# American Institute of Mathematical Sciences

January  2010, 13(1): 1-25. doi: 10.3934/dcdsb.2010.13.1

## Theoretical assessment of avian influenza vaccine

 1 Department of Mathematical Sciences, Federal University of Technology Akure, P.M.B. 704, Akure, Nigeria 2 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

Received  November 2008 Revised  May 2009 Published  October 2009

This study presents a deterministic model for theoretically assessing the potential impact of an imperfect avian influenza vaccine (for domestic birds) in two avian populations on the transmission dynamics of avian influenza in the domestic and wild birds population. The model is analyzed to gain insights into the qualitative features of its associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction number and a measure for vaccine impact. A sub-model without vaccination is first considered, where it is shown that it has a globally-asymptotically stable disease-free equilibrium whenever a certain reproduction threshold is less than unity. Unlike the sub-model without vaccination, the model with vaccination undergoes backward bifurcation, a phenomenon associated with the co-existence of multiple stable equilibria. In other words, for the model with vaccination, the classical epidemiological requirement of having the associated reproduction number less than unity does not guarantee disease elimination in the model. It is shown that the possibility of backward bifurcation occurring decreases with increasing vaccination rate (for susceptible domestic birds). Further, the study shows that the vaccine impact (in reducing disease burden) is dependent on the sign of a certain threshold quantity (denoted by $\nabla_{\mathcal P}$). The vaccine will have positive or no impact if $\nabla_{\mathcal P}$ is less than or equal to unity. Numerical simulations suggest that the prospect of effectively controlling the disease in the avian population increases with increasing vaccine efficacy and coverage.
Citation: Folashade B. Agusto, Abba B. Gumel. Theoretical assessment of avian influenza vaccine. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 1-25. doi: 10.3934/dcdsb.2010.13.1
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