On the Time Shift Phenomena in Epidemic Models - Peker-Dobie, Ayse; Demirci, Ali; Bilge, Ayse Humeyra; Ahmetolan, Semra (Frontiers in Physics, 2020)
In the standard Susceptible-Infected-Removed (SIR) and Susceptible-Exposed-Infected-Removed (SEIR) models, the peak of infected individuals coincides with the inflection point of removed individuals. Nevertheless, a survey based on the data of the 2009 H1N1 epidemic in Istanbul, Turkey \citep{bib19} displayed a time shift between the hospital referrals and fatalities. An analysis of recent Covid-19 data \citep{ahmetolan2020can} and the records for Spanish flu (1918-1919) and SARS (2002-2004) epidemics confirm this observation. We use multistage $SIR$ and $SEIR$ models \citep{bib14} to provide an explanation for this time shift. Numerical solutions of these models present strong evidence that the delay between the peak of $R'(t)$ and the peak of $J(t)=\sum_i I_i(t)$ is approximately half of the infectious period of the epidemic disease. In addition, we use a quadratic approximation to show that the distance between successive peaks of $I_i$ is $1/\gamma_{i}$ where $1/\gamma_{i}$ is the infectious period of the $i$th infectious stage and we present numerical calculations that confirm this approximation.