The dynamical model for COVID-19 with asymptotic analysis and numerical implementations

Abstract

The 2019 novel coronavirus (COVID-19) emerged at the end of 2019 has a great impact on China and all over the world. The transmission mechanism of COVID-19 is still unclear. Except for the initial status and the imported cases, the isolation measures and the medical treatments of the infected patients have essential influences on the spread of COVID-19. In this paper, we establish a mathematical model for COVID-19 transmission involving the interactive effect of various factors for the infected people, including imported cases, isolating rate, diagnostic rate, recovery rate and also the mortality rate. Under the assumption that the random incubation period, the cure period and the diagnosis period are subject to the Weibull distribution, the quantity of daily existing infected people is finally governed by a linear integral-differential equation with convolution kernel. Based on the asymptotic behavior and the quantitative analysis on the model, we rigorously prove that, for limited external input patients, both the quantity of infected patients and its variation ratio will finally tend to zero, if the infected patients are sufficiently isolated or the infection rate is small enough. Finally, numerical performances for the proposed model as well as the comparisons between our simulations and the clinical data of the city Wuhan and Italy are demonstrated, showing the validity of our model with suitably specified model parameters.