The first case of coronavirus disease 2019 (COVID-19) in Japan was reported on 15 January 2020 and the number of reported cases has increased day by day. The purpose of this study is to give a prediction of the epidemic peak for COVID-19 in Japan by using the real-time data from 15 January to 29 February 2020. Taking into account the uncertainty due to the incomplete identification of infective population, we apply the well-known SEIR compartmental model for the prediction. By using a least-square-based method with Poisson noise, we estimate that the basic reproduction number for the epidemic in Japan is $R_0=2.6$ ( $95\%$ CI, $2.4$ – $2.8$ ) and the epidemic peak could possibly reach the early-middle summer. In addition, we obtain the following epidemiological insights: (1) the essential epidemic size is less likely to be affected by the rate of identification of the actual infective population; (2) the intervention has a positive effect on the delay of the epidemic peak; (3) intervention over a relatively long period is needed to effectively reduce the final epidemic size.

In this paper, we evaluate the effect of the state of emergency for the first wave of COVID-19 in Japan, 2020 from the viewpoint of mathematical modelling. In Japan, it was announced during the period of the state of emergency from April 7 to May 25, 2020 that the 80% reduction of the contact rate is needed to control the outbreak. By numerical simulation, we show that the reduction rate seems to have reached up to 86%. Moreover, we estimate the control reproduction number Rc during the period of the state of emergency as Rc=0.36 (95%CI, 0.34–0.39), and show that the effective reproduction number Re after the lifting of the state of emergency could be greater than 1. This result suggests us that the second wave of COVID-19 in Japan could possibly occur if any effective intervention will not be taken again.

In this paper, we study the global dynamics of an susceptible, vaccinated, infectious and recovered epidemiological model with infection-age structure. Biologically, we assume that effective contacts between vaccinated individuals and infectious individuals are less than that between susceptible individuals and infectious individuals. Using Lyapunov functions, we show that the global stability of each equilibrium is completely determined by the basic reproduction number $\Re _0$ : if $\Re _0\leqslant 1,$ then the disease-free equilibrium is globally asymptotically stable; while if $\Re _0 > 1$ , then there exists a unique endemic equilibrium which is globally asymptotically stable.