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.

中文翻译：

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具有环境传播的两种禽流感模型：稳定性分析和最优控制策略

世界卫生组织合理地担心，一旦禽流感病毒到达人群并维持人际传播，就会出现变异，因此我们制定了一个假设的数学模型，用于解释禽流感病毒的环境传播和变异。一种禽流感病毒，有能力传播到人群中并成为高致病性毒株。我们计算模型的基本再生数并用它来研究平衡点的存在性和稳定性。我们推导出这三个均衡中任何一个的全局渐近稳定性的条件。该模型经过扩展，纳入了六个相关的时间依赖性控制，并使用庞特里亚金极大值原理得出最佳疾病控制的必要条件。最后，对最优控制问题进行数值求解，以显示每个控制参数及其组合的效果。计算增量成本效益比，以研究控制策略的所有可能组合的成本效益。这项研究表明，隔离感染者可能是控制病毒突变的禽流感传播的最具成本效益的策略。