Optimal timing for social distancing during an epidemic

Oscar Patterson-Lomba

The idea of an optimal social distancing timing is investigated in the context of a classic SIR type model [3] . Individuals are classified based on their infectious status as susceptible (S), infected (I) and recovered (R). The force of infection at time t is given by β(t)I(t), where β(t) is the transmission probability. This term is the product of the number of contacts per capita per time unit, c(t), and the probability of transmission given a contact, p; that is β(t) = c(t)p. The social distancing measure affects the c(t) parameter, such that when social distancing is in effect c(t) is reduced. The corresponding reduction in β(t) due to a temporary social distancing measure can be modeled as a "rectangular well" function, as shown in Figure (1) . From the start of the epidemic until time t 0 (the time at which social distancing is initiated), the transmission rate is β. From time t 0 to t 0 + T , a social distancing measured is implemented such that the transmission rate is reduced to β r = rβ, with 0 ≤ r ≤ 1, and T being the length of the period when social distancing measures are in place. Then, after time t 0 + T (when social distancing ends), the transmission rate comes back up to β, until time t I , at which point a fully effective intervention (e.g., vaccine) is applied such that transmission is completed halted; that is, the effective reproduction number is zero for t > t I (for simplicity we model such scenario by assuming that β = 0).