An optimal control theory approach to non-pharmaceutical interventions

Lin, Feng; Muthuraman, Kumar; Lawley, Mark

. There is no well-defined performance measure to evaluate the NPI policy, especially when the policy is defined in a 2-dimensional state space. We Figure 3 Epidemic curves of infectious and dead population with and without NPI implementation. Figure 3 shows the impact of optimal control on pandemic severity, peak, and total deaths, when NPIs are triggered at different initial states. (a) compares the epidemic curves with and without NPIs, starting from a state 99% susceptible and 1% infected when b = 0.4. (b) compares the epidemic curves with and without NPIs, starting from a state 99% susceptible and 1% infected when b = 0.6. (c) compares the epidemic curves with and without NPIs, starting from a state 67% susceptible and 33% infected when b = 0.4. (d) compares the epidemic curves with and without NPIs, starting from a state 50% susceptible and 50% infected when b = 0.6. chose ω because it captures the overall intensiveness of NPI implementations. In addition, we investigated the effect of these parameters on the outcome of applying the control policy, defined as the mean cumulative death, d T . We simulated the SIRD system under the optimal policy starting from all state (s 0 , i 0 , r 0 , d 0 ), where s 0 > 80%, i 0 <20%, and d 0 = 0. The simulation was terminated at a randomly selected exponential terminal time, and we recorded and analyzed the cumulative number of deaths. The mean cumulative death was calculated by taking the average of cumulative deaths over all tested initial states. Table 3 summarizes the estimated probability distribution functions (PDFs) of five input parameters, assuming the input parameters are statistically independent. The PDFs of influenza transmission characteristics (R 0 , 1/g, and 1/τ) are estimated based on the 1918 pandemic [25] . Note that the infection rate b can be written as R 0 (g + τ). The effect of NPIs, b, was found to reduce the infection rate, b, by up to 30-50% in 1918 and in many cases the effect of NPIs was very limited [11] ; thus we assume the impact of NPI implementation, b, follows Uniform(0,50%). We are not able to find any empirical data on the cost of NPI, c. This value is a relative cost, which depends on decision makers' perceptions of saving lives versus maintenance of daily societal functions. In our analysis we let c take the value from 0 to 0.25 uniformly, which would imply that the cost of four sessions of maximal NPI implementation is equivalent to the cost of one death.