Extinction Times and Phase Transitions for Spatially Structured Closed Epidemics

Abstract

This paper considers the time to extinction for a stochastic epidemic model of SEIR form without replacement of susceptibles. It first shows how previous rigorous results can be heuristically explained in terms of the more transparent dynamics of an approximating deterministic system. The model is then extended to include a host population structured into patches, with weak nearest-neighbour mixing of infection. It is shown, by considering the approximating deterministic system, that the expected time to extinction in a population ofn+ 1 patches each of sizeNis of the formalogN + bn, provided thatN>NcwhereNcis a critical patch size below which transits are unlikely to occur. This corresponds to the simple decomposition of the time of an epidemic into the time it takes to spread through one patch plus the time it takes to transit to each ofnsuccessive patches. Expressions for this threshold and the coefficients of the time to extinction are given in terms of the transmission parameters of infection and the coupling strength between patches. These expressions are compared with numerical results using parameters relevant to a study of phocine distemper virus in North Sea seals, and the agreement is found to be good for large and smallN. In the region whenN≈Nc, where transits may or may not occur, interesting transitional behaviour is seen, leading to a non-monotonicity of the extinction time as a function ofN.