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

Marina Voinson; Alexandra Alvergne; Sylvain Billiard; Charline Smadi

In this section, we focus on the effect of the reservoir transmission rate (τ ) on the number of outbreaks when the infection is subcritical (R0 < 1). The idea is the following: first, as the excursions of a subcritical branching process are 410 small, we can make the approximation that, at the beginning, the infection rate 27 . CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder.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 by the reservoir is constant equal to τ N , and that the direct transmission rate is equal to βN I. Making this approximation allows us to treat the two processes of infection (by contact and by the reservoir) independently. Second, using a result of Singh and Myers (2014), we get the mean number of independent 415 infections by the reservoir during an outbreak under our approximation [19] . As the infections by the reservoir during an outbreak describe a Poisson process, we know that their times are uniformly distributed during the outbreak. As a consequence, we make the approximation that the successive local maxima of the number of infected individuals during the outbreak corresponds to the 420 successive maxima of the outbreaks generated by the independent infections by the reservoir. However, we have to take into account the fact that there is at least one infected individual remaining in the population when a new individual is infected by the reservoir (otherwise the new infected individual would generate a new outbreak). Hence we will make the approximation than the event 'the 425 excursion does not hit the treshold c' and the event 'no excursion generated by an infection by the reservoir and with initially two infected individuals, hits the treshold c' are equivalent. Notice that by doing that we underestimate the probability of an excursion to reach any threshold we may explain why we overestimate the spillover transmission τ maximizing the number of outbreaks. 28 . CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder.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 Hence we make the approximation that P(excursion is not an outbreak) ∼ (1 − p c ) m .