Using a real-world network to model the trade-off between stay-at-home restriction, vaccination, social distancing and working hours on COVID-19 dynamics

Real-world social network data description and classification

We use the BBC documentary, Contagion (Soderberg, 2011), to make our simulation more realistic of human interaction patterns. The BBC (2017) demonstrates the social network data of 469 volunteers from Haslemere, England (Klepac, Kissler & Gog, 2018). The data is not categorized as where the interactions occur. Our main goal is to order each interaction, whether it occurs in workplaces, social environments, or households. By classifying the edges of the graph, we can understand which of the settings and behaviors of staying in that environment would affect the COVID-19 dynamics.

The Haslemere dataset consists of the pairwise distances of up to 50m resolution between 469 volunteers from Haslemere, England, at five-minute intervals over three consecutive days (Thursday 12 Oct–Saturday 14 Oct 2017). It gives users’ data for sixteen daytime hours only, between 07:00:00am and 22:55:00pm, excluding the hours between 11 pm and 7 am. There are 576-time points for each user. According to the 2011 UK census, volunteers of the Haslemere dataset established a sample of 4.2% of the total population of Haslemere. Participants downloaded the BBC Pandemic mobile phone app and then went about their daily business with the app running in the background. The study was restricted to volunteers of at least 16 or 13 ages having their parental consent. The pairwise distances between volunteers were calculated using the Haversine formula for great-circle geographic distance and are based on the most accurate GPS coordinates that the volunteers’ mobile phones could provide (Kissler et al., 2019; Kissler et al., 2018).

We want to categorize this network into households, workplaces, and social environments. In other words, we classify each edge and contact into three categories. We choose contacts with an average pairwise distance of 20m or less, making 3245 unique contacts. Our primary analysis identifies contacts whose average pairwise distances are 5m or less because we aimed to capture household encounters and understand their daily routine and behavior. To investigate the behavior of everyone during these three days, we developed a visualization method (see Fig. S2). After investigation, we developed procedures (see Table S1) to categorize the network edges and contacts in the household, workplace, and social environment. There are 1,350 unique contacts, which occurred between 1m and 5m. We use the visualization method to classify 123, 514, and 713 of them as household, workplace, and social environment contacts, respectively. 

We established an automated classification algorithm to classify the remaining 1,895 unique encounters, which occurred between 6m and 20m distances. We developed the algorithm using procedures from the visualization method, see Fig. S1 for algorithm pseudocode. From 1,895 unique contacts, 52, 790, 1,003 of them occurred in the household, workplaces, and social environments. We create a confusion matrix to evaluate the algorithm’s performance by choosing the classified data points (using the visualization method) as actual data and the algorithm classified data points as predicted data. Overall, 91.73% of the predictions of our algorithm are correct, and 8.27% are incorrect, according to the confusion matrix of the algorithm concerning data points classified by the visualization method (see Fig. S2). The distribution of household, workplace and social environment interactions after classification is illustrated in Fig. 1B.

Agent based model

We have developed a discrete-time stochastic agent based model, parameterized to simulate distinct types of COVID-19 outbreaks across the Haslemere data set. An agent in our simulation can be in the following states: E (t) (exposed), PS (t) (pre-symptomatic, documented), A (t) (asymptomatic, undocumented), S (t) (symptomatic, documented) H (t) (hospitalized) and R (t) (recovered) (Fig. 1C). In our model, symptomatic individuals are those who show symptoms; those who are not considered undocumented (asymptomatic). The agent based model starts with an exposed individual. Initially, the j th individual is exposed to the virus, and he/she cannot infect others during his/her latent period for d ₁ =2.7 days (see Table S2). After the latency period finishes, the j th individual ends up in one of the two branches: pre-symptomatic with a ratio of s or asymptomatic with (1 −s). When the period of delay from the onset finishes, we generate a random number ɛ, and there are two probabilities for the j th individual to proceed; the first one, if ɛ<1-s, he/she proceeds to the asymptomatic stage. The infectious period of the asymptomatic stage is d ₃ = 5.4 days (see in Table S2). Through the infectious period of the asymptomatic stage, when the j th individual comes in contact with the i th individual, then we infect the i th individual with probability µ_H Ph_j,i if the edge is classified as household and µ_o Poj,i if non-household. The µ_H and µ_o are the reduction factor for asymptomatic transmission in households and non-household. Ph_j,i and Po_j,i are the probability that j th individual infects i th individual of his/her contacts in household and non-household. When the infectious period of the asymptomatic stage finishes, the j th individual proceeds to the recovered stage.

Alternatively, the j th individual can be in the pre-symptomatic stage. The infectious period of the pre-symptomatic stage is d₂ = 2.4 (Ding et al., 2021). The infection probabilities are Ph_j,i and Po_j,i in household and non-household. When the pre-symptomatic stage finishes, the j th individual proceeds to the symptomatic stage and stays for d ₄ = 3 days (Faes et al., 2020). The infection probabilities are the same as in the pre-symptomatic stage. When the delay period from symptomatic finishes, j th individual proceeds to the hospitalization stage where he/she will recover or die. The hospital stage does not necessarily mean that the agent is hospitalized. All reported cases end up in the hospital stage, regardless of severity (which is not modelled). The default parameter values for the simulations with their sources were presented in Table S2.

Constructing contact matrices and simulation with real-world social network data

We construct classification matrices for households, H_i,j, workplaces, W_i,j, and social environment, S_i,j, which we classified from Haslemere data. H_i,j =1 if the contact of j th and i th agent occurs in a household environment, and H_i,j = 0 otherwise. The simulations are made for 5 min time slot. Each contact is checked at each time step if one of the agents is infectious; also, if the other is susceptible, then it is infected by the precomputed probability. There are three days in the Haslemere data: Thursday, Friday, and Saturday. Our simulations take 14 days, so we construct the more prolonged contact network using real data. The weekday contacts are taken from Thursday to Friday. We repeat each day sequentially and in whole. We do not mix the data that is in a single day. The Sunday contacts are replicated from Saturday contacts of the Haslemere data. We start the simulation with two exposed individuals. The first 14 days are designed as a warmup run. After 14 days, we randomly chose two individuals whose states are exposed, presymptomatic, asymptomatic, or symptomatic. Furthermore, the repetitive use of the Haslemere data can be problematic. However, we simulate for a very short period and for small outbreaks. The goal is for our simulations to be independent of one another.

We start with a more realistic initial sample of states by this method. Then, we restart the simulation for another fourteen days. We calculate the effective reproduction number (R_e), infection occurrence ratio, and secondary attack rate (SAR) of households from the second set of 14-day simulations. We run five hundred trial simulations for each single scenario of our research (see Table S3). The default parameter values for the simulations were present in Table S2. By starting with only two infected individuals, we aimed that the total cases at the end of 14 days do not exceed 10% of the total population. If the susceptible population size is not high enough, it will cause the reproduction number to be overestimated (Delamater et al., 2019). Therefore, we chose a small portion of the population to be infected. In addition, the number of infections should not be too low for reproduction number predictability to be unaffected by stochasticity. For independent events the standard deviation is the root square of the mean. Therefore, although the maximum number that we decided to infect is a small percentage of the population, its square root should be smaller than its mean. Hence, we set the maximum infection occurrence percentage at 10% of the population, corresponding to 40 people. In line with our decision, even after two years, COVID-19 cases in countries did not reach a substantial percentage (Mathieu et al., 2020; World Health Organization, 2022).

Calculating infection probability (P) and basic reproduction number (R_o)

We specify the probability that a susceptible agent i th becomes infected by a j th infectious agent in a 5-minute interval is given as a function of distance in Eq. (1) for household and in Eq. (2) for non-household (workplace, social environment).

(1)${\displaystyle P{h}_{j,i}\left(t\right)=\left\{\begin{array}{c}{\beta }_h{e}^{-\left(\alpha di{s}_{j,i}\right)}di{s}_{j,i}\le \theta \hfill \\ 0di{s}_{j,i}>\theta \hfill \end{array}\right.}$ (2)${\displaystyle P{o}_{j,i}\left(t\right)=\left\{\begin{array}{c}{\beta }_o{e}^{-\left(\alpha di{s}_{j,i}\right)}di{s}_{j,i}\le \theta \hfill \\ 0di{s}_{j,i}>\theta \hfill \end{array}\right..}$

Ph_j,i(t) is the probability of infection if the contact occurs in a household. Po_j,i(t) is the probability of infection if the contact occurs in the workplace or social environment. Also, dis_j,i is the distance in meters between individuals i and j at time t; α , β_h, β_o are distance scaling parameters, transmission probability per 5 min in and out of households, respectively. θ defines the cutoff distance, after which the infection probability is assumed to be zero. Our social network consists of a pairwise distance of up to 50m. Since we choose only those interactions which occurred less than or equal to 20 m, so θ=20. Infection probability ranges between 0 and 1. Our network consists of 102,831 interactions in 5 min time intervals for three consecutive days (Tuesday, Friday, and Saturday). We calculate the infection probability for each interaction separately. Effective reproduction number (R_e) is estimated directly by counting the descendants of a discovered agent after the simulation is finished, then averaged for 14 days. Percentage reduction of transmission probability is computed as a ratio of the effective-outside transmission probability to WT (default, estimated) transmission rate. Thus (3)${\displaystyle D=\frac{{\beta }_o}{{\beta }_{WT}}100.}$

Estimating infection occurrence ratio

Throughout the simulations, we track down antecedents and descendants of infected agents. Each infection pair is an edge of the classification matrices, H_i,j, W_i,j, S_i,j. Then each classified edge is calculated.

Calculating household secondary attack rate (SAR)

The household secondary attack rate is defined as the probability of transmission per susceptible household member when a single infected individual is in the house (Nande et al., 2021; Moreland et al., 2020). The SAR of households calculates the number of cases among contacts of primary cases divided by the total number of primary cases. Our model starts the simulations with ten exposed individuals executed with a daily life contact matrix, but we calculate the SAR of households from the second set of 14-day simulations which start with 2 infected individuals. After that, we tract the infection occurrence environments (household, workplace, social environment). We identify those who are members of different households. We accept those initially exposed individuals who are members of different households as our primary case in each household. At the end of the simulation, we calculate the SAR of each household by the following formula: (4)${\displaystyle \frac{\text{Number of cases among contacts of primary cases}}{\text{Total number of contacts of primary cases}}\times 100.}$

Calculating scaling parameters of distance (α) and transmission rate (β) of COVID-19

We estimate scaling parameters of distance and transmission rate by sampling to obtain a baseline Ro = 2.87 (Billah, Miah & Khan, 2020) and 0.46 ≤ SAR ≤ 0.72 (Kuwelker et al., 2021), which are the basic reproduction number and secondary attack rate of households of COVID-19, respectively. Figure S4 in supplementary information shows the sampling results for calculating α and β constants for COVID-19. To do this, we calculate the absolute error between the estimated model R_o and the exact COVID-19. We also compute the absolute error between the estimated and the exact household SAR model for COVID-19. Finally, we add these two errors and choose α and β values that give the smallest error. The estimated values are present in Table S2.

Changing number of work hours during weekdays

We alter work hours during weekdays by changing the work edges of the network. According to our classification methodology of the network, individuals work for 9 h from 9:00 AM up to 6:00 PM during weekdays. To decrease work hours from 9 h to 9 − i (1 ≤ I ≤ 9) hours during weekdays, we identify a home, work, and social environment edges between 06:00 PM and 9 − i hour before 6:00 PM of each weekday. Secondly, we randomly change only work edges with work, home, and social environment edges from edges between 6:00 PM and 11:00 PM of each weekday. For example, if we want to decrease work hours from 9 h to 8 h (i = 1) during weekdays. Firstly, we identify every workplace, social environment, and household contact that occurred between 06:00 PM hour and 05:00 AM hour. Then we changed only workplace contact with the workplace, social environment and household contacts between 6:00 PM and 11:00 PM.

Simulating stay-at-home restrictions

We investigate the role of stay-at-home restriction on the epidemic dynamics. Systematically, restricting all days and their combinations is possible. However, we have simulated the most probably scenarios. For example, most countries including Turkey (Daily Sabah, 2021; United States of America Department of State, 2021), India (Nayak, 2021), and France (The Connexion, 2022; France 24, 2021) have implemented a stay-at-home restriction only on Saturday and Sunday. We do all that and additionally implement stay-at-home restrictions on Thursday, Friday. Therefore we simulate four stay-at-home restrictions scenarios: restrictions on Sunday, restrictions on Sunday and Saturday, restrictions on Sunday, Saturday and Friday and restrictions on Sunday, Saturday, Thursday, and Friday. To implement stay-at-home restrictions, we replace all non-household (workplace and social environment) contacts that occurred in a 5-minute time step with a household connection that also occurred in that same day.

Lowering transmission probability, increasing social distancing

The infection probability is parameterized by two parameters, β and α. The α indicates the probability of decay with the distance of contacts, whereas β is the maximum transmission probability (at a distance of 0). We have obtained these parameters by sampling many simulations that fit the COVID-19’s Ro and secondary attack rate. We use two β parameters, β_h, and β_o, to distinguish between the infectiousness in households and outside. In our simulations, we reduce the infectious probability to simulate a population where people reduce the probability of infection by personal social distancing measures. The parameter β could be the total virus present near an infectious agent. In contrast, α is the “loss parameter” (due to diffusion and other effects) for distance. Thus, we have chosen to alter the parameter β on the total number of virus changes when a person engages in protective measures such as wearing masks. The parameter α is left untouched since the events that lead to the distance loss do not change. Alternative explanations can be made; however, this is how our simulation has been conducted. Whenever we mention a reduction in infectiousness, we reduce the parameter β.

People do not conduct social distancing in their homes (Nande et al., 2021). To simulate this phenomenon, we reduce the total transmission rate by decreasing the non-household transmission (β_o) while fixing the household transmission rate (β_h). However, in some simulations, we alter both β_h, and βo when agents at home are also engaging in personal protection.

Vaccination and the delta variant

The simulations with delta variants use different values for β and α parameters. Again, we have left α untouched. We have changed β according to the literature  (CDC , 2021). It is assumed that the infectious probability of the delta variant is double of the wild type of virus. We have used β for the delta variant as 0.33, whereas the wild type was 0.167.

The agents that are vaccinated are chosen randomly. We investigate six vaccination scenarios: no vaccination, 50% of the population vaccinated, 60% of the population vaccinated, 70% of the population vaccinated, 80% of the population vaccinated, and 90% of the population vaccinated. The vaccinated agents have reduced β value. We have assumed a 93% and 88% reduction in infectiousness for the wild type and delta variants, respectively (Lopez Bernal et al., 2021). So, β for the vaccinated, wild type and delta variant is 0.012 and 0.067, respectively, whilst α is unchanged.

Parameters

General: d ₁ =2.7 days, d ₂ =2.4 days, d ₃ =5.4 days, s = 0.83, p = 3 days, α =0.9841, µ_H =0.696 µ_o =0.42, g = 192.

Figures 2A–2B: For the first scenario where people implement social distancing measure only in non-household environment, β_h = 0.1672, 0.001 ≤β_o ≤ 0.1672. For the second scenario where population implement social distancing measure in both household and non-household, 0.001 ≤ β_h ≤ 0.1672, 0.001 ≤ β_o ≤ 0.1672.

Figure 3: β_h = β_o = 0.1672 for R_o = 2.87, β_h = β_o = 0.0214 for R ₀ = 1, β_h = 0.1672, β_o = 0.0165 for R ₀ = 1, 88% reduction applied to β_o. Same parameter for random networks.

Figures 5A–5B: β_h = 0.1672, 0.001 ≤ β_o ≤ 0.1672.

Figures 7A–7B: β_h =0.1672, 0.001 ≤β_o ≤ 0.1672.

Estimating infection occurrence ratio in the household, workplace and social environment

It is crucial to know where infections occur in relation to the effective reproduction number (R_e). Here, we count the infection occurring in three environments: households, work, and social.

Our agent based model simulates R_e by independently varying household and non-household transmission rates β_h and β_o, respectively. Squares with a line in Fig. 2A show R_e’s dependence on varying β_o when β_h is constant. Additionally, triangles with a line show the dependence of R_e on varying β_o when β_h is also changing, β_o = β_h. The latter simulations are made to assess the effect of making the distinction between β_o and β_h. Figure 2A show that, interestingly, until R_e<1, Re does not depend on the decrease of β_h. Only after R_e<1 further decreasing in β_h decreases R_e more than when β_h is kept at its maximum.

The household, workplace, and social environment infection occurrence ratio has been counted for R_e (for real networks). Figures 2B and 2C demonstrate that household infections are dominant when R_e<1, and most infections occur in the social environment when R_e<2.5.

We capture the only difference between Fig. 2B (β_h kept constant) and Fig. 2C (β_h decreases) was in the slight increase of household infection occurrence ratio when β_h was kept constant, R_e<1. At R_e<1, the infections’ environments do not change significantly. These results are further supporting the results in Fig. 2A. The decrease in household transmission rate only affects the dynamics at low R_e.

Additional simulations have been made with random networks. Two separate results are plotted: (1) the range of β_o and β_h are the same with the real-network, line with circles in Fig. 2A; (2) the maximum R_e is taken as the 2.87 (same as the maximum of the real-network), line with diamonds in Fig. 2A.

For case 1, the random networks show much higher R_e. Except, when the transmission rate is reduced by more than 88%, the random network shows lower R_e than the simulations with the real network (β_h fixed), Figure 2A. Thus, the real network used is limited by the number of unique connections. The interactions are more clustered and local. On the other hand, the frequency of interactions among the selected agents is significantly higher.

R_e’s decreasing is much faster than the real networks (Fig. 2A). Thus, decreasing R_e in real networks is harder and retains infection chains.

Estimating contribution of household and non-household in stabilizing of COVID-19

To better understand the effect of household and non-household transmission on the spread of disease, we study the transmission chain of infections. In the earlier results, Fig. 2, we counted the place of infections in the transmission chain, namely, the household (H), social environment (S), and workplace (W). This can be regarded as the 0th order knowledge on the transmission chain—we first group S and W to non-household transmission (O) to get further information. We then study the third-order knowledge by counting three consecutive places of infections, e.g., HOH (the first infection is in household, the infected person infected another person in O, and that person infected another person in H).

There are eight combinations of H and O: HHH, HHO, HOH, HOO, OHH, OHO, OOH, OOO. The combination OOO is dominant for Ro at 2.87, the default case Fig. 3. HHH is the least frequent. Thus, a decrease in β_o affects the OOO-chain three times, whereas β_h has a minor influence. Therefore, the spread of the disease (Ro at 2.87) on our real network depends much more sensitively on β_o than β_h.

Similar analysis can be made for the real network when R_e =1, β_h is fixed or not. The combination OOO is significantly lower. The combinations with Hs are much more frequent. Therefore, the system becomes more sensitive to β_h.

Besides the real-world network, a random network was also simulated for R_e at 2.87. For the correct comparison, the number of H and O edges is kept the same in random and real networks. The number of H connections is more than O connections. It is natural to assume that infections should occur more frequently on H-edges. Indeed, At R_o = 2.87, the frequencies of the combinations are the opposite for the random networks and the real network, Fig. 3. The combination of OOO dominates the real network, while the HHH is the highest random network. The discrepancy is that the unique pairs for H connections are scarce, but their frequency is significantly higher.

Overall, at high β_h and β_o, both H and O infections are high, but the higher number of unique O pairs contributes more to the spread. In contrast, when the β decrease, the infection at O diminishes faster than H because the high frequency of connections for the H-edges provides a higher infection probability.

Calculating the influence of stay-at-home restriction during weekends in the spread of COVID-19

After investigating the effects of contagion probability, we have investigated the effect of stay-at-home restrictions on weekends. To understand the impact of stay-at-home restriction versus transmission probability, we have simulated three cases: free weekend without restrictions, restriction on Sunday, and restriction on Saturday and Sunday. We have simulated each restriction scenario when social distancing is applied in the workplace and social environment (β_h fixed). The edge frequencies of the altered networks are given in Figs. 4A, 4B, and 4C. As detailed in the methods, the household edges replace the social environment edges to implement the weekend restrictions.

We have found that stay-at-home restrictions during the weekends cause a decrease in R_e. Figure 5A shows that R_o drops from 2.95 to 2.71 for restrictions on Saturdays and to 2.56 when restrictions are on Saturdays and Sundays. Figures 5B, 5C and 5D show the ratio of infection occurrence ratio for increasing R_e when implementing stay-at-home restrictions during weekends. When stay-at-home restrictions increase, the social environment’s infection decreases, and work infections increase.

Expectedly, decreasing the social environment edges decreases infections in the social environment. However, the increase in household edges does not increase the number of household infections due to saturation. Overall, it leads to a decrease in R_o  (Moreland et al., 2020; Czeisler et al., 2020).

Estimating impact of the daily working hours, stay-at-home restriction and transmission reduction level during weekends and weekdays on COVID-19

The subsequent alteration to the network is on working hours. We alter working hours by changing working edges. For example, if the working hours decreased from nine to eight hours, the work edges between 5 pm to 6 pm are converted to a sample from the edges between 6 pm to 11 pm, as detailed in the methods. The edge frequencies of the altered networks are given in supplementary information Figs. 6A, 6B, 6C, 6D, and 6E.

The last alteration is the vaccination. Different vaccination levels have been simulated. At the beginning of the simulations, a certain percentage of randomly selected individuals are vaccinated.

We simulate combinations of different vaccination levels, working hours, social distancing measures (transmission probability), and weekend restrictions: thirty thousand simulations were made. The number of parameters hinders us from understanding the interplay of parameters. One would like to understand the following functions Re_WT (DW, SDM, SH, Vac), Re_delta (DW, SDM, SH, Vac), where DW is the decrease in working hours, the SDM is the social distancing measure, the SH is the stay-at-home restriction (in weekends and in weekdays), Vac is the vaccination ratio.

Plotting for all simulations is hindered due to the multidimensionality of the results. Deducing a closed-form solution is also almost impossible. Instead, we wanted to approximate the solutions by fitting a multidimensional linear surface.

(5)${\displaystyle R{e}_{WT}={\alpha }_0+{\alpha }_1\cdot DW+{\alpha }_2\cdot SH+{\alpha }_3\cdot Vac+{\alpha }_4\cdot SDM}$ (6)${\displaystyle R{e}_{delta}={\beta }_0+{\beta }_1\cdot DW+{\beta }_2\cdot SH+{\beta }_3\cdot Vac+{\beta }_4\cdot SDM.}$

The linear surfaces yielded a poor fit when we plotted against simulated data. Therefore, the mathematical expressions are made more complicated systematically. Firstly, the square term of SDM, Vac, SH, and DW are added to (5) (see Table S4 equ. B, C, D, and E) and Eq. (6) (see Table S5 equ. B, C, D, and E). Then, the root mean square errors (RMSE) of updated equations are calculated based on the simulated data and compared with the RMSE of Eqs. (5) and (6). Among these, the equation with the square of SDM reduces RMSE, but not sufficiently.

Secondly, the mathematical expressions are updated by adding a multiplication term (see Table S4 for WT and S5 for delta variant, equ. F, G, H, and j). Accordingly, the RMSE of the updated equation with multiplication term of the SDM parameter is less than the equation with multiplication term of the Vac, DW, and SH. Additionally, it has a smaller RMSE than the equation with the square root of SDM. Consequently, it has been observed that the SDM is a sensitive parameter in Eqs. (5) and (6).

Thirdly, the mathematical expressions are updated by multiplying the four parameters (see Table S4 for the WT and S5 for delta variant, Equation K). However, it has approximately the same RMSE as the equation, which is updated by adding the multiplication term of the SDM parameters.

Next, the mathematical expressions are updated by adding a multiplication term with a square term (see Table S4 for WT and S5 for delta variant, equ. M, N, O, and P). Interestingly, the updated equation with multiplication and square terms of the SDM parameter has a smaller RMSE than the others. Therefore, it is predicted that updating the equations with the multiplication term and power term of the SDM parameter will reduce the RMSE. Considering this feature, finally, the equations are updated by adding the third power of the SDM parameter. Consequently, a substantial RMSE is obtained (see Table S4 for WT and S5 for delta variant, equ. S). The updated Eqs. (7) and (8) has a good fit with the simulated reproduction number of WT COVID-19 (see Fig. S5) and delta variant COVID-19 (see Fig. S6), respectively. It is crucial to remember that our goal is to discover a trustworthy and simple expression that would allow us to examine the trade-off between our variables rather than the best fit.

(7)${\displaystyle R{e}_{WT}=\left({\alpha }_0+{\alpha }_1\cdot DW+{\alpha }_2\cdot SH+{\alpha }_3\cdot \frac{Vac}{100}\right)\cdot \left({\alpha }_4\cdot \frac{SDM}{100}+{\alpha }_5\cdot {\left(\frac{SDM}{100}\right)}^2+{\alpha }_6\cdot {\left(\frac{SDM}{100}\right)}^3\right)}$ (8)${\displaystyle R{e}_{delta}=\left({\beta }_0+{\beta }_1\cdot DW+{\beta }_1\cdot SH+{\beta }_2\cdot \frac{Vac}{100}\right)\cdot \left({\beta }_3\cdot \frac{SDM}{100}+{\beta }_4\cdot {\left(\frac{SDM}{100}\right)}^2+{\beta }_5\cdot {\left(\frac{SDM}{100}\right)}^3\right).}$

The expressions are valid for the range of the simulations, 0 to 4 h for the DW, 0 to 3 for the SH, 0 to 90 for the Vac (90 per cent vaccination), and 0 to 100 for the SDM. The scatter plots for the simulated and estimated values are given in Figs. 7A–7B. The coefficients of determination (R2) are 0.9722, and 0.9951 for WT and delta variant, respectively. After rearranging the Eqs. (7) and (8), the expressional relations are as follows:

(9)${\displaystyle R{o}_{WT}=2.76\cdot \left(1-0.0125\cdot DW-0.072\cdot SH-0.54\cdot Vac\right)\cdot \left(1-0.56\cdot SDM+0.31\cdot SD{M}^2-0.64\cdot SD{M}^3\right)}$ (10)${\displaystyle R{o}_{delta}=3.44\cdot \left(1-0.015\cdot DW-0.074\cdot SH-0.36\cdot Vac\right)\cdot \left(1-0.69\cdot SDM+1.21\cdot SD{M}^2-1.41\cdot SD{M}^3\right).}$

According to Eqs. (5) and (6), the trade-offs are easily measured. Changing working hours has a minor effect for both the WT and delta variants, while vaccination has the most effect. More specifically, for the WT, one day of the weekend restriction equals more than 4 h of work per weekday. In comparison, one day of restriction is equal to 13.6% vaccination. For the delta variant, the effect of the decrease in working hours and weekend restrictions act the same, while one day of restriction is equal to 20.2% per cent of vaccination. However, the relative effect of vaccination is 19.6% less effective than the WT.

It is important to note that the validity of functions is limited to the simulated ranges of parameters. Of course, full vaccination is expected to lower R_e below 1. However, this is not the case for functions in Eqs. (5) and (6). This is expected because our simulated range is 0 to 90 per cent. After 90% vaccination Re depends non-linearly on Vac until reaching R_e = 0. The same discussion can be done for other parameters as well.

The SDM (transmission probability) varies from 0 to 100 per cent, unlike other parameters. As seen in Fig. 2A, the reduction in transmission probability reduces R_e significantly after an 80% reduction.

The parameters that we have simulated, DW and SH, vary marginally. For example, a decrease in work hours (DW) only changes from 0 to 4 h. In terms of working hours, the DW varies from 9 h to 5 h of work. In the simulated range, R_e is not significantly affected. Perhaps simulating for the remaining hours, from 9 to 0 h of work, would show the non-linearity that we have suspected. However, the point we want to make is that a marginal decrease in working hours (several hours) leads to a minimal decrease in R_e. The working hours must be decreased substantially to decrease R_e significantly. However, its burden on economic activity would be profound.

The range for SH (stay-at-home restrictions) has been simulated from 0 to 4 days. The meaningful range is 0 to 2 days, the weekend. Two days of restrictions are equal to 26.8% vaccination for the WT, 41% for the delta type. However, it should be noted that the simulations on the delta variant with 90% vaccination and weekend restrictions have about R_e = 2 as the delta variant starts with a higher R_e. The weekend restriction on the wild type is more meaningful. The simulations with 90% vaccination with weekend restrictions have R_e = 0.98.