A simple model of epidemic dynamics with memory effects

TL;DR

This paper presents a modified SIR epidemic model incorporating memory effects through random immunity durations, revealing persistent oscillations in infection levels that align with real-world epidemic observations.

Contribution

The study introduces a novel SIR model with stochastic immunity durations, demonstrating how memory effects influence epidemic dynamics and cause sustained oscillations.

Findings

Memory effects induce persistent oscillations in infection numbers.

Finite immunity duration prevents system relaxation to steady state.

Model aligns with real-life epidemic oscillations.

Abstract

We introduce a modified SIR model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a finite time interval. In the first part, we introduce a random life time or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and cannot be captured by standard SIR models.