Abstract
With the vaccination against Covid-19 now available, how vaccination campaigns influence the mathematical modeling of epidemics is quantitatively explored. In this paper, the standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to a fourth compartment, V, of vaccinated persons. This extension involves the time t-dependent effective vaccination rate, v(t), that regulates the relationship between susceptible and vaccinated persons. The rate v(t) competes with the usual infection, a(t), and recovery, μ(t), rates in determining the time evolution of epidemics. The occurrence of a pandemic outburst with rising rates of new infections requires k+b<1−2η, where k=μ(0)/a(0) and b=v(0)/a(0) denote the initial values for the ratios of the three rates, respectively, and η≪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 η completely determine the reduced time evolution of the SIRV-quantities Q(τ). 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 in Israel, this can happen in all countries considered.
Highlights
Statistical physics of vaccination has a rich history in physics research [1]
An implicit exact solution τ = τ (ψ) parameterized by ψ is derived, while all SIRV quantities can be expressed in ψ as well
Provided the reduced vaccination rate b exceeds a critical bc, for which the explicit expression (92) is provided, it is shown that the explicit solution of the SIRV equation can be written as follows
Summary
Statistical physics of vaccination has a rich history in physics research [1]. The model proposed here is an extension of an often employed model in epidemics research, the susceptible-infectious-recovered/removed (SIR) model. The purpose of the present paper is to analytically and quantitatively investigate, for a given ratio b(t) = v(t)/a(t) of the vaccination rate to infection rate a(t), the effect on the time evolution of the ongoing epidemic waves. In the case of negligible vaccination, assuming a constant ratio between infection and recovery rate, considerable improvements of the analytical modelling of epidemics with this compartment model have been achieved [109,110]. In what follows, these improvements are of a frequent use. In order to keep the analysis as simple and transparent as possible, such complicating issues as age grouping, vital dynamics and/or spatial spread effects, recently investigated in the literature [118,119] with numerical solutions, are ignored here
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