key: cord-0745500-hkdzyo0u
authors: Acuña-Zegarra, Manuel Adrian; Santana-Cibrian, Mario; Velasco-Hernandez, Jorge X.
title: Modeling behavioral change and COVID-19 containment in Mexico: A trade-off between lockdown and compliance
date: 2020-05-06
journal: Math Biosci
DOI: 10.1016/j.mbs.2020.108370
sha: 50c5f3aea7de540ff9ab2d1869d59e51af5851e4
doc_id: 745500
cord_uid: hkdzyo0u

Sanitary Emergency Measures (SEM) were implemented in Mexico on March 30th, 2020 requiring the suspension of non-essential activities. This action followed a Healthy Distance Sanitary action on March 23rd, 2020. The aim of both measures was to reduce community transmission of COVID-19 in Mexico by lowering the effective contact rate. Using a modification of the Kermack-McKendrick SEIR model we explore the effect of behavioral changes required to lower community transmission by introducing a time-varying contact rate, and the consequences of disease spread in a population subject to suspension of non-essential activities. Our study shows that there exists a trade-off between the proportion of the population under SEM and the average time an individual is committed to all the behavioral changes needed to achieve an effective social distancing. This trade-off generates an optimum value for the proportion of the population under strict mitigation measures, significantly below 1 in some cases, that minimizes maximum COVID-19 incidence. We study the population-level impact of three key factors: the implementation of behavior change control measures, the time horizon necessary to reduce the effective contact rate and the proportion of people under SEM in combating COVID-19. Our model is fitted to the available data. The initial phase of the epidemic, from February 17th to March 23rd, 2020, is used to estimate the contact rates, infectious periods and mortality rate using both confirmed cases (by date of symptoms initiation), and daily mortality. Data on deaths after march 23rd, 2020 is used to estimate the mortality rate after the mitigation measures are implemented. Our simulations indicate that the most likely dates for maximum incidence are between late May and early June, 2020 under a scenario of high SEM compliance and low SEM abandonment rate.

• A mathematical model for CoVID-19 disease is formulated considering that a proportion of the population is subject to suspension of non-essential activities.

• The model explores the effect of behavioral changes required to reduce transmission by lowering the contact rate and infection probability.

• Key parameters that determine the initial growth of the epidemic are estimated using Bayesian inference. This is used to quantify uncertainty for each scenario.

• Results show that there is an optimum value of the lockdown proportion which minimizes maximum incidence.

• Date of maximum incidence for the optimal scenario is consistent with Mexican Government projections.

a straightforward calculation shows that this number amounts to an average daily attack rate of 21.5 per million people. On March 19th, 2020 the Heath Secretariat informed [3] that 140,367 (56% of the expected cases) would be mild not requiring hospitalization, 24,564 (9.8%) will require hospitalization, but will not be critical, and 10,528 (4.2%) would be critical patients.

Thus, the remaining 75,197 (30% of the expected cases) are asymptomatic carriers. This percentage agrees with the percentage reported by [4] which estimated the size of this population in 30.8%. On the same date, March 19th, it was informed [3] that the Mexican Health Federal Sector at that time had 4,291 intensive care unit (ICU) beds and 2,053 ventilators ready for the contingency. On March 23rd, 2020 schools were closed and on March 30th, 2020 the general suspension of non-essential activities was announced [1] . All the recommendations pertained to so-called non-pharmaceutical interventions such as washing hands, keeping cough/sneeze etiquette, avoiding handshakes, keeping distance from other people, working at home. These activities require changes in behavior and, therefore, have a learning curve.

Importantly, they can be forgotten, relaxed or only partially implemented in any given population.

We take into account, together with the suspension of non-essential activities, a time-varying effective contact rate. We model the efficacy of the Sanitary Emergency Measures (SEM) using two parameters, namely, q, the parameter representing the proportion of the population that strictly follows the required actions, and the parameter ω where 1/ω represents the average time that an average resident of Mexico keeps his/her low-risk community contact behavior. With respect to the time-varying effective contact rate, 3 J o u r n a l P r e -p r o o f we introduce three parameters: the proportion of contact rate reduction desired to achieve (α), the expected time to achieve that reduction from its original value to the desired fraction (θ), and the time lapse between the identification of the first cases to the implementation of the anti-COVID-19

control and mitigation measures (T θ ). For the incubation period distribution we use the infection-to-onset time reported in [5] [6] [7] that postulate a Gamma distribution with mean 5.1 days, and coefficient of variation 0.86 (but also see [8] [9] [10] [11] for similar estimates).

The model is parametrized with the available information on confirmed cases and mortality. It allows the exploration of scenarios dictated by the adopted public health control and mitigation measures vis a vis the transmission dynamics of the COVID-19 pandemic in Mexico. We have used realistic parameter values and estimated contact rates and reproductive numbers using the publicly available incidence and mortality data for Mexico City [12] .

In Section 2 we present a modification of the Kermack-McKendrick SEIR model used and describe the parameter estimation for Mexico City. In Section 3 we assess the population level impact of the implementation of containment measures for community contact rate reduction. Here we also analyze the effect of contact rate reduction in combination with suspension of nonessential activities, exemplifying our findings for the case of Mexico City.

Finally, we present our discussion and conclusions in Section 4.

A flow diagram of the mathematical model considered in this study is depicted in Figure A. We also assume that the population following the mitigation indicatives, do so strictly in such a way that the two populations do not mix. We see the parameter q as a measure of the efficacy of suspension, for instance q = 0.9

if 90% of the population is following the sanitary actions.

In the first part of our model, we keep record of the number of people in each of the SEIR compartments up to day T θ when the sanitary actions are

implemented. The contact rates before T θ are b s and b a for the symptomatically and asymptomatically infected populations, respectively. The equations for the first part (pre-dating the application of SEM ) of the model are given by the following deterministic system of non-linear differential equations:

where x s , x e , x ya , x ys , x r and d represent respectively the number of susceptible, exposed, asymptomatically infected, symptomatically infected, recovered, and dead individuals. After the sanitary actions are implemented, the population is split into two as described above with a proportion q, of individuals following the actions and, the remaining proportion 1 − q not adhering to the sanitary actions. Thus, the dynamics of the population adhering to 6 J o u r n a l P r e -p r o o f SEM is given by the following differential equations:

Similarly the dynamics of the the non-adhering subpopulation is given by the following differential equations:

where c s , c e , c ya , c ys , and c r represent, respectively, the number of susceptible, exposed, asymptomatically infected, symptomatically infected, and recovered individuals. Deaths from both subpopulations are described by the following differential equation:

For this second instance of the model, we define two time dependent contact rates. The first for the population under SEM, given by:

and the second for the non-adhering subpopulation given by:

Here, k = a or k = s are used for asymptomatically and symptomatically infectious individuals, respectively. The form of these contact rates is shown in demic was computed as the spectral radius of the next-generation operator [14] of the equation and it is given as:

It is clearly a weigthed average determined by the proportion of asymptomatically infected individuals, ρ. In summary, for t < T θ the pandemic follows the standard SEIR Kermack-McKendrick formulation with two infectious compartments. Once t = T θ is reached, the population is split into two.

In Mexico City, the COVID-19 outbreak started with imported cases.

Mitigation measures announced on March 23rd and March 30th, 2020 attempted to reduce the effective contact rate. We consider February 17th, indicates the fraction q of the population that is under SEM (i = 1), and the fraction 1 − q that is not (i = 2). The suspension of non-essential activities of March 30th, 2020 is seen as a reinforcement of the already ongoing reduction in contact rate.

The initial phase of the pandemic in Mexico covers from February 17th, 2020 to March 22nd, 2020. SEM was implemented on March 23rd, 2020. We use the number of daily confirmed COVID-19 cases in Mexico City by date of symptoms onset, and the daily number of deaths [12] . The initial (pre-SEM) effective contact rates, b a , b s , the infectious periods, η a , η s , and mortality rate µ were estimated using a Bayesian inference approach. Technical details can be found in Appendix A. We also estimated the mortality rate after March 23rd, 2020 (post-SEM) denoted by µ * . Table 1 shows the median values and 95% intervals for each parameter. Figure A. 3 shows the trajectory of the epidemic and the observed data. Using the median values presented in Table 1 for the effective contact rates and infectious periods, our median basic reproduction number works out to be R 0 = 3.87, with a 95% credible interval 9 J o u r n a l P r e -p r o o f 

In this section we present the main results of our analysis.

At the time of writing, there were no estimates of the learning times for contact rate reduction. Therefore, we set θ 1 = 150 days and θ 2 = 150 days as the average learning times for the population fractions q and 1 − q, respectively. With these values, the contact rate reduction by April 30th, 2020, the announced initial date of lifting of SEM measures, are, respectively, 23% and The parameter 1/ω represents the average time an individual is committed to all the behavioral requirements of effective social distancing (not leaving home, washing hands, frequently cleaning common surfaces, using masks, etc). All these assumptions imply that the mitigation measures are always leaky, not every member of the community follows them but also that a substantial part of the population under SEM will keep a low-risk behavior even beyond the formal termination of SEM.

Letting the pandemic run its course, we measure the maximum prevalence and maximum incidence, as well as the dates when they occur. Figure A.4 shows the median and 95% credible interval for the dates where maximum prevalence and maximum incidence are projected to occur as a function of q.

On the other hand, Figure A .5 shows the same statistics for the values of the maximum prevalence and maximum incidence, as a function of q. Observe that there is a value of q that appears to be optimum (produces minimum prevalence and earlier peak date). We will come back to this issue later in show 95% credible intervals and median estimates for the symptomatic incidence. Here 50% and 95% of the population follows the mitigation indicatives, respectively. Note that the symptomatic incidence level is higher when q is greater (compliance of 95%) than when it is only of 50%.

As we vary q, the more likely dates for the incidence peaks also change.

This is shown in Figure A Figure A.9(a) ) and, when compliance is 95%, they occur in early June, 2020 (median value of Figure A.9(b) ).

In Figure A .10, we present the scenario for q = 0.7, when maximum incidence is minimized. Under this scenario, our likely dates for maximum incidence coincide with the projection of middle to late May, 2020 suggested by the Mexican Secretariat of Health [16] . However, in this scenario, the interval of dates for the maximum incidence is wide ( Figure A.10(a) ). This effect is due to (i), the existence of different infectious periods for asymptomatically and symptomatically infectious individuals (η a , η s ), and (ii) the size of the confidence interval of η s compared to that of η a (Table 1) . When both infectious periods are the same, the empirical distribution of dates for the maximum incidence has less dispersion ( Figure A.10(b) ).

To end this section we show in Figure A .11 the trend of the instantaneous reproduction number [17] . Since there is no estimation of the serial interval for the pandemic in Mexico City or Mexico, we adopt the results in [18] where a Gamma distribution is used with median of 4.7 and standard deviation of 2.9 days. It can be appreciated that, at least in terms of the instantaneous reproduction number under the assumptions made on the serial interval descriptors, the SEM is at least maintaining the epidemic curve under control (low attack rate).

The control, containment, mitigation and possible elimination of the coronavirus pandemic requires rapid and consistent implementation of control and mitigation strategies as described in [6] . Mathematical models are cen-

J o u r n a l P r e -p r o o f Journal Pre-proof tral to this effort but certain issues have to be considered and measured for a increased efficacy in their application. Mexico has the lowest testing rate among the OCDE countries [19] . As this same study points out, a high testing rate is recommended to adequately plan the lifting of mitigation restrictions already in place in many countries, including Mexico. Testing is necessary to have a good estimate of the true size of the epidemic. In Mexico, several hundred Health Units constitute the country's sentinel surveillance system.

Here cases are detected and then a methodology is followed to identify contacts and other relevant information [20] . However, the identification of a suspected case detected by symptomatic surveillance has to be confirmed by testing. Limitations in testing correlate with a higher number of confirmed cases [21] . The positivity test rate in Mexico was around 20% on April [22] , comparable to that of the US [21] . This high rate may mislead conclusions on the true growth rate of the epidemic and, in particular, on the date where the maximum incidence is expected.

The model presented in this work, using the official data on confirmed cases [12] , is presented to evaluate the mitigation actions implemented by the We estimate key parameters of the model using publicly available data from the Mexican Secretariat of Health [12] . We use the reports on the daily confirmed COVID-19 cases and deaths in Mexico City from February 17th, 2020 to April 25th, 2020.

Case fatality rates are estimated as the ratio

represents the cumulative number of confirmed COVID cases at time t, D(t)

is the number of deaths at time t and T is the average time from symptom onset to death. This formula is a correction of the naive estimator D(t)/C(t).

We calculate two case fatality rates, µ = 0.018 before SEM (before March 23rd, 2020), and one after this date µ * = 0.12. Table 1 shows the 95% intervals for this parameters. It is important to point out that the pandemic in Mexico City before SEM implantation had 475 confirmed cases but only 2 deaths. This is the reason why µ is considerably smaller than µ * . Now, assume that the proportion of susceptible, exposed, asymptomatically and symptomatically infected, recovered and dead individuals in a closed population of size N can be described by equation (1), where 1/η a and 1/η b are the infectious periods, b a and b s are the effective contact rates before the mitigation measures, respectively, with subscripts a and s denoting asymptomatically and symptomatically infected populations; γ denotes the incubation rate, ρ is the proportion of exposed cases that become asymptomatic carriers and µ is the percentage of symptomatically infected that die. We set ρ = 0.45 and γ = 1/5.1. The parameters η a , η s , b a , b s , µ are 

We use a Bayesian inference approach were ψ is treated as a random vector and inferences are drawn from its posterior distribution. To construct this function, we first assign prior probability distributions to each parameter in ψ. We assume that infectious periods 1/η a < 1/η s are between 5 and 25 days. with the posterior distribution of the parameters of interest given as π(ψ|y 1 , . . . , y n ) ∝ π(y 1 , . . . , y n |ψ)π(ψ).

This expression does not have an analytical form because the likelihood function depends on the solution of the ODE system. The posterior distribution is analyzed through simulation using an MCMC algorithm that does not require tuning called t-walk [25] . This algorithm generates samples from the posterior distribution that can be used to estimate marginal posterior densities, as well as the mean, variance and quantiles of the parameters. We run the t-walk for 500,000 iterations, discard the first 10,000 and use 5,000

samples to generate estimates of the parameters. Table 1 shows the median posterior estimates for each parameter and 95% probability intervals.

The samples generated above are also used to generate the credibility intervals shown in this work. To generate the histograms, for each sample of (η a , η s , b a , b s ) we calculate the solution of the differential equation given in (1).

This results in a set of solutions that can be used to calculate the distribution of the maximum prevalence and incidence at each time, the distribution of the dates when these maxima occur, and any other statistic of interest. We and η varying within their corresponding 95% credible intervals as reported in Table 1 . 

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