Estimating the impact of non-pharmaceutical interventions and vaccination on the progress of the COVID-19 epidemic in Mexico: a mathematical approach
Hugo Flores-Arguedas, José Ariel Camacho-Gutiérrez, Fernando Saldaña
EEstimating the impact of non-pharmaceuticalinterventions and vaccination on the progress of theCOVID-19 epidemic in Mexico: a mathematicalapproach
Hugo Flores-Arguedas a , Jos´e Ariel Camacho-Guti´errez b , Fernando Salda˜na c a Centro de Investigaci´on en Matem´aticas, 36023 Guanajuato, Guanajuato, Mexico b Facultad de Ciencias, Universidad Aut´onoma de Baja California, 22860 BajaCalifornia, Mexico c Instituto de Matem´aticas, Campus Juriquilla, 76230, Universidad Nacional Aut´onomade M´exico, Qu´eretaro, Mexico
Abstract
Non-pharmaceutical interventions have been critical in the fight against theCOVID-19 pandemic. However, these sanitary measures have been partiallylifted due to socioeconomic factors causing a worrisome rebound of the epi-demic in several countries. In this work, we assess the effectiveness of themitigation implemented to constrain the spread of SARS-CoV-2 in the Mex-ican territory during 2020. We also investigate to what extent the initialdeployment of the vaccine will help to mitigate the pandemic and reduce theneed for social distancing and other mobility restrictions. Our modeling ap-proach is based on a simple mechanistic Kermack-McKendrick-type model.To quantify the effect of NPIs, we perform a monthly Bayesian inferenceusing officially published data. The results suggest that in the absence ofthe sanitary measures, the cumulative number of infections, hospitalizations,and deaths would have been at least twice the official number. Moreover,for low vaccine coverage levels, relaxing NPIs may dramatically increase thedisease burden; therefore, safety measures are of critical importance at theearly stages of vaccination. The simulations also suggest that it may be moredesirable to employ a vaccine with low efficacy but reach a high coverage thana vaccine with high effectiveness but low coverage levels. This supports thehypothesis that single doses to more individuals will be more effective thantwo doses for every person. 1 a r X i v : . [ q - b i o . P E ] F e b eywords: COVID-19, Mathematical model, Disease modeling,Non-pharmaceutical interventions, Vaccination
1. Introduction
Since the beginning of the pandemic, the scientific community has actedfast to better understand several aspects of COVID-19, including epidemi-ological, biological, immunological, and virological features. Mathematicalmodeling has been crucial in helping public health officers make informeddecisions [23]. In particular, there is a growing literature on epidemiologicalmodeling papers that have been mainly used to forecast the epidemic dynam-ics in specific countries or cities, see, for example, [3, 5, 13, 14, 20, 22, 29, 33].Mathematical models have also been central to evaluate the procedures in-volved in the containment of the pandemic [4]. Many governments worldwidehave implemented national lockdowns as extreme measures to stop diseasespread. Lockdowns in addition to other non-pharmaceutical interventions(NPIs) such as mask-wearing, social distancing, temperature screening, clo-sure of schools, restaurants, bars, and other places for social gathering havebeen of paramount importance in the fight against the pandemic. However,although such measures have had a significant impact in reducing the num-ber of deaths and infections, helping to decrease the risk of health servicesbeing overwhelmed, their cost to society and economic life have been huge[15]. Hence, public health authorities are constantly monitoring the currentstate of the epidemic to evaluate lockdown exit strategies, and once againthe use of mathematical models becomes a valuable tool to study the impactof partial mobility restrictions and the optimal time to relax the imposedrestrictions [3].There have been many modeling efforts presented to analyze and under-stand the COVID-19 pandemic in Mexico [2, 7, 11, 16, 24, 25, 27, 28, 31].The study of the early phase of the pandemic together with estimations ofthe basic and effective reproduction numbers is presented in [2, 16, 24]. In[7], the authors present a forecasting model aiming to predict hospital oc-cupancy. Using both hospital admittance confirmed cases and deaths, theyinfer the contact rate and the initial conditions of the dynamical system,
Email address: [email protected] (Hugo Flores-Arguedas)
Preprint submitted to Advances in applied mathematics February 23, 2021 onsidering breakpoints to model lockdown interventions and the increase ineffective population size due to lockdown relaxation. In [28], the authors usea mathematical model to characterize the impact of short duration transmis-sion events. They showed that super-spreading events have been one of themain drivers of the epidemic in Mexico.Beyond NPIs, there has been a vast-scale effort by researchers and phar-maceutical companies to develop an effective and safe vaccine to prevent in-fection with the SARS-CoV-2 and now several vaccines have been approvedby national regulatory authorities. The availability of a vaccine represents animportant positive step towards the control of the pandemic and the hope toreturn to normality. Nevertheless, the implementation of mass vaccinationworldwide involve several financial, logistic, and social challenges [30]. More-over, even for a very effective vaccine, the immunization coverage needed toreach herd immunity levels and successfully control the pandemic may bevery high and potentially difficult to achieve.The first goal of this work is to evaluate the impact of the sanitary mea-sures implemented in the mitigation of the COVID-19 pandemic in Mexicoduring the year 2020. We also investigate to what extent the initial deploy-ment of the vaccine will help to mitigate the pandemic and reduce the needfor social distancing and other mitigation measures. To this end, we fit asimple mechanistic Kermack-McKendrick-type model using official data ofthe COVID-19 epidemic in Mexico. We use Bayesian inference to calibratethe state variables and estimate how key parameters have been changingalongside the epidemic. As in [7], our modeling approach assumes that aslockdown measures as relaxed, more individuals become in contact with theoutbreak. In other words, lockdown-relaxations not only cause a change inthe transmission rates but also causes changes in the effective size of thepopulation at risk.The rest of the paper is structured as follows. In the next section, wepresent the mathematical model and use officially published data on the dailynumber of confirmed cases, hospitalizations, and cumulative deaths duringthe year 2020 to perform a monthly parameter inference. In Section 3.1, suchresults are used to assess the role of NPIs in the mitigation of the COVID-19pandemic. We also explore several vaccination scenarios depending on theimmunization coverage, delivery time, and vaccine efficacy. A discussion ofthe results is presented in Section 4. 3 . Methods
The model presented here is based on the mathematical model first in-troduced in a previous work [25]. The model is an extension of the classicalSEIR Kermack-McKendrick-type model tailored to incorporates the mostimportant features of the COVID-19 disease and the population-level impactof vaccination.The disease dynamics are described by the following system of differentialequations S (cid:48) = − λS − φS,V (cid:48) = − (1 − ψ ) λV + φS,E (cid:48) = λS − kE, ˜ E (cid:48) = λ (1 − ψ ) V − k ˜ E,A (cid:48) = (1 − p ) kE + (1 − ˜ p ) k ˜ E − γ A A,I (cid:48) = pkE + ˜ pk ˜ E − γI − ηI − µI,H (cid:48) = ηI − γ H H − mH,R (cid:48) = γ A A + γI + γ H H,D (cid:48) = µI + mH, (1)where S, V, E, ˜ E, A, I, H and R represent the number of susceptible, vac-cinated susceptible, exposed, vaccinated exposed, asymptomatic infectious,symptomatic infectious, hospitalized, and recovered individuals, respectively.In this study, the asymptomatic class A includes infected individuals with nosymptoms but also considers mild symptomatic infections. The symptomaticclass I consists of individuals who develop severe disease and therefore areexpected to have fewer contacts in comparison with individuals in the A class.Considering disease-induced deaths, the total population size, denoted N ( t ),is N ( t ) = S ( t ) + V ( t ) + E ( t ) + ˜ E ( t ) + A ( T ) + I ( T ) + H ( t ) + R ( t ) + D ( t ).The force of infection (FOI) λ = ( β A A + β I I ) /N in model (1) representsthe classical standard incidence, where β A and β I are the effective contactrates for the asymptomatic and symptomatic infectious classes, respectively.Note that we are assuming that the hospitalized class is effectively isolatedand does not contribute to the FOI. After a mean latent period of 1 /k , aproportion p of the exposed class E transition to the symptomatic infectiousclass I , while the other proportion 1 − p enter the asymptomatic infectious4lass A . The parameters γ A , γ , and γ H are the recovery rates of the classes A , I , and H , respectively. The parameter η denotes the rate of transitionfrom the I class to the hospitalized class H . Individuals in the symptomaticand hospitalized classes experience disease-induced death at rates µ and m ,respectively. We incorporate vaccination in the susceptible class with a rate φ . In our model, vaccination not only prevents SARS-CoV-2 infection butalso prevents severe symptomatic COVID-19 disease. The parameter ψ is thevaccine efficacy to prevent infection and 1 − ˜ p is the fraction of vaccinatedindividuals who after infection do not develop severe disease. Therefore, 1 − ˜ p may be interpreted as the vaccine efficacy to prevent severe disease. Thebasic qualitative properties of the model (1) and the reproduction numbersare studied in S-I. In this section, we perform a parameter inference using a Bayesian ap-proach. Our main objective is to estimate the impact of the sanitary emer-gency measures implemented to control the spread of the virus. However,such measures have been relaxed or only partially implemented alongsidethe epidemic depending on several circumstances. The initial phase of theepidemic in Mexico covers from February 17 to March 22, 2020. On March23, 2020, phase 2 was declared, which primarily includes the suspension ofcertain non-essential economic activities, the restriction of massive congrega-tions, and the recommendation of home quarantine to the general population.On March 30, 2020, a sanitary emergency was declared and the public healthauthorities implemented a national lockdown until May 31, 2020. After this,the lockdown was lifted and other mitigation measures were only partiallyimplemented. These changes have had a significant impact on the value ofthe transmission parameters [7, 24]. We take this into account making aparameter inference by periods. We consider the estimation of parametersby month using three sets of data obtained from the daily report of the Mex-ican Federal Health Secretary [26]: (i) new daily reported infections, (ii) newdaily hospitalizations, and (iii) cumulative deaths in Mexico. We remarkthat this data corresponds to the confirmed cases on the date that the pa-tient approached the medical center and not on the day its symptoms began.Moreover, the testing rate in Mexico is the lowest among the OECD coun-tries [16], so the data on the confirmed cases corresponds to symptomaticinfections, see S-II for the details of the inference.5 a r c h A p r il M a y J un e J u l y A u gS e p t O c t N o v D e c I (a) M a r c h A p r il M a y J un e J u l y A u gS e p t O c t N o v D e c (b) M a r c h A p r il M a y J un e J u l y A u gS e p t O c t N o v D e c m (c) M a r c h A p r il M a y J un e J u l y A u gS e p t O c t N o v D e c S (d) M a r c h A p r il M a y J un e J u l y A u gS e p t O c t N o v D e c R e (e) A p r il M a y J un e J u l y A u g S e p t O c t N o v D e c S / N (f) Figure 1: Mean and variance (vertical black lines) of (a) the effective contact rate of thesymptomatic class β I , (b) the hospitalization’s rate η , (c) the mortality rate from hospital-ized patients m , (d) the effective susceptible population S , (e) the effective reproductionnumber, and (f) the susceptible fraction .
6s in [7, 13, 14, 16], we assume that partial lifting of lockdowns andNPIs not only affects the transmission rate but also the effective popula-tion size, N e , that usually satisfies N e (cid:28) N . We remark that deterministicKermack-McKendrick-type models predict a single epidemic wave. How-ever, our modeling approach allows us to capture the possibility of multiplewaves induced by NPIs relaxation. Hence, for the inference, we consider N e = S + E + A (0) + I + H (0) + R (0) + D (0). The vector of parameterto estimate in the Bayesian formulation is x = ( S , E , I , β I , η, m ). The in-ference process is performed in a monthly way from March to December (seeFigure 1).As expected, the highest value of the effective contact rate in the symp-tomatic class is reached at the beginning of the pandemic (March), beforethe national lockdown and NPIs implementation. The maximum for thehospitalization’s rate is reached in April, one month after the start of thepandemic in Mexico, and the maximum in the mortality rate from hospital-ized patients reached in May, respectively (see Figure 1). At the beginningof the pandemic, the implementation of the lockdown in Mexico produced asmall effective susceptible population for March and April. The progressionof the pandemic and the relaxation of the measures produced a monthly in-crease until September. After this, an increase in the new daily infectionsand the contact rates in October and November produced measures that re-sulted in a decrease in the corresponding S . In December a new increaseis noted probably due to the end of the year festivities. Moreover, observethat the effective reproduction number R e was very high in the early phaseof the epidemic. The national lockdown reduced significantly the value of R e achieving its lowest value in August. After lockdown-relaxation, June 1, thevalue of R e has been oscillating close to unity. The susceptible fraction S/N is practically decreasing along the period from April to December. This be-havior was broken in September, same month where the effective susceptiblepopulation reaches its maximum value. Performing a parameter inference byperiods allow us to obtain accurate estimations on the number of daily infec-tions, hospitalizations, and cumulative deaths and make a reliable short-termprediction of such outcomes per period (see Figure 2).7 a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r N e w D a il y I n f e c t i o n s (a) M a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r N e w D a il y H o s p i t a li z a t i o n s (b) M a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r C u m u l a t i v e D e a t h s (c) Figure 2: Monthly estimation for new daily infections (a), new daily hospitalizations (b)and cumulative deaths (c) in the Mexican population. Vertical lines represent official datawhile solid lines represent our model predictions for the respective month. Observe thatwe extended our model predictions to overlap some unseen data of the next month.
3. Results
The national lockdown in Mexico was officially lifted on May 31. How-ever, after this date, NPIs were issued through mass media by public healthauthorities and are still partially implemented in the population. In this sec-8ion, we investigate the impact of such NPIs in the control of the transmissiondynamics of SARS-CoV-2 for the first year of the pandemic in Mexico. Forthis, we compare with a theoretical case in which no mitigation measureswere implemented. We must remark that after the national lockdown waslifted, NPIs were state-specific in Mexico. Nevertheless, for simplicity, we areconsidering the total data for the whole country. M a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r C u mm u l a t i v e I n f e c t i o n s With measuresNo measures (a) M a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r C u mm u l a t i v e H o s p i t a li z a t i o n s With measuresNo measures (b) M a r c h M a r c h A p r il M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r C u mm u l a t i v e D e a t h s With measuresNo measures (c)
Figure 3: Monthly worst-case scenario for cumulative number of infections (a), hospital-izations (b) and deaths (c) predicted in the absence of NPIs or any sanitary measures(dotted lines). Vertical lines represent official published data by the Mexican Secretary ofHealth since the beginning of the pandemic until December 30, 2020.
In Fig. (3) we evaluate a counterfactual scenario that reflects a monthly9orst-case scenario, in which no sanitary measures were implemented dur-ing the whole year of 2020. For each month, the no-measures scenario wasobtained by using the same initial conditions from the Bayesian inference,but changing the value of the estimated β for that month to the value of β at the beginning of the pandemic. Such simulations allow us to quantify theimpact of NPIs in the reduction of the burden caused by COVID-19 in theMexican population. The outcomes of interest are the cumulative number ofinfections, hospitalizations, and deaths predicted in the worst-case scenarioin comparison with the official data. The results suggest that in the absenceof the sanitary measures, the cumulative number of infections by the end of2020 would have been above 3 million cases which is more than twice theofficial number. We observe a similar pattern for the cumulative number ofhospitalizations and deaths, that is, they would have presented at least a two-fold increase in the absence of NPIs. From these results, it is evident thatthe implementation of NPIs has been of paramount importance to reducethe transmission of SARS-CoV-2 and the morbidity caused by COVID-19disease. Since a vaccination program is starting in Mexico, we would like to knowif there are vaccination scenarios that allow similar savings to those obtainedby the implementation of NPIs. Results from leading vaccine developers haveshown that their vaccines are more than 90% effective to prevent infectionwith SARS-CoV-2 (see [25] and the references therein). Nevertheless, thesuccess of a vaccination program depends not only on the vaccine efficacybut also on the immunization coverage, C , and the time needed to achievesuch a coverage τ .The population-level impact of a vaccination program on the number ofnew daily infections, hospitalizations, and cumulative deaths is shown in Fig-ure 4 (a), (b), and (c), respectively. To have a better understanding of theeffect of vaccination, we consider its interaction with the impact of imple-menting or partially lifting NPIs. The no-NPIs scenario, that is, the case inwhich no mitigation measures are implemented at all, is shown in solid redlines. On the other hand, the disease burden for the case in which NPIs areimplemented is shown in solid blue lines (see Figure 4). Moreover, dashedlines correspond to the respective scenario considering the introduction of avaccination program. Vaccination parameters are as follows: coverage time10
10 20 30 40 50 60 70 80 90Days D a il y I n f e c t i o n s (a) D a il y H o s p i t a li z a t i o n s no-NPIno-NPI (Vacc.)NPINPI (Vacc.) (b) C u m u l a t i v e D e a t h s (c) Figure 4: Impact of a vaccination program on the number of (a) new daily infections, (b)hospitalizations, and (c) cumulative deaths (dashed lines). The no-NPIs scenario is shownin solid red lines and the scenario under NPIs implementation is shown in solid blue lines.The initial deployment of the vaccine starts on October 1. Vaccination parameters are asfollows: coverage time τ = 90 days, vaccination coverage C = 0 .
20, efficacy ψ = 0 .
90, andsymptomatic fraction ˜ p = 0 .
30. Bar plots represent officially reported data. = 90 days, vaccination coverage C = 0 .
20, efficacy ψ = 0 .
90, and symp-tomatic fraction ˜ p = 0 .
30. The vaccination rate φ is obtained from theapproximation 1 − exp {− φτ } = C . The initial deployment of the vaccinestarts on October 1, so we employ our estimated parameters for October,November, and December. The results shown in Figure 4 point out that,in the absence of NPIs, the immunization coverage needed to control thepandemic is very high.We further explore the joint impact vaccine introduction and implemen-tation of NPIs in Figure 5. The contour plots show the percentage scaleof the pandemic burden under vaccination concerning the scenario withoutNPIs for (a)-(b) new daily infections, (c)-(d) new daily hospitalizations, and(e)-(f) cumulative deaths. In Figure 5, a value of 60% means that the corre-sponding curve under vaccination is 60% the value of the same curve underthe scenario without NPIs. We explore the effects of varying vaccine efficacy ψ , and vaccination coverage C . We also explored the effect of varying thesymptomatic fraction ˜ p , and we found that this parameter does not affectsignificantly the outcomes under study. On the other hand, the vaccinationcoverage C has a greater effect in terms of reducing the disease burden. How-ever, the immunization coverage needed to see a significant reduction in thenumber of infections should be close to 40%.In Figure 6, we show the maximum value for new daily infections, newdaily hospitalizations, and cumulative deaths in a window of 30 days pre-dicted for a vaccination program starting on November 1 with variationsin vaccine-associated parameters. In Figure 6(a), we explore the impact ofvaccination target coverage, C , against the contact rate β I . As expected, in-creasing coverage is a very effective way to control the pandemic. However,assuming realistically low coverage levels (close to 10% within the first weeksof vaccine introduction), we can observe that relaxing NPIs altogether withlow vaccination coverage may dramatically increase the disease burden [1].However, if the implementation of NPIs manages to maintain a low contactrate, the introduction of the vaccine improves notably the control of the dis-ease burden. In Figure 6(b), we show the effects of varying vaccine efficacy ψ along with coverage C . The results imply that it may be more desirable toemploy a vaccine with low efficacy but reach a high coverage than a vaccinewith high effectiveness but low coverage levels [19].12 a) (b)(c) (d)(e) (f) Figure 5: Countour plots that show the percentage scale of the pandemic burden undervaccination with respect to the scenario without NPIs for (a)-(b) new daily infections,(c)-(d) new daily hospitalizations and (e)-(f) cumulative deaths. For all plots, vaccinationstarts on November 1. When fixed, the vaccine-associated parameters are coverage time τ = 90 days, target coverage C = 40%, effectiveness ψ = 0 .
90 and symptomatic fraction˜ p = 0 . a)(b) Figure 6: Impact of vaccine-associated parameters on the maximum value of new dailyinfections, new daily hospitalizations and cumulative deaths for November. (a) The im-munization coverage C and the contact rate β I are varied (the rest of parameters are fixed ψ = 0 .
90, ˜ p = 0 .
40 and τ = 30 days). (b) The vaccine efficacy ψ and coverage C are varied(the rest of parameters are fixed ˜ p = 0 .
40 and τ = 30 days).
4. Discussion
Since the emergence of SARS-CoV-2, health authorities worldwide haveimplemented unprecedented mobility restrictions and other NPIs in an at-tempt to control the epidemic. Nevertheless, due to socio-economic reper-cussions, many countries decided to lift or at least partially relax such re-strictions causing a worrisome rebound of the epidemic. Due to constantmodifications of NPIs according to epidemiological risk factors, it is verydifficult that the deterministic dynamics of a compartmental model may fitlong-term data or multiple waves of the epidemic. Moreover, the forecastin the absence of data is limited to conditions remaining constant which isunrealistic due to the changes needed to reactivate the economy. To cir-cumvent this situation, we consider a monthly parameter inference. Thisstrategy allows us to observe and compare the progression of transmission14arameters alongside the epidemic. Besides, Bayesian inference allowed usto calibrate key model features and evaluate the effects of NPIs given data.In particular, we used official epidemiological data at the early phase of theepidemic in Mexico, before lockdowns and NPIs implementation, to establishthe effective contact rate without mitigation measures. This allowed us toassess the effects of NPIs in the following months of the epidemic. Our modelsimulations show that the number of daily infections, hospitalizations, andcumulative deaths would have presented at least a two-fold increase in theabsence of NPIs. Hence, the implementation of NPIs in Mexico has been ofparamount importance to reduce the transmission of SARS-CoV-2 and themorbidity caused by COVID-19 disease.After model calibration, we explored how a vaccination program affectsthe control of the transmission dynamics in comparison with NPIs. Theresults suggest that vaccination alone is not enough to control disease spreadif NPIs are abandoned prematurely and if the immunization coverage is lowas expected in some countries or regions. In other words, lifting mitigationmeasures completely at the early stages of vaccination may lead to a dramaticincrease in the disease burden. Therefore, even though mass vaccinationprograms have already started all around the world, mobility restrictions andother NPIs are still of principal importance in the control of the COVID-19pandemic. Furthermore, the simulations also suggest that it may be moredesirable to employ a vaccine with low efficacy but reach a high coveragethan a vaccine with high effectiveness but low coverage levels. This supportsthe hypothesis that delaying the second dose and prioritizing giving the firstdoses of vaccine to more individuals will be more optimal to mitigate theCOVID-19 epidemic.
Acknowledgements
Fernando Salda˜na acknowledges support from DGAPA-PAPIIT-UNAMgrant IV 100220 (proyecto especial COVID-19). Ariel Camacho thanks SEP-SES-PRODEP-UABC for his Postdoctoral Fellowship support. We thank Dr.Jorge X. Velasco-Hern´andez for helpful discussions on a previous version ofthis work. 15 eferences [1] Abo, S. M. C. and Smith?, S. R. (2020). Is a COVID-19 Vaccine Likelyto Make Things Worse?
Vaccines , 8(4):761.[2] Acu˜na-Zegarra, M. A., Santana-Cibrian, M., and Velasco-Hernandez,J. X. (2020). Modeling behavioral change and covid-19 containment inmexico: A trade-off between lockdown and compliance.
Mathematical Bio-sciences , page 108370.[3] Aguiar, M., Ortuondo, E. M., Van-Dierdonck, J. B., Mar, J., and Stol-lenwerk, N. (2020). Modelling covid 19 in the basque country from intro-duction to control measure response.
Scientific reports , 10(1):1–16.[4] Anirudh, A. (2020). Mathematical modeling and the transmission dy-namics in predicting the covid-19-what next in combating the pandemic.
Infectious Disease Modelling , 5:366–374.[5] Bertozzi, A. L., Franco, E., Mohler, G., Short, M. B., and Sledge, D.(2020). The challenges of modeling and forecasting the spread of covid-19.
Proceedings of the National Academy of Sciences , 117(29):16732–16738.[6] Byrne, A. W., McEvoy, D., Collins, A., Hunt, K., Casey, M., Barber, A.,Butler, F., Griffin, J., Lane, E., McAloon, C., et al. (2020). Inferred dura-tion of infectious period of sars-cov-2: rapid scoping review and analysisof available evidence for asymptomatic and symptomatic covid-19 cases. medRxiv .[7] Capistran, M. A., Capella, A., and Christen, J. A. (2021). Forecast-ing hospital demand in metropolitan areas during the current covid-19 pandemic and estimates of lockdown-induced 2nd waves.
PloS one ,16(1):e0245669.[8] Christen, J. A., Fox, C., et al. (2010). A general purpose samplingalgorithm for continuous distributions (the t-walk).
Bayesian Analysis ,5(2):263–281.[9] Delamater, P. L., Street, E. J., Leslie, T. F., Yang, Y. T., and Jacobsen,K. H. (2019). Complexity of the basic reproduction number (r0).
Emerginginfectious diseases , 25(1):1. 1610] Diekmann, O., Heesterbeek, J. A. P., and Metz, J. A. (1990). On the def-inition and the computation of the basic reproduction ratio r 0 in modelsfor infectious diseases in heterogeneous populations.
Journal of mathemat-ical biology , 28(4):365–382.[11] Hernandez-Vargas, E. A. and Velasco-Hernandez, J. X. (2020). In-hostmathematical modelling of covid-19 in humans.
Annual reviews in control .[12] Hethcote, H. W. (2000). The mathematics of infectious diseases.
SIAMreview , 42(4):599–653.[13] Ku, C. C., Ng, T.-C., and Lin, H.-H. (2020). Epidemiological bench-marks of the covid-19 outbreak control in china after wuhan’s lockdown: amodelling study with an empirical approach.
Available at SSRN 3544127 .[14] Maier, B. F. and Brockmann, D. (2020). Effective containment explainssubexponential growth in recent confirmed covid-19 cases in china.
Science ,368(6492):742–746.[15] Mandel, A. and Veetil, V. (2020). The economic cost of covid lock-downs: An out-of-equilibrium analysis.
Economics of Disasters and Cli-mate Change , 4(3):431–451.[16] Mena, R. H., Velasco-Hernandez, J. X., Mantilla-Beniers, N. B.,Carranco-Sapi´ens, G. A., Benet, L., Boyer, D., and Castillo, I. P. (2020).Using the posterior predictive distribution to analyse epidemic models:Covid-19 in mexico city. arXiv preprint arXiv:2005.02294 .[17] Nogrady, B. (2020). What the data say about asymptomatic covid in-fections.
Nature .[Oran and Topol] Oran, D. P. and Topol, E. J. The proportion of sars-cov-2infections that are asymptomatic: A systematic review.
Annals of InternalMedicine .[19] Paltiel, A. D., Schwartz, J. L., Zheng, A., and Walensky, R. P. (2020).Clinical Outcomes Of A COVID-19 Vaccine: Implementation Over Effi-cacy.
Health Aff. [20] Petropoulos, F. and Makridakis, S. (2020). Forecasting the novel coro-navirus covid-19.
PloS one , 15(3):e0231236.1721] Ponciano, J. M. and Capistr´an, M. A. (2011). First principles modelingof nonlinear incidence rates in seasonal epidemics.
PLoS Comput Biol ,7(2):e1001079.[22] Ribeiro, M. H. D. M., da Silva, R. G., Mariani, V. C., and dosSantos Coelho, L. (2020). Short-term forecasting covid-19 cumulativeconfirmed cases: Perspectives for brazil.
Chaos, Solitons & Fractals ,135:109853.[23] Roda, W. C., Varughese, M. B., Han, D., and Li, M. Y. (2020). Why isit difficult to accurately predict the covid-19 epidemic?
Infectious DiseaseModelling , 5:271–281.[24] Salda˜na, F., Flores-Arguedas, H., Camacho-Guti´errez, J. A., and Bar-radas, I. (2020). Modeling the transmission dynamics and the impact ofthe control interventions for the covid-19 epidemic outbreak.[25] Saldana, F. and Velasco-Hernandez, J. X. (2020). The trade-off betweenmobility and vaccination for covid-19 control: a metapopulation modelingapproach. medRxiv .[26] Salud, S. (2020). Datos covid-19 m´exico. https://datos.covid-19.conacyt.mx/. Accessed 12-11-2020.[27] Santamar´ıa-Holek, I. and Casta˜no, V. (2020). Possible fates of thespread of sars-cov-2 in the mexican context.
Royal Society open science ,7(9):200886.[28] Santana-Cibrian, M., Acu˜na-Zegarra, M. A., and Velasco-Hernandez,J. X. (2020). Lifting mobility restrictions and the effect of superspreadingevents on the short-term dynamics of covid-19.
Mathematical Biosciencesand Engineering , 17(5):6240–6258.[29] Sarkar, K., Khajanchi, S., and Nieto, J. J. (2020). Modeling andforecasting the covid-19 pandemic in india.
Chaos, Solitons & Fractals ,139:110049.[30] Su, Z., Wen, J., McDonnell, D., Goh, E., Li, X., ˇSegalo, S., Ahmad, J.,Cheshmehzangi, A., and Xiang, Y.-T. (2021). Vaccines are not yet a silverbullet: The imperative of continued communication about the importance18f covid-19 safety measures.
Brain, Behavior, & Immunity-Health , page100204.[31] Torrealba-Rodriguez, O., Conde-Guti´errez, R., and Hern´andez-Javier,A. (2020). Modeling and prediction of covid-19 in mexico applyingmathematical and computational models.
Chaos, Solitons & Fractals ,138:109946.[32] Van den Driessche, P. and Watmough, J. (2002). Reproduction numbersand sub-threshold endemic equilibria for compartmental models of diseasetransmission.
Mathematical biosciences , 180(1-2):29–48.[33] Zhao, H. and Feng, Z. (2020). Staggered release policies for covid-19control: Costs and benefits of relaxing restrictions by age and risk.
Math-ematical biosciences , 326:108405. 19 upplementary Material: Estimating the impact ofnon-pharmaceutical interventions and vaccination onthe progress of the COVID-19 epidemic in Mexico: amathematical approach
S-I. Mathematical properties of the model
The biologically feasible region for model (1) isΩ = (cid:110) ( S, V, E, ˜ E, A, I, R, D ) ∈ R : S + V + E + ˜ E + A + I + R + D = N (cid:111) . Clearly, the region Ω is positively-invariant, that is, for a well-defined initialcondition that starts in Ω, the solution remains in Ω for all t >
0. Therefore,the model is both epidemiologically and mathematically well posed [S12].System (1) presents a continuum of disease-free equilibria of the form E = ( S, V, E, ˜ E, A, I, H, R, D ) = ( S ∗ , V ∗ , , , , , , , , where S ∗ , and V ∗ are the proportions of non-vaccinated and vaccinated sus-ceptible at the initial time. An straightforward computation allow us to ob-tain the basic reproduction number R = (1 − p ) β A /γ A + pβ I / ( γ + η + µ ). Notethat, by definition, R assumes a fully susceptible population and, hence, itcannot be modified through vaccination campaigns. To examine the effectsof vaccination and non-pharmaceutical interventions, the more appropriatemeasure to use is the effective reproduction number R e [S9].We obtain R e taking a next-generation approach [S10]. The matrix ofnew infections F and the matrix of changes in the infection status V aregiven by: F = β A S ∗ /N ∗ β I S ∗ /N ∗ β A (1 − ψ ) V ∗ /N ∗ β I (1 − ψ ) V ∗ /N ∗ and V = k k − (1 − p ) k − (1 − ˜ p ) k γ A − pk − ˜ pk γ + η + ν . K = FV − and the effective reproductionnumber is the spectral radius R e = ρ ( K ). If the symptomatic fractionssatisfy p = ˜ p , it is easy to see that R e = ( S ∗ /N ∗ + (1 − ψ ) V ∗ /N ∗ ) R , and inthe absence of vaccination R e = ( S ∗ /N ∗ ) R . Therefore, we approximate thetime-varying effective reproduction number as the product of the proportionof the susceptible among the effective population size at the beginning ofeach month, i.e. R e = ( S /N e ) R (see also [S13]). Let us note that using anexplicit formula for R e based on the dynamic of the model means that thesame assumptions are taken into account. That is, the value of R e includesthe postulates about the non observed dynamic. Although it may producediscrepancies with other ways of calculating it, the monthly computation of R e allows us to assess the efficacy of NPIs implemented. In section 2.2, weobtain a predictive marginal for R e based on the marginal posterior for S and the predictive marginal of R . As a consequence of the Theorem 2 in[S32], we establish the following result regarding the local stability of thedisease-free equilibrium. Theorem S-I.1.
The continuum of disease-free equilibrium of system (1) given by E is locally asymptotically stable if the effective reproduction numbersatisfies R e < and unstable if R e > . S-II. Bayesian Inference
To perform the parameter inference, we first retrieved some of the modelparameters from the literature and epidemiological data on COVID-19. Theincubation period of COVID-19 is on average 5-6 days, but can be as longas 14 days, we choose the average estimation 1 /k = 5 . . − . /γ A = 1 /γ = 7 . m = 10 µ , so the mortality rate is higher in the hospitalizedclass than in the symptomatic class. Recent studies suggest that at least onethird of SARS-CoV-2 infections are asymptomatic [SOran and Topol], thus,we postulate the value p = 0 .
5. With respect to the asymptomatic effectivecontact rate, we assume the relation β A = 2 β I since symptomatic individualsare the ones that develop severe conditions and therefore, on average, are2xpected to have reduced mobility compared with asymptomatic cases. Weremark that some studies have found that completely asymptomatic individ-uals will transmit the virus to significantly fewer people than a symptomaticcase [S17]. However, in our model the asymptomatic class A includes both,completely asymptomatic and mild infections who do not attend a medicalcenter.The initial value of the state variables (corresponding to March, 2020)are fixed as follows: A (0) = 20, and R (0) = H (0) = D (0) = 0. The rest ofinitial conditions, S (0) , E (0) , and I (0), are included in the inference process.We assume the following model for new infections and new hospitalizationsdata y i y i ∼ Poisson ( I j ( x )) , i = 0 , . . . , k (S1)where I j ( x ) denote the predicted number of new cases between times j − j , see [S21]. For the case of the infections in our model, I Ij ( x ) = (cid:90) t j t j − pkE ( x ) dt (S2)with E the exposed individuals given by system (1) at time t i and x the vectorof parameters to estimate. By assuming independence on the observations,the likelihood function L ( x ) satisfies: L ( x ) ∝ j = k (cid:89) j =0 e −I Ij ( x ) ( I Ij ( x )) y j y j ! . (S3)Analogously, we consider I Hj ( x ) = (cid:90) t j t j − ηI ( x ) dt (S4)the incidence for the new hospitalizations. In the case of deaths, as at thebeginning of the epidemic there were only a few, we consider cumulative casesand propose a Gaussian likelihood to fit this data.We define by π ( x ) the prior distribution for x . We assume independenceof the parameters, hence π ( x ) = π ( S ) π ( β I ) π ( E ) π ( I ) π ( η ) π ( m ) . (S5)We consider gamma distributions for each one. Recall that the gamma dis-tribution is denoted by Γ( α, β ) with α the shape parameter and β the inverse3
50 200 E (a)
25 50 75 I (b) I (f) (d) m (e) S (c) Figure S1: Marginal posterior for the parameters in x : (a) E , (b) I , (c) S , (d) η , (e) m and (f ) β I . scale parameter. If Z ∼ Γ( α, β ) then E [ Z ] = α/β and V ar [ Z ] = α/β . Wepropose S ∼ Γ( a, b ) , β I ∼ Γ(1 , , E ∼ Γ(20 , ,I ∼ Γ(20 , , η ∼ Γ(0 . , , m ∼ Γ(0 . ,
1) (S6)where a and b are chosen such that E [ S ] = 5 . e x are obtained using the t-walk [S8] and are shown inFigure S1. It is natural to expect a lower contact rate for April and May asa consequence of the lockdown. Since there was no vaccination in the earlyphase of the pandemic, compartments V and ˜ E are not considered in theinference.The inference allows us to obtain a predictive marginal for state variables4 S, E, A, I, H, R, D ) at the final date of March. These distributions will pro-vide information to perform the inference in the next month. The mean ofthese values will be used as initial condition of the model for the quantities
A, H, R, D and the marginal predictive of