Non-Markovian SIR epidemic spreading model

Abstract

We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete-and continuous-time versions. The incubation period, delayed infectiousness and the distribution of the recovery period are modeled with general functions. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, in the spring, 2020.

Endemic state equivalence between non-Markovian SEIS and Markov SIS model in complex networks

“…Consequently, Γ(0) = 0, Γ(0) = 1, Γ(τ ) = 1 − (1 − γ) τ , Γ(τ ) = (1 − γ) τ . Acting similar as in [16], we obtain: The last equation is the equation of the Markov SIS model (18). To summarize, for arbitrary values of parameters β and γ, the discrete-time SIS model, may be represented as non-Markovian SEIS model, providing B(τ ) and γ(τ ) satisfy the relations defined in the introduction of this Subsection.…”

“…In that sense the dynamical stability of the system (6) reduces to the dynamical stability of this set of equations only: from (6) follows that, if lim t→∞ p ′I i (t) → 0, then lim t→∞ p ′E i (t) → 0. By conducting a Laplace transform on both sides of each of the equations from the second set of N equations in (6), following the methodology in [16], one obtains:…”

“…By considering B(τ ) = 1, consequently p E i (t) = p I i (t), and Γ(τ ) = exp(−γτ ) and T → ∞ (see [16] for details) one obtains: The last relation is identical with the classical formulation of the Markov SIS model occurring on complex network, in continuous time (25). The relation confirms that, for a given Markov SIS, defined by parameters β and γ, exists a non-Markovian SEIS model defined with the identical parameter β and CTPFs B(τ ) = 1 and Γ(τ ) = exp(−γτ ), such that the non-Markovian SEIS mimics the behaviour of the Markov SIS model on complex networks.…”

“…This equivalence is only vaguely mentioned and numerically illustrated in the Conclusions of [15]; here we show this feature trough a rigorous mathematical procedure. One should note that similar analysis was conducted in respect to the non-Markovian and Markov SIR model in [16].…”

Analysis of COVID-19 infection waves in Tokyo by Avrami equation

“…We have been attempting to analyze the wave of COVID-19 in Japan [1,2] by the Avrami (or JMAK) equation [3] that describes the phase transformation dynamics and is another approach than the conventional SIR model and its expanded models [4,5,6,7,8]. As its entity model, the physical process of the linear growth of nuclei forming randomly in the parent phase was assumed.…”