PLoS ONE plos plosone PLOS ONE 1932-6203 Public Library of Science San Francisco, CA USA 10.1371/journal.pone.0281224 PONE-D-23-01297 Research Article Medicine and health sciencesMedical conditionsInfectious diseasesViral diseasesCOVID 19 Biology and life sciencesImmunologyVaccination and immunization Medicine and health sciencesImmunologyVaccination and immunization Medicine and health sciencesPublic and occupational healthPreventive medicineVaccination and immunization Medicine and health sciencesDiagnostic medicineVirus testing Medicine and health sciencesEpidemiology Medicine and health sciencesEpidemiologyPandemics Medicine and health sciencesMedical conditionsInfectious diseasesInfectious disease controlSocial distancing Biology and life sciencesOrganismsVirusesRNA virusesCoronavirusesSARS coronavirusSARS CoV 2 Biology and life sciencesMicrobiologyMedical microbiologyMicrobial pathogensViral pathogensCoronavirusesSARS coronavirusSARS CoV 2 Medicine and health sciencesPathology and laboratory medicinePathogensMicrobial pathogensViral pathogensCoronavirusesSARS coronavirusSARS CoV 2 Biology and life sciencesOrganismsVirusesViral pathogensCoronavirusesSARS coronavirusSARS CoV 2 Biology and life sciencesImmunologyImmunity Medicine and health sciencesImmunologyImmunity Trajectories of COVID-19: A longitudinal analysis of many nations and subnational regions Dynamics of COVID-19 epidemiology https://orcid.org/0000-0002-0941-3827 Burg David Conceptualization Data curation Investigation Methodology Writing – original draft Writing – review & editing 1 2 3 4 * https://orcid.org/0000-0001-9568-9880 Ausubel Jesse H. Conceptualization Writing – original draft Writing – review & editing 4 Tel Hai Academic College, Qiryhat Shemona, Israel Hemdat Academic College, Netivot, Israel Ahskelon Academic College, Ashkelon, Israel Program for the Human Environment, The Rockefeller University, New York, NY, United States of America Lal Rajnesh Editor Fiji National University, FIJI

The authors have declared that no competing interests exist.

 * E-mail: david@model-lab.ne 23 6 2023 2023 18 6 e0281224 17 1 2023 7 6 2023 2023 Burg, Ausubel This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The COVID-19 pandemic is the first to be rapidly and sequentially measured by nation-wide PCR community testing for the presence of the viral RNA at a global scale. We take advantage of the novel "natural experiment" where diverse nations and major subnational regions implemented various policies including social distancing and vaccination at different times with different levels of stringency and adherence. Initially, case numbers expand exponentially with doubling times of ~1–2 weeks. In the nations where interventions were not implemented or perhaps lees effectual, case numbers increased exponentially but then stabilized around 102-to-103 new infections (per km2 built-up area per day). Dynamics under effective interventions were perturbed and infections decayed to low levels. They rebounded concomitantly with the lifting of social distancing policies or pharmaceutical efficacy decline, converging on a stable equilibrium setpoint. Here we deploy a mathematical model which captures this V-shape behavior, incorporating a direct measure of intervention efficacy. Importantly, it allows the derivation of a maximal estimate for the basic reproductive number Ro (mean 1.6–1.8). We were able to test this approach by comparing the approximated "herd immunity" to the vaccination coverage observed that corresponded to rapid declines in community infections during 2021. The estimates reported here agree with the observed phenomena. Moreover, the decay (0.4–0.5) and rebound rates (0.2–0.3) were similar throughout the pandemic and among all the nations and regions studied. Finally, a longitudinal analysis comparing multiple national and regional results provides insights on the underlying epidemiology of SARS-CoV-2 and intervention efficacy, as well as evidence for the existence of an endemic steady state of COVID-19.

 The author(s) received no specific funding for this work. Data Availability Data relevant to this study are available from GitHub at https://github.com/Model-Lab-Net/COVID-19. Outbreaks COVID-19 Introduction

Quantitative studies of viral infection in human severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infected subjects have been enabled by the massive global deployment of sensitive and rapid PCR testing for detecting viral RNA in infected persons. Data obtained with these procedures have allowed for extensive mathematical modeling of infection dynamics and viral expansion [1]. Indeed, epidemiological modeling of this pandemic has exploded, though results have been mixed and show how difficult it can be to provide accurate information and predictions, especially in the early stages of the pandemic [2].

COVID-19 cases initially grew exponentially in every nation. Reduction of community infection was initially achieved by non-pharmaceutical and social distancing interventions [3, 4]. The early and drastic social distancing measures undoubtedly curbed viral expansion [5]. However, the underlying biological, environmental and social dynamics were not fundamentally modified, and viral circulation was only temporarily inhibited. National vaccination programs deployed during 2021 were also aimed to block person-to-person infection. These interventions were enacted at different times, with different levels of enforcement, compliance and extent among nations and in major regions within nations. This global "natural experiment" makes the COVID-19 pandemic a unique opportunity to longitudinally model epidemiological dynamics.

COVID-19 modeling is primarily based on the standard SIR model as the foundational tool of mathematical epidemiology and attempts to capture the main characteristics of the complex interplay among the virus, its host and the environment [6]. The theoretical SIR model’s solution converges on a logistic-like S-curve trajectory with rapid expansion reaching a peak and declining in one wave [7]. Many much more elaborate models were deployed to study COVID-19 dynamics [8, 9]; however, complexity invokes problems such as overfitting, global optimization, and interpretability. An important feature not reproduced in these models is the existence of a non-trivial dynamical equilibrium setpoint.

The large amount of publicly available quantitative data amassed allowed a surge of mathematical modeling papers and reports during the COVID-19 pandemic. Researchers have published more than 1,100 peer-reviewed papers in less than two years [10], mostly based on alterations to the SIR model, e.g., the SEIR model and other more complex derivatives [11]. Much work has been performed on modeling the waves of infection which spread across the globe [12]. Saldaña et al. reviewed the main types of epidemiological modeling during COVID-19 [13]. In our extensive reading we found no mention of the V-shaped kinetics observed in the infection data during intervention programs, nor attempts to model the possible endemic steady-state. We aim to show that these are critical characteristics of the virus and the social and pharmaceutical reactions to it, and can be exploited to better understand the observed dynamics of the pandemic. We will show that valuable information about the epidemic can be extracted directly from the kinetics observed in the infection data.

A key criterion of epidemic expansion is the basic reproductive number (Ro) which represents a disease’s transmissibility. Specifically, it is the average number of productive secondary infections arising from one active infectious individual [14]. It is derived from the ratio between the infection and removal rate constants in the SIR or similar models [15]. A bifurcation threshold condition for the occurrence of a sustained epidemic is Ro≥1, meaning that as Ro<1 the infection will converge on the disease-free state. This is also an indication for "herd immunity" [16, 17]. In contrast to the outcome of a disease-free state, most models in the context of COVID-19 lack the capacity to depict sustained endemic levels of infection.

Estimation of the value of Ro is commonly based on the initial exponential growth rate [18] and the median infectious period [19, 20]. This is clearly an overestimate as it disregards the removal rate of cases [21]. Another problem is it ignores the distinctive infection peak and inherent inevitable negative second derivative predicted by SIR models. Other approximations treat reproductive rates as a function of time during the epidemic and Wallinga & Lipsitch [22] summarize the main methods to calculate this time-dependent "effective" R (Re). A recent review demonstrated that Cori et al. [23] derived an accurate estimate for this parameter [24]. It has also been suggested that a simple Dirac delta distribution can be used as a proxy for Re [25]. These are important though Re will fluctuate as a function of the changes in infection rates as the epidemic develops [26], but further discussion is beyond the scope of this paper. While these track changes in infection rates change over time (e.g., the first derivative) they do not capture the underlying fundamental biological and social interactions.

This paper highlights applicability of mathematical modeling based on the viral dynamics paradigm [27–29]. A notable characteristic of these methodologies is an endemic-like non-trivial, non-zero, infection dynamical steady or equilibrium state. Further, they can directly model the effects of interventions to block transmission of the pathogen throughout the population. Its major advantage is the ability to derive estimations for the values of model parameters directly from the data [30].

We refrain from exploring the dynamics of the COVID-19 virus itself. SARS-CoV-2, the virus that causes COVID-19, is continuously changing and accumulating mutations in its genetic code. Some variants emerge and disappear, while others emerge, spread, and replace previous variants. For the USA, variant proportions are tracked at https://covid.cdc.gov/covid-data-tracker/#variant-proportions. Obviously, the strategies for suppression can interact with the evolution of the virus. We simply assume a virus which is able to evolve so that it can reinfect previously infected individuals.

Publicly available data for COVID-19 were used to characterize the epidemiological dynamics of community infection. The implementation of efficacious social distancing and lockdown interventions instituted across many nations allows the modeling of the dynamics of infection decay and subsequent rebound as interventions were lifted or lose effectiveness. A longitudinal comparison among nations and major subnational regions provides insights into pathogenesis that would be difficult or impossible to obtain in past pandemics.

 Materials and methods Epidemiological data

Data for confirmed active infected cases, COVID-19-associated mortality and PCR tests were retrieved from [31]. For most purposes we stop in September 2021 when the widespread availability of self-testing changed the testing regimes and reduces the reliability of some of the relevant time series. Preliminary review shows that the data exhibit two artifacts. First, a weekly cycle is clearly observed with a tendency for more reporting in the middle of the week and less during weekends, sometimes with orders-of-magnitude differences. Second, large inter-day fluctuations are reported, sometimes with differences spanning multiple orders-of-magnitude. While it is common to smooth the data with a moving average, the resulting estimates are highly sensitive to the fitting window, especially with small numbers and the extremely noisy data (up to an order-of-magnitude between days). Therefore, weekly averages were adopted here and calculated from the geometric mean of the daily measurements to stabilize the variance in the data [32].

There is clearly a delay between time of infection and reporting. Incubation times for COVID-19 are 6.2 days and the mean generation interval is 6.7 days, with a concurrent latent period of 3.3 days [33]. Further, there is a lag between infection and detection by lab test with a skewed distribution [34, 35]. While the exact value is unknown, it will only offset the data in time and does not affect the shape of the infection trajectories. Therefore, a ten-day delay is applied here to all confirmed case numbers, only shifting them left in time and not affecting the shape of the data.

 Inclusion criteria

Analyses were performed for nations and major subnational regions with 10-fold mean difference between PCR tests and positive confirmed cases, high GDP (PPP) per capita [36] indicating the ability to perform an extensive testing program, and approximately one log decrease in infections from peak to minimum rates during interventions. The 45 qualifying units include 24 European nations, Australia and New Zealand, the UK and the four nations constituting the UK, 10 USA states, and four Asian nations.

 Interventions, mobility and vaccination coverage

Dates for national policy intervention initiation and termination are available and collated from numerous sources and the COVID-19 stringency index was accessed from [37]. Even so, compliance was imperfect, and mobility was used as a minimal estimate for the cumulative efficacies of the intervention polices to block community infection. With data downloaded from [38, 39], the magnitude decrease in mobility was calculated between the average weekly mobility pre-intervention and the minimum mobility observed within six weeks. This difference was used in the model fit to provide an initial estimate for the intervention efficacy parameter during the first V-shape decrease in early 2020. The number of doses of vaccines were retrieved from Mathieu et al. [40] and population data from the World Bank [41]; these enable calculation of the percent of the populace vaccinated. To compare countries and regions, data are commonly normalized to population size, such as "per million". However, COVID-19 is strongly dependent on population density [42].Therefore, to alleviate the population density bias, the data were normalized to the built-up area [43, 44].

 Mathematical modeling of COVID-19

The epidemiology of COVID-19 was analyzed here using a mathematical model of viral dynamics, attempting to capture the mechanism of the virus infecting susceptible individuals. The three model compartments include susceptibles (S), COVID-19-confirmed individuals (I), and free virus particles (V). The model assumes that uninfected people are being made available at a constant rate (σ) and the virus productively infects them with probability βVS. Detected infected individuals are removed by quarantine at rate δI. Viral particles are released from infected individuals at rate pI and are inactivated at rate cV. These assumptions lead to the coupled nonlinear ordinary differential eqs: dSdt=σ−(1−η(t))βVS dIdt=(1−η(t))βVS−δI dVdt=pI−cV

This is the simplest epidemiological model which affords a non-trivial non-zero infection steady state. A global stability analysis can be found here [45]. Table 1 summarizes the model parameters.

10.1371/journal.pone.0281224.t001 Model parameter summary.
Parameter Symbol Units
Susceptible influx σ S∙wk-1
Infection rate constant β V-1∙wk-1
Infected removal rate constant δ wk-1
Intervention efficacy η %
Virus production rate constant p wk-1
Virus decay rate constant c wk-1
Time of intervention implementation t0 time
Time of intervention efficacy cessation t1 time
Infection decay rate r wk-1
Infection rebound rate r1 wk-1
Basic reproductive number Ro --

Intervention efficacy to block infection, via social distancing, lockdowns and/or vaccination, is parameterized here by η(t). Assuming partial and incomplete effectiveness, e.g., 0<η<1, the system will converge on a new lower steady state. The parameter σ is usually interpreted as the repopulation rate of S, though here it can also indicate the constant availability of new susceptibles to the virus as it diffuses through the population. While σ can be expanded in more elaborate terms, e.g., as a function of time or recovered individuals, we demonstrate that a constant value suffices as a first approximation to provide a dynamical endemic steady state. The mean infectious time is 1/δ. The average number of virus particles produced during the infectious interval of a single infected person (the burst size) is given by p/c. While asymptomatic carriers are thought to be efficient spreaders, they are not included here as no information is available for this group, and we assume as a first approximation that their dynamics are similar with I and probably change in tandem with the confirmed cases.

COVID-19 associated deaths can be thought of as a subset of infected persons. Indeed, death rates appear to be in a quasi-steady state with the infection rates, being consistently 1-2log lower though lagging by 4–6 weeks throughout the period studied. A Granger causality test provides statistical evidence for this observation (r = 0.95, p<0.01). For analytical simplicity, they are not modeled here explicitly.

Sustained viral propagation ensues if, and only if, the average number of secondary infections that arise from one productively infected person is larger than one (Ro >1). This is the basic reproductive number and for Eq (1) it is defined by Ro = βσp/(δc). The intrinsic growth rate constant, r, is solved for by the dominant root of the eq: r2+(δ+c)r+δc(1−Ro)=0

However, if c>>δ and r, then this can be simplified to: r = δ(Ro−1). When Ro>1, then infection rates will initially experience an exponential increase [46].

The model predicts that as the infection grows it decelerates. The infection will converge in damped oscillations to the non-trivial equilibrium: S¯=δc/βp,I¯=(Ro−1)δc/(βp),V¯=(Ro−1)(β). This dynamical steady state is obtained when the number of new infections equals the number of recovering individuals, where every productive infection generates, on average, only one more new secondary infection.

Assuming a quasi-steady state, i.e., the viral dynamics are much more rapid than the epidemiological phenomenon (p>>c), then Eq (1) can be reduced to: dSdt=σ−(1−η)β'IS dIdt=(1−η)β'IS−δI β'=βp/c with no loss of generality for the major trajectories of infection dynamics [47]. This functional form has the advantage to decrease model complexity, especially because the viral compartment is less relevant at the community-scale. Exponential decay under interventions to block infection is given by r0 = δ−(1−η)β’S0, where S0 are the number of susceptibles before interventions are implemented (t0). Under highly efficient interventions, i.e., η→1, then a minimal estimate for δ can be derived directly from the observed decay half-life of t½ = ln(2)/δ [48, 49]. This model is capable of simulating the dynamics shown in Fig 1.

10.1371/journal.pone.0281224.g001 Epidemiological dynamics under interventions to block infection.

Initially, infections rise exponentially (though national COVID-19 testing programs were also ramping up). During stringent intrvention and effective cessation of viral transmission, between t0 and t1, infection decays exponentially with a half-life of t½ = ln(2)/r0, where r0 is derived from the slope of the ln-transformed infection data. This provides a minimal estimate for the value of parameter δ, assuming partial intervention efficacy (0<η<1). This decay will decelerate reaching a lower steady state. Infections will naturally rebound upon lifting of interventions and/or loss of vaccine efficacy with a doubling time of t2 = ln(2)/r and r also calculated from the exponential up-slope. The system will converge with damped oscillations to an elevated infection steady state. This basic pattern will recur as interventions are deployed at different times. Parameter values: σ = 104 S∙wk-1, β = 10−5 V-1∙wk-1, δ = 0.64∙wk-1, η = 70%, t0 = 16, t1 = 28.

When interventions are withdrawn, lockdowns are rescinded, other NPI become lax or vaccines become ineffectual, at time t1 then infections will rebound at an exponential rate given by r = β’S1−δ, where S1 is the level of available susceptibles at t1. Crucially, r can be obtained directly from the observed slope on the semi-log graph, and its doubling-time is t2 = ln(2)/r. This expansion in infections will continue in damped oscillations returning to the steady-state.

 Estimation of the basic reproductive ratio

The basic reproductive number is based on a ratio among all five model parameters. However, the paucity of independent knowledge and accurate values for them precludes adequate approximations of Ro. To alleviate this, the relationship between the basic reproductive ratio (Ro) and the exponential growth rate (r) can be recovered such that Ro = 1+r(r+δ+c)/δc. If r+δ is small compared to c, then this approaches: Ro=1+r/δ which can be calculated directly from the exponential slopes, r0 and r, as described above.

 Parameter values and statistical analysis

To determine the initial values for model parameters, half-life decay during interventions and rebound doubling-times were calculated from the logn-transformed data of confirmed cases (weekly geometric means). Optimized values were generated by nonlinear fitting [50], minimizing the objective function J=∑i=1nlog(Oi/Pi) where Oi and Pi are the observed and expected values, for n datapoints, with the advantage of stabilizing the variance during the fit [32]. Many functional forms for intervention efficacy (η) can be used but for simplicity, generalizability and as a first approximation: η(t)={η,t0<t<t10,otherwise for each intervention wave. The observed decrease in mobility is used here be used as a proxy to estimate its value for each country [51]. Trivially, the proportion of the population needed to be vaccinated in order to block community spread, known as "herd immunity" threshold is [52, 53]: H=1−1/Ro

Longitudinal comparisons on the parameter values are performed using the Mann-Whitney u test. 95% confidence intervals, along with their statistical significance, are calculated as appropriate. Model errors (RMS) are reported. Data, simulations and results are available online at: https://github.com/Model-Lab-Net/COVID-19.

 Results Dynamics of COVID-19 epidemiology

A preliminary analysis of confirmed COVID-19 cases from 12 nations which did not implement stringent intervention policies, or were unsuccessful at their implementation, indicates widely varying rates and infection levels (Fig 2). By the end of February 2020 these nations had initial infection levels of ~10° cases per km2 with sustained infection doubling times of 1.2–1.7 weeks. Levels increased exponentially for 20±8 weeks and stabilized around a dynamical steady state with fluctuations no larger than 0.5log. Setpoints among these countries fluctuated around 100–400 cases per km2 built-up area per day. Interestingly, South Africa and Armenia exhibited spontaneously oscillating kinetics with an amplitude of approximately one order-of-magnitude, perhaps alluding to the existence of a ’limit cycle’. India exhibited one of the largest differences in infection over time, increasing to 102.5, declining to 101.5 then peaking at 103 before declining spontaneously again to 102 cases per km2 built-up area per day. Because there were no observed effective measures to block COVID-19 spread, the number of confirmed cases attained a dynamical equilibrium around which case numbers fluctuated.

10.1371/journal.pone.0281224.g002 <p>A) COVID-19 case levels for 10 nations with no or ineffective interventions increased nearly-exponentially then spontaneously stabilized around 100 cases per km2 built area. B) The Republic of South Africa and Armenia exhibit cycling infection dynamics with spontaneous orbits around a setpoint of approximately 100 cases per km<sup>2</sup> built area for 24 months. Data are normalized to built-up area to account for density effects in infection rates.</p> </caption> <graphic mimetype="image" position="float" xlink:href="info:doi/10.1371/journal.pone.0281224.g002" xlink:type="simple"/> </fig> </sec> <sec id="sec011"> <title>Dynamics during effective lockdowns

COVID-19 positive case turnover allows analysis of effective social distancing through population-level lockdowns. Non-pharmaceutical means to block new rounds of infections were initially rapid and effectively implemented. Infections begin to decay exponentially 7–10 days after the lockdown policies are implemented, with down slopes of 0.5±0.3 per week and corresponding to half-life values of 2.0±1.1weeks. Infection rates attained nadir within 4–6 weeks with average efficacy of 68% (range: 46–93%), declining 1-2log lower than pre-lockdown case numbers. Confirmed cases rebounded exponentially with doubling times of 2.3-2.6 weeks following the end of severe lockdowns. The trajectory then converged on an empirical equilibrium steady state of approximately 102-103 cases per km2 built area and with fluctuations less than 0.5log. See Fig 3.

10.1371/journal.pone.0281224.g003 COVID-19 positive confirmed cases between February 2020 and September 2021.

Data are normalized to built-up area to account for density effects in infection rates. On this scale the recurring patterns become apparent. The exponential decay during lockdowns and following vaccination is clear, as are the geometric rebound trajectories. On this scale the recurring patterns in COVID-19 community diffusion kinetics are undoubtedly evident. Shaded areas indicate the duration of aggressive interventions such as social lockdowns.

The UK as a whole had, on average, similar dynamical characteristics as its neighbors. However, the observed decay rates during lockdowns were significantly less rapid, leading to differences that will be expanded upon later. While the initial doubling times before lockdowns were similar to other nations and regions, half-lives during lockdowns were nearly twice as rapid, 1.3±0.5 vs. 2.0±1.1 weeks. Asian nations, generally, had somewhat different COVID-19 trajectories probably due to the unique measures induced in the included countries here. The Asian rebound rates differed less relative to other countries, though they were more prolonged with some clear oscillatory effects. Additionally, the setpoint infection rates in Japan and South Korea were an order-of-magnitude lower than in Europe. See Fig 3.

The USA is composed of distinct political entities, with large inter-state variation. SARS-CoV-2 surged and waned differently, peaking and ebbing at different times among the various states. Therefore, analyses of COVID-19 for the USA have been done at the state level. Ten states conformed to the inclusion criteria. The US state COVID-19 dynamics were less extreme with lockdown declines of less than 2log in most states, albeit the up- and down-slopes during lockdowns were comparable with European nations. Four states suffered elevated steady-states approximately one order-of-magnitude higher (103.2–103.5 cases per built-up area per day). See Fig 4.

10.1371/journal.pone.0281224.g004 COVID-19 positive confirmed cases in ten US states conforming to inclusion criteria from February 2020 to September 2021.

More rural and less dense populations have lower COVID-19 infection rates, in general. Data are normalized to built-up area to account for density effects in infection rates.

The earliest, most stringent and prolonged restrictions were implemented in Australia and New Zealand. Confirmed case rates were perturbed to extremely low levels and kept at about 0.5log below the lowest rates achieved in Europe for 35 months, until July 2021. Even so, these strict "Zero COVID" policies were insufficient to completely snuff out community spread. As limits were relaxed, infections surged exponentially with doubling times and equilibrium states comparable to elsewhere, even in the milieu of high vaccination coverage. See Fig 5.

10.1371/journal.pone.0281224.g005 COVID-19 positive confirmed cases for Australia and New Zealand from February 2020 to September 2021.

The strict "Zero COVID" policies implemented for 35 months kept infection levels at low rates but they rebounded when restrictions were lifted and achieved levels similar to those in Europe. Shaded areas indicate the duration of aggressive interventions. Data are normalized to the built-up area to account for density effects in infection rates.

 Modeling of early COVID-19 infection dynamics

The frequent and robust PCR testing for COVID-19 deployed in nations and regions included here allow for the mathematical analyses of infectious persons. Results of the modeling and the parameter values obtained are found in Table 2. The infection dynamics parameter values were obtained from the exponential slopes directly from the data. Initial infection expansion rate constants were 0.5-0.7 per week during February-March 2020, with corresponding doubling times of 1.2-1.6 weeks. Social distancing, lockdowns, and other such interventions resulted in exponential decay of infection rates from the pre-intervention peak values of 0.4-0.5 per week with half-life values of 1.7-2.3 weeks. This provides a maximal estimate for the case recovery rate constant parameter (δ).

10.1371/journal.pone.0281224.t002 COVID-19 kinetic characteristics in countries with no effective interventions.
Country Initial growth Time to steady state Steady State
rate t2 weeks logI ± SD
Argentina 0.2 2.8 25 2.2 ± 0.3
Armenia* 0.4 2.0 11 2.2 ± 0.4
Brazil 0.6 1.1 13 2.4 ± 0.2
Chile 0.4 1.6 13 2.4 ± 0.3
Colombia 0.3 2.4 21 2.4 ± 0.3
Costa Rica 0.5 1.3 26 2.4 ± 0.5
Ecuador 1.3 0.5 19 1.9 ± 0.1
El Salvador 0.4 1.7 17 1.8 ± 0.3
India 1.1 0.6 20 3.9 ± 0.5
Iran 0.2 4.1 45 2.3 ± 0.4
Iraq 0.4 1.7 22 2.5 ± 0.2
Mexico 0.8 0.8 17 1.7 ± 0.2
Oman 0.5 1.5 11 2.1 ± 0.4
Pakistan 0.3 2.6 25 2.3 ± 0.4
Peru 0.8 0.8 16 2.5 ± 0.3
S. Africa* 0.4 1.9 16 1.8 ± 0.4
mean 0.5 1.7 20 2.3
CI95% 0.4–0.7 1.2–2.2 15–24 2.0–2.6

*) limit cycle dynamics

Infection rates rebounded with doubling times of 2.6–3.7 weeks (range:0.6–4.4 weeks) upon lifting of the extreme social distancing measures. These represent a minimal estimate for r0. This is four-fold less rapid than the initial pre-intervention exponential growth rates. Finally, after 4–12 weeks infections reached a relatively stable setpoint level with values ranging among countries ranging between 101.3–103.4 (CI95%: 102.3–102.6) cases per km2 built-up area per day. Notably, initial pre-intervention infection rates are significantly correlated with steady state infection levels (PPMCC = 0.41, P = 0.037) alluding to the importance of the intrinsic infection rate and extent of very early viral expansion in the infective dynamical and endemic steady state.

Similar patterns were observed for 10 states in the USA and five nations in Asian regions. Israel implemented a second lockdown intervention during September to November 2020 leading to infections decaying with a half-life of 1.5 weeks and a subsequent rebound with a doubling time of 2.0 weeks; values which are only 15 and 43% more rapid than those during the primary lockdown, respectively. Markedly, not only were decay and rebound slopes among countries of similar magnitude, but they were also similar among infection waves within countries.

 Basic reproductive number (R<sub>o</sub>)

The analytical approach here contributes insight on the basic reproductive ratio for the community spread of SARS-CoV-2. In the literature reporting on COVID-19, and other epidemics, this is approximated from the initial exponential growth phase [18] and, as noted previously, represents an overestimation because it ignores the β/δ ratio. Here the "natural experiment" of the efficient impedance of viral community spread during the initial phase of the SARS-CoV-2 pandemic allows the use of the empirical rebound up-slope (r) and values for the recovery/removal rate constant (δ). The estimates for the basic reproductive number are provided in Table 3. Using experimentally established values for δ (0.4–0.5) from the decay slope during interventions to block viral expansion and ranges for r (0.2–0.3) leads to basic reproductive numbers ranging between 1.4–2.3, narrowing for a CI95% to 1.6–1.8. From this perspective, active COVID-19 infected individuals would generate approximately 1.7 new secondary infections, on average.

10.1371/journal.pone.0281224.t003 Optimized COVID-19 model parameter values.
Country Initial growth Decay Slope Intervention efficacy Rebound trajectory Steady state infection rate Reproductivenumber Herd immunity RMS
Europe rate t2 δ t½ eta r t2 logI ± SD Ro obs exp
wk-1 wks wk-1 wks % wk-1 wks number % %
Australia 0.3 2.3 0.8 1.6 53 0.41 1.7 1.8 ± 0.1 1.2 52 --* 0.17
Austria 0.5 1.4 0.4 1.7 71 0.3 2.6 2.0 ± 0.2 1.6 38 25 0.19
Belgium 0.3 1.8 0.3 2.3 66 0.3 2.3 2.3 ± 0.1 1.8 44 35 0.13
Cyprus 0.2 4.1 0.4 1.9 50* 0.3 2.6 2.2 ± 0.4 1.7 41 40 0.33
Czechia 0.6 1.1 0.5 1.5 60 0.3 2.2 2.9 ± 0.2 1.7 41 33 0.23
Denmark 0.3 2.1 0.2 3.9 69 0.2 4.6 2.0 ± .02 1.9 47 --* 0.21
Estonia 1.3 0.5 0.4 1.8 64 0.2 3.0 2.4 ± 0.2 1.6 38 26 0.19
Finland 1.0 0.7 0.3 2.1 51 0.3 2.7 1.2 ± 0.2 1.8 44 22 0.15
France 0.7 2.0 0.6 1.2 79 0.3 2.5 2.3 ± 0.2 1.5 33 33 0.18
Germany 0.5 1.3 0.4 2.0 57 0.3 2.2 2.4 ± 0.2 1.9 47 28 0.20
Greece 0.4 1.6 0.2 3.0 80 0.2 3.0 1.9 ± 0.2 2.0 50 25 0.26
Hungary 0.9 0.8 0.5 1.5 75 0.4 1.9 2.7 ± 0.3 1.8 44 37 0.26
Iceland 0.7 0.9 0.9 0.7 58 0.4 2.0 1.9 ± 0.3 1.4 29 --* 0.33
Ireland 0.5 1.5 0.4 1.6 76 0.3 2.7 2.0 ± 0.3 1.6 38 --* 0.27
Israel 0.5 1.3 0.7 1.0 70 0.3 2.2 2.6 ± 0.3 1.5 33 37 0.30
Italy 0.5 1.3 0.3 2.7 93 0.3 2.2 2.4 ± 0.3 1.8 44 29 0.17
Luxembourg 0.6 1.2 0.4 1.7 71 0.4 1.9 2.5 ± 0.1 1.9 47 30 0.15
Netherlands 0.6 1.2 0.7 1.0 67 0.3 2.0 2.7 ± 0.1 1.5 33 27 0.20
New Zealand 0.6 1.1 1.5 0.5 50 0.2 3.1 2.0 ± 0.3 1.7 40 --* 0.33
Norway 0.4 1.8 0.4 1.8 56 0.2 3.3 1.3 ± 0.2 1.5 33 22 0.19
Slovenia 0.7 0.9 0.6 1.1 73 0.3 2.6 2.6 ± 0.1 1.4 29 36 0.19
Slovakia 0.8 0.9 1.1 0.6 60 0.4 1.9 2.9 ± 0.2 1.4 29 28 0.20
Spain 0.4 1.7 0.3 2.7 93 0.3 2.0 2.3 ± 0.3 2.3 57 63 0.29
Switzerland 0.4 1.7 0.5 1.4 57 0.3 2.8 2.6 ± 0.3 1.5 33 27 0.18
UK 0.5 1.5 0.2 2.9 72 0.3 2.5 3.3 ± 0.2 2.2 55 22 0.14
England 0.6 1.1 0.3 2.4 74 0.3 2.3 3.5 ± 0.2 2.1 52 22 0.12
Wales 0.4 1.8 0.3 2.3 75 0.4 2.0 3.2 ± 0.2 2.1 52 38 0.14
Scotland 0.8 0.9 0.4 1.6 72 0.2 3.7 2.7 ± 0.3 1.4 29 31 0.17
N. Ireland 1.1 0.6 0.4 2.0 67 0.3 2.4 2.8 ± 0.2 1.8 44 16 0.18
mean 0.6 1.4 0.5 2.0 68 0.3 2.5 2.4 1.7 41 28 0.21
SD 0.3 0.7 0.3 1.1 11 0.1 0.7 0.3 0.3 8 12 0.06
Asia Japan 0.6 1.3 0.5 1.3 80 0.4 1.9 1.3 ± 0.4 1.8 44 46 0.12
Malaysia 0.5 1.5 0.6 1.2 72 0.1 6.5 2.1 ± 0.2 1.2 14 51 0.27
Singapore 0.7 1.0 0.4 1.9 73 0.3 2.2 3.8 ± 0.4 1.8 44 --* 0.27
S. Korea 2.0 0.4 1.0 0.7 46 0.2 3.5 1.4 ± 0.1 1.2 16 --* 0.23
mean 1.0 1.0 0.6 1.3 68 0.3 3.5 2.3 1.6 39 29 0.21
SD 0.7 0.5 0.3 0.5 15 0.1 2.1 0.3 0.4 13 11 0.07
US Connecticut 0.6 1.1 0.3 2.2 56 0.1 6.5 2.7 ± 0.2 1.6 36 34 0.22
Hawaii 0.8 0.9 0.9 0.8 85 0.3 2.5 1.9 ± 0.2 1.9 47 32 0.32
Illinois 0.7 1.1 0.3 2.3 63 0.1 8.8 2.7 ± 0.4 1.4 29 26 0.19
Massachusetts 0.6 1.2 0.3 2.3 69 0.1 6.9 2.8 ± 0.2 1.5 35 35 0.17
Montana 0.7 1.0 1.0 0.7 64 0.4 1.7 1.4 ± 0.4 1.4 31 23 0.29
N. Hampshire 0.7 1.0 0.4 1.7 55 0.3 2.2 2.4 ± 0.2 1.8 50 36 0.15
New Jersey 0.4 1.9 0.3 2.8 68 0.2 4.3 3.5 ± 0.1 1.7 40 35 0.14
New York 0.3 2.3 0.2 2.8 70 0.2 3.3 3.4 ± 0.2 2.0 50 34 0.11
Pennsylvania 0.4 1.6 0.3 4.4 60 0.1 8.0 3.2 ± 0.2 1.8 44 32 0.11
Rhode Island 0.6 1.1 0.2 3.3 69 0.1 6.4 3.4 ± 0.3 1.8 44 34 0.20
mean 0.6 1.3 0.4 2.3 66 0.2 5.1 2.7 1.7 41 32 0.19
SD 0.2 0.5 0.3 1.1 9 0.1 2.6 0.2 0.2 8 4 0.07
mean 0.6 1.2 0.5 2.0 67 0.3 3.2 2.5 1.7 40 0.20
CI95% 0.5–0.7 1.2–1.6 0.4–0.5 1.7–2.3 64–71 0.2–0.3 2.6–3.27 2.3–2.6 1.6–1.8 37–43 0.19–0.21

*) No rapid decreases in cases observed following vaccination.

**) Data for the fits come from several sources. See Methods section.

 Herd immunity and inhibition of infection by vaccination

Herd immunity is a threshold value at which new infections cannot perpetuate within the community and is derived from the basic reproductive number. Indeed, nearly all countries which had rapid vaccine rollouts experienced a delayed but rapid exponential decline in case numbers with efficacies of 44–99% (CI95%: 64–72). These half-life values following the distribution of SARS-CoV-2 vaccinations (CI95%: 1.3–1.7. Table A1 in S1 Appendix) are similar to those during the early NPI and lockdown interventions. The observed percent of the population vaccinated concomitant with decay in confirmed cases is between 44–55%, based on the nations and regions included here (Table 3). Now it is possible to test the previous calculation of Ro, which should be smaller than the observed values. Indeed, the observed "herd immunity" was slightly above the values derived mathematically, as expected from Eq (4), thereby supporting our earlier estimates for the basic reproductive number. Finally, these are clearly lower than reported values for Ro in other studies which seem extremely high.

 Delta variant wave rebound

In June 2021, after the large decrease in COVID-19 following national vaccination programs, COVID-19 cases rebounded spontaneously. The wave was apparently driven by the Delta variant, which became the dominant variant. This rebound was characterized by doubling times of 1.1–1.3 weeks (Table A1 in S1 Appendix). Infections attained average rates similar to those observed prior to vaccination deployment. The decay due to vaccinations and this resurgence both correspond to the trajectories observed in early 2020.

 Discussion

Infection doubling times (t2) and half-life (t½) values reveal consistent rates with extremely small variance and narrow range, longitudinally, among all countries analyzed here (Table 2). Mean doubling times for infection levels during the initial exponential phase of the pandemic were 1.0 weeks (CI95%: 0.5–2.0). These were quite robust with a caveat about the rate of deployment of testing regimes.

SIR-based models assert that infection is acquired by S in physical contact with I. Here the full model (Eq 1) includes the viral compartment which we interpret as the amount of virions being expelled by the infecteds (pI) which infect the susceptibles. However, assuming that V turns over more rapidly than I, then Eq (1) reduces to Eq (3) (i.e., with p>>c the viral compartment dynamic is limited to the slower dynamics of the infectious compartment). This is the quasi-steady state discussed earlier. Indeed, this is reasonable since COVID-19 generates large amounts of virus and is believed to be short-lived outside the host [54, 55]. Moreover, here we have the advantage to explicitly track the infected compartment vis-à-vis the confirmed cases positive for SARS-CoV-2, which is not usually possible in in vivo viral dynamics studies.

Lockdown interventions were extremely effective by inhibiting physical contact and blocking the virus from circulating. Countries with no effective social distancing measures rapidly reached a setpoint equilibrium state. Limiting movement of the population was related to intervention efficacy. Restrictions to travel of 45–93% decreased infection rates by 10-fold or more, leading to an exponential decay of >90% in confirmed cases. Importantly, this was uncorrelated with the minimal infection numbers. More stringent lockdowns do not appear to confer further inhibition to stop viral diffusion and may signify the existence of an optimum in interventions to block COVID-19. The mean associated half-life value during lockdown interventions was 2.0 weeks (CI95%: 1.7–2.4) with no statistically significant difference among the nations and regions studied here. The epidemiological interpretation of this measure is the maximal value for the recovery rate of infected individuals.

As distancing policies were lifted, infections rebounded exponentially as viral diffusion over the social network is no longer perturbed. Intrinsic doubling times can, therefore, be determined empirically by the up-slope on a semi-log graph. The observed doubling time was consistently 2.5±0.7 weeks in European countries. Asian nations included here had values of 3.5±2.1 weeks, perhaps owing to their stricter regulations and higher compliance. In the states of the United States the value was even higher at 5.1±2.6 weeks, perhaps alluding to lower compliance.

Taken together, we are able provide a maximal estimate for the basic reproductive number from analysis of the rebound and decay rates in the V-shape dynamics during perturbations on the system to block infections. Ro is overall quite consistent with a mean value of 1.7 (CI95%: 1-6-1.8), due to the invariance of the model parameters. Spain, Greece, and Britain (i.e., England and Wales) were areas of relatively elevated infectivity with values of 2.3, 2.0 and 2.1, respectively. An important outcome of this calculation is the elucidation of the epidemiological "herd immunity" threshold and the novel ability to verify it empirically from the vaccination coverage.

During emergent pandemics, estimates of the basic reproductive number tend to be overestimated. Early COVID-19 studies reported very high values [56, 57]. Our estimates for Ro pertaining to SARS-CoV-2 vary only slightly during waves of COVID-19, which would make sense if the dynamical properties of the infection did not appreciably change. Interestingly, they are comparable to historical influenza pandemics [58] and commensurate with seasonal influenza outbreaks [59]. Although these estimates are substantially lower than those reported elsewhere for COVID-19, they agree with other studies [60].

Vaccination deployment against SARS-CoV-2 had a dramatic effect on infection rates. Confirmed cases decayed exponentially with a mean half-life value with similar rates as during the interventions of social distancing and lockdowns, after achieving the herd immunity threshold. For example, Israel with its early and rapid vaccination program experienced a half-life of 1.0 weeks in confirmed cases once 45% of the population was immunized. This agrees with the prediction given by the approximations for R0 based on Eq (3).

Following the achievement of herd immunity, after approximately 30 weeks, infections spontaneously rebounded again as the delta-variant emerged. The observed escape trajectory was empirically equivalent to the rebound trajectories following the interventions and with doubling times approximately every 1.2±0.3 weeks, similar to the post-intervention rebound doubling times. Interestingly, the Delta variant emerged in every nation included here within 4 weeks, surprising due to the low volume of international travel. Finally, infection rates returned to similar levels as the pre-vaccination setpoint and invariant among the sampled countries.

Although infection rates tended to initially increase exponentially when numbers were low, they quickly saturated to a level of 102−103 confirmed cases per km2 built-up area per day. This was reached in nearly all nations and regions within 4–6 weeks, even in absence of interventions. Even New Zealand and Australia with strict and highly effective lockdowns rapidly reached this level of infections with the lifting of social distancing measures. Such observations, seen everywhere, suggest a basic, perhaps fundamental, shared epidemiological dynamic and the importance of population density for the spread of SARS-CoV-2 [42, 61, 62].

As we have shown, waves of both infection and suppression can define COVID-19. Our concluding perspective views the infection data decomposed into their wavelet phases and modeled with the generalized multi-logistic model [63]. This approach allows derivation of the saturation level of cases as well as the "characteristic time" (F044t) denoting how long the infection takes to increase from 10% to 90% of its extent. While data for many nations and regions resolve neatly into a succession of waves, Israel is unusual in having excellent data for seven waves of infection (so far), as well as companion data about societal responses and suppression for the first five waves. Fig 6 shows the first five infection waves and their durations ranging from 4.4 to 10.6 weeks. The sequence of waves suggests the extremely dynamic interaction of COVID-19, generating new variants, with the social and medical context, including lockdowns, distancing, and vaccines. Predicting new waves remains an unsolved challenge.

10.1371/journal.pone.0281224.g006 Logistic curves for first five waves of COVID-19 in Israel and the number of weeks each waves took to run its course.

To conclude, the dynamical properties of COVID-19 epidemiology are conserved with consistent kinetic patterns with little variation during multiple waves of infection and globally among nations and subnational regions. Nations and regions which implemented interventions sufficient to block community spread effectively experienced a rapid decline in confirmed cases. However, with lifting of interventions, rates rebounded to the previous high infection rates and attained a relatively stable empirical steady state. For COVID-19, societies so far appear to face a choice between relatively high oscillations involving waves of suppression and infection and lesser oscillations around an endemic setpoint. The approach presented here, based on the viral dynamics paradigm, allows derivation of fundamental measures vital to policy such as the basic reproductive number and the magnitude of intervention efficacies. Values for Ro derived here of 1.6–1.8 are maximal estimates and lower than other reports. Information on variables of interest for policy normally difficult to obtain is available through this approach and may suggest monitoring strategies efficient for accurate determination of the dynamical properties of future pandemics.

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Technol Forecast Soc Change. 1994;47: 89–102. doi: 10.1016/0040-1625(94)90042-6 10.1371/journal.pone.0281224.r001 Decision Letter 0 Lal Rajnesh Academic Editor 2023 Rajnesh Lal This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 0

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PONE-D-23-01297Trajectories of COVID-19: a longitudinal analysis of many nations and subnational regionsPLOS ONE

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We will update your Data Availability statement to reflect the information you provide in your cover letter.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Partly

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

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3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: No

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors examined transmission of COVID-19 using data from various countries. They observed the reduction, rebound, and convergence of confirmed cases according to government policies and suggested a mathematical model that could simulate these policies. It has been shown that the estimated values from this model agree with the observed phenomena. In particular, it was a notable result that, contrary to my expectations, the decay rates d and rebound rates r0 were similar in all countries and regions studied during the pandemic. The author has did a good work. hope my comments would help the authors to improve their work.

Major:

1. First, in the introduction section, it would be good to clarify the differences and novelties of this study from related studies. The model used in this study is a basic model used in virus dynamics. This model represents the intracellular spread of the virus in humans. It would be nice to have a detailed reason why this model was applied to the COVID-19 spread simulation. (I recommend introducing related studies, summarizing the results, and writing more prominently the advantages of using this model.)

2. It would be easier to see if there was a table that summarized the description of the parameters, the values, and references used in the model.

3. Regarding comment 2, it would be useful to explain what role a group of free virus particles('V' compartment) play in the COVID-19 spread model in this compartment model. In the general SIR model, it is assumed that an individual belonging to group S is infected by contact with an individual belonging to group I. In this study, however, it is not. The strength of this study is the use of the virus dynamics model for the spread of COVID-19, so it is necessary to explain how the compartment used in the model applies to the spread of COVID-19 in order to emphasize this. (It is a basic model, but it would be nice to describe the dynamics of this model in the Appendix or provide related references.)

4. Line 119 “uninfected people are being made available at a constant rate (\\sigma)”.

Why is the number of uninfected people continuously increasing in the sentence above? (In general, sir-based models assume population growth at birth.)

5. How did you get the lockdown efficacy from Table 2 on page 12? This appears to be related to \\eta in eq(1). However, this \\eta represents not only lockdown, but overall intervention efficacy such as social distancing and vaccination. If the intervention efficacy is interpreted by limiting it to Lockdown, the estimated values in Table 2 are the optimal control solutions for \\eta?

Minor:

1. It would be good to add a diagram representing Eq(1) (lines 124-126).

2. Github address on line 191 is not connected.

3. In Table A1 of Appendix, the post-vax steady state is shown as blank.

4. Since the figures in appendix are in black and white, it is difficult to distinguish between countries.

Positive:

1. I like table2, which summarizes the results for many countries.

2. Figure 6 shows the waves of COVID-19 in Israel well, and at the same time it is good to grasp the "characteristic time" at a glance.

Reviewer #2: The manuscript considers an interesting topic about COVID-19 for several countries in the world. However, this paper is not easy to understand due to errors in writing. The quality of the figures is not high standard and hard to get the information described. The following are some of the errors or unclear descriptions needed to improve.

Remove the dot at the end of title.

Line 120 on page 5: “Deaths can be thought of as a subset …and neglected for the purpose of this study.” If the deaths are significant, I do not think we can neglect. Can you provide evidence from the data you have about this?

Line 136 on page 5: r^2+(δ+c)r+δc(1-R_0 ) should be an equation following the text.

Line 158 on page 6: No Figure 1 is shown here but it is shown at the end. What are the initial values used in this figure?

Line 152: what is t_0? From Figure 1, it is the time exponential decay starting. Please give an explanation.

Line 153 on page 5: insert a space between δ and “can”.

Line 180 on page 7: what is “(Berkeley Madonna v8)”?

Line 196, page 8: the lines used in Fig. 2 for different countries are not readable and time intervals are not the same. The first two are between March 2020 and March 2021 but the last one is between March 2020 and February 2022. When we compare the varying rates, we may use the same time intervals.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

Submitted filename: Comments to the author.docx

Submitted filename: Review report for PONE-D-23-01297.pdf

 10.1371/journal.pone.0281224.r002 Author response to Decision Letter 0 Submission Version 1

5 May 2023

We thank the reviewers for many helpful comments. Below we respond to each issue.

The file "Response to Reviewers" is formatted for better clarity.

Reviewer #1:

Major:

1. First, in the introduction section, it would be good to clarify the differences and novelties of this study from related studies. The model used in this study is a basic model used in virus dynamics. This model represents the intracellular spread of the virus in humans. It would be nice to have a detailed reason why this model was applied to the COVID-19 spread simulation. (I recommend introducing related studies, summarizing the results, and writing more prominently the advantages of using this model.)

Response: Thank you for this important comment. We added a paragraph to the introduction to distinguish better the contribution of our research to the literature. We added references to 4 papers which deal with the large modeling endeavor during COVID-19. Our aim, however, is not a review paper (and PLOS One is not for review papers), so our summary is brief.

2. It would be easier to see if there was a table that summarized the description of the parameters, the values, and references used in the model.

Response: We added a table summarizing the parameters.

3. Regarding comment 2, it would be useful to explain what role a group of free virus particles ('V' compartment) play in the COVID-19 spread model in this compartment model.

In the general SIR model, it is assumed that an individual belonging to group S is infected by contact with an individua l belonging to group I. In this study, however, it is not. The strength of this study is the use of the virus dynamics model for the spread of COVID-19, so it is necessary to explain how the compartment used in the model applies to the spread of COVID-19 in order to emphasize this.

Response: We agree and added a paragraph to the beginning of the Discussion section to highlight this. To recap, the virus compartment is important for viral diffusion. However, under certain assumptions (e.g., p>>c is large), then the main model dynamics can be reduced to a function of S and I. This is the quasi-steady state.

(It is a basic model, but it would be nice to describe the dynamics of this model in the Appendix or provide related references.)

Response: We agree. We therefore cite in the text:

De Leenheer P, Smith HL (2003). Virus Dynamics: A Global Analysis. SIAM J Appl Math.;63: 1313–1327, which provides a full analysis of the dynamics of Eq.1, and we moved the reference to a more prominent place in the text (directly after Eq.1).

4. Line 119 “uninfected people are being made available at a constant rate (\\sigma)”.

Why is the number of uninfected people continuously increasing in the sentence above? (In general, sir-based models assume population growth at birth.)

Response: The reviewer is astute to recognize tan issue here. Sigma is indeed a constant rate of influx of susceptible persons. One interpretation is: The virus may be regional and geographic diffusion (not analyzed here) could allow a constant availability of persons susceptible to the virus. Alternately, sigma could be a function. As an initial approach to the problem, we believe a constant parameter which allows a steady state suffices.

5. How did you get the lockdown efficacy from Table 2 on page 12? This appears to be related to \\eta in eq(1). However, this \\eta represents not only lockdown, but overall intervention efficacy such as social distancing and vaccination. If the intervention efficacy is interpreted by limiting it to Lockdown, the estimated values in Table 2 are the optimal control solutions for \\eta?

Response: Another good point. We updated the text to rename “lockdown efficacy” as “intervention efficacy.” The phrase was used as a shorthand for infection suppression during the large decreases of infections concomitant with the lockdowns in early 2020. We have made the needed changes to the manuscript.

Indeed, eta is the efficacy of whatever intervention is implemented. For an initial estimate for the interventions which decreased infections in early 2020, which appear concomitant with the lockdowns, we used public data downloaded from Apple and Google [34,35]. The magnitude decrease in mobility was calculated between the average weekly mobility pre-lockdown and the minimum mobility observed within 6 weeks. This is a minimal estimate for eta in this interval.

In early 2021 infections decreased exponentially in many countries, concomitant with vaccinations, and the value for eta was estimated by the fitting algorithm.

Finally, our interest is in obtaining values of r0 and r1 to capture the main trajectories and calculate R0 rather than in the specific mechanisms and policies of interventions and their relative efficacies.

In the section on "Methods- Lockdown interventions, mobility and vaccination coverage" we use mobility data as a minimal estimate for the cumulative effects of all interventions to block community infection early in 2020 (line 109-110). However, it seems reasonable that eta is in large part due to severe lockdowns because the exponential decays began concomitantly with those lockdowns in all countries analyzed here, and viral diffusion recommenced following the lifting of the lockdowns. While other interventions (distancing, masking, hygiene, etc.) may impede contraction of the virus, the decay rates suggest that none would have an effect of the same magnitude. When social distancing policies were rescinded in 2022, no exponential increases of >1log were observed. Much research will surely try to dissect the effectiveness of different measures.

Minor:

1. It would be good to add a diagram representing Eq(1) (lines 124-126).

Response: Figure 1 was a qualitative illustration for the dynamics of Eq.1. We replaced it with a numerical simulation and provide the parameter values used to derive it.

2. Github address on line 191 is not connected.

Response: We neglected to make the repo public. It is now available. Apologies.

3. In Table A1 of Appendix, the post-vax steady state is shown as blank.

Response: Fixed

4. Since the figures in appendix are in black and white, it is difficult to distinguish between countries.

Response: Fixed, now in colors.

 

Reviewer #2:

The manuscript considers an interesting topic about COVID-19 for several countries in the world. However, this paper is not easy to understand due to errors in writing.

Response: No other readers have commented to us (both native English speakers) about language difficulties. We hope that the 20+ specific improvements made because of the two reviews help readers. It is clear that both reviewers understood the central arguments in the paper.

The quality of the figures is not high standard and hard to get the information described.

Response: Improved, including addition of colors. Please note that we cannot be responsible for changes performed by PLOS servers. Therefore, we have not removed the figures from the text in the version.

The following are some of the errors or unclear descriptions needed to improve:

Remove the dot at the end of title.

Response: Fixed.

Line 120 on page 5: “Deaths can be thought of as a subset …and neglected for the purpose of this study.” If the deaths are significant, I do not think we can neglect. Can you provide evidence from the data you have about this?

Response: Deaths are observed to be 1-2log lower and lag 4-6 weeks behind infections. They seem to be in a quasi-steady state with infections. A Granger causality analysis provides the statistical evidence for this. We do not mean that deaths are unimportant, only that the focus on cases suffices for our argument. We added a few words to explain better our focus on cases.

Line 136 on page 5: r^2+(δ+c)r+δc(1-R_0 ) should be an equation following the text.

Response: Fixed.

Line 158 on page 6: No Figure 1 is shown here but it is shown at the end. What are the initial values used in this figure?

Response: We have moved the reference for Figure 1 closer to the Figure itself. Also, we have replaced it with a numerical simulation, and the parameter values are included in the caption.

Line 152: what is t_0? From Figure 1, it is the time exponential decay starting. Please give an explanation.

Response: We added "…before social distancing measures are implemented (t0)."

Line 153 on page 5: insert a space between δ and “can”.

Response: There is a space there but added one more.

Line 180 on page 7: what is “(Berkeley Madonna v8)”?

Response: It is a nonlinear ODE fitting software package. We replaced it with a citation.

Line 196, page 8: the lines used in Fig. 2 for different countries are not readable and time intervals are not the same. The first two are between March 2020 and March 2021 but the last one is between March 2020 and February 2022. When we compare the varying rates, we may use the same time intervals.

Response: We have tried to make the Figure clearer. For clarity we split the first group of countries from South Africa and Armenia (there was also a typing mistake fixed now). The calculated rates are presented in Table 1 for comparison.

Submitted filename: PLOSone responses.docx

 10.1371/journal.pone.0281224.r003 Decision Letter 1 Lal Rajnesh Academic Editor 2023 Rajnesh Lal This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 1

31 May 2023

PONE-D-23-01297R1Trajectories of COVID-19: a longitudinal analysis of many nations and subnational regionsPLOS ONE

Dear Dr. Burg,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please address a minor correction as pointed out by Reviewer 2- "Line 158 on page 6: “The intrinsic growth rate constant, r, is solved for by the dominant root of the equation: r^2+(δ+c)r+δc(1-R_0 )” is incorrect.

Since it is referred to as an equation, it should read "r^2+(δ+c)r+δc(1-R_0 )=0".

Please submit your revised manuscript by Jul 15 2023 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

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We look forward to receiving your revised manuscript.

Kind regards,

Rajnesh Lal, Ph.D

Academic Editor

PLOS ONE

Journal Requirements:

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Additional Editor Comments:

Please address a minor correction as pointed out by Reviewer 2- "Line 158 on page 6: “The intrinsic growth rate constant, r, is solved for by the dominant root of the equation: r^2+(δ+c)r+δc(1-R_0 )” is incorrect.

Since it is referred to as an equation, it should read "r^2+(δ+c)r+δc(1-R_0 )=0".

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors wonderfully addressed my concerns and have completed a analysis across many countries. I believe it can be published in PLOS ONE.

Reviewer #2: The manuscript considers an interesting topic about COVID-19 for several countries in the world. The revised version is improved in explanations and graphs. Except for the following issue, I did not find anything else needed to be fixed.

Line 158 on page 6: “The intrinsic growth rate constant, r, is solved for by the dominant root of the equation: r^2+(δ+c)r+δc(1-R_0 )” is incorrect. This was one of the comments for the initial submission, but this problem has not been fixed.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Youngho Min

Reviewer #2: No

**********

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Submitted filename: Review report for PONE-D-23-01297 R1.pdf

 10.1371/journal.pone.0281224.r004 Author response to Decision Letter 1 Submission Version 2

7 Jun 2023

We thank the reviewer for his comment. Below is our response.

Reviewer #2:

Minor:

1. Line 158 on page 6: “The intrinsic growth rate constant, r, is solved for by the dominant root of the equation: r^2+(δ+c)r+δc(1-R_0 )” is incorrect. This was one of the comments for the initial submission, but this problem has not been fixed.

Response: We thank the reviewer and apologies for missing this on the first round. It is now corrected:

r^2+(δ+c)r+δc(1-R_0)=0

Submitted filename: PLOSone responses2.docx

 10.1371/journal.pone.0281224.r005 Decision Letter 2 Lal Rajnesh Academic Editor 2023 Rajnesh Lal This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 2

7 Jun 2023

Trajectories of COVID-19: a longitudinal analysis of many nations and subnational regions

PONE-D-23-01297R2

Dear Dr. Burg,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Rajnesh Lal, Ph.D

Academic Editor

PLOS ONE

 10.1371/journal.pone.0281224.r006 Acceptance letter Lal Rajnesh Academic Editor 2023 Rajnesh Lal This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

13 Jun 2023

PONE-D-23-01297R2

Trajectories of COVID-19: a longitudinal analysis of many nations and subnational regions

Dear Dr. Burg:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Rajnesh Lal

Academic Editor

PLOS ONE