Stochastic compartment model with mortality and its application to epidemic spreading in complex networks

• Main Authors: Granger, Teo, Michelitsch, Thomas M, Bestehorn, Michael, Riascos, Alejandro P, Collet, Bernard A
• Format: Journal Article
• Language: English
• Published: 18-03-2024
• Subjects: Mathematics - Dynamical Systems; Physics - Physics and Society; Quantitative Biology - Populations and Evolution
• DOI: 10.48550/arxiv.2403.11774

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 Barab\'asi-Albert
(BA), Erd\"os-R\'enyi (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. 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.