Berhe Hailay Weldegiorgis, Makinde Oluwole Daniel
Department of Mathematics, Mekelle University, Mekelle, Ethiopia.
Faculty of Military Science, Stellenbosch University, Cape Town, South Africa.
Biosystems. 2020 Apr;190:104102. doi: 10.1016/j.biosystems.2020.104102. Epub 2020 Feb 7.
Measles is an awfully contagious acute viral infection. It can be fatal, causing cough, red eyes, followed by a fever and skin rash with signs of respiratory infection. In this paper, we propose and analyze a model describing the transmission dynamics of a measles epidemic in the human population using the stability theory of differential equations. The model proposed undergoes a backward bifurcation for some parameter values. Sensitivity analysis is carried out on the model parameters in order to determine their impact on the disease dynamics. We extend the model to an optimal control problem by including time-dependent control variables: prevention, treatment of infected people and vaccination of the susceptible humans. In an attempt to minimize the infected people and the cost applied we design the cost functional. Next, we show that optimal control exists for the system, and the Pontryagin maximum principle is employed to characterize the continuous controls. Numerical simulation is performed to justify the analytical results and discussed quantitatively.
麻疹是一种极具传染性的急性病毒感染。它可能致命，会引发咳嗽、眼睛发红，随后出现发烧和皮疹，并伴有呼吸道感染症状。在本文中，我们提出并分析了一个使用微分方程稳定性理论来描述人群中麻疹疫情传播动态的模型。所提出的模型在某些参数值下会出现反向分岔。对模型参数进行了敏感性分析，以确定它们对疾病动态的影响。我们通过纳入随时间变化的控制变量：预防、对感染者的治疗以及对易感人群的疫苗接种，将该模型扩展为一个最优控制问题。为了尽量减少感染者数量和所投入的成本，我们设计了成本泛函。接下来，我们证明该系统存在最优控制，并采用庞特里亚金极大值原理来刻画连续控制。进行了数值模拟以验证分析结果并进行定量讨论。
