Department of Mathematics and Institute of Applied Mathematics (IAM), University of British Columbia, Vancouver, British Columbia, Canada.
Faculty of Mathematics, Technion Israel Institute of Technology, Haifa 32000, Israel.
Math Biosci Eng. 2023 Jan 12;20(3):5379-5412. doi: 10.3934/mbe.2023249.
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been spreading worldwide for over two years, with millions of reported cases and deaths. The deployment of mathematical modeling in the fight against COVID-19 has recorded tremendous success. However, most of these models target the epidemic phase of the disease. The development of safe and effective vaccines against SARS-CoV-2 brought hope of safe reopening of schools and businesses and return to pre-COVID normalcy, until mutant strains like the Delta and Omicron variants, which are more infectious, emerged. A few months into the pandemic, reports of the possibility of both vaccine- and infection-induced immunity waning emerged, thereby indicating that COVID-19 may be with us for longer than earlier thought. As a result, to better understand the dynamics of COVID-19, it is essential to study the disease with an endemic model. In this regard, we developed and analyzed an endemic model of COVID-19 that incorporates the waning of both vaccine- and infection-induced immunities using distributed delay equations. Our modeling framework assumes that the waning of both immunities occurs gradually over time at the population level. We derived a nonlinear ODE system from the distributed delay model and showed that the model could exhibit either a forward or backward bifurcation depending on the immunity waning rates. Having a backward bifurcation implies that $ R_c < 1 $ is not sufficient to guarantee disease eradication, and that the immunity waning rates are critical factors in eradicating COVID-19. Our numerical simulations show that vaccinating a high percentage of the population with a safe and moderately effective vaccine could help in eradicating COVID-19.
严重急性呼吸综合征冠状病毒 2（SARS-CoV-2）已在全球范围内传播超过两年，报告的病例和死亡人数已达数百万。在抗击 COVID-19 的过程中，数学模型的应用取得了巨大成功。然而，这些模型大多针对疾病的流行阶段。针对 SARS-CoV-2 的安全有效的疫苗的开发为学校和企业的安全重新开放以及恢复 COVID-19 之前的正常生活带来了希望，直到像 Delta 和 Omicron 变体等更具传染性的突变株出现。大流行几个月后，有报道称疫苗和感染诱导的免疫力减弱的可能性，这表明 COVID-19 可能会比之前预期的持续时间更长。因此，为了更好地了解 COVID-19 的动态，有必要使用地方病模型来研究这种疾病。在这方面，我们使用分布时滞方程开发并分析了一种包含疫苗和感染诱导免疫减弱的 COVID-19 地方病模型。我们的建模框架假设，两种免疫的减弱都是在人群水平上随着时间的推移逐渐发生的。我们从分布时滞模型推导出一个非线性 ODE 系统，并表明该模型可能根据免疫减弱率表现出前向或后向分岔。后向分岔意味着 $R_c < 1$ 不足以保证疾病的根除，并且免疫减弱率是根除 COVID-19 的关键因素。我们的数值模拟表明，用安全且中度有效的疫苗为高比例的人群接种疫苗有助于根除 COVID-19。
