Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

Version 1 : Received: 29 March 2021 / Approved: 30 March 2021 / Online: 30 March 2021 (09:40:33 CEST)

A peer-reviewed article of this Preprint also exists.

Schlickeiser, R.; Kröger, M. Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations. Physics 2021, 3, 386–426, doi:10.3390/physics3020028. Schlickeiser, R.; Kröger, M. Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations. Physics 2021, 3, 386–426, doi:10.3390/physics3020028.

## Abstract

With the now available vaccination against Covid-19 it is quantitatively explored how vaccination campaigns influence the mathematical modeling of epidemics. The standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to the fourth compartment V of vaccinated persons and the vaccination rate v(t) that regulates the relation between susceptible and vaccinated persons. The vaccination rate v(t) competes with the infection (a(t)) and recovery (\mu(t)) rates in determining the time evolution of epidemics. In order for a pandemic outburst with rising rates of new infections it is required that k+b<1-2\eta, where k=\mu_0/a_0 and b=v_0/a_0 denote the initial ratios of the three rates, respectively, and \eta << 1 is the initial fraction of infected persons. Exact analytical inverse solutions t(Q) for all relevant quantities Q=[S,I,R,V] of the resulting SIRV-model in terms of Lambert functions are derived for the semi-time case with time-independent ratios k and b between the recovery and vaccination rates to the infection rate, respectively. These inverse solutions can be approximated with high accuracy yielding the explicit time-dependences Q(t) by inverting the Lambert functions. The values of the three parameters k, b and \eta completely determine the reduced time evolution the SIRV-quantities Q(\tau). The influence of vaccinations on the total cumulative number and the maximum rate of new infections in different countries is calculated by comparing with monitored real time Covid-19 data. The reduction in the final cumulative fraction of infected persons and in the maximum daily rate of new infections is quantitatively determined by using the actual pandemic parameters in different countries. Moreover, a new criterion is developed that decides on the occurrence of future Covid-19 waves in these countries. Apart from Israel this can happen in all countries considered.

## Supplementary and Associated Material

https://www.complexfluids.ethz.ch/SIRV: Real-time analysis of the 2nd Covid-19 wave (world-wide)

## Keywords

coronavirus; statistical analysis; extrapolation; parameter estimation; pandemic spreading

## Subject

Biology and Life Sciences, Biochemistry and Molecular Biology