流行病动态的最优控制策略：整合SIR-SI模型和Lotka-Volterra模型
Optimal Control Strategies for Epidemic Dynamics: Integrating SIR-SI and Lotka--Volterra Models
AI总结：
本文通过整合SIR-SI模型和Lotka-Volterra模型，提出一种最优控制策略，利用自然捕食者调节媒介种群以控制疾病传播，并通过数学方法优化干预措施以减少感染和成本。
摘要：
本文提出一个数学模型，将媒介传播疾病的流行病动态（SIR-SI）与Lotka-Volterra捕食者-猎物生态相互作用相结合。研究分析了自然捕食者如何作为生物控制机制调节媒介种群，从而影响宿主的疾病传播。我们引入了生态繁殖数的概念，这是一个阈值，将捕食者-猎物循环的振幅与疾病持续性联系起来，表明自然控制严重依赖于媒介最大密度与捕食者最小密度的比值。在自然控制不足的情况下，我们基于捕食者释放制定了一个最优控制问题。利用Pontryagin极大值原理，我们刻画了最优策略，该策略最小化受感染个体的累积数量和干预成本，同时在时间跨度结束时最大化易感宿主种群。数值模拟验证了模型的有效性，显示外部干预可以减轻流行病峰值并稳定系统，使其免受生物种群的自然振荡影响。
英文摘要：
In this work we present a mathematical model that integrates the epidemiological dynamics of a vector-borne disease (SIR-SI) with Lotka Volterra predator prey ecological interactions. The study analyzes how the presence of natural predators acts as a biological control mechanism to regulate the vector population and, consequently, disease transmission in host. We introduce the concept of the ecological reproduction number, a threshold that links the amplitude of predator prey cycles to disease persistence, showing that natural control depends critically on the ratio between the maximum vector density and the minimum predator density. In scenarios where natural control is insufficient, we formulate an optimal control problem based on the release of predators. Using the Pontryagin Maximum Principle, we characterize the optimal strategy that minimizes the cumulative number of infected individuals and intervention costs, while simultaneously maximizing the susceptible host population at the end of the time horizon. Numerical simulations validate the effectiveness of the model, showing that external intervention mitigates the epidemic peak and stabilizes the system against the natural oscillations of biological populations.
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