A Markovian random walk model of epidemic spreading

Continuum Mech. Thermodyn. https://doi.org/10.1007/s00161-021-00970-z O R I G I NA L A RT I C L E Michael Bestehorn · Alejandro P. Riascos · Thomas M. Michelitsch · Bernard A. Collet A Markovian random walk model of epidemic spreading Received: 16 October 2020 / Accepted: 4 January 2021 © The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2021 Abstract We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time Markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then, we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node, there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations, we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics. Keywords Markovian random walks · Ergodic networks · Epidemic spreading 1 Introduction Within the last two decades, network science has become a huge interdisciplinary field [1–3] recently driven by the significant upswing of online (social) networks and search engines with a burst of works focusing on human mobility and encounter networks [4]. It turned out that random walks in networks are especially powerful to cover spreading and diffusion phenomena widely observed in nature. These diffusion phenomena include so-called anomalous diffusion which have been successfully described by space-time fractional partial differential diffusion equations [5]. On the other hand within the last two decades, an impressive amount of scientific work has been devoted to epidemic spreading models. For an introduction in epidemic modeling and state-of-the-art models such as the ‘SIR model’ (S = susceptible, I = infected, R = recovered), we refer to [6]. It is natural that the present Communicated by Marcus Aßmus, Victor A. Eremeyev, and Andreas Öchsner. M. Bestehorn Institut für Physik, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Germany E-mail: A.P. Riascos Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de Mexico, Mexico E-mail: T. M. Michelitsch · B. A. Collet (B) Institut Jean le Rond d’Alembert, Sorbonne Université, CNRS UMR 7190, 4 place Jussieu, 75252 Paris, cedex 05, France E-mail: · E-mail: M. Bestehorn et al. worldwide pandemic context of COVID-19 is boosting an additional interest to this topic [7,8]. Epidemic spreading in complex networks was studied by several authors [9–11] and the epidemic dynamics in scale-free networks was analyzed in [12]. In a recent paper, the effects of quarantine measures to epidemic spreading in activity-driven adaptive temporal networks were studied [13] including percolation effects in epidemic spreading in small-world networks [14,15]. A renormalization group approach has been employed to model the second COVID-19 wave in Europe [16], just to quote a few examples. Strongly driven by the present world-wide COVID-19 spreading there is a huge and urgent need of reliable models that are able to capture essential aspects of the space-time dynamics of infectious diseases allowing to develop preventive strategies. For an overview of the present world-wide COVID-19 situation as far known, we refer to [17]. Infectious diseases such as measles, mumps, and rubella can be studied in the framework of nonlinear dynamical systems. For the most simple case of spatially homogeneous infection rates, SIR models have been applied successfully in the past [18,19]. As mentioned, SIR stands for the three compartments susceptible– infected–recovered into which the individuals are grouped, depending on their state. A susceptible individual (S) can be infected and become ill (I). After a certain time τ1 , it will recover and be removed from the system (R) in the subsequent computer simulation model. During time τ1 , it can infect other susceptible individuals. The mathematical description in the SIR model is achieved by an ordinary first-order differential equation for each rate. If spatial effects are taken into account, the rates can be assumed space-dependent and a set of three nonlinear coupled diffusion equations can be derived [6]. Instead of using partial differential equations, the individuals or particles can be considered as independent random walkers on a discrete network with a given architecture. Our model is based on the following assumptions. The particles perform random jumps from one node to another connected node of the network. If on the same node an infected particle meets a susceptible one, the susceptible walker may be infected with a given probability P. To describe the process of recovery, each particle has an inner variable parametrizing its state. This variable changes in course of time. If ’time’ is assumed to be discrete, the whole dynamics on the network and of the inner variable can be formulated as a (nonlinear) mapping from one-time step to the next. The system has no memory, its state is uniquely defined by the positions of the particles and the values of their inner variables at a certain time step (Markov process). Our paper is organized as follows. In the subsequent Sect. 2, we give a brief general introduction into the dynamics of Z independent Markovian random walkers on finite connected (ergodic) graphs. Without loss of generality, we confine us here to undirected graphs. We utilize the Markovian walk approach to derive an upper bound for the so-called basic reproduction number R0 which is defined subsequently. In this part, we consider the situation when there is a single infected walker and Z − 1 susceptible walkers in the network. We derive explicit formulae for the expected number of times the infected walker meets a susceptible one which defines an upper bound for R0 . In Sect. 3, we perform numerical simulations employing above mentioned assumptions to generate spacetime patterns of the susceptible/infected walkers where we consider Z independent walkers on a finite 2D square lattices with variable adjacency matrices and connectivity. In this way, we explore how the architecture of a network affects the space-time dynamics of the epidemic spreading and identify pertinent parameters governing the space-time patterns in order to establish predictive measures such as confinement and social distance rules. 2 Multiple random walkers model 2.1 Some basic features In the present section, we rec (...truncated)