simulation of the Spread of HBV Disease with Infectious Latent

This paper studies the global behavior of the spread of HBV using a SEIR model with a constant vaccination rate. The infectivity during the incubation period is considered as a second way of transmission. The basic reproduction number R0 is derived as a function of the two contact rates β1 and β2 . There is a disease free equilibrium point (DFE) of our model. When R0 < 1, the (DFE) is asymptotically stable. On the other hand, if R0 > 1, there is a unique endemic equilibrium. We proved that the endemic equilibrium was globally asymptotically stable when R0 > 1 and that the disease persisted in the population. These results are original for our model with vaccination and two contact rates.

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Moneim, I. and Khalil, H. (2015) Modelling and Simulation of the Spread of HBV Disease with Infectious Latent. Applied Mathematics, 6, 745-753. doi: 10.4236/am.2015.65070.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Abdulrahman, S., Akinwande, N.I., Awojoyogbe, O.B. and Abubakar, U.Y. (2013) Sensitivity Analysis of the Parameters of a Mathematical Model of Hepatitis B Virus Transmission. Universal Journal of Applied Mathematics, 1, 230-241. [2] Saher, F., Rahman, K., Quresh, J.A., Irshad, M. and Iqbal, H.M. (2012) Investigation of an Inflammatory Viral Disease HBV in Cardiac Patients through Polymerase Chain Reaction. Advances in Bioscience and Biotechnology, 3, 417-422. http://dx.doi.org/10.4236/abb.2012.324059 [3] Centers for Disease Control and Prevention (2012) http://www.cdc.gov/hepatitis/HBV [4] Moneim, I.A., Al-Ahmed, M. and Mosa, G.A. (2009) Stochastic and Monte Carlo Simulation for the Spread of the Hepatitis B. Australian Journal of Basic and Applied Sciences, 3, 1607-1615. [5] Li, G. and Jin, Z. (2005) Global Stability of an SEI Epidemic Model with General Contact Rate. Chaos, Solitons and Fractals, 23, 997-1004. [6] Li, G. and Jin, Z. (2005) Global Stability of a SEIR Epidemic Model with Infectious Force in Latent, Infected and Immune Period. Chaos, Solitons and Fractals, 25, 1177-1184. http://dx.doi.org/10.1016/j.chaos.2004.11.062 [7] Li, G., Wang, W. and Jin, Z. (2006) Global Stability of an SEIR Epidemic Model with Contact Immigration. Chaos, Solitons and Fractals, 30, 1012-1019. http://dx.doi.org/10.1016/j.chaos.2005.09.024 [8] Li, X. and Fang, B. (2009) Stability of an Age-Structured SEIR Epidemic Model with Infectivity in Latent Period. Applications and Applied Mathematics: An International Journal (AAM), 4, 218-236. [9] Kapur, J.N. (1990) Mathematical Models in Biology and Medicine. Affiliated East-West Press, New Delhi. [10] Korobeinikov, A. and Wake, G.C. (2002) Lyapunov Functions and Global Stability for SIR, SIRS, and SIS Epidemiological Models. Applied Mathematics Letters, 15, 955-960. http://dx.doi.org/10.1016/S0893-9659(02)00069-1 [11] Zhuo, X. (2011) Global Analysis o f a General HBV Infection Model. IEEE International Conference on Systems Biology (ISB), Zhuhai, 2-4 September 2011, 978-1-4577-1666-9/11. [12] Wiah, E.N., Dontwi, I.K. and Adetunde, I.A. (2011) Using Mathematical Model to Depict the Immune Response to Hepatitis B Virus Infection. Journal of Mathematics Research, 3, 157-116. http://dx.doi.org/10.5539/jmr.v3n2p157 [13] Zou, L., Zhang, W. and Ruan, S. (2010) Modeling the Transmission Dynamics and Control of Hepatitis B Virus in China. Journal of Theoretical Biology, 262, 330-338. http://dx.doi.org/10.1016/j.jtbi.2009.09.035 [14] Greenhalgh, D. and Moneim, I.A. (2003) SIRS Epidemic Model and Simulations Using Different Types of Seasonal Contact Rate. Systems Analysis Modelling Simulation, 43, 573-600. http://dx.doi.org/10.1080/023929021000008813 [15] Kimbir, A.R., Aboiyar, T., Abu, O. and Onah, E.S. (2014) Simulation of A Mathematical Model of Hepatitis B Virus Transmission Dynamics in the Presence of Vaccination and Treatment. Mathematical Theory and Modeling, 4, 44-59.