COVID-19 pandemics modeling with SEIR(+CAQH), social distancing, and age stratification. The effect of vertical confinement and release in Brazil.

The ongoing COVID-19 epidemics poses a particular challenge to low and middle income countries, making some of them consider the strategy of vertical confinement. In this strategy, contact is reduced only to specific groups (like age groups) that are at increased risk of severe disease following SARS-CoV-2 infection. We aim to assess the feasibility of this scenario as an exit strategy for the current lockdown in terms of its ability to keep the number of cases under the health care system capacity. We developed a modified SEIR model, including confinement, asymptomatic transmission, quarantine and hospitalization. The population is subdivided into 9 age groups, resulting in a system of 72 coupled nonlinear differential equations. The rate of transmission is dynamic and derived from the observed delayed fatality rate; the parameters of the epidemics are derived with a Markov chain Monte Carlo algorithm. We used Brazil as an example of middle income country, but the results are easily generalizable to other countries considering a similar strategy. We find that starting from 60% horizontal confinement, an exit strategy on May 1st of confinement of individuals older than 60 years old and full release of the younger population results in 400 000 hospitalizations, 50 000 ICU cases, and 120 000 deaths in the 50-60 years old age group alone. The health care system avoids collapse if the 50-60 years old are also confined, but our model assumes an idealized lockdown where the confined are perfectly insulated from contamination, so our numbers are a conservative lower bound. Our results discourage confinement by age as an exit strategy.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) outbreak has been 2 ongoing for 5 months now [1]. Since it was first reported in Dec 2019 in China [2], the 3 virus rapidly made its way to other parts of the world taking pandemic proportions [3].
influenza [13], contributing to the understanding of the dynamics of disease and 23 providing useful predictions about the potential transmission of a disease and the 24 effectiveness of possible control measures, which can provide valuable information for 25 public health policy makers [14]. SIR-type models, also known as Kermack-McKendrick 26 model [15], consists of a set of differential equations and has been applied to a variety of 27 infectious diseases. Although considered as simple, SIR models have been of great help 28 to stop epidemics in the past such as putting in place effective vaccination protocols [16]. 29 Here we develop an SIR type compartmental models for COVID-19 including both 30 symptomatic and asymptomatic, quarantined, and hospitalized while taking into 31 consideration differences by age groups. We also analysed at the effect of confinement that is actually in confinement. However, it is estimated to be around is 56% according 46 to satellite data 1 ,

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Given socio-economic consequences of a lockdown, particularly on a middle income 48 country, decision makers are considering a vertical confinement as an exit strategy to 49 the regular lockdown. Vertical confinement is understood as reducing contact to a 50 specific age group that is more at risk of contracting and developing SARS-CoV-2 [18], 51 as opposed to horizontal (or general) confinement that does not discriminate between 52 age groups. Throughout this manuscript, we will present the model which we validated 53 with data on other countries. We then applied the model to the specific SARS-CoV-2 54 scenario in Brazil. We finally tested the effect of both general confinement and the causalities.

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The model 60 We used a modified version of an SIR-type deterministic compartmental model to trace 61 COVID-19 epidemic evolution in an isolated population of N individuals 2 . We assumed 62 that a population could be subdivided into the following compartments:  We split that the population in subcategories by age (range, 0-10, 10-20, 20-30, 30-40, 75 40-50, 50-60, 60-70, 70-80, and 80+ years old) and that rates should vary with age [18]. 76 Taking into consideration the 8 compartments and the 9 age groups, the model is The software is written in python 3.7, and is made public at https://github.com/wlyra/covid19 April 9, 2020 3/17 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint .
For each compartment X the age sub-bins add up to X = i X i and compartments 79 are such that S + C + E + A + I + Q + H + R = N , with N = i N i being the total 80 population; N i is the population in each age bin.

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Eqs. (1)-(8) describe a compartmentalization of the population and the flow between 82 the compartments. Contact with infected individuals removes a fraction of the 83 susceptible (S) population at a rate given by λ, referred to as infection force, making 84 them exposed (E) to (SARS-CoV-2) . Exposed (E) becomes infectious at the rate σ; a 85 fraction p of them becoming symptomatic (I) and (1 − p) asymptomatic (A). The  The timescales corresponding to σ, γ, θ, ξ, and η are the incubation time t σ = σ −1 92 the infectious interval t γ = γ −1 , the remission time t θ = θ −1 , the time to hospitalization 93 t ξ = ξ −1 , and the average length of hospital stay t η = η −1 .

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The infection force is driven by the infected, both symptomatic (I) and where we use the shorthand notation and β is the infection rate, related to the reproduction number Lock-down consists of having a fraction of the susceptible population removed from 99 the epidemic dynamic by moving them from S i to C i at a rate ψ i .Similarly, lifting the 100 lock-down is done by placing C i into S i at the rate φ i . We consider these functions to 101 be Dirac deltas 102 April 9, 2020 4/17 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint  Fatality rate as a function of median age. The fatality rate by age bins is taken from [18] and the population pyramids from UN data.
where t lock and t lift are the time (in days) of lock-down and of lifting of the lock-down, 103 respectively. To allow for partial demographic lock-downs (e.g., 80% lock-down of the 104 population are in complete lock-down ), a i and b i are allowed to vary by age (e.g., 80% 105 lock-down of the 40's age group population are in complete lock-down ). The flow chart 106 between compartments is shown in Fig. 1.

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Other diagnostic quantities are the numbers U i of people in need of an intensive care 108 unit (ICU) bed where ζ i is the fraction of hospitalized patients that need critical care. Both ζ i and the 110 hospitalization fraction q i are age-stratified.

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For integration, we use a standard Runge-Kutta algorithm, with timesteps

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In this section we present details about the model validation and strategies to verify 114 characteristic timescales and other parameters.

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April 9, 2020 5/17 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.  research on COVID-19 [19]. The age-dependent parameters fatality rate µ i , fraction of 120 infectious that are hospitalized q i , and fraction of hospitalized that need critical care are 121 shown in Table 2.   [18].
Because all these timescales are much smaller than a human lifetime, aging of the 123 population is ignored and no upward flow between the age sub-compartments 124 (i → i + 1) is considered. Population pyramids are taken from UN data 3 , and split into 125 the pre-defined age bins. 126 We derive R(t) from the available statistics since knowledge on the real number of 127 infected is not clear. The most reliable indicator in this situation is the number of 128 deaths. Given a fatality rate µ and an average time τ between exposure and death, the 129 number of dead at a time t + τ will equal the fatality rate times the number of people 130 that got exposed at time t. Assuming that confinement dynamics do not play a role 131 (although it is trivial to include it), the equation is the following: Taking the continuous limit and substituting Eq. (1) where we also write t r = t + τ for the retarded time. Summing over all age bins 134 D = i D i we have the cumulative death rate on the LHS, which is an observable and µS = i µ i S i . We can then substitute Eq. (9) and solve for R(t) as a function of 136 time 137 Since death occurs an average of τ days after infections, we setup the integration to 138 start τ days before the first reported COVID-19 death, i.e., t = 0 means t r = τ . Finally, 139 to start the integration we need to define the initial number of exposed individuals.

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This should be where t 0 is the time of the first death and µ = N −1 i µ i n i is the age-weighted fatality 142 rate. According to current knowledge of the epidemics, τ ≈ 14 days [18]. 143 We compared our model predictions with official data on cases and deaths for rate for a number of countries, which corresponds to the left hand side of Eq. (18). We 147 apply Eq. (19) to convert this data into R(t), feeding this value into Eq. (1)-Eq. (8) to 148 start the SEIR evolution. The populations I(t) and S(t) that enter in Eq. (19) are then 149 calculated to update R(t). The resulting values are plotted in the right-hand-side of

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The timescales σ, γ, θ, and ξ, as well as the fractions p and w, are found by Markov 152 chain Monte Carlo (MCMC) fitting, with priors as given in Table 1 Fig. 4a shows the evolution of the compartments of exposed (E), asymptomatic (A), 166 symptomatic (I), and hospitalized (H), in linear scale. Fig. 4b shows the same curve of 167 H but also the fraction of hospitalizations needing ICU (U ), in log scale. The epidemic 168 is starting at March 1st and number of symptomatic is predicted to end at July 1st.  To not overwhelm the health care system capacity (≈ 3 × 10 4 ) ICU beds, the level of social distancing should be over 70%. Brazil is managing 56%.
April 9, 2020 9/17 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
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is the (which was not peer-reviewed) The copyright holder for this preprint  . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint .
higher degree, at 90% (solid blue line) and 99% (dashed blue line). The cyan line marks 204 the same model as the upper plots, where 60% of the population is confined, irrespective 205 of age. The 3rd, 4th, and 5th row of plots show the same analysis but confining 60% up 206 50, 60, and 70 years old, respectively. As seen in the cyan line, the number of 207 hospitalized rises from 30-60 and falls for 70 onwards. That is because even though 70+ 208 are more likely to be hospitalized, the number of 30-60 is much higher in the population. 209 Fig. 8 shows the same results for the fraction of hospitalized that needs ICU. Fig. 9 210 shows results from the same suite of models but for the number of fatalities. For the 211 number of ICU cases, there is no significant difference past age 60, with only a minor 212 uptick at the 70-80 age range. Collapse of health care system can be avoided if vertical 213 confinement is instored on people who are 60 or older, but at the expense of a significant 214 number of extra ICU cases for the 50-60 age bin. At 60% confinement, hundred of 215 thousands of deaths are seen in the 60-70, 70-80, and 80+ age bins. The number lowers 216 to 50 000 in the 90% confinement. As noted before, vertical confinement for 60 years old 217 and older, leads to a significant number of deaths for the 50-60 age bin (over 50 000).

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Vertical confinement at 50 years old leads to much lower death rate for this age segment. 219 Finally, we look at vertical confinement as an exit strategy. In Fig. 10 we model a  Our model estimate hundreds of thousands of infected people in Brazil at April 1st. 247 This is more than the number of expected cases in the country while we write this 248 article, considering the estimated under notification of cases [20] and do nothing to 249 control the infection. It is possible that the actual number be less than that although it 250 is also important to notice Brazil has not done a real lockdown so far.

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The model also ignores mobility, in the sense that it does consider travel to and from 252 the country. Given that right now we are the stage of community tranmission, this limitation should not be of significance to the results. 254 Conversely, and more importantly, the model assumes that the confined population 255 is completely safe from infection, whereas in reality a vertical lockdown may not be 256 feasible to implement as the elderly are not adequately distanced from the younger in 257 their family and/or social circle, and infection cannot be avoided if the younger is 258 exposed to  Finally, the analysis assumes that the data on fatalities is accurate. Underreported 260 deaths should lead to an unknown source of error in the present study. Also, the 261 MCMC produces error bars in the parameters that we did not take into account in the 262 forward modeling.

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Is this study we examine the strategy of vertical confinement as currently debated in 265 Brazil. Since the fatality rate of COVID-19 is disproportionate among the elderly, the 266 federal government has suggested confining only the at-risk age groups. Such a strategy 267 would limit the economic impact of the pandemic while at the same time minimizing  An exit strategy of vertical confinement of individuals older than 60 year old by May 279 1st would see a second wave disproportionally affect the 50-60 age bin. The ICU cases 280 in this age range alone would bring the health care system to collapse and result in over 281 100 000 deaths. If vertical confinement is contemplated, it should be of the population 282 over 50 years old. However, this age range, 50-60, is also a part of the workforce, and 283 thus defeats the purpose of a vertical confinement. Moreover, we emphasize that our 284 model assumes an idealized lockdown where the confined are perfectly insulated from 285 contamination, while in reality there would be several practical barriers to it as the 286 confined elderly would depend on the young for most essential activities, and a perfect 287 lockdown would not be achieved in a multi-generational household, especially in close 288 quarters such as found in the low and even middle income neighborhoods common in 289 Brazil. Our results therefore discourage vertical confinement as an exit strategy and 290 point toward enforcing a higher degree of horizontal confinement than currently 291 practiced. We urge Brazilian authorities to take action to prevent virus dissemination in 292 the critical coming weeks.

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Markov Chain Monte Carlo. To fit the best value to w and p, and to better 295 constrain σ −1 , γ −1 , θ −1 , ξ −1 , we use the affine-invariant ensemble sampler for Markov 296 chain Monte Carlo (MCMC) [21] to sample the parameter space around the solutions 297 and evaluation of the parameter uncertainties. For the priors input, we use the values 298 taken from the [18]. To search for the minimization of cumulative hospitalization H c , 299 we generated a cumulative error C err on the reported confirmed cases C c . 300 April 9, 2020 14/17 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.09.20060053 doi: medRxiv preprint As the JHU-CSSE reports on the confirmed cases are given daily with some 301 fluctuations, we need to take this into account while weighing all solutions by adding a 302 1-day error matrix together with the confirmed cases (being conservative). In an ideal 303 scenario, the cumulative number of hospitalization would be the same as the number of 304 confirmed cases. In real life, not all confirmed cases are hospitalized so we do not expect 305 to fit the H c with C c . Rather, we weigh the C c array with the H c array using: H c is the weighed cumulative hospitalizations and n is the length of the data.

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Following we get the residual between C c and H c , and we used the negative binomial 308 distribution to calculate each likelihood [22]:  (10 20 ) to discard as a bad fit. 312 We limit each parameter using a range cutoff in when feeding the probability 313 function to restrict parameter space. That way, we do not run models with unrealistic 314 physical parameters (such as e.g. symptomatic going to the hospital in −2 days), and 315 also constrain the known range for the other parameters. The MCMC function feeds on 316 6 free parameters, 4 fixed parameters and 2 predetermined arrays as presented in 317  . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.