A mathematical model of malaria dynamics with naturally acquired transient immunity in the presence of protected travellers is presented. The qualitative analysis carried out on the autonomous model reveals the existence of backward bifurcation, where the locally asymptotically stable malaria-free and malaria-present equilibria coexist as the basic reproduction number crosses unity. The increased fraction of protected travellers is shown to reduce the basic reproduction number significantly. Particularly, optimal control theory is used to analyse the non-autonomous model, which incorporates four control variables. The existence result for the optimal control quadruple, which minimizes malaria infection and costs of implementation, is explicitly proved. Effects of combining at least any three of the control variables on the malaria dynamics are illustrated. Furthermore, the cost-effectiveness analysis is carried out to reveal the most cost-effective strategy that could be implemented to prevent and control the spread of malaria with limited resources.

中文翻译：

---

通过部分免疫和受保护的旅行者为疟疾动态建模：最佳控制和成本效益分析。

提出了在受保护的旅行者存在下具有自然获得的瞬时免疫力的疟疾动态的数学模型。在自治模型上进行的定性分析揭示了反向分叉的存在，其中局部渐近稳定的无疟疾和存在疟疾的均衡与基本繁殖数越过并存。显示出受保护旅行者的分数增加会大大减少基本繁殖数量。特别地，使用最优控制理论来分析非自治模型，该模型包含四个控制变量。明确证明了最佳控制四元组的存在结果，该控制组可最大程度地减少疟疾感染和实施成本。说明了至少结合三个控制变量对疟疾动态的影响。此外，进行了成本效益分析以揭示可以实施的最具成本效益的战略，以预防和控制资源有限的疟疾传播。