Fig. 2: Steady-state and transient-state equivalence between Markovian and non-Markovian dynamics.

a–c The solid brown, blue, and green curves represent the theoretical results of the susceptible, infected, and removed fractions, while the solid orange, red, and purple curves show the corresponding results of 100 independent Monte Carlo simulations (\({\alpha }_{\inf }=3\), \({\beta }_{\inf }=4\), α_rem = 2.5, β_rem = 3.6, where α and β respectively represent the shape and scale parameters of time distributions of Weibull types, and subscripts “\(\inf\)” and “rem” indicate the time distributions of infection and removal, respectively; R₀ = 1.86). d The orange and green curves, respectively, depict the removed fractions from the memory-dependent and memoryless Monte Carlo simulations of 100 independent realizations with steady-state equivalence (\({\alpha }_{\inf }=1.5\), \({\beta }_{\inf }=2.63\), α_rem = 2, β_rem = 2.4 for memory-dependent simulations; γ = 0.11 and μ = 0.14 for memoryless simulations; identical basic reproduction number as R₀ = 1.86 for the two types simulations). e Red + and blue × markers, respectively, represent the steady-state removed fractions of memory-dependent and memoryless Monte Carlo simulations for different values of R₀, where each marker is the result of averaging 100 independent simulations. The orange curve is the numerical calculations from Eq. (7), and the vertical dashed line denotes the critical point R₀ = 1. f–g For T_gen = T_rem in the non-Markovian theory (\({\alpha }_{\inf }=1.57\), \({\beta }_{\inf }=5.57\), α_rem = 1.57, β_rem = 7.79), the blue and green curves in (f) denote the susceptible and removed fractions, while the black × markers represent the inferred susceptible fractions calculated by substituting removed fractions in Eq. (11), which agrees with the susceptible curve calculated from Eq. (1). The red and green curves in (g) denote the non-Markovian infected and removed fractions, while the orange and purple dashed curves are the corresponding curves of the Markovian transmission (γ = 0.20, μ = 0.12) obtained from Eqs. (13, 14), which agree with the non-Markovian results. They all have the same basic reproduction number, i.e., R₀ = 1.87. (The Euler-Lotka equation assumes exponential growth of a disease outbreak during the initial stage. As a result, the Markovian curves in (g) slightly deviate from the non-Markovian ones as the cumulative infections increase.) h–i For T_gen ≠ T_rem in the non-Markovian theory (\({\alpha }_{\inf }=0.74\), \({\beta }_{\inf }=4.16\), α_rem = 3.32, β_rem = 7.80), the inferred susceptible curve in (h) does not match the numerical result, and the infected and removed curves of the Markovian transmission in (i) (γ = 0.41, μ = 0.28) obtained from Eqs. (13, 14) do not match the corresponding non-Markovian results. They all have the same basic reproduction number, i.e., R₀ = 1.87. j Five scenarios for the non-Markovian time-distribution setting (within each scenario, \({T}_{\inf }\) and T_rem are fixed): Weibull, \({T}_{\inf }=5\), T_rem = 7 (blue+); Weibull, \({T}_{\inf }=5\), T_rem = 5 (red × ); Weibull, \({T}_{\inf }=7\), T_rem = 7 (purple □); log-normal, \({T}_{\inf }=5\), T_rem = 7 (green Δ); gamma, \({T}_{\inf }=5\), T_rem = 7 (orange ♢). The value of T_gen is modified to adjust \(\ln \eta\) for better visualization.