On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates

Abstract

The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection (a(t)) and recovery (μ(t)) rates, regulating the transitions between the compartments S→I and I→R, respectively, the time-dependent vaccination rate v(t) accounts for the transition between the compartments S→V of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections J(t)≪1. As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit J(t)≪1 is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections J˚(t)=a(t)S(t)I(t), the corresponding cumulative fraction J(t), and V(t), respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction J∞ after infinite time allows us to check the validity of the original assumption J(t)≤J∞≪1.

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD and SIR Compartment Models

“…which apart from a slight change in notation (R tot and µ tot instead of R and µ before) agree exactly with the earlier considered Equations ( 13) - (17) in [21].…”

“…Only if the temporal dependence of I(t) is not drastically different from the temporal dependence of J(t), the a-posteriori approach is justified. As has been emphasized before [21], for COVID-19 outbursts in many countries the monitored cumulative fraction of infected persons J(t) 1 has been much smaller than unity at all times, so that then in the SIR model indeed I(t) j(t).…”

“…In the following sections we will derive a class of special exact solutions (Sections 3-4) and derive approximate analytical solutions of Equations ( 21) and ( 25) in the limit of small J(τ) 1 (Sect. 5) following our earlier approach [21]. This approximation then leads to analytical approximations of all fractions of the SIRVD epidemics model as a function of time for given time dependencies of the three ratios (10).…”

“…Recently, it has been demonstrated that an accurate analytical approximation to the SIR and SIRV equations [21,26] exists if the cumulative fraction of new infections J(t) = J(τ) at all times is much smaller than unity. For the COVID-19 outbreaks in many countries this assumption is well fulfilled.…”

“…First we will investigate new classes of exact solutions to the SIRVD and SIRD equations. Secondly we will apply the recently developed analytical approximation [20,21] for the SIR and SIRV models also to the more general SIRVD model. These accurate analytical approximations have been derived for all epidemic quantities of interest such as the rate of new infections J(t) and the corresponding cumulative fraction of infections J(t).…”