Imran Mudassar, Rafique Hassan, Khan Adnan, Malik Tufail
Lahore University of Management Sciences, Sector U DHA, Lahore, Pakistan,
Theory Biosci. 2014 Jun;133(2):91-109. doi: 10.1007/s12064-013-0197-0. Epub 2013 Dec 30.
In this paper, we present a rigorous mathematical analysis of a deterministic model for the transmission dynamics of hepatitis C. The model is suitable for populations where two frequent modes of transmission of hepatitis C virus, namely unsafe blood transfusions and intravenous drug use, are dominant. The susceptible population is divided into two distinct compartments, the intravenous drug users and individuals undergoing unsafe blood transfusions. Individuals belonging to each compartment may develop acute and then possibly chronic infections. Chronically infected individuals may be quarantined. The analysis indicates that the eradication and persistence of the disease is completely determined by the magnitude of basic reproduction number R(c). It is shown that for the basic reproduction number R(c) < 1, the disease-free equilibrium is locally and globally asymptotically stable. For R(c) > 1, an endemic equilibrium exists and the disease is uniformly persistent. In addition, we present the uncertainty and sensitivity analyses to investigate the influence of different important model parameters on the disease prevalence. When the infected population persists, we have designed a time-dependent optimal quarantine strategy to minimize it. The Pontryagin's Maximum Principle is used to characterize the optimal control in terms of an optimality system which is solved numerically. Numerical results for the optimal control are compared against the constant controls and their efficiency is discussed.
在本文中，我们对丙型肝炎传播动力学的确定性模型进行了严格的数学分析。该模型适用于丙型肝炎病毒两种常见传播方式（即不安全输血和静脉注射吸毒）占主导地位的人群。易感人群被分为两个不同的部分，即静脉注射吸毒者和接受不安全输血的个体。每个部分的个体都可能发展为急性感染，然后可能发展为慢性感染。慢性感染个体可能被隔离。分析表明，疾病的根除和持续完全由基本再生数R(c)的大小决定。结果表明，当基本再生数R(c) < 1时，无病平衡点是局部和全局渐近稳定的。当R(c) > 1时，存在地方病平衡点，疾病是一致持续的。此外，我们进行了不确定性和敏感性分析，以研究不同重要模型参数对疾病流行率的影响。当感染人群持续存在时，我们设计了一种时间依赖的最优隔离策略以使其最小化。利用庞特里亚金极大值原理，根据一个通过数值求解的最优性系统来刻画最优控制。将最优控制的数值结果与常数控制进行比较，并讨论了它们的效率。
