Abstract
Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast. In particular, it is possible to test lockdown and lifting interventions because the new solutions hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections.
Highlights
The Susceptible-Infectious-Recovered (SIR) model has been developed nearly hundred years ago [1, 2] to understand the time evolution of infectious diseases in human populations
To calculate the rate and cumulative number in real time according to Eq 2 we adopt as time-dependent infection rate the integrable function known from shock wave physics aLD(t)
In countries where the peak of the first Covid-19 wave has already passed such as e.g. Germany, Switzerland, Austria, Spain, France and Italy, we may use the monitored fatality rates and peak times to check on the validity of the SIR model with the determined free parameters
Summary
The Susceptible-Infectious-Recovered (SIR) model has been developed nearly hundred years ago [1, 2] to understand the time evolution of infectious diseases in human populations. In paper A it has been shown that, for arbitrary but given infection rates a(t), apart from the peak reduced time τ0 of the rate of new infections, all properties of the pandemic wave as functions of the reduced time are solely controlled by the inverse basic reproduction number k. The dimensionless peak time τ0 is controlled by k and the value ε −lnS(0), indicating as only initial condition at the observing time the fraction of initially susceptible persons S(0) e−ε. This suggests to introduce the relative reduced time Δ τ − τ0 with respect to the reduced peak time. In real time t the adopted infection rate a(t) acts as second parameter, and the peak time tm, where J_(t) reaches its maximum must not coincide with the time, where the reduced j reaches its maximum, i.e., τm ≡ τ(tm) ≠ τ0, in general
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