Stochastic extinction of epidemics: how long would it take for Sars-Cov-2 to die out without herd immunity?
Worldwide, we are currently in an unprecedented situation with regard to the SARS-Cov-2 epidemic, where countries are using isolation and lock-down measures to control the spread of infection. This is a scenario generally not much anticipated by previous theory, and in particular, there has been little attention paid to the question of extinction as a means to eradicate the virus; the prevailing view appears to be that this is unfeasible without a vaccine. We use a simple well-mixed stochastic SIR model as a basis for our considerations, and calculate a new result, using branching process theory, for the distribution of times to extinction. Surprisingly, the distribution is an extreme value distribution of the Gumbel type, and we show that the key parameter determining its mean and standard deviation is the expected rate of decline ρe = γ(1-Re) of infections, where γ is the rate of recovery from infection and Re is the usual effective reproductive number. The result also reveals a critical threshold number of infected I† = 1/(1-Re), below which stochastic forces dominate and need be considered for accurate predictions. As this theory ignores migration between populations, we compare against a realistic spatial epidemic simulator and simple stochastic simulations of sub-divided populations with global migration, to find very comparable results to our simple predictions; in particular, we find global migration has the effect of a simple upwards rescaling of Re with the same Gumbel extinction time distribution we derive from our non-spatial model. Within the UK, assuming no migration, using recent estimates of I0≈37000 infected and Re= 0.9, this model predicts a mean extinction time of 616±90 days or approximately ~2 years, but could be as short as 123±15 days, or roughly 4 months for Re = 0.4; in practice, while there is prevalence of infections globally, it is likely that nations can only at best achieve quasi-extinction, a state of temporary extinction, when immigration of cases is rare. This highlights the importance of a global decline in infections, and here the theory predicts extinction in less than 200 days, if the reproductive number is restricted to Re < 0.5. Overall, these results highlight the extreme sensitivity of extinction times when Re approaches 1 and the necessity of reducing the effective reproductive number significantly (Re<≈0.5) for relatively rapid extinction of an epidemic or pandemic.
