key: cord-289550-b8f4a7o3
authors: Neuwirth, C.; Gruber, C.; Murphy, T.
title: Investigating duration and intensity of Covid-19 social-distancing strategies
date: 2020-04-29
journal: nan
DOI: 10.1101/2020.04.24.20078022
sha: 
doc_id: 289550
cord_uid: b8f4a7o3

The exponential character of the recent Covid-19 outbreak requires a change in strategy from containment to mitigation. Meanwhile, most countries apply social distancing with the objective to keep the number of critical cases below the capabilities of the health care system. Due to the novelty and rapid spread of the virus, an a priori assessment of this strategy was not possible. In this study, we present a model-based systems analysis to assess the effectiveness of social distancing measures in terms of intensity and duration of application. Results show a super-linear scaling between intensity (percent contact reduction) and required duration of application to have an added value (lower fatality rate). This holds true for an effective reproduction of R > 1 and is reverted for R < 1. If R is not reduced below 1, secondary effects of required long-term isolation are likely to unravel the added value of disease mitigation. We recommend an extinction strategy implemented by intense countermeasures.

This article is written in mid-April 2020 where globally the number of confirmed in the long-term, school closure and home confinement will negatively affect children's 20 health [10] and the global economy, to name only two big drawbacks of these measures. 21 In this light, it is of particular interest, for what duration these exceptional 22 interventions must remain in place. According to recent estimates, we are probably at 23 least 1 year to 18 month away from large-scale vaccine production [5] . Independent of 24 the time it takes to develop a vaccine, the epidemic spread will also come to an end, if 25 sufficient people have been infected to establish herd immunity. Studies on the 26 effectiveness of the concept of disease mitigation with the objective to establish herd 27 immunity shows some potential in the case of pandemic influenza [11] . 28 In this study, we present an exploratory and model-based systems analysis that is 29 aimed at investigating the application of social distancing strategies to Covid-19. 30 Specific objectives of this research are: 1) to investigate the effectiveness of contact 31 reduction policies with respect to intensity and duration and 2) to estimate the amount 32 of time to establish herd immunity by considering the national health care systems of 33 Austria and Sweden, which are very different in terms of critical care capabilities. A 34 detailed description of model equations, assumptions as well as uncertainty of currently 35 available data are presented in the following section. Data uncertainty is addressed by 36 the analysis of alternative scenario runs to enhance robustness of model results. In a 37 concluding section, we compare our results to similar studies, discuss current limitations 38 of data availability and give recommendations based on exploratory results. 39 Method 40 Adapted SIR model 41 The current scenario of novel pathogen emergence includes considerable uncertainty 42 [12] . This means that a reliable scientific evidence base on Covid-19 is yet to be 43 established. Under these preconditions, the use of models for exploratory rather than 44 predictive purposes is more appropriate [13] . Accordingly, the simulation model 45 presented in this study was designed to identify and systematically explore important 46 qualitative behavior of this dynamic system that remains unchanged irrespective of 47 parameter variations. An adaptation of the popular susceptible-infected-recovered (SIR) 48 model turned out to be most suitable for this purpose (see Fig. 1 ). In order to meet the 49 specific requirements of a simulation model on Covid-19 mitigation, the structure of the 50 original model was adapted accordingly. For instance, pathological findings of Covid-19 51 indicate that there is a considerable number of cases that develop mild or no symptoms 52 [14] . To account for this characteristic, we separated the infected population into those 53 that are asymptomatic and those that are not, which in the latter case leads to isolation 54 or hospitalization. The asymptomatic infected get resistant without prior isolation.

Exponential growth in numbers of infected poses a challenge to health care facilities. 56 In Italy, specialists are already considering denying life-saving care to the sickest and 57 giving priority to those patients most likely to survive [15] . This will inevitably cause 58 potentially avoidable deaths. In the model, deaths caused by a lack of intensive care is 59 considered independently.

The calculation of population quantities in respective compartments (S, IU, RA, II, 61 D, DL and R) is in line with the logic of the standard SIR model. Initially everyone in 62 the total population T P is susceptible. The number of susceptible is reduced over time 63 by infections i

where i r is the infection rate (rate of contacts between uninfected and infected that 65 result in infections) and c ui is the number of contacts between infected and uninfected, 66 which is calculated as 67 April 22, 2020 2/15 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 29, 2020. . 

where c d is the personal contacts per day, S t is the susceptible at time t and IU is 68 the number of unknown infections. To take account for a lower infection rate of 69 asymptomatic infected, the unknown infections IU in equation 2 is substituted by

with IU c being the unknown infected corrected for asymptomatic infected, a f the 71 fraction of asymptomatic among infected and a p the asymptomatic population's 72 potential to infect.

The flows from compartment IU to r A and II -i.e. asymptomatic cases getting 74 resistant r A and isolation of infected iso -are calculated by

Parameter i t is the time between infection and isolation and d a is the duration of 76 asymptomatic infection. The flows from compartment II are given by

where parameter d s is the duration of distinct symptomatic sickness, c f r is the case 78 fatality rate and ICU d and ICU s is the intensive care demand and supply respectively. 79 The intensive care demand ICU d is calculated by taking the critical fraction c f (see 80  Table 1 of infected in isolation II. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 29, 2020. reproduction, case fatality rates and proportion of asymptomatic cases is quite 84 substantial and growing. The wide range of suggested parameter values, however, poses 85 a considerable challenge to model parametrization. For instance, estimates of the basic 86 reproduction number R0 vary within a range from 1.4 [16] to 4.71 [17] . Part of this 87 variation is explained by geographic variation of population densities and social habits. 88 Moreover, there is uncertainty in the percentage of asymptomatic cases. The outbreak 89 in a smaller isolated population is an opportunity to derive representative numbers by 90 applying comprehensive and repeated laboratory testing. One such example is the 91 outbreak of Covid-19 on board of the Diamond Princess cruise ship. However, given 92 that most of the passengers were 60 years and older, the nature of the age distribution 93 may lead to underestimation of asymptomatic cases if older individuals tend to 94 experience more symptoms [18] . In a normal population higher ratios of up to 50% 95 asymptomatic carriers of Covid-19 are expected ( [19] . The question whether or not 96 asymptomatic carriers are able to infect others is still controversial (e.g. [20] ).

The severity of the disease does also play an important role in estimating the ratio of 98 critically ill patients who need intensive care. According to Chinese statistics, 5% of 99 positively tested patients are admitted to intensive care [21] . This number was adopted 100 by the World Health Organization [22] and other studies (e.g. [23] , whereas national 101 statistics show significant deviations; e.g. 9 to 11% in Italy [15] and 2.2% in Austria 102 [24] . A potential explanation for these considerable differences is that in Italy a lot of 103 the older population were infected [25] .

The specific age distributions of affected communities may also show some biasing 105 effect on estimated case fatality rates. Another factor that contributes to regional 106 differences in case fatality is the occupation or over-occupation of available intensive 107 care beds (ICU beds). In a few instances, national critical care capabilities are exceeded 108 by the number of critically ill patients (e.g. Italy and France), which drastically elevates 109 fatality rates. By contrast, the true case fatality rates are lower if theoretically all cases 110 were found by testing the entire population. Accordingly, a lower case fatality rate 111 (CFR) was reported by countries who were effective in extensive testing and 112 maintaining the prevalence of critical cases below critical care capabilities like South 113 Korea [26] . A higher CFR was reported by countries who refrain from extensive testing 114 and/or are overwhelmed by the pace of new infections like Iran, Italy and others [27] . 115 In the model, we use the more reliable South Korean figures and simulate the additional 116 fatalities due to the critical care limit based on capability limits of national critical care 117 units (see Eq. 6 and Eq. 7).

Among the parameters in Table 1 , the basic reproduction number R0 is the only 119 parameter without explicit representation in the model equations. This parameter is the 120 number of secondary cases, which an infected person produces in a completely 121 susceptible population [31] . In the model, R0 is defined as the arithmetic product of it 122 the time between infection and isolation, c d the personal contacts per day and i r the 123 infection rate.

In response to the outbreak of an epidemic disease, changes in contact behavior 125 diminish the reproduction. We refer to this modified reproduction as effective 126 reproduction number R. In model scenarios where contact behavior is not constant, 127 effective reproduction is denoted as R t .

The choice of appropriate scenarios is based on parameter uncertainty and model 129 sensitivity. Sensitivity analysis indicate a linear response in model output to variations 130 in c f r and c f , and interestingly non-linear effects in response to variations in R, a f and 131 a p . Accordingly, the latter variables were selected as scenario parameters (see Table 2 ). 132 Dependent on the research objectives 1 and 2 (see Introduction); prolonged and 133 April 22, 2020 4/15 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. . [16] was applied in respective scenarios (see Table 2 ).

Whereas prolonged social distancing is defined by constants c r and d m , intermittent 135 social distancing is implemented by dynamic adaptation of contact reduction c r during 136 simulation runtime dependent on the amount of ICU beds available. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. . Fig. 2 ). Consequently, social distancing flattens the curve of daily 151 infections, while higher proportions of asymptomatic cases elevate the peak.

This flattening effect can be expressed analytically. The daily infections resemble a 153 normal distribution, which is defined by a mean µ (days between outbreak and peak of 154 April 22, 2020 6/15 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. daily infections) and a standard deviation σ. A lower R will lead to a higher µ (see Fig. 155 3) and a broader distribution σ (see Fig. 4 ). Additionally, the number of initial infected 156 people reduces µ (see Fig. 5 ), whereas σ is independent of it. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. . . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. . measures in terms of contact reduction, the longer the duration needs to be to have an 161 added value; i.e. a relatively lower fatality rate. In other words, the harder you break, 162 the longer it takes. 163 For instance, 40% contact reduction needs to be applied for additional 600 days to 164 outperform a 30% contact reduction in scenario 2 (see Fig. 6 ). The lower the basic 165 reproduction in the scenarios, the larger the time lag associated with an intensification 166 of social distancing (see Fig. 6, scenarios 1, 2, 4, 5 and 6 ). This is in line with 167 above-mentioned relationships that show increased effects of R on µ and σ with lower R. 168 Given the trade-offs associated with required long-term lockdown, the effectiveness 169 of additional social distancing decreases with R close to 1. The secondary effects of lock 170 down have not been modelled, but it is speculated that reductions in social contacts will 171 increase mortality (e.g. social isolation and homicide; obesity and cardiovascular 172 diseases etc.) making moderate contact reduction more adequate.

Interestingly, if social distancing is intense enough to drop R below one, a further 174 increase in intensity removes the pandemic earlier (see Fig. 6, scenario 3) . This is 175 contrary to the case of R > 1 where the effectiveness of more intense measures is in 176 danger to be unraveled by the super-linear increase in duration.

The curve flattening effect of social contact reduction also explains why drastic 178 contact reduction may cause more deaths than mild contact reduction, if measures are 179 applied for too short time. In the worst case, intense social distancing will hardly have 180 any effect (see Fig. 7 ).

Duration to establish herd immunity 182 Intensity and duration are also closely related in the intermittent social distancing and 183 herd immunity scenario (see Fig. 8 ). The strategic objective in this scenario is to keep 184 the demand for ICU beds within the bounds of ICU supply until herd immunity is 185 established.

Independent of what values the policy thresholds have (see section Model inputs and 187 parameters), the demand for the ICU beds behaves like a damped oscillation (see Fig. 188 8). This is explained by the delay in the system, diminishing number of susceptible 189 people and the negative feedback between number of available ICU beds and social 190 contacts.

In the early phase of the outbreak, the number of patients exceeds the number of 192 available ICU beds due to high reproduction potentials. Higher basic reproduction R0 193 results in additional over-occupation of ICU capabilities (Fig. 8, scenario 9 and 10 ).

Moreover, the variation of the constant of availability of intensive care brings about 195 a shift in the time needed to achieve the strategic objective of herd immunity (compare 196 Sweden and Austria in Fig. 8 ). This relationship exhibits an almost linear scaling.

Independent of national health care capabilities, results show that social distancing and 198 herd immunity strategies require extraordinary endurance.

This is also the case under more favorable conditions. In Austria, for instance, it is 200 estimated that only 2.2% of confirmed cases are admitted to ICU [24] . Combined with 201 Austria's high performance health care system and low effective reproduction, the time 202 to establish herd immunity is still estimated to be about 2 years (see Fig. 9 ). Given 203 that the ICU beds are also needed for patients other than Covid-19, an even longer 204 period has to be expected. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. . is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 29, 2020. infections. As a consequence of this delay, measure intensity scales super-linearly with 211 the required duration of application to show added value; i.e. a relatively lower fatality 212 rate. Given the large scale of temporal delay (up to multiple years for a 10% increment 213 of additional contact reduction), secondary effects of long-term social isolation such as 214 psychological distress, depression [32] , and increased mortality [33] are likely to 215 unravel the added value of disease mitigation. This holds true for effective reproduction 216 numbers above one. Below this threshold, more intense measure applications are 217 associated with earlier termination of viral spread.

In the absence of a vaccination, mitigation strategies are crucial to keep the number 219 of severe and critical cases below the capabilities of the health care system. If the use of 220 mitigation interventions is well balanced against capability limits, the time required to 221 establish herd immunity linearly scales with available capabilities of the health care 222 system (defined by the number of ICU beds in the simulation). Other important factors 223 are the reproduction number and the severity of the disease (expressed by the fraction 224 of cases that need ICU admission). Depending on the calibration of those factors, it is 225 estimated that herd immunity on a national level will be established in more than 2 226 years from now. This is in line with an agent-based simulation study by [34] , who 227 indicate a duration of 2 years and 4 months for the Netherlands. According to a 228 deterministic simulation by [16] in the United States the epidemic could last into 2022 229 under current critical care capabilities.

It is important to mention that assumptions and policies implemented in models are 231 not exactly reproducible in reality. For instance, Bock et al. [35] argue that mitigation 232 measures imposed by state authorities can hardly be fine-tuned enough to hit the 233 narrow feasible interval of epidemiologically relevant parameters with which a successful 234 mitigation is possible. Given those constraints, as well as trade-offs associated with 235 required long-term lockdown, we conclude that the success of a strategy based on social 236 distancing, delay and herd immunity is unrealistic under known preconditions.

According to [35] , an extinction strategy implemented by intense countermeasures seems 238 promising. This is supported by our low effective reproduction scenario (R < 1). To . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 24.20078022 doi: medRxiv preprint date, a more differentiated assessment of alternative countermeasures such as the 240 selective isolation of vulnerable individuals or approaches of contact tracing and 241 isolation are limited by data scarcity and in part data inconsistency.

For instance, there is little reliable information about age-stratified asymptomatic 243 ratios. There is also few studies on secondary effects of social distancing and isolation in 244 the case of a global pandemic. Up until now, the impact of country-based measures has 245 hardly been empirically assessed by methods of inferential statistics. While such studies 246 will shed light on important system dependencies, large-scale investment into health 247 care and medical research is essential to spawn game-changing innovation such as the 248 development of vaccines, drugs and affordable test kits.

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