Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. Entropy2024, 26, 362.
Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. Entropy 2024, 26, 362.
Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. Entropy2024, 26, 362.
Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. Entropy 2024, 26, 362.
Abstract
We study epidemic spreading in complex networks by a multiple random walker approach. Each walker
performs an independent simple Markovian random walk on a complex undirected (ergodic) random
graph where we focus on Barabási-Albert (BA), Erdös-Rényi (ER) and Watts-Strogatz (WS) types. Both,
walkers and nodes can be either susceptible (S) or infected and infectious (I) representing their states of
health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be
infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes)
is possible. This model mimics a large class of diseases such as Dengue and Malaria with transmission
of the disease via vectors (mosquitos). Infected walkers may die during the time span of their infection
introducing an additional compartment D of dead walkers. Infected nodes never die and always recover
from their infection after a random finite time. This assumption is based on the observation that infectious
vectors (mosquitos) are not ill and do not die from the infection. The infectious time spans of nodes and
walkers, and the survival times of infected walkers, are represented by independent random variables.
We derive stochastic evolution equations for the mean-field compartmental populations with mortality of
walkers and delayed transitions among the compartments. From linear stability analysis, we derive the
basic reproduction numbers R M , R 0 with and without mortality, respectively, and prove that R M < R 0 .
For R M , R 0 > 1 the healthy state is unstable whereas for zero mortality a stable endemic equilibrium
exists (independent of the initial conditions) which we obtained explicitly. We observe that the solutions
of the random walk simulations in the considered networks agree well with the mean-field solutions for
strongly connected graph topologies, whereas less well for weakly connected structures and for diseases
with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, among many others.
Keywords
Epidemic spreading; compartment model with mortality; memory effects; random walks; random graphs
Subject
Physical Sciences, Other
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
