Analytical solution of the SIR-model for the temporal evolution of epidemics: part B. Semi-time case

The earlier analytical analysis (part A) of the susceptible–infectious–recovered (SIR) epidemics model for a constant ratio k of infection to recovery rates is extended here to the semi-time case which is particularly appropriate for modeling the temporal evolution of later (than the first) pandemic waves when a greater population fraction from the first wave has been infected. In the semi-time case the SIR model does not describe the quantities in the past; instead they only hold for times later than the initial time t = 0 of the newly occurring wave. Simple exact and approximative expressions are derived for the final and maximum values of the infected, susceptible and recovered/removed population fractions as well the daily rate and cumulative number of new infections. It is demonstrated that two types of temporal evolution of the daily rate of new infections j ( τ ) occur depending on the values of k and the initial value of the infected fraction I (0) = η : in the decay case for k ⩾ 1 − 2 η the daily rate monotonically decreases at all positive times from its initial maximum value j (0) = η (1 − η ). Alternatively, in the peak case for k < 1 − 2 η the daily rate attains a maximum at a finite positive time. By comparing the approximated analytical solutions for j ( τ ) and J ( τ ) with the exact ones obtained by numerical integration, it is shown that the analytical approximations are accurate within at most only 2.5 percent. It is found that the initial fraction of infected persons sensitively influences the late time dependence of the epidemics, the maximum daily rate and its peak time. Such dependencies do not exist in the earlier investigated all-time case.