### Network model

The constructed network is a combination of scale-free and random networks with multiple cliques of sizes 1–4. Hence, in the resulting network most of the nodes have a low degree (< 10) but there also are some hubs (with degree > 100). As illustration we show a 100-node network constructed by this method (Fig. 1). This sample network has average degree of ~ 5 with maximum degree of 21. It has 10 connected components, the largest of which contains 86 nodes.

The centrality measures of a network generated using this method vary as a result of the randomness and stochasticity in the generation process, however the network still has some common features. The degree distribution of the network shows two peaks, one at low degrees (6–7) and the other centered around 25, the average degree of the essential workplace network; it also has a significantly long tail, see Fig. 2. The average degree of a 10,000-node network is ~ 10 and the maximum degree is 184. The network has nearly 50 connected components, the largest of which contains 9950 nodes.

### Simulations of spread

In the absence of mitigation measures, the disease either dies off (in 73% of the simulations) or spreads in the population. In the simulations where the epidemic takes off, the effective reproduction number starts at a high value (larger than 4), as expected from the theory of epidemic spreading (see Fig. 3). In this case, the disease completes its course in ~ 80 days, infecting almost everyone. The peak of infection is around day 30, when ~ 45% of the population is infected at the same time. This disease time course leads to ~ 400 deaths, with very few deaths after day 80. The reproduction number peaks at ~ 7 around day 3 and goes to below 1 around day 20. Nearly 60% of the population becomes immune to the disease but over 30% of the population remains susceptible. Since herd immunity needs that at least 70%-80% of the population be immune (D’Souza and Dowdy 2020), the remaining susceptible population could be affected by a second wave of the disease sparked from outside of the network. In conclusion, letting the disease run its course unmitigated is not expected to benefit society in any way. We note that the number of deaths is only valid if the assumption of a constant death rate (of 5%) holds. It is likely that after day 10, when the hospitals start getting overwhelmed, the patients do not get all the medical attention they need, thus the death rate may increase.

### Lockdown is fixed to 3 months

If a lockdown is implemented for 3 months, which is longer than the natural duration of the unmitigated outbreak, we find that it almost completely avoids the possibility of a second outbreak. We evaluate 90 day lockdown periods with different starting times: day 5 (when ~ 1% of the population is infected), day 10 (when ~ 6% of the population is infected), and day 15 (when ~ 20% of the population is infected). All these cases lead to more than 150 deaths, with the number of deaths being higher if the lockdown starts later. Shown in Fig. 4a, when the lockdown is from day 5 to day 95, the infection peaks at ~ 20% of the population. In nearly 1% of the simulations, there is a second wave of infection which peaks at less than 5% of the population infected around the same time. This wave of infection peaks nearly 50 days after the end of the lockdown. After 240 days, this scenario leads to a total of 150 deaths which is 1.5% of the population. Shown in Fig. 4b, when the lockdown is from day 10 to day 100, the infection peaks to a little over 20% of the population. This scenario leads to nearly 0.7% of the simulations showing a small second wave of infection that affect a very small fraction of the population. After 240 days, this scenario leads to a total of 155 deaths. Figure 4c shows the scenario when lockdown is from day 15 to day 105. Nearly 100% of the simulations in this case lead to only the first wave of infection which peaks to ~ 25% of the population. After 240 days, this leads to ~ 175 deaths.

A high peak of infection implies that the medical facilities will be overwhelmed, which is likely to cause a surge in the number of deaths. While these simulations show that a 90-day lockdown is beneficial for mitigating a widespread effect of the pandemic, they also show that the earlier a lockdown starts, the less the number of deaths and the less pressure on medical facilities. It is also interesting to note that an earlier lockdown leads to a slightly higher possibility of a second wave of infection since a smaller fraction of the population becomes immune. However, despite two waves of infection, the number of deaths is still less if the lockdown starts earlier. In all of these cases, the reproduction number peaks at ~ 6 around day 3–4, denoting the high infection spread in the following few days and it goes to below 1 around day 20. These results show that a 90-day lockdown starting the earliest possible would be the most efficient way to mitigate the effects of the disease through social distancing measures. However, due to the adverse economic effects of a lockdown, it is useful to find equally effective strategies that are shorter than a 90-day period. Hence, we next explore the effect of a 60-day lockdown.

### Lockdown is fixed to 2 months

When the lockdown is fixed to 60 days and the starting time is varied, we find that a second wave of infection becomes more likely than in the 90-day lockdown case. Like the 90-day lockdown case, we explore 60-day lockdown for starting times of day 5 (infection at 1%), day 10 (infection at 6%) and day 15 (infection at 20%). The simulation of the lockdown from day 5 to day 65 shows that there is a second wave of infection in ~ 40% of the cases (Fig. 5a) and only one wave of infection in 24% of the cases (Fig. 5b); the remaining 36% of cases show no spread. The first wave of the infection in response to a day 5 to day 65 lockdown peaks around day 20 and infects nearly 20% of the population at the same time – this can be seen in both Fig. 5a, b. On day 65, when the lockdown is lifted, around ~ 1% of the population is infected. As shown in Fig. 5a, the second wave of infection peaks around day 110 and it is highly varying across simulations. On average, the peak of this wave infects another 10% of the population at the same time. Simulations with two waves of infection lead to a total of ~ 300 deaths which is 3% of the population. Simulations with only one wave of infection lead to a total of ~ 150 deaths. On average, the day 5 to day 65 lockdown scenario leads to ~ 250 deaths after 240 days.

We next explore the case when the lockdown is from day 10 to day 70. This results in ~ 33% of simulations showing a second wave of infection (Fig. 6a), ~ 39% of simulations show only one wave of infection (Fig. 6b) and the remaining 28% of the simulations show no spread. The second wave of infection peaks around day 110 and infects another 10% of the population at the same time. Two waves of infection lead to ~ 310 deaths and only one wave of infection leads to ~ 155 deaths. On average, the day 10 to day 70 lockdown scenario leads to ~ 220 deaths after 240 days. The day 15 to day 75 lockdown scenario leads to two waves of infection in ~ 16% of the simulations (Fig. 7a), only one wave of infection in ~ 58% of the simulations (Fig. 7b) and no spread in the remaining 26% of the simulations. In this case, the second wave of infection peaks around day 120 and infects another ~ 7% of the population at the same time. Two waves of infections lead to ~ 300 deaths and only one wave of infection leads to ~ 180 deaths. On average, the day 15 to day 75 lockdown scenario leads to 200 deaths.

In all these results, there is at least one peak for the effective reproduction number and this first peak is ~ 6 around day 3–4. In the results that show a second wave of infection, R_{t} goes to below 1 during the lockdown but increases again after the end of lockdown; it peaks ~ 20 days before the peak in the number of infectious people. From these simulations, we can see that the probability of a second wave of infection is lower if the lockdown starts later. However, the first wave of the infections gets increasingly worse if the lockdown starts later. It makes sense to start the lockdown earlier to minimize the effects of the first wave of infection, but we need other mechanisms to lower the effect of the second wave of infection. This can be done in various ways. For example, impose a second lockdown in response to the second wave of infection and so on until a consequent wave of infection is too small to affect the population adversely. Alternately, instead of lifting the lockdown after a fixed time-period, the lockdown can be lifted when the number of infected cases goes below a fixed threshold. If the number of cases crosses this threshold again, the lockdown is imposed again. In the next section, we consider another similar scenario where after the end of the first lockdown, the population resumes normal life gradually.

### Phasing out of the lockdown

Here, we consider a slow phasing out of the lockdown. That is, facilities reopen slowly after the end of the lockdown and the population strictly adhere to social distancing, wearing masks and washing hands, etc. We implement this by restoring the network to its original state, that is, including all edges but reducing the transmission probability after the lockdown ends. The transmission probability then gradually increases to the original transmission probability in the window between the end of the lockdown and the day normalcy is resumed. We assume that the lockdown starts on day 5 and ends on day 65 and we consider three rates of increase in the transmission probability following the end of the lockdown.

In the first case, shown in Fig. 8a, b, the network follows a linearly increasing transmission probability from 0.01 on day 65 to 0.05 on day 80. This results in ~ 20% of the simulations having a second wave of infection, shown in Fig. 8a. This second wave peaks around day 125 and infects nearly 10% of the population at the same time. The plot in this case (Fig. 8a) is very similar to the plot in Fig. 5a. However, the benefit of the gradual decrease of mitigation is reflected in much lower likelihood of the existence of a second wave (20%) compared to the likelihood when we resume to normalcy right after the end of the lockdown (40%). This case leads to an average of ~ 300 deaths. Figure 8b shows the simulation results when there is only one wave of infection; this case leads to an average of ~ 150 deaths. The gradual resumption to normalcy on day 80 results in a lower overall death count at ~ 200 compared to the ~ 250 with sudden resumption of the original transmission rate (Fig. 5a, b).

In the case shown in Fig. 9a, the lockdown is from day 5 to day 65 and we resume to complete normalcy on day 95. The network follows a linearly increasing transmission rate from day 65 to day 95; on day 95, the transmission rate reaches the normal 5%. This case results in ~ 6% of simulations with a second wave of infection shown in Fig. 9a. These simulations result in an average of ~ 300 deaths on average. The majority of the simulations have only one wave of infection and result in an average of ~ 150 deaths, shown in Fig. 9b. This scenario results in ~ 160 deaths overall.

We also try the case where lockdown is from day 5 to day 65 and we resume to unrestricted interaction on day 110 (Fig. 10). In this case, the probability of a second wave of infection is ~ 3%—this case is shown in Fig. 10a. These simulations result in an average of ~ 300 deaths. The majority of the simulations, shown in Fig. 10b have only one wave of infection and result in an average of ~ 150 deaths. Overall, this scenario results in ~ 150 deaths. Similar to the case presented in the previous subsection, the effective reproduction number for all of these cases has at least one peak with value ~ 6 occurring around day 3–4 and has another peak for the results that show a second wave of infection. The R_{t} peak is around 20 days before the peak of the second wave of infection. With the much lower probability of a second wave of infection and fewer deaths, we can conclude that a 60-day lockdown with an extended period of gradual decrease of mitigation would be the best approach.