key: cord-0732085-rjvdaifb
authors: Santana-Cibrian, M.; Acuna-Zegarra, M. A.; Velasco-Hernandez, J. X.
title: Lifting mobility restrictions and the induced short-term dynamics of COVID-19
date: 2020-07-24
journal: nan
DOI: 10.1101/2020.07.23.20161026
sha: 420730f8f702dcc8d0784ab8a2f21283adc01ac7
doc_id: 732085
cord_uid: rjvdaifb

SARS-CoV-2 has now infected 15 million people and produced more than six hundred thousand deaths around the world. Due to high transmission levels, many governments implemented social-distancing measures and confinement with different levels of required compliance to mitigate the COVID-19 epidemic. In several countries, these measures were effective, and it was possible to flatten the epidemic curve and control it. In others, this objective was not or has not been achieved. In far to many cities around the world rebounds of the epidemic are occurring or, in others, plateau-like states have appeared where high incidence rates remain constant for relatively long periods of time. Nonetheless, faced with the challenge of urgent social need to reactivate their economies, many countries have decided to lift mitigation measures at times of high incidence. In this paper, we use a mathematical model to characterize the impact of short duration transmission events within the confinement period previous but close to the epidemic peak. The model describes too, the possible consequences on the disease dynamics after mitigation measures are lifted. We use Mexico City as a case study. The results show that events of high mobility may produce either a later higher peak, a long plateau with relatively constant but high incidence or the same peak as in the original baseline epidemic curve, but with a post-peak interval of slower decay. Finally, we also show the importance of carefully timing the lifting of mitigation measures. If this occurs during a period of high incidence, then the disease transmission will rapidly increase, unless the effective contact rate keeps decreasing, which will be very difficult to achieve once the population is released.

are seeing that the epidemic has entered a plateau, and when the plateau ends it gives place to a 23 new epidemic growth phase. We exemplify our model applying it to the case of Mexico City. We 24 analyze the interrelation between the date of maximum incidence and heightened incidence events 25 taking place before the projected maximum. The manuscript is organized as follows. In Section 2, 26 we formulate the mathematical model and give some general results arising from it; in Section 3, 27 we describe the Mexico City data that is used to exemplify our model results and present a brief 28 analysis of the impact of mitigation measures. In Section 4, we use our proposed model to gen-29 erate scenarios that describe the impact of superspreading events associated with holidays, on the 30 shape of the epidemic curve, and explore the probable consequences of lifting mitigation measures. 31 Finally, the discussion and conclusions are presented in Section 5. Figure 1 : S , E, I a , I s , I r , R, D represent the populations of susceptible, exposed, asymptomatically infected, symptomatically infected, reported infected, recovered and dead individuals, respectively. Previous to the mitigation measures, the epidemic follows the dynamics represented by the blue diagram. Once the mitigation measures are implemented (on March 23 in the case of Mexico), the population splits into two: those who comply with the control measures (green box) and those who do not (orange box). The dashed line connecting the green and orange boxes represent the compliance-failure rate ω(t).

following system of ordinary differential equations:

S N * − γE, I a = ργE − η a I a , I s = (1 − ρ)γE − (η s + δ s ) I s , I r = δ s I s − δ r I r , R = η a I a + η s I s + (1 − µ) δ r I r , D = µδ r I r ,

(2. 1) where N * = S + E + I a + I s + R. Note that the I r compartment does not participate in transmission since, it is assumed, once confirmed, the cases are effectively isolated. b represents the effective contact rate, 1/γ the incubation period, ρ the proportion of asymptomatically infected individuals, 1/η a and 1/η s the periods from symptoms onset to recovery for symptomatically and asymptomatically infected individuals, δ s the rate at which a symptomatically infected individuals becomes a reported (confirmed) case, 1/δ r the time from confirmation to recovery or death and µ is the proportion of those reported cases that die. The basic reproductive number of the system 2.1 is given by:

where the first term represents the number of new cases produced by asymptomatically infected individuals and the second term represents the number produced by symptomatically infected individuals. After mitigation measures are implemented (T ), the population is split into two groups: one constituted by those individuals who comply with the measures that is referred to as the confined group, and another constituted by those individuals who do not, that is called the unconfined group, either because they disobey the social-distancing guidelines or because they belong to an strategic sector of the economy. The dynamics in the second time interval are given by:

The index i = 1 gives the population that complies with the control measures, while i = 2 indicates those who do not. Note that the compartments I r , R, 4 . CC-BY-NC-ND 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 July 24, 2020. . and D are common to both groups. Following [5] , the effective contact rates are different for each group and are given by:

3) assumes decreasing contact rates where the parameters q 2i ≤ q 1i , i = 1, 2, represent a reduction in the baseline effective contact rate b. T is the time at which the effective contact rates start to act and θ j , j = 1, 2, are the duration of the contact reduction process. Finally, we define the compliance-failure rate ω(t) as a step function, that is,

where κ > 1 is the increase in the compliance-failure rate relative to the baseline value ω 0 (κ = 1), 1 T * is the time at which the perturbation starts and θ ω is the duration of the perturbation. This model 2 admits several periods during which ω(t) may act but here we illustrate only one period.

The inclusion of short-term superspreading events into equations 2.3 and 2.4 renders scenarios 4 in which a plateau-like behavior appears. We show three scenarios: (I) long plateau with slight 5 decrease and rebound; (II) shorter plateau with a later decreasing trend in the epidemic-curve; 6 (III) long plateau with a later decreasing epidemic-curve. Each scenario is subdivided into three 7 sub-cases defined as follows: 8 • Scenario X.a: Increase κ = 3 of the baseline value ω 0 .

• Scenario X.b: Increase κ = 5 of the baseline value ω 0 .

• Scenario X.c: Increase κ = 7 of the baseline value ω 0 .

where X = I, II, III. Fixed parameter values are given in Table 1 

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The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20161026 doi: medRxiv preprint 2.1 (I) Long plateau with slight decrease and rebound 1 Setting q 1i as in Table 1 , in (2.3) we put for both groups, θ 1 of approximately two weeks and θ 2 of 2 about 3 months with q 21 = 0.3 (decrease of 70% for the confined group), q 22 = 0.4 (decrease of 3 60% for the unconfined group). 4 Figure 2 illustrates the case where the perturbed epidemic curve shows a slight decrease imme-5 diately after the perturbation relative to the baseline curve. Here, it is observed that as κ increases, 6 the incidence increases again. Scenario Ic is the worst-case scenario where the growth rate keeps 7 growing. Scenario Ia illustrates the appearance of a plateau-like behavior followed first by a slight 8 decline and then growth again. The length of plateau is preserved for several weeks. Setting q 1i as in Table 1 , in (2.3) we put, as before, θ 1 of about two weeks and θ 2 of about three 11 months with q 21 = 0.3 (decrease of 70% for the confined group), q 22 = 0.3 (decrease of 70% for 12 the unconfined group).

13 Figure 3 illustrates the case where the epidemic curve decreases after the end of the interval 14 of maximum incidence. Scenario IIc is the worst-scenario since where after the perturbation, the 15 epidemic peak is pushed higher and later in time. Scenario IIa, on the other hand, is comparatively 16 benign. The peak is still reached as projected on the baseline case, and then the epidemic curve 17 decreases at a slower rate than the baseline. Finally, scenario IIb shows the appearance of a plateau-18 like behavior. In this case, when the peak is reached, the incidence curve does not show a significant 19 decay but rather, enters a sustained phase of maximum incidence that lasts several weeks. 20 6 . CC-BY-NC-ND 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 July 24, 2020. Figure 3 : Impact of high mobility, for 12 days, on the baseline epidemic curve (blue line). The increase period starts 23 days before of the baseline epidemic curve peak. (A) Newly reported cases proportion per day, and (B) daily deaths proportion per day.

1 Set q 1i as in Table 1 , and, in (2.3) we put, for both groups, θ 1 of about two weeks but now θ 2 of 2 about four months with q 21 = 0.2 (confined group), q 22 = 0.3 (non-confined group).

3 Figure 4 illustrate the case where the epidemic curve decreases after the peak. Scenario IIIa 4 shows the appearance of a plateau that lasts some months after the date of the baseline peak. On 5 the other hand, scenarios IIIb and IIIc shows a runaway epidemic with a much higher peak than 6 that of the baseline curve. In this section, we exemplify our model results with the case study of Mexico City. We start the 2 analysis with some context. On March 23, 2020, social distancing measures where implemented 3 in Mexico to slowdown the spread of COVID-19 pandemic, mainly focused in closing schools 4 and some non-essential activities. One week later, on March 30, 2020, a Sanitary Emergency was 5 declared to last until April 30, 2020, a date that was later extended to June 01, 2020. These mea-6 sures aimed to "flatten the curve", meaning to lower the incidence to ensure that critical cases 7 would remain under manageable levels. On April 16, 2020, the federal government announced 8 that for Mexico City, the day of maximum incidence would occur on May 08-10, 2020 [8, 9] , and 9 consequently, the date for lifting mitigation measures was announced to start on June 01, 2020. 

This plateau may have appeared as a consequence of two important holidays, children's day, and 18 mother's day, occurring approximately 10-14 days before the expected date for the peak of maxi-19 mum incidence . We postulate that these events are an important factor that explains the observed 20 quasi-stationary epidemic trend that the data shows at present (mid-July 2020). In this section 21 we attempt an explanation of the probable causes of such behavior. Furthermore, we explore the Richards model is an extension of a simple logistic growth model that is an standard tool commonly used to predict cumulative COVID-19 cases in China (see, for example, [11] ). In this model the curve of cumulative cases, C(t), is described by the solution of

where r is the epidemic growth rate, K is the final epidemic size, and a is a parameter that accounts 29 for the asymmetry of the epidemic curve. We estimate the Richards curve for two different periods, April 30, 2020. The parameters a, r, and K for each period are estimated using Bayesian inference 32 8 . CC-BY-NC-ND 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 July 24, 2020. . https://doi.org/10.1101/2020.07.23.20161026 doi: medRxiv preprint [12, 13] . Technical details can be found in Appendix A. We limit the fitting of the Richards curve 1 to April 30 since on that date an important event of increased transmission started that violated the 2 hypothesis that population conditions for transmission were essentially unchanged since March 23, 3 2020. In terms of our model, unchanged conditions mean ω(t) = ω 0 for all t. both effective contact rates are q 11 = 0.6 (40% reduction in the confined group), q 12 = 0.7 (30% 26 reduction in the unconfined group), respectively. The duration θ 2 and magnitudes (q 2i ) of the 27 second reductions (q 2i ) depend on the study scenario as described in the next section.

Once under confinement, individuals may abandon the isolation with a compliance-failure rate 29 ω(t). This is the parameter we use as a proxy for population mobility. ω(t) is a time-dependent 30 rate: we assume that increased mobility lasts only for a period of θ ω days, with a background 31 compliance-failure rate ω 0 that in atypical events is increased by a factor κ (equation 2.4). We take 32 ω 0 = 0.005/day as a baseline value [5] , that is, given the exponential nature of this rate, this means 33 that 50% of the confined populations will stay there at least for 4 months, while the other 50% will 34 abandon confinement earlier. CC-BY-NC-ND 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 July 24, 2020. . In Mexico, there were two important holidays (in terms of population mobility) within the 10 period of confinement: April 30, 2020, children's day, and May 10, 2020, mother's day. Population 11 mobility increased these days as evidenced by the intantaneous reproduction number (Figure 6 ). 12 Note that R t shows a slight increase just around April 30, 2020, and May 01, 2020. We center 13 our attention on the effect of increased mobility within the period from April 29, 2020, to May 10, 14 2020. 15 10 . CC-BY-NC-ND 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 July 24, 2020. Mexico City epidemic has shown a plateau-like behavior since middle-May ( Figure 8) . As 1 Figure 6 shows, a significant increase in mobility occurred in the period from April 29, 2020, 2 to May 10, 2020. Therefore, we set the start of the perturbation, T * , as April 29, 2020, and its Table 1 . Figure 7 shows simulations for κ = 3. We compare our results with the reported confirmed 6 cases and deaths per day. Our projections go until July 01, 2020. Blue and yellow bars represent 7 confirmed and suspect+confirmed cases, respectively. Observe that the mortality is not well de-8 scribed by our model (Figure 7 ). This observed decrease in mortality is intriguing. It could be due 9 to enhanced treatment of grave cases, shifting of the morbidity towards age classes with a lower 10 risk of death or, perhaps, incomplete mortality records and reporting time delays. Since we are 11 comparing with the absolute number of deaths, testing has little impact in this case. measures. The concern is that an increase in the number of cases may occur as it has occurred in 15 the USA. In this section, we explore the possibility of this scenario for Mexico City.

Mexico's federal government developed an epidemiological panel to oversee the reactivation 17 11 . CC-BY-NC-ND 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 July 24, 2020. We model this process by updating the number of individuals at a fixed time (T * ) in equa-12 tion (2.2), as x 1 (t i ) = (1 − q)x 1 (t i−1 ) and x 2 (t i ) = qx 1 (t i−1 ) + x 2 (t i−1 ) where t i = T * , x 1 and x 2 13 represent the susceptible, exposed, asymptomatically infected and symptomatically infected for 14 the confined and unconfined groups, respectively. change from red to orange light). We explore the short and medium-term effects of the lifting of the 19 mitigation measures on the epidemic curve. For each case, we project three scenarios: 25%, 50%, 20 or 75% of the confined population abandoning confinement. We present three curves that have a 21 good fit to Mexico City data. These curves are obtained for the parameters shown in Table 2 . CC-BY-NC-ND 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 July 24, 2020. . Table 2 : Parameters for the lifting of mitigation measures scenarios. described by q 11 = 0.6, q 21 = 0.3 (for the confined group), and q 12 = 0.7, q 22 = 0.4 (for the 1 not-confined group).

2 Figure 8 shows the behavior of reported cases I r . Solid lines are the baseline, while dotted 3 lines represent the perturbed system (after lifting restrictions on June 29, 2020). Note that, as the 4 percentage of the confined population is decreased, the incidence increases to higher levels. Also, 5 for the scenario of 25% of the population leaving confinement (i.e., increasing its effective contact 6 rate), we obtain a good data fit. 13 . CC-BY-NC-ND 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 July 24, 2020. .

Lifting on June 29, 2020, with decreasing effective contact rate 1 We explore the scenario where the effective contact rate continues to decline even after partially lift-2 ing mitigation measures. Here, we consider that the effective contact rates in both groups decrease 3 until August 01, 2020, and then both remain constant. For this scenario, we set q 11 = 0.6, q 21 = 0.1 4 (confined group), and q 12 = 0.7, q 22 = 0.2 (unconfined group). Other parameter values are similar 5 to those employed in the above subsection. Figure 9 : Lifting of the mitigation measures on June 29, 2020, but both effective contact rates (for lockdown and non-lockdown environments) decrease until August 01, 2020. A) Dynamics, when nobody leaves confinement, B) dynamics when 25% of the confined population leaves confinement, C) if 50% of the confined population leaves confinement, and D) when 75% of the confined population leaves confinement.

14 . CC-BY-NC-ND 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 July 24, 2020. . https://doi.org/10.1101/2020.07.23.20161026 doi: medRxiv preprint

1 Figures 8 and 9 show that, as the percentage of the population under confinement decreases, the 2 number of new reported cases also increases. A similar behavior is observed for the dynamics of 3 daily deaths. City, the federal government originally forecasted the peak of maximum incidence to occur by May 11 8-10 , 2020 and the date for lifting mitigation measures was set to start on June 1, 2020. The peak, 12 however, occurred much later in the month as the data and independent models have shown [5, 6, 7].

The epidemic in Mexico City shows that it entered a period of constant incidence since around 

A characteristic of our model that we think is worth underlining, is that the effective contact rates 25 are time-varying which, we think, constitutes a realistic approximation since we are looking at 26 population averages. For example, adopting safe behaviors, such as wearing face masks, involves 27 a learning process; it does not suddenly occur, rather it takes time for the face masks to be adopted 28 by a significant proportion of the general population. Our results indicate that after lifting mobility 29 restrictions, a decrease of the effective contact rate should continue in order to force the epidemic 30 curve to make a downward turn. We interpret this continuing decrease in the effective contact rate 31 as related to the use of face masks, social-distancing, washing hands, etc.

Mathematical models are essential in the fight against COVID-19. They are tools for evaluating 33 mitigation measures, estimating mortality and incidence, and projecting scenarios to help public 34 health decision-makers in their very difficult and important task of controlling the epidemic. In this 35 paper, we have used mathematical models to evaluate and generate scenarios. Although precise 36 forecasting is not our aim, we consider that these results can be helpful for decision-makers. 37 15 . CC-BY-NC-ND 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 July 24, 2020. CC-BY-NC-ND 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 July 24, 2020. . 

. CC-BY-NC-ND 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 July 24, 2020. . Let Y j , for j = 1, 2, ...n, be the number of observed new daily cases at time t j , with t j given in days since the first reported case started symptoms. We assume that Y j follows a Negative Binomial distribution with mean value X(t j |a, r, K) = C(t j |a, r, K) − C(t j−1 |a, r, K) and dispersion parameter α. Here, C(t|a, r, K) is the solution of Richards model presented in (3.1). Assuming that, given the parameters, the observations Y 1 , Y 2 , . . . , Y n are conditionally independent, then

Var[Y j |a, r, K, α] = X(t j |a, r, K) + αX(t j |a, r, K) 2 (A.

2)

The Negative Binomial distribution allows to control the variability of the data by considering 4 over-dispersion which is common for epidemiological data. If α = 0, then we return to the Poisson 5 model which is often used in this context.

Let θ = (a, r, K, α) be the vector of parameters to estimate. The inclusion of the parameter α, which is related to the variability of the data, not to the Richards model, is necessary since in practice this variability is unknown. Then, the likelihood function, which represents how plausible is the data under the Negative Binomial assumption and Richards model if we knew the parameters, is given by π(y 1 , . . . , y n |θ) = n j=1 Γ(y j + τ) Γ(y j )Γ(τ) τ τ + C(t j |a, r, K) τ C(t j |a, r, K) τ + C(t j |a, r, K)

Consider that parameters a, r, K and α as random variables. Assuming prior independence, the joint prior distribution for vector θ is π(θ) = π(a)π(r)π(K)π(α),

where π(a) is the probability density function (pdf) of a Uniform(0,2) distribution, π(r) is the 7 pdf of a Uniform(0, 2), π(K) is the pdf of a Uniform(K min , K max ), and π(α) is the pdf of a 8 Gamma(shape=2, scale=0.1). To select the prior for parameter r, we consider that previous es-9 timations of r are close to 0.3 [11], and a Uniform(0,2) represents a weekly informative prior as it 10 allows for a wide range of values of r. Also, there is no available prior information regarding the 11 final size of the outbreak K. This is a critical parameter in the model and, in order to avoid bias, 12 we assume a uniform prior over K min and K max . To set these last two values, we consider that the 13 minimum number of confirmed cases is the maximum number of observed cumulative cases Y(t n )

(for the studied period) times two, i.e. K min = y n * 2. To set the upper bound for K, we consider a 15 fraction of the total population K max = N * 0.05, where N is the population size of Mexico. This 16 18 . CC-BY-NC-ND 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 July 24, 2020. Table 3 : Parameter median estimates and 95% posterior probability intervals for Richards model. Two periods are considered, from February 22, 2020 to March 22, 2020, and for two different periods, from March 23, 2020 to April 30, 2020. Here, r is the growth rate, K is the final size of the outbreak and a is a scaling factor. K is shown by completeness.

fraction was determined based on the observations of other cities such as the New York where the 1 total population size is similar to Mexico City. mean, variance, quantiles, etc. We refer the reader to [12] for more details on MCMC methods and 7 to [13] for an introduction to Bayesian inference with differential equations. 8 We estimate vector θ using data from Mexico City for two different periods, from February 22, 9 2020 to March 22, 2020, and from March 23, 2020 to April 30, 2020. Table 3 shows the median 10 posterior estimates and 95% probability intervals for the parameters in each period.

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