在本文中,我们提出了COVID-19的数学模型,其中病毒的多个变种处于最优控制之下.数学建模已被用来更深入地了解COVID-19的传播,并实施了各种预防和控制策略来减轻其传播。我们的模型是基于SEIR的多菌株COVID-19模型,具有7个隔室。我们还考虑了导致COVID-19免疫终止的循环结构。根据解的积极性和有界性以及平衡点的存在性建立了模型,解的局部稳定性。由于加纳的COVID-19数据与模型拟合,原始病毒和Delta变体的基本繁殖数估计为1.9396,Omicron变体的基本繁殖数估计为3.4905,是其1.8倍。我们观察到,具有相同初始传播率的两种菌株的潜伏期和恢复期即使有很小的差异,也会导致受感染个体数量的巨大差异。在COVID-19的情况下,由Omicron变体引起的感染比原始病毒引起的感染多1.5至10倍。在最优控制策略方面,我们制定了三种关注社会距离的控制策略,疫苗接种,和测试治疗。我们使用Pontryagin的最大原理为上述多应变模型的三种策略开发了最优控制模型。通过数值模拟,我们分析了每种应变的三种最优控制策略,并考虑了两种控制策略的组合。作为模拟的结果,所有控制策略都能有效减少疾病传播,特别是,疫苗接种策略比其他两种控制策略更有效。此外,与实施一种策略相比,两种策略的组合还使感染人数减少了1/10,即使实施轻度水平。最后,我们表明,如果测试-治疗策略没有得到正确实施,无症状和身份不明的感染人数可能会激增。这些结果有助于指导政府干预水平和预防策略的制定。
In this paper, we suggest a mathematical model of COVID-19 with multiple variants of the virus under optimal control. Mathematical modeling has been used to gain deeper insights into the transmission of COVID-19, and various prevention and control strategies have been implemented to mitigate its spread. Our model is a SEIR-based model for multi-strains of COVID-19 with 7 compartments. We also consider the circulatory structure to account for the termination of immunity for COVID-19. The model is established in terms of the positivity and boundedness of the solution and the existence of equilibrium points, and the local stability of the solution. As a result of fitting data of COVID-19 in Ghana to the model, the basic reproduction number of the original virus and Delta variant was estimated to be 1.9396, and the basic reproduction number of the Omicron variant was estimated to be 3.4905, which is 1.8 times larger than that. We observe that even small differences in the incubation and recovery periods of two strains with the same initial transmission rate resulted in large differences in the number of infected individuals. In the case of COVID-19, infections caused by the Omicron variant occur 1.5 to 10 times more than those caused by the original virus. In terms of the optimal control strategy, we formulate three control strategies focusing on social distancing, vaccination, and testing-treatment. We have developed an optimal control model for the three strategies outlined above for the multi-strain model using the Pontryagin\'s Maximum Principle. Through numerical simulations, we analyze three optimal control strategies for each strain and also consider combinations of the two control strategies. As a result of the simulation, all control strategies are effective in reducing disease spread, in particular, vaccination strategies are more effective than the other two control strategies. In addition the combination of the two strategies also reduces the number of infected individuals by 1/10 compared to implementing one strategy, even when mild levels are implemented. Finally, we show that if the testing-treatment strategy is not properly implemented, the number of asymptomatic and unidentified infections may surge. These results could help guide the level of government intervention and prevention strategy formulation.