SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates

Abstract

The temporal evolution of pandemics described by the susceptible-infectious-recovered (SIR)-compartment model is sensitively determined by the time dependence of the infection (a(t)) and recovery (μ(t)) rates regulating the transitions from the susceptible to the infected and from the infected to the recovered compartment, respectively. Here, approximated SIR solutions for different time dependencies of the infection and recovery rates are derived which are based on the adiabatic approximation assuming time-dependent ratios, k(t)=μ(t)/a(t), varying slowly in comparison with the typical time characteristics of the pandemic wave. For such slow variations, the available analytical approximations from the KSSIR-model, developed by us and valid for a stationary value of the ratio k, are used to insert a posteriori the adopted time-dependent ratio of the two rates. Instead of investigating endless different combinations of the time dependencies of the two rates a(t) and μ(t), a suitably parameterized reduced time, τ, dependence of the ratio k(τ) is adopted. Together with the definition of the reduced time, this parameterized ratio k(τ) allows us to cover a great variety of different time dependencies of the infection and recovery rates. The agreement between the solutions from the adiabatic approximation in its four different studied variants and the exact numerical solutions of the SIR-equations is tolerable providing confidence in the accuracy of the proposed adiabatic approximation.

Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates

“…Additionally, analytical solutions for arbitrary but given time dependencies of the infection rate a(t) have been derived for the infinite [4] and semitime (t ≥ t 0 ) [5] time domains for the case of a stationary ratio k = µ(t)/a(t), thereby implying that the recovery rate has exactly the same time dependence as the infection rate. Analytical approximations have been developed [50] for slowly varying ratios of k(t) in comparison with the typical time characteristics of the epidemic wave. Below, we will derive approximate analytical solutions of the SIR model equations for the limit of not-too-late times, where the cumulative number of new infections J is much smaller than unity.…”

Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past Covid-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution

“…Recently [35,36] this integral solution has been generalized to arbitrary time-dependent infection rates a(t) assuming a stationary value of the ratio k(t) = k 0 so that the recovery rate has the same time dependence as the infection rate. A further generalization to slowly time-dependent ratios k(t) is also possible [37].…”