Aldila Dipo, Chávez Joseph Páez, Nugroho Bayu, Omede Benjamin Idoko, Peter Olumuyiwa James, Kamalia Putri Zahra
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok, Indonesia.
Innovative Mathematics and Predictive Analytics for Complex System and Technology Laboratory (IMPACT Lab), Universitas Indonesia, Depok, Indonesia.
PLoS One. 2025 Jul 31;20(7):e0328488. doi: 10.1371/journal.pone.0328488. eCollection 2025.
A co-infection model between HIV and COVID-19 that takes into account COVID-19 vaccination and public awareness is discussed in this article. Rigorous analysis of the model is conducted to establish the existence and local stability conditions of the single-infection models. We discover that when the corresponding reproduction number for COVID-19 and HIV exceeds one, the disease continues to exist in both single-infection models. Furthermore, HIV will always be eradicated if its reproduction number is less than one. Nevertheless, this does not apply to the single-infection COVID-19 model. Even when the fundamental reproduction number is less than one, an endemic equilibrium point may exist due to the potential for a backward bifurcation phenomenon. Consequently, in the single-infection COVID-19 model, bistability between the endemic and disease-free equilibrium may arise when the basic reproduction number is less than one. From the co-infection model, we find that the reproduction number of the co-infection model is the maximum value between the reproduction number of HIV and COVID-19. Our numerical continuation experiments on the co-infection model reveal a threshold indicating that both HIV and COVID-19 may coexist within the population. The disease-free equilibrium for both HIV and COVID-19 is stable only if the reproduction numbers are less than one. Additionally, our two-parameter continuation analysis of the bifurcation diagram shows that the condition where both reproduction numbers equal one serves as an organizing center for the dynamic behavior of the co-infection model. An extended version of our model incorporates four different interventions: face mask usage, vaccination, and public awareness for COVID-19, as well as condom use for HIV, formulated as an optimal control problem. The Pontryagin's Maximum Principle is employed to characterize the optimal control problem, which is solved using a forward-backward iterative method. Numerical investigations of the optimal control model highlight the critical role of a well-designed combination of interventions to achieve optimal reductions in the spread of both HIV and COVID-19.
本文讨论了一个同时考虑新冠病毒疫苗接种和公众意识的HIV与新冠病毒合并感染模型。对该模型进行了严格分析，以确定单一感染模型的存在性和局部稳定性条件。我们发现，当新冠病毒和HIV的相应繁殖数超过1时，两种单一感染模型中的疾病都会持续存在。此外，如果HIV的繁殖数小于1，它将始终被根除。然而，这不适用于单一感染新冠病毒模型。即使基本繁殖数小于1，由于可能出现反向分岔现象，仍可能存在地方病平衡点。因此，在单一感染新冠病毒模型中，当基本繁殖数小于1时，地方病平衡点和无病平衡点之间可能会出现双稳性。从合并感染模型中，我们发现合并感染模型的繁殖数是HIV和新冠病毒繁殖数中的最大值。我们对合并感染模型进行的数值延拓实验揭示了一个阈值，表明HIV和新冠病毒可能在人群中共存。只有当繁殖数小于1时，HIV和新冠病毒的无病平衡点才是稳定的。此外，我们对分岔图进行的双参数延拓分析表明，两个繁殖数都等于1的条件是合并感染模型动态行为的组织中心。我们模型的一个扩展版本纳入了四种不同的干预措施：佩戴口罩、接种疫苗、提高公众对新冠病毒的认识以及使用避孕套预防HIV，将其表述为一个最优控制问题。运用庞特里亚金极大值原理来刻画最优控制问题，并使用向前向后迭代法求解。对最优控制模型的数值研究突出了精心设计的干预措施组合对于实现HIV和新冠病毒传播最优减少的关键作用。
