A two-strain avian-human influenza model with environmental transmission: Stability analysis and optimal control strategies

On the basis of the WHO legitimated fear that there will be an avian influenza virus strain capable of mutating once it reaches the human population and sustains human-to-human transmissions, we formulate an hypothetical mathematical model which accounts for the environmental transmission and mutation of an avian influenza virus having the ability to spill over into the human population and become a highly pathogenic strain. We compute the basic reproduction number of the model and use it to study the existence and stability of equilibrium points. We derive conditions for the global asymptotic stability of any of the three equilibrium. The model is extended to incorporate six relevant time-dependent controls, and use the Pontryagin's maximum principle to derive the necessary conditions for optimal disease control. Finally, the optimal control problem is solved numerically to show the effect of each control parameter and their combination. The incremental cost-effectiveness ratios are calculated to investigate the cost-effectiveness of all possible combinations of the control strategies. This study suggests that quarantine infected humans might be the most cost-effective strategy to control avian influenza transmissions with the virus mutation.