New model suggest that aggressive lockdown to contain COVID-19 must be maintained for at least 44 days

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People around the world are wondering how long COVID-restrictions have to last in order to curtail the pandemic.

A research study on 36 countries and 50 U.S. states has found that aggressive intervention to contain COVID-19 must be maintained for at least 44 days.

The study is co-authored by Professor Gerard Tellis of USC Marshall School of Business, Professor Ashish Sood of UC Riverside’s A. Gary Anderson Graduate School of Management, and Nitish Sood, a student at Augusta University studying Cellular & Molecular Biology.

The paper is published in the open sources journal SSRN and is titled, “How Long Must Social Distancing Last”.

The authors identify two simple, intuitive, and generalizable metrics of the spread of disease: daily growth rate and time to double cumulative cases.

Daily growth rate is the percentage increase in cumulative cases.

Time to double, or doubling time, is the number of days for cumulative cases to double at the current growth rate.

Time to double in disease spread is the opposite of half-life in drug metabolism.

“Counts of total or new cases can be misleading and difficult to compare across countries,” Professor Tellis said.

“Growth rate and Time to double are critical metrics for an accurate understanding of how this disease is spreading.”

Given these two metrics, the researchers defined three measurable benchmarks for analysts and public health managers to target:

  • Moderation: when growth rate stays below 10% and doubling time stays above seven days.
  • Control: when growth rate stays below 1% and doubling time stays above 70 days.
  • Containment: when growth rate remains 0.1% and doubling time stays above 700 days.

“These simple, intuitive, and universal benchmarks give public health officials clear goals to target in managing this pandemic,” Professor Sood said.

Preliminary results using this model to analyze the data suggest that once aggressive interventions are in place, large countries take almost three weeks to see moderation, one month to get control, and 45 days to achieve containment.

With less aggressive intervention, it can take much longer. Important differences exist by size of country.

Public health administrators should note larger countries take longer to see moderation.

The authors defined aggressive intervention as lockdowns, stay-at-home orders, mass testing and quarantines.
“Singapore and South Korea adopted the path of massive test and quarantine, which seems to be the only successful alternative to costly lockdowns and stay-at-home orders,” Nitish Sood said.

Preliminary results using this model to analyze the data suggest that once aggressive interventions are in place, large countries take almost three weeks to see moderation, one month to get control, and 45 days to achieve containment.

Their research focuses on diffusion of innovations, new products, and new technologies. The same concepts and tools can be applied to analyze the spread of COVID-19 and the effects of measures to stop it.

“Even though huge differences exist among countries, it’s striking to see so many similarities from aggressive intervention to moderation, control, and containment of the spread of the disease,” Professor Sood said.

Professor Tellis added, “Besides size of country, borders, cultural greetings (bowing versus handshaking and kissing), temperature, humidity, and latitude may explain these differences.”

The researchers say their analysis bolsters the case for adopting aggressive measures, whether it’s the aggressive lockdowns of Italy or California, massive testing and quarantine of South Korea or Singapore, or a combination of both as seen in China.

However, the U.S. may have a unique challenge because of its federal constitution.

Only half of the states have adopted aggressive intervention and that at varying times. Should these states achieve control or containment, they may be vulnerable to contagion from states that were late to do so, the researchers say.


The novel coronavirus, COVID-19, originated in Wuhan and has spread rapidly across the globe. The World Health Organization has declared it to be a pan- demic. In the absence of a vaccine, social distancing has emerged as the most widely adopted strategy for its mit- igation and control [1].

The suppression of social contact in workplaces, schools and other public spheres is the target of such measures. Since social contacts have a strong assortative structure in age, the efficacy of these measures is dependent on both the age structure of the population and the frequency of contacts between age groups across the population.

As these are geographi- cally specific, equal measures can have unequal outcomes when applied to regions with significantly differing age and social contact structures.

Quantitative estimates of the impact of these measures in reducing morbidity, peak infection rates, and excess mortality can be a significant aid in public-health planning. This requires mathemat- ical models of disease transmission that resolve age and social contact structures.

In this paper we present a mathematical model of the spread of the novel coronavirus that takes into account both the age and social contact structure [2]. We use it to study the impact of the most common social distancing measures that have been initiated to contain the epidemic in India: workplace non-attendance, school clo- sure, “janata curfew” and lockdown, the latter two of which attempt, respectively, complete cessation of public contact for brief and extended periods.

We emphasise that models that do not resolve age and social contact structure cannot provide information on the differential impact of each of these measures. This information is vital since each of the specific social distancing measures have widely varying economic costs.

Our model allows for the assessment of the differential impact of social distancing measures. Further, both morbidity and mortality from the COVID-19 infection have significant differences across age-groups, with mortality increasing rapidly in the elderly.

It is necessary therefore to estimate not only the total number of infections but also how this num- ber is distributed across age groups Our model allows for the assessment of such age-structured impacts of social distancing measures.

The remainder of our study is organized as follows. In Section (II) we compare the age and social contact struc- ture of the Indian, Chinese, and Italian populations. Age distributions are sourced from the Population Pyramid website [3] and social contact structures from the state- of-the-art compilation of Prem et. al. [2] obtained from surveys and Bayesian imputation.

We show that even with equal probability of infection on contact, the differences in age and social contacts in these three countries translate into differences in the basic reproductive ratio R0.

In Section (III) we study the progress of the epidemic in the absence of any mitigation to provide a base- line to evaluate the effect of mitigation. In Section (IV) we investigate the effect of social distancing measures and find that the three-week lockdown that commenced on 25 March 2020 is of insufficient duration to prevent resurgence.

Alternative protocols of sustained lockdown with periodic relaxation can reduce the infection to levels where social contact tracing and quarantining may become effective. Estimates of the reduction in morbid- ity and mortality due to these measures are provided. We conclude with a discussion on the possibilities and limi- tations of our study.

An appendix provide details of our mathematical model and the social contact structure.
It has been known from retrospective analyses of the 1918–19 pandemic that delays in introducing social distancing measures are correlated with excess mortality [4, 5].

Our study confirms the urgency and need for sus- tained application of mitigatory social distancing.

Figure 1. Age and contact structures of the populations of India, China and Italy. The first column shows population pyramids by age and gender. The second and third columns show the contact structures in households and workplaces with darker colours representing greater contacts. The diagonal dominance of these matrices shows strong assortative mixing in all three countries. Significant differences appear in the off-diagonals. In India, the pentadiagonal character of the household contacts reflects the prevalance of three-generation households, which are smaller in China and negligible in Italy.

EPIDEMIC WITHOUT MITIGATION

We fit our mathematical model, described in detail in Appendix, to case data to estimate the probability of infection on contact β. Though our model allows for infectives to be both asymptomatic and symptomatic, given the large uncertainty in estimating asymptomatic cases, we assume all cases to be symptomatic.

A possible effect of this is to underestimate the severity of the outbreak. We then run the model forward in time to forecast the progress of the epidemic with results shown in Fig. (2).

Panel (a) shows the fit to case data available upto 25th March 2020 and a three-week forecast, in the absence of social distancing measures. The basic reproductive ratio is R0 = 2.10. Panel (b) shows a five month forecast, again, in the absence of social distancing.

The peak infection is reached at the end of June 2020 with in excess of 150 million infectives. The total number infected is estimated to be 900 million. Panel (c) shows the time-dependent effective basic reproductive ratio R0^eff(t) which gives the dominant contribution to the linearised growth at any point in time.

This number is greater than unity before peak infection and smaller than unity beyond peak infection. The serves as a useful measure of the local rate of change of infectives at any point in time. In Fig. (3) we provide estimates of (a) the morbidity and (b) the excess mortality from the unchecked spread of the epidemic.

The fraction infected across age groups is the largest for the 15-19 year olds and least amongst the 75-79 year olds. However, due to the strong age-dependence in death rates, mortality is amongst the least for the 15-19 year olds and greatest for the 60-64 year olds.

We emphasise that these numbers, alarming as they are, are counterfactuals, as mitigation measures are already in place of this writing. They do, however, point to the unbearable cost in human life that must be paid for the any lack of, or delay in, mitigatory action.

IMPACT OF SOCIAL DISTANCING

We now investigate the impact of social distancing measures on the unmitigated epidemic. We assume that social distancing in any public sphere, which in our model is partitioned into workplace, school and all others, re- moves all social contacts from that sphere. This, of course, transfers the weight of these removed contacts to the household, where people must now be confined.

We ignore this in the first instance. We interpret the lock- down imposed from 25 March 2020 to remove all social contacts other than the household ones. This is an op- timistic interpretation but it does allow us to assess the most favourable impact of such a measure. The results that follow, then, are an expected best-case scenarios. Then, the time-dependent social contact matrix at time t is

contacts comprising of contributions from the household, workplace, schools and all others, with obvious super- scripts. The control function, described in Appendix, is constructed to reflect a social distancing measure that is initiated at t = ton and suspended at ton = toff.

The measure has a lag tw to be effective which we choose to be shorter than a day. The function varies smoothly from zero to one in the window ton-toff .

For repeated initi- ations and suspensions, the control function is a sum of such terms with times adjusted accordingly. It is possi- ble, of course, to have differentiated controls which apply distinct social distancing measures at different times and for different durations.

We do not explore these here as the general setting for such an investigation would be within the framework of optimal control theory [6] with an appropriate cost function. We postpone this to future work.

Our results are show in in the four panels of Fig. (4) for four different control protocols. Panel (a) shows the effect of the three-week lockdown. While this immedi- ately changes the sign of the rate of change of infectives, it does not reduce their number sufficiently to prevent a resurgence at the end of the lockdown period.

Panel(b) shows the effect a suspension of the lockdown by 5 days followed by a further lockdown of 28 days. This too, does not reduce the number of infectives sufficiently to prevent resurgence.

Panel (c) shows a protocol of three consecutive lockdowns of 21 days, 28 days and 18 days spaced by 5 days of suspension. This brings the num- ber of infective below 10 where explicit contact tracing followed by quarantine may be successful in preventing a resurgence.

Panel (d) shows a single lockdown period to reach the same number of infectives which our model predicts to be 49 days.
Table (II) show the excess mortality that can be ex- pected for each of the social distancing measures above.

While we emphasise, again, that these are likely to be best-case scenarios, the substantive message is that of the crucial importance of rapid and sustained social dis- tancing measures in reducing morbidity and mortality.

[1] N. M. Ferguson et al., “Impact of non-pharmaceutical interventions (npis) to reduce covid-19 mortality and healthcare demand,” London: Imperial College COVID-19 Response Team, March 16 (2020), 10.25561/77482.
[2] K. Prem, A. R. Cook, and M. Jit, “Projecting social contact matrices in 152 countries using contact surveys and demographic data,” PLoS Comp. Bio 13, e1005697 (2017).
[3] “https://www.populationpyramid.net/” .

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[5] Richard J. Hatchett, Carter E. Mecher, and Marc Lipsitch,“Public health interventions and epidemic intensity 9 during the 1918 influenza pandemic,” Proc. Natl. Acad.Sci. 104, 7582–7587 (2007).
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