key: cord-0789216-q6j2njm8 authors: Tocto-Erazo, M.; Espindola-Zepeda, J. A.; Montoya-Laos, J. A.; Acuna-Zegarra, M. A.; Olmos-Liceaga, D.; Reyes-Castro, P. A.; Figueroa-Preciado, G. title: Lockdown, relaxation, and ACME period in COVID-19: A study of disease dynamics on Hermosillo, Sonora, Mexico date: 2020-08-22 journal: nan DOI: 10.1101/2020.08.20.20178509 sha: 59f0c4f80f447136953f479910423625964dc6c2 doc_id: 789216 cord_uid: q6j2njm8 Lockdown and social distancing measures have been implemented for many countries to mitigate the impacts of the COVID-19 pandemic and prevent overwhelming of health services. However, success on this strategy depends not only on the timing of its implementation, but also on the relaxation measures adopted within each community. At the request of Sonoran Health Ministry, we developed a mathematical model to evaluate the impacts of the lockdown implemented in Hermosillo, Mexico. We compared this intervention with some hypothetical ones, varying the starting date and also the population proportion that is released, breaking the confinement. For this purpose, a Monte Carlo study was performed by considering three scenarios to define our baseline dynamics. Results showed that a hypothetical delay of two weeks, on the lockdown measures, would result in an early ACME around May 9 for hospitalization prevalence and an increase on cumulative deaths, 42 times higher by May 31, when compared to baseline. On the other hand, in respect of relaxation dynamics, the ACME levels depend on the proportion of people who gets back to daily activities or the individual behavior regarding prevention measures. It is important to stress that, according to information provided by health authorities, the ACME occurring time was closed to the one given by our model. Hence, we considered that our model resulted useful for the decision-making assessment, and that an extension of it can be used for the study of a potential second wave. The comprehension of this pandemic has grab the interest of many scientific areas, mainly with 23 the aim of providing ideas that could reduce the severity of the disease. In particular, the area of 24 mathematical modeling has drawn the attention during this epidemic mostly due to its usefulness 25 in providing information about the evolution of transmissible diseases. Current work is focused 26 on parameter estimation that serves as a basis for more complex studies [12] , the evaluation of 27 non-pharmacological interventions during the epidemic, such as social distancing or lockdown 28 [13, 14, 15, 16, 17] and forecast short term trends of the disease [18] . In general, one of the main 29 purposes of mathematical models has been the evaluation of the effects of different governmental 30 interventions and also providing to decision-makers with more elements for responding to a need, susceptible, exposed, asymptomatically infected, symptomatically infected, hospitalized, quarantined, recovered and dead individuals, respectively. Protected individuals (P) get involved in the disease dynamics when mitigation measures are implemented, whereas released population (P R ) does so when relaxation of these measures occurs. Our manuscript is organized as follows. Initially, we present our proposed mathematical model. 1 Then, statistical analyses of different parameter scenarios, that validate the data, are presented. A 2 discussion about the results obtained with the adjusted models is included. Our results arise from 3 the statistical and modeling perspectives and are related to the ACME occurring time, implications 4 of lockdown occurrence time, and consequences of the lifting mitigation measures. Finally, we end 5 up with a discussion section. 6 2 Methods 7 2.1 Compartmental mathematical model 8 We formulate a compartmental mathematical model, whose diagram can be observed in Fig 1, 9 where susceptible (S ), exposed (E), asymptomatic infectious (I a ), symptomatic infectious (I s ), 10 recovered (R), quarantined (Q), hospitalized (H), and dead individuals (D) are considered. P rep- 11 resents a proportion of individuals in the population that decided to stay at home in order to protect 12 themselves from illness, and P R are those released from the P class, when certain proportion of 13 protected individuals needed or decided to break control measures. 14 To formulate the mathematical model, we considered that susceptible individuals are moved 15 to the protected class when they obey the mitigation measures implemented by the government 16 and some become infected when interacting with an infectious individual. Dynamics of protected 17 individuals is similar; that is, they either can become infected or moved to the protected released while the latter one is a result of a mitigation measures break up (a proportion of the protected pop-1 ulation returns to their usual activities). On the other hand, protected released people only leave 2 the class by the interplay with symptomatic or asymptomatic individuals (becoming infected). The 3 exposed class represents individuals that are infected but not infectious. After a while, an exposed 4 individual can become infectious, asymptomatic, mildly symptomatic, or severe symptomatic. As 5 a first approximation and to analyze data of a specific Mexican state, we considered that the stages 6 previously mentioned are grouped into two classes: i) asymptomatic people (I A ), and severe symp-7 tomatic people (I S ). We assumed that mildly symptomatic people can be distributed in both classes. 8 People from I A class are recovered with a mean time equal to 1/η a . In contrast, individuals from 9 I S class are identified as infected after 1/γ s days (on average), after which they are reported and 10 become hospitalized or quarantined/ambulatory. We considered that ambulatory individuals might 11 recover or worsen their condition, being then hospitalized. This happens after 1/ψ days (on av-12 erage). Finally, we assumed that only hospitalized individuals may die, and that occurs after 1/µ 13 days, on average. 14 Following the hypotheses previously stated, the mathematical model is given bẏ where N * = S +E+I A +I S +R+P+P R . It is important to emphasize that the infection contact rates of 15 released protected people are less or equal than the infection contact rates of susceptible individuals. Parameter Definition α a , (α a ,α a ) Transmission contact rates for susceptible (protected, protected released) class linked to asymptomatic individuals α s , (α s ,α s ) Transmission contact rates for susceptible (protected, protected released) class linked to symptomatic individuals δ Incubation rate θ Proportion of symptomatic individuals η a Recovery rate for asymptomatic individuals γ s Output rate from the symptomatic class by register β Proportion of hospitalized individuals ψ Output rate from the quarantined class by hospitalization/recovery µ Output rate from the hospitalized class by recovery/death The implementation of social distancing resulted in a proportion of the population being protected by staying at home. For that reason, we modeled this event considering that susceptible individuals moved to the protected class during some period. This phenomenon occurs until a certain percentage of the population is reached. We represent this period by T L 1 , T U 1 . The mathematical description of the dynamics is given by and parameter w 10 represents the protection rate of susceptible individuals per unit of time. On the other hand, at the moment of writing this paper, it has been observed that many people who were initially obeying mitigation measures have now broken the confinement, going back to their usual activities. For that reason, we consider that certain proportion of protected people become protected released people. We model this phenomenon in a similar way to the one presented in previous function. Thus Here, period from T L 2 to T U 2 represents the time in which a percentage of the population that breaks 2 the confinement is reached. the COVID-19 outbreak in Hermosillo, Sonora, like the ACME value and ACME date, but also we 10 were able to explore different intervention schemes such as: changes in the beginning and lifting 11 restriction dates, variation in the population proportions that return to usual activities on June 01, 12 2020 (a date fixed by Federal Government), and also the possibility of exploring some other periods 13 where people break the confinement. 14 The Monte Carlo method that was considered here for exploring epidemic characteristics of the 15 COVID-19 outbreak consists of the following steps. in Sonora, a mandatory confinement was declared by the state governor. This statewide 25 stay-home directive was intended to avoid the spread of this coronavirus. Nevertheless, even 26 on May 6, 2020, Sonora State government divulged a video message asking citizens for 27 remaining in quarantine and taking social distancing seriously, since a considerable increase 28 in the number of cases were occurring. 29 Considering the above information, we assumed that the period from March 16 to April (which was not certified by peer review) 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 August 22, 2020. . parameters, as well as their selected distributions, are shown in Table 2 . Table 2 : Model parameter distributions. Here, N a,b (µ 0 , σ 0 ) is the truncated normal distribution with a truncation range (a, b), where µ 0 and σ 0 are the mean and variance of this distribution; IG(α 0 , β 0 ) is the inverse gamma distribution with shape and scale parameters α 0 , β 0 , respectively; B(a, b) is the Beta distribution with parameters a and b; U(min, max) is a uniform distribution on an interval that goes from min to max. Heuristic analysis was used by three different researchers, in order to propose the scenarios 2 presented in Table 2 . Basically, the strategy was to delimit the support of these distributions, 3 either by considering a wide range for parameter values or by selecting these ranges based 4 on a bibliographic review. The fit of the model solutions, to the initially reported data, was 5 done either manually (visual-fit), through a shiny app created in Rstudio (script available on 6 a Github repository [22] ), and also by minimizing the sum of squared errors. The values and 7 ranges of model parameters that were taken as a starting point to specify the support of these 8 distributions, are shown in Table 3 . Some of these can be found on COVID-19 literature and 9 some others have been assumed by the authors. portions are achieved within 30 and 15 days, respectively. The values obtained for w 10 and 1 w 20 allowed us to propose the corresponding distributions given in Table 2 as well as the 2 parameters ranges shown in Table 3 . A similar procedure was carried out to obtain the 3 distributions for parameters w 10 and w 20 in the second scenario, except that a U(0.05, 0.2) 4 distribution is considered for the proportion of people who have broken the confinement. 5 For the third scenario, a B(0.8, 0.05) distribution was used to describe the protected propor- distributions, respectively. A strategy, similar to the one described in the first two scenarios, 10 was considered to obtain the distributions given in Table 2 and parameters ranges shown in 11 Table 3 : Initial parameter ranges and values, taken from current literature or assumed (*) by the researcher. • Empirical constraint on prevalence: A study carried out in Spain shed some light about 13 the highest prevalence percentage in that country, with an estimation around 21.6% [40] . 14 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. . Considering this result, we decided to include solutions where the cumulative number of 1 infected people, since the first case until day 200, were at most 21.6% of the total population 2 in Hermosillo. 3 • Data: The dataset used here is the latest public data on COVID-19, available at the offi-4 cial website of the Mexican Federal Government [41] , updated at July 19, 2020. Taking 5 into consideration some decisions adopted by the Mexican government, regarding to lift- 6 ing confinement measures, the study covered a period spanning from March 11 to May 31. 7 COVID-19 positive cases considered in this study included Hermosillo residents who were 8 registered in a medical unit in the Sonora state. Variables under study were the symptom 9 onset case date, hospitalized and ambulatory cases by date of admission to a health service 10 unit, and also daily deaths. • Empirical restriction on epidemic curves: In order to ensure reasonable solution curves, we considered an inclusion criterion that consists on selecting a curve when the sum of squared errors about the data was smaller than an upper bound. The reason for adopting this criterion was the fact that epidemiological characteristics were not only determined by the selected scenarios but also were linked to the actual behavior of the epidemic in Hermosillo, Sonora. Next, we briefly describe the steps that were performed to get the upper bounds for the sums of squared errors. First, we obtained m = 1000 parameter sets from a scenario and then we used them to calculate daily incidence of symptomatic infections, daily incidence of hospitalized cases, daily incidence of ambulatory cases, and daily incidence of deaths. In order to obtain this information we defined the following variables with respect to the model: later to explore other dynamics related to dates of lockdown implementation and levels of 23 relaxation. 24 9 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. . In this section, we applied the methodology previously explained to compute three parameter sets 2 that are used to define our baseline scenarios. Then, the strengths and weaknesses of our results are 3 discussed. Finally, we explored some scenarios regarding possible consequences of i) change of 4 dates for implementing mitigation measures, and ii) lifting mitigation measures on June 01, 2020. 5 3.1 About the ACME occurring time 6 Based on the three scenarios previously considered, we obtained the quantile-based intervals shown 7 in Table 4 . Fig 2 increased our knowledge about the parameters behavior, providing, for each pa-8 rameter and each scenario, some interesting complementary information. In these plots we can 9 observe that in some cases, there is no intersection between these empirical distributions, while in 10 others a considerable overlap occurs. This illustrates a well known problem of parameter identifia-11 bility where basically, for the same data set, different parameter regions could validate the data; this 12 can actually be observed in Fig 3. It follows then that solutions obtained throughout System 2.1, 13 under three different scenarios, should not be expected always to be close solutions; so therefore, 14 it could be risky to assign a predictive nature to our model. Table 4 : Median and 95% quantile-based intervals for parameters in System 2.1. 10 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. To complement the fact previously exposed, we included Fig 4, where the distribution of the max-1 imum number of daily new reported, hospitalized, and deaths can be observed. As expected, the 2 three scenarios provided solutions that in general do not coincide in the ACME levels. However, 3 our study gave us some certainty in another aspect. For the three scenarios, Fig 5 shows the dis-4 tributions of the estimated date of the ACME for the daily new reported, hospitalized, and death 5 variables. Here, we can clearly observe that Scenarios 2 and 3 presented very similar distributions 6 for these three variables. In contrast, histograms for Scenario 1 are flattened, their beginning is 7 too early, and they ended almost at the same dates of Scenarios 2 and 3. Actually, quantile-based 8 intervals for Scenario 1 will almost contain the ones corresponding to Scenarios 2 and 3. These re-9 sults indicate that even when parameters do not provide consistent information about the intensity 10 of the outbreak, it did preserve the property of having an ACME occurring time in a specific time 11 interval. 12 3.2 Implications of Lockdown occurrence time. 13 Based on System 2.1 and the parameter ranges and values obtained in previous section, we eval-14 uated implications on the magnitude of the variables of interest, if the lockdown had been im- 15 plemented one or two weeks later than our real scenario. This exploration intends to analyze the 16 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Figure 4 : Histograms of ACME levels for different epidemic curves. Black, red and blue bars are related to Scenarios 1, 2, and 3, respectively. A) ACME of daily new reported cases. B) ACME of daily hospitalizations. C) ACME of daily deaths. possible consequences of a late decision making. For our simulations, we used 5000 parameter 1 combinations of scenario 2, and calculated the quantile 0.5 of all these solutions. Here, we ba-2 sically present hospitalized prevalence, in order to relate this with bed saturation and cumulative 3 deaths. 4 It is important to have in mind that in real setting, lockdown took place from March 16 to April 15 5 (Baseline). Therefore, we carried out these simulations, for Scenario 2, considering that lockdown new hospitalizations and deaths would occur if distancing measures were taken two weeks after 10 the original date, exhibiting the importance of timely decision making. 11 12 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. Figure 5 : Histograms of ACME dates for different epidemic curves. Black, red and blue bars are related to Scenarios 1, 2, and 3, respectively. A) ACME of daily new reported cases. B) ACME of daily hospitalizations. C) ACME of daily deaths. deaths. 9 13 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. . approximately 16%, 33%, and 66% of the population, that fulfilled with social-distancing mea- 5 sures, returned to their usual activities on June 01, 2020, respectively. Here, we can deduce that the 6 number of people returning to their usual activities, is directly affecting the ACME level of these 7 epidemic curves, which also depend on the adopted social-distancing measures. The latter will few available data to come up with a good estimate and answer such a question. For that reason, 10 our results were merely qualitative [42] . 11 In this work, we proposed a mathematical model to study COVID-19 dynamics in Hermosillo, 12 Sonora, Mexico. Here, we assessed the timing to implement different social-distancing scenarios 13 during COVID-19 epidemic and explored different levels of mitigation-measure relaxation. We 14 followed a two-step approach to achieve our aims. First, we conducted a Monte Carlo study, and 15 under three different scenarios, some epidemic curves are fitted. As a result, we obtained a baseline 16 with adequate properties when providing an approximation to the occurring time of the epidemic 17 peak. According to our findings, the median dates for the ACME of incidence cases would occur 18 between July 18 and August 6. These results were consistent with ongoing surveillance data pro- 19 vided by local health authorities, which reported the incidence peak by the 31st epidemiological 20 week (July 27 -August 2). Since then, decrements have been observed for incidence cases and 21 hospitalizations [43] . In summary, our model described well the epidemic dynamic and the impact 22 of lockdown intervention measures throughout time. 23 On the other hand, some differences were observed between Scenarios 1, 2 and 3, when estimating 24 the number of cases. Specifically, the three scenarios fitted the data properly, but they did not 25 coincide in the magnitude of the outbreak (Fig 3) . Overall, it is not clear which one provides the 26 better fitting. In that sense, it is meaningless to talk about predictability on the intensity of the 27 outbreak, with a model like the one considered here. Nevertheless, the three scenarios agreed on 28 what is called the ACME occurring time, so there might be information contained in the structure 29 of the epidemic and in the model itself that might lead us to observe these results. However, as 30 we know, mathematical models can serve to understand some properties of the evolution of the 31 disease, qualitatively. In this work, we have obtained three-parameter distribution sets that adjust 32 the data. In all scenarios, α a ,α a andα a , which are the infection rates for the Susceptible, Protected 33 and Released Protected individuals, respectively, the order relationα a ≤α a ≤ α a is satisfied. Thus, 34 the three scenarios are valid in the way we conceive the model. However, the intensity described 35 by each scenario tells us a different story. In Scenario 3, the results claim that released protected, 36 and protected people infect at the same rate. In other words, people that have returned to their 37 daily activities are protecting themselves as if they still were in the protected class. Whether or not 38 this situation really happened, a conclusion is that specific pieces of information, from a particular 39 place, could be used to discriminate spurious solutions and head toward having a predictive nature 40 of these results. 41 Once we acknowledged the properties of the model and its limitations, we were able to use it to 1 assess and compare our baseline with different lockdown scenarios. Our findings suggest that a hy-2 pothetical delay of two weeks (intervention B) for the implementation of the lockdown measures 3 would result in an early peak (May 9). Moreover, the two weeks delay considered in intervention B, 4 would increment in about 42 times the cumulative deaths, when compared to the ones observed un-5 der baseline, by May 31. In the absence of a vaccine or an effective treatment, the implementation 6 of social distancing measures at the early stages of this pandemic, helped to delay and slowdown 7 the epidemic dynamic, allowing to gain time to strengthen healthcare capacities, avoiding being 8 overwhelmed by an excessive demand. 9 Lifting mitigation measures showed considerable changes in daily cases, hospitalizations, and 10 deaths, depending on the proportion of people released to the public space, on June 01. Fig 7A 11 shows that ACME levels varied from 11% to 35% at the peak of the outbreak compared to base-12 line. Our conclusion regarding this issue must be conservative, since these results clearly depend 13 on the population proportion who returned to usual activities. An important factor that influences 14 on the magnitude of this proportion, is the poverty level. According to official data, 35% of the 15 occupied labor force have informal jobs [44] , and 19% lives in poverty condition [45] . Economic 16 inequalities contribute to impeding that a significant proportion of people could maintain a rig-17 orous lockdown, since their conditions force them to return to work. Improvements not just in 18 surveillance but social data, at local level, will benefit future estimations. 19 As a final note, the inclusion of the vital dynamics in the model can be useful when studying the 20 evolution of the disease for longer periods. For example, it can be helpful to provide qualitative 21 information on a possible second outbreak that might occur during the flu season, which runs from 22 September to January. A. Acuña-Zegarra, Daniel Olmos-Liceaga. 35 Project administration: José A. Montoya-Laos. 36 16 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. . https://doi.org/10.1101/2020.08.20.20178509 doi: medRxiv preprint All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 August 22, 2020. . https://doi.org/10.1101/2020.08.20.20178509 doi: medRxiv preprint A new coronavirus associated with human respiratory disease in China A novel coronavirus outbreak of global health concern An interactive web-based dashboard to track COVID-19 in real time. 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