A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale

Abstract

We define and study an open stochastic SIR (Susceptible -Infected -Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be densitydependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finistère) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin-Wentzell theory of large deviations from a dynamical system. Keywords multitype SIR model · epidemic and demography over the same time scale · continuous-time multitype branching processes · Markovian process · major outbreak probability · basic reproduction number · real network · epidemic extinction time · epidemic total size · endemicity

Propagation of cyber incidents in an insurance portfolio: counting processes combined with compartmental epidemiological models

“…Zhang and Wang (2013) study the asymptotic behavior of a stochastic SIR model with jumps perturbation. Montagnon (2019) points out the role of the population renewal in the persistence of an endemic disease, and thus proposes a model coupling epidemiological multi-type stochastic processes with demographic models.…”

Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19

“…The SIR model compartmentalizes or divides the population into Susceptible (S), Infected (I), and Removed (R). This compartmentalization allows for analyzing the population with these states and is useful to determine projections in relation to the total number of patients and the duration of the disease [ 11 ]. The SIR model approach is eminently deterministic; however, it has also been used from a stochastic perspective, improving the representation of the dynamics of infectious diseases through the probability of the appearance of epidemic outbreaks [ 11 ].…”

“…This compartmentalization allows for analyzing the population with these states and is useful to determine projections in relation to the total number of patients and the duration of the disease [ 11 ]. The SIR model approach is eminently deterministic; however, it has also been used from a stochastic perspective, improving the representation of the dynamics of infectious diseases through the probability of the appearance of epidemic outbreaks [ 11 ]. There are other mathematical models in epidemiology that have been developed from the SIR model, adding variables such as exposure and the effect of quarantine measures such as the SEIR and SEQIJR model, respectively [ 12 ].…”

Competing control scenarios in probabilistic SIR epidemics on social-contact networks

“…In this model individuals are allocated either to susceptible (S), infected (I), or removed (R) meta-population. The extensions of simple epidemic models for populations towards structured networks of individual agents (or meta-populations) [5,6], have successfully been employed in many fields to study phenomena for which interrelationships matter [7] and, include, biological demographic dynamics [8], international trade [9], technology diffusion [10], information spreading [11,12], and contagion in financial markets [13]. In all these cases the adoption of network models has led to new perspectives and novel insights.…”