Lockdowns

While vaccination is the optimal response to an epidemic, recent events have obliged to explore new strategies for containing worldwide epidemics via lockdown strategies, where the contacts among the population are strongly reduced in order to slow down the propagation of the infection. Lockdown strategies are non-pharmaceutical interventions exploiting the fact that the transmission coefficient β can be thought as the product Cλ of a contact rate C (related to social habits and interactions) times a disease-dependent transmission probability λ. A lockdown strategy aims to decrease the contact rates, resulting in a S(E)IR dynamic with a reduced basic reproduction number R0lock<R0. A natural measure of the lockdown strength is the parameter δlock

that is can be interpreted as the decrease of contact rates needed to reach a given lockdown level.

For S(E)IR dynamics, a lockdown corresponds to a straight line of slope -R0lock in the r-lns plane. Following the same reasoning of "Herd immunity" section, such dynamics will intersect the boundary beyond a point rlock∗=1-1/R0lock. Since rlock∗<r∗, a lockdown dynamics can end with a fraction rend of immune individuals that is unstable respect the unconstrained epidemics, i.e. rlock∗≤rend≤r∗. In such a case, releasing the lockdown can result in a new epidemic outburst.

A vaccination policy aims to bring the fraction of recovered individuals above the herd immunity threshold r∗ by inoculating vaccines and avoiding the people experiencing a dangerous disease course. However, r∗ normally corresponds to an high fraction of the population: for new-born epidemics, it can be the case that it is not possible to produce enough vaccine before the epidemic ends. Let’s assume that, by non-pharmaceutical interventions, epidemic has been dampened out and that an amount of vaccines of efficacy ϵ<1 useful to immunize a fraction v of the population has been produced; however, let’s also assume that the vaccines produced are not sufficient to reach herd immunity threshold since ϵv<r∗. It it possible to imagine to introduce a lockdown that reduces the herd immunity threshold to available vaccination capabilities, i.e. rlock∗=ϵv? This goal can be accomplished by noticing that for a lockdown implements a given herd immunity threshold rlock∗ if R0lock=1/(1-rlock∗); thus, indicating with R0vax the lockdown level corresponding to a vaccine immunization of a fraction v of the population, we have that R0vax=1/(1-ϵv). Thus, the strength δvax of a lockdown to reach herd immunity in presence of partial vaccination is