Stochastic dynamics of an epidemics with recurrent spillovers from an endemic reservoir

Marina Voinson; Alexandra Alvergne; Sylvain Billiard; Charline Smadi

A Stochastic Susceptible-Infected-Recovered (SIR) compartmental transmis-85 sion model [20] with recurrent introduction of the infection into an incidental host by a reservoir is considered ( Figure 2 ). Our goal here is not to study a S I R A Figure 2 : Representation of the stochastic model with transition rates. A reservoir (A) has been added to a classical SIR model where the pathogen is persistent. Individuals are characterized by their epidemiological status in the incidental host (S: suceptible; I: infected; R: recovered). A susceptible individual becomes infected through the transmission by contact at rate βSI or through the reservoir at rate τ S. An infected individual recovers at rate γ. Stochastic simulations are performed with the following values: the transmission by contact, expressed as the basic reproductive ratio, 0 < R 0 < 10, the spillover transmission, 10 −1 < τ < 10 −10 and the recovery rate γ = 0.1. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/213579 doi: bioRxiv preprint of the basic reproductive ratio of the pathogen, R 0 , which is widely used in epidemiology. R 0 corresponds to the average number of secondary infections 105 produced by an infected individual in an otherwise susceptible population. The interest of this ratio is mostly the notion of threshold: in a deterministic model, for a pathogen to invade the population, R 0 must be larger than 1 in the absence of reservoir. In a stochastic model, the higher the R 0 the higher the probability for the pathogen to invade the population. In a SIR model, the basic reproduc-110 tive ratio R 0 equals to βN γ . Individuals in the recovered compartment do not contribute anymore to the transmission process. Since we assume that demographic processes are slower than epidemic processes, the number of susceptible individuals decreases during the epidemic due to the consumption of susceptibles by the infection until the extinction of the population.