Fig. 1

Flow diagram of the age-structured mathematical model for COVID-19 in NYC. The total population in NYC were divided into 5 age groups (0–17, 18–44, 45–64, 65–74 and 75–100 years). The sub-population in each age group \(i\) in NYC were further divided into eight compartments: susceptible individuals \(S_{i} (t)\); vaccinated individuals \(V_{i} (t)\); exposed individuals \(E_{i} (t)\); infected but asymptomatic individuals \(A_{i} (t)\); infected and symptomatic individuals \(I_{i} (t)\); confirmed individuals who stayed at home \(C_{i} (t)\); hospitalized cases \(H_{i} (t)\) and recovered cases \(R_{i} (t)\). The details of the force of infection \(\Lambda\) were provided in the Supplementary Text 1 (model formulation). The susceptible individuals \(S_{i} (t)\) would become vaccinated individuals when they were vaccinated. The vaccination coverage rate was \(p(t)\) and we assumed the vaccination coverage rate was a logistic function, i.e., \(p(t) = \frac{{p_{\max } p_{0} }}{{p_{0} - (p_{0} - p_{\max } )\exp ( - rt)}}\), where \(p_{\max }\) was the maximum vaccination coverage rate, \(p_{0}\) was the initial vaccination coverage rate and \(r\) was the growth rate of vaccination in NYC. The effectiveness of vaccine for COVID-19 was \(q\). The incubation period of exposed individuals was \({1 \mathord{\left/ {\vphantom {1 \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }\). The recovery rate of asymptomatic infections in the free environment, confirmed cases and hospitalized cases were \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\), respectively. The proportion of symptomatic infections in age group \(i\) was \(\eta_{i}\), the transfer rate from symptomatic individuals to confirmed cases in age group \(i\) was \(\delta_{i}\), the transfer rate from confirmed cases to hospitalized cases in age group \(i\) was \(\alpha_{i} (t)\), and the death rate in age group \(i\) was \(\mu_{i} (t)\). Here, we assumed that \(\alpha_{i} (t)\) and \(\mu_{i} (t)\) were exponentially decreasing functions. More details were provided in the Additional file 1: Text 1 (model formulation)