Effect of sojourn time distributions on the early dynamics of COVID-19 outbreak

Compartmental models are commonly used in practice to investigate the dynamical response of infectious diseases such as the COVID-19 outbreak. Such models generally assume exponentially distributed latency and infectiousness periods. However, the exponential distribution assumption fails when the sojourn times are expected to distribute around their means. This study aims to derive a novel S (Susceptible)-E (Exposed)-P (Presymptomatic)-A (Asymptomatic)-D (Symptomatic)-C (Reported) model with arbitrarily distributed latency, presymptomatic infectiousness, asymptomatic infectiousness, and symptomatic infectiousness periods. The SEPADC model is represented by nonlinear Volterra integral equations that generalize ordinary differential equation-based models. Our primary aim is the derivation of a general relation between intrinsic growth rate r and basic reproduction number R_0 with the help of the well-known Lotka–Euler equation. The resulting r-R_0 equation includes separate roles of various stages of the infection and their sojourn time distributions. We show that R_0 estimates are considerably affected by the choice of the sojourn time distributions for relatively higher values of r . The well-known exponential distribution assumption has led to the underestimation of R_0 values for most of the countries. Exponential and delta-distributed sojourn times have been shown to yield lower and upper bounds of the R_0 values depending on the r values. In quantitative experiments, R_0 values of 152 countries around the world were estimated through our novel formulae utilizing the parameter values and sojourn time distributions of the COVID-19 pandemic. The global convergence, R_0=4.58 , has been estimated through our novel formulation. Additionally, we have shown that increasing the shape parameter of the Erlang distributed sojourn times increases the skewness of the epidemic curves in entire dynamics.