Multi-Generational Sir Modeling: Determination of Parameters, Epidemiological Forecasting and Age-Dependent Vaccination Policies

MULTI-GENERATIONAL SIR MODELING: DETERMINATION OF PARAMETERS, EPIDEMIOLOGICAL FORECASTING AND AGE-DEPENDENT VACCINATION POLICIES

EDUARDO LIMA CAMPOS, RUBENS PENHA CYSNE, ALEXANDRE L. MADUREIRA, AND GÉLCIO L.Q. MENDES

Abstract. We use an age-dependent SIR system of equations to model the evolution of the COVID-19. Parameters that measure the amount of interaction in diﬀerent locations (home, work, school, other) are approximated using a random optimization scheme, and indicate changes in social distancing along the course of the pandemic. That allows the estimation of the time evolution of the classical and age-dependent reproduction numbers

R0. With those parameters we predict the disease dynamics, and compare our results with data from several locations in Brazil. We also provide a preliminary investigation regarding age-based vaccination policies, indicating connections between the age of those immunized, contagious parameters and vaccination schedules.

1. Introduction

Several months after the onset of the COVID-19 pandemic, it became clear how powerful numerical models can be useful to understand the dynamics of the disease. In particular, simple compartmental epidemiological models are convenient, depending on few parameters and still capturing with reasonable precision essential aspects of the dynamics of infectious diseases. The Susceptible-Infected-Recovered SIR modeling in its simplest form depends on two parameters, the average number of suﬃcient contacts (those suﬃcient for transmission) and the mean waiting period during which patients are infectious. It assumes that the population is homogeneous both in terms of behavior and biological susceptibility. It also assumes, in its simplest instance, that demographic aspects as age structure, birth and death rates do not matter for the dynamics of the disease.

Date: January 20, 2020. The authors would like to acknowledge the financial support of CNPq and FAPERJ. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. 1 2 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES Some of the above assumptions appear to hold well in the case of the COVID-19 pandemic. Demographic changes appear to be of less importance due to difference in time scales, com- pared with the evolution of the pandemic. Also, the period that one is infectious seems to be independent of age, to some reasonable extent. However, dropping the homogeneity hypothesis allows a more realistic model because age and location are certainly important in understanding the dynamics of the disease. Mathematical epidemiology has a somewhat long history, with Hamer (1906); M’Kendrick (1925); Kermack et al. (1927) being among the earliest contributions. See Brauer (2017); Hethcote (2000) for a thorough review of mathematical modeling of infectious diseases, and Klepac and Caswell (2011); Klepac et al. (2009); Iannelli and Milner (2017); Li et al. (2020); Sun (2010); Towers and Feng (2012) for multi-group models. Some other models consider spatial aspects of infectious diseases, as Bertaglia and Pareschi (2020); Lang et al. (2018); Paeng and Lee (2017); Peixoto et al. (2020); Takács and Hadjimichael (2019); Viguerie et al. (2020). Theoretical aspects of models are covered in Kuniya and Wang (2018); Wen et al. (2018); Wu and Zou (2016). For the control related models for COVID-19 see for instance Perkins and España (2020) and references therein. After the onset of the COVID-19 pandemic, a massive amount of articles related to the topic materialized. Several articles considered compartmental models to forecast different scenarios, as in Albani et al. (2020); Amaku et al. (2021); Atkeson (2020); Coelho et al. (2020); Prem et al. (2020); Komatsu and Menezes-Filho (2020); Viguerie et al. (2020); Walker et al. (2020). Socioeconomic costs of imposing social distancing are combined with compartmental models in Acemoglu et al. (2020); Alvarez et al. (2020); Atkeson (2020), and the interplay between economics and individual decisions are considered in Borelli and Góes (2020); Brotherhood et al. (2020); Eichenbaum et al. (2020). See Kremer (1996); d’Onofrio and Manfredi (2020); Funk et al. (2015); Manfredi and d’Onofrio (2013); Perrings et al. (2014); Soofi et al. (2020) for a description of behavioral- and economic- epidemiology ideas. Also, macro-economic aspects of the pandemic appears, e.g., in Guerrieri et al. (2020). Population heterogeneity and its influence on herd immunity is considered in Aguas et al. (2020); Britton et al. (2020); Cui et al. (2019); Gomes et al. (2020). Vaccination is considered in several of the above references and also, for instance, in Chen and Toxvaerd (2014); d’Onofrio and Manfredi (2020); González and Villena (2020); Jia et al. (2020); May and Anderson (1984). Finally, alternative approaches, as agent-based and statistical models and a combination of these techniques with compartmental models (specially to gather data), are also commonly EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 3

employed, as in Amaral et al. (2021); Bendtsen Cano et al. (2020); Calvetti et al. (2020); Donnat and Holmes (2020); Ferguson et al. (2020); Roda et al. (2020).

In the present paper, we consider a modification of the age-structured SIR model used by Towers and Feng (2012). As in Prem et al. (2017), we allow for age-dependent probabilities of one being sub-clinical or clinical, and use their contact matrices which differentiates not only age but also location. Random searches approximate some key parameters that are related to social distancing in different locations, computing their values for some Brazilian regions. Following the approximation of those parameters, we are able to forecast long term best/worst case scenarios, and we gauge the efficiency of the scheme comparing our predictions with available data using in-sample/out-of-sample testing.

We next use the model to compare age-dependent vaccination policies. In a simple setting, we compare vaccinating the young or the older population. The model is consistent with vaccination policies prioritizing either the elderly or the young population, depending on the values of the parameters, including how late after the start of the pandemic the vaccinations take place. In general, for a short term period, vaccination prioritizing the elderly population is the best option with respect to the total number of deaths. But that might not be the case in the long run if extra vaccine supplies are not provided.

Disclaimer: It is not our intention to propose real-life epidemiological policies or to provide predictions upon which policies could be developed. Our intention is solely to investigate possible patterns of the disease evolution and raise possible alternatives in terms of vaccination policies.

2. The model

In the spirit of compartmental modeling, we divide the whole population among susceptible (S), infected (I), removed (R). Due to the short time scale of the disease, we can assume that demographic changes are not relevant, and deaths are estimated from the removed, R. The above quantities are 16-dimensional vectors, stratiﬁed by age, running from 0-4 (ﬁrst components), 5-9 (second components), etc, up to 75-above (16th components). There are P16 Pi individuals for each age group, and the total population P = i=1 Pi is constant. 4 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES The equations governing the dynamics of the disease is as follows1, for i = 1 ..., 16 and t ∈ N:

Si(t + 1) = Si(t) − βi(I)Si(t),

(1) Ii(t + 1) = Ii(t) + βi(I)Si(t) − γIi(t),

Ri(t + 1) = Ri(t) + γIi(t), plus initial conditions. The model is “conservative” in the sense that Si + Ii + Ri = Pi is constant for all ages i.

The term βiSi is the rate of new infections of the i-population, and we remark that β depends on the number of infected of all ages, making the model nonlinear. Those infected are removed from their condition at rate γIi (they either recover or die). We further divide sc the infected into two groups, those that are asymptomatic or sub-clinical Ii = (1 − ρi)Ii in c the sense that they do not require medical care. The clinical group Ii = ρiIi corresponds to those that get ill, requiring medical attention. The age-dependent fractions of those who become sub- or clinical are ρi ∈ [0, 1]. To fully describe the model, we ought to deﬁne β and the other parameters. (i) the components of the incidence function β are given by the formula

16 sc sc c c 16 X α Ij + α Ij X Ij β (I) = Ce = αsc(1 − ρ ) + αcρ Ce , i P i,j j j P i,j j=1 j j=1 j where Ce is an effective contact matrix that measures the number of contacts between age groups. It depends on the amount and choices of lock-down imposed, as we describe further ahead. The age-independent parameters αsc and αc correspond to the fraction of those infected that have the potential to infect others, due to social behavior or biology. For convenience, we also write the vector β(I) = CeDI, where D is a diagonal matrix defined by sc c α (1 − ρj) + α ρj Djj = . Pj (ii) The time dependent effective contact matrix Ce is defined by

e home work school other (2) C (t) = βh(t)C + βw(t)C + βs(t)C + βo(t)C ,

home work school other ` where the exogenous data C , C , C , C are 16 × 16 matrices, where Cij indicates the average number of diﬀerent people from the age group j that someone from the age group i contacted per day at ` ∈ {home, work, school, other}, as compiled

1Note that we are not using the Einstein’s summation convention of summing up repeated indices. EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 5

by Mossong et al. (2008); Prem et al. (2017). The parameters β` control the fraction of contacts that are “adequate”, in the sense that they lead to infections. (iii) Each component of ρ indicates whether an infected individual will become a sub-clinical

or a clinical case. Since COVID-19 symptoms are more aggressive for older patients, ρi grows with i.

−1/dI (iv) γ = 1 − e indicates the daily probability that someone stays infected, where dI is the average duration of infection, as in Prem et al. (2017). (v) The contagious parameters αsc and αc correspond to fractions of sub-clinical and clinical infected patients that still spread the virus. In Acemoglu et al. (2020), a SIR model that takes into account age is also used, and three groups are considered. The authors include the possibility of using a family of scaling for the incidence function β, but none of them reproduces our own choice.

2.1. Reproduction and replacement numbers. The basic reproduction number R0 is understood as the secondary cases produced by an infected individual introduced in a totally susceptible population, see Barril et al. (2020); Delamater et al. (2019); Diekmann et al. (1990); Hethcote (2000). Hence, suppose that the whole population with distribution P is susceptible (S = P ), and we represent the infected individual by Iˆ, a vector belonging to the canonical basis {e1,..., e16}. The number of infected individuals in the ﬁrst day belonging to the i-th is

16 16 16 X ˆ X ˆ X ˆ δikβk(I)Pk = δikPkCkjDjlIl = PimCmjDjlIl k=1 k,j,`=1 m,j,`=1 where δik indicates the Kronecker delta, which equals one if i = j and zero otherwise, and we deﬁne P as the diagonal matrix such that Pkk = Pk. Let A = PCD. Then the vector containing the contaminated individuals in the ﬁrst day is AIˆ, and we can bound its Euclidean norm as

ˆ ˆ (3) kAIk2 ≤ λmaxkIk2 = λmax,

d−1 ˆ where λmax is the largest eigenvalue of A. Note that only the fraction (1 − γ) I of the infected outsider remains infected on the d-th day. Those individuals infect less than (1 − d−1 γ) λmax, as in (3), and adding up all days, we gather that the total number of infected agents is bounded by the basic reproduction number

∞ X 1 (4) R := (1 − γ)d−1λ = λ . 0 max γ max d=1 6 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES Reasoning in a similar fashion, we deﬁne the replacement number 1 (5) R := λ (t), t γ max where λmax(t) is deﬁned as the maximum eigenvalue of the matrix SCD and S is the diagonal matrix with the diagonal given by Sj for j = 1,..., 16.

The above deﬁnitions of R0 and Rt rely on the Euclidean and spectral norms. However, if one is interested in considering age-dependent quarantine or vaccination policies, it might be interesting to consider the inﬁnity norm. An infected individual of the age group ` ∈ {1,..., 16} would contaminate, on a single day,

16 e X e (6) β(e`) · P = C De` · P = Ci`D``Pi. i=1 As before, adding up the days and considering that at the jth day there will be a fraction d−1 (1 − γ) e` of infected outsider, we gather that the total number of infected agents is the age-dependent basic reproduction number ∞ X 1 (7) R := (1 − γ)d−1(P T CD) = (P T CD) . 0,` ` γ ` d=1 Similarly, we deﬁne the corresponding age-dependent replacement number 1 (8) R := (ST CD) . t,` γ `

The above number Rt,` indicates how many people someone from the age group ` would contaminate at time t.

2.2. Modeling the lethality. Mortality is not part of the dynamics2, resulting from the number of infected patients. We postulate that only those clinically infected die, at an c age-dependent rate. So, among the “recovered” patients, d(t) = γµ(t)wd · I (t) die, where the constant vector wd is the lethality weight of the disease and the lethality strength µ is a scalar that captures how lethality varies with time. Note that the lethality proportion between diﬀerent ages is constant.

2.3. Data. The transmission parameters αsc and αc are such that the transmission rate of asymptomatic is 60% larger than those symptomatic. This ad hoc choice is based on the assumption that symptomatic patients are more likely to quarantine themselves. We stipulate that the proportion of symptomatic patients are as in Figure 1 (left), assuming that the proportion of symptomatic patients follows the same pattern of the infected that

2We assume that the recovered patients do not get reinfected. EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 7

Risk of becoming clinical 1 0.2

0.9 0.18

0.8 0.16

0.7 0.14

0.6 0.12

0.5 0.1

0.4 0.08

0.3 0.06

0.2 0.04

0.1 0.02

0 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 age

Figure 1. Probability of being clinical among infected patients ρ (left) and

lethality weight wd (right).

were hospitalized, as reported in Verity et al. (2020). However, the proportion of asymptomatic infected patients is far from being settled, with estimates ranging from 5% to 80%, as described in Heneghan et al. (2020). Similarly, we assume that the lethality rates are as Figure 1 (right), and we assume that the proportions follow Brazeau et al. (2020); Verity et al. (2020). The contact matrices Chome, Cwork, Cschool, Cother are from Prem et al. (2017), and population data, as seen in Figure 6 is from the Instituto Brasilerio de Geograﬁa e EstatÃstica

(IBGE). The average number of days dI that a patient is infectious is set to 12. We use the available data (number of new cases and deaths) from each location to estimate e the remaining functions βh(t), βw(t), βs(t), βo(t) used to define the effective matrix C in (2), and µ(t) defined in Section 2.2. These functions are transient since not only individuals change behavior with time, as in Brotherhood et al. (2020), but also lethality rates decay as the medical expertise increases with time. For a simple SIR model, the available data uniquely determines the parameters, as in Bakhta et al. (2020). In general, that is not the case for our model, for instance if one of the four contact matrices vanish or if Cwork = Cother. We reduce possible indeterminations of the parameters by assuming that βw = βo, i.e., the behavior at work is equal to the behavior at other locations. We also make then some further simplifying hypotheses. First, we assume that βh is constant, as there is no reason to expect that the amount of contacts at home varies significantly with time. The second assumption is that βs(t) is identically zero, since schools were closed during the period of data gathering. data Consider that data is available for the period {1,...,T } ⊆ N, and let and NI (j) and data ND (j) be the number of new cases and new deaths at the j -th day. For simplicity, assume 8 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES

N that T = Nδt where δt = 10, and consider the partition Tδt of {1,...,T } = ∪j=1Ij where Tδt = Ij : j = 1,...,N ,Ij = (j − 1)δt + 1, jδt .

Let P0(Tδt) be the space of piecewise constant functions with respect to the above partition.

To determine βw we pose the problem of ﬁnding (βh, βw) that solves

sir data kN (βˆ , βˆ ) − N k T ˆ ˆ ˆ ˆ I h w I R (9) min J(βh, βw), where J(βh, βw) = , ˆ ˆ data (βh,βw)∈R×P0(Tδt) kNI kRT T sir ˆ ˆ and k·kRT is the Euclidean norm in R . Above, NI (βh, βw)(t) is the number of new infected clinical patients that the SIR model (1) yields at the t-th day if one replaces βh and βw by ˆ ˆ βh and βw in (2).

The above problem has ﬁnite dimension N + 1 since any function β ∈ P0(Tδt) can be written as N X (10) βw(t) = bjχIj (t) j=1

for some unknown coeﬃcients b1, . . . , bN ∈ R, where χIj (t) is the characteristic function of

Ij, which equals one if t ∈ Ij, and zero otherwise. The coeﬃcients uniquely deﬁne a function

in P0(Tδt) and vice-versa. To solve (9) we employ a random optimization approach

(i) Choose a initial guess (βh, βw) (ii) While some convergence criteria is not reached ˆ ˆ (a) Deﬁne (βh, βw) from (βh, βw) by adding noise ˆ ˆ ˆ ˆ (b) If J(βh, βw) < J(βh, βw), then (βh, βw) ← (βh, βw) The estimation of the lethality strength µ is based on the number of daily deaths data data c ND . The total number of deaths simulated by the SIR model is given by γµ(t)wd · I (t). Equating both quantities we compute

data ND (t) (11) µ(t) = c . γwd · I (t) Of course, parameter determination cold be pursued using more sophisticated methods as in Albani et al. (2020) for instance.

3. Simulation of the dynamics of the disease

In what follows we show the results coming out of parameter estimations. We then show results addressing how well the model predicts the number of new cases. We compare with EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 9 real daily data for new cases and deaths3. To smooth out oscillations, we present the data using a 7-day moving average. One of the hurdles for models as ours is the definition of initial conditions, since when the virus started to spread and how many were the first infected individuals are not known. As a initial condition of our model, we assume that the first infected case occurred 30 days before the official recordings. Indeed, the first day of the official data logs 489 cases, and the real initial condition of the system is not known; cf. Lourenço et al. (2020), who speculates that the epidemic in Italy and UK started one month before the first reported death. We also stipulate that there were 10 infected patients at each age group at day one. Such uncertainty of initial conditions makes the results unreliable at the initial periods of simulations. In particular, the computation of (11) is not feasible for the lack of data. We make then µ(j) = 1 at the initial stages.

3.1. Parameter estimations, and prediction analysis. We consider results for the City of Rio de Janeiro (see the Appendix for other locations), ﬁrst presenting the computed results for all variables, adding up the values for all ages, i.e. 16 16 16 16 X sc X sc c X c X S(t) = Sj(t),I (t) = Ij (t),I (t) = Ij (t),R(t) = Rj(t). j=1 j=1 j=1 j=1 We recall the assumption that the contacts might lead to infections at home are time independent, and the contacts at work and other locations have a similar pattern, i.e., βw = βo.

Also, schools are closed, thus βs = 0. We display in Figure 2 the results of our parameter estimations; see the caption for a detailed description of each individual plot. The in-sample predictions from the model seem reasonable, approximating well the data and capturing the dynamics of the disease. In particular we see that βw (top-right plot) oscillates following the “macroscopic” oscillations of daily cases (top-left plot). Regarding the lethality, we note the decrease of its strength after day 90, in line with what was presented by Dennis et al. (2020); Horwitz et al. (2020). Two important pieces of information that comes out of the modeling are the value of the reproduction and replacement numbers R0 and Rt, and these computations follow easily from (4) and (5). We display the results in Figure 3 (left ﬁgure). Note that values of R0 is characterized only by the β’s and the data, while Rt depends also on the dynamics of the disease through the susceptible population. That is important since even if R0 > 1, having

Rt smaller than one is enough to have a declining number of infected patients.

3All data were from taken from https://covid.saude.gov.br/, an oﬃcial site from the Brazilian’s Government Department of Health. 10 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES

New daily cases (SIR and data) Acumulated cases Beta 0.025 1.6 0.014 SIR: Infected clinical SIR clinical data data 1.4 0.012 0.02 1.2 0.01

1 0.015 0.008

0.8 β 0.006 0.01 0.6

0.004 0.4 0.005 percentage of total population percentage of total population 0.2 0.002

0 0 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 days days days

×10 -3 Daily deaths - faixa etaria > 0 Acumulated deaths Letality 3 0.18 4 SIR R0 data Rt (L infinito) 0.16 3.5 2.5 Rt (espectral) ponto de recuperacao 0.14 3

2 0.12 2.5 0.1

1.5 µ 2 0.08 1.5 1 0.06

1 0.04 percentage of total population percentage of total population 0.5 0.5 0.02 SIR data 0 0 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 days days days

Figure 2. Results for the City of Rio de Janeiro. On the ﬁrst couple of ﬁgures on the top we plot the new cases for each day, and the accumulated results. The red plot corresponds to real data, and the black plot depicts clinical cases modeled by SIR. On the top right we plot the estimated values

of βw, except for the ﬁrst (that has a value roughly ten times bigger than the others). For visualization purposes we interpolated the discontinuous function

βw using cubic splines. The figures on the bottom are related to fatalities. The first and the second figures show daily and accumulated cases. The third figure displays the values of the lethality strength µ, defined in (11). The blue curve is the least square quadratic curve.

Another issue that is worth discussing is that we fit the number of clinically infected patients Ic to the data. However, we discussed nothing thus far about the sub-clinical patients Isc. It turns out that the number of sub-clinical patients is, in this case, roughly ten times the number of registered patients, as shown in Figure 3 (right figure). Such difference resonates with often claims that there is a significant sub-notification. For instance, Havers et al. (2020) reports that the number of cases in the US might be at least 10 times of what is registered. It would not come as a surprise if this difference is even greater in Rio. Also, the EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 11

SIR: R0 and Rt New daily cases (SIR (subclinical) and 10 X data) 5 0.3 R0 SIR: Infected subclinical 4.5 Rt (espectral) 10 X data Recovery threshold 0.25 4

3.5 0.2 3

2.5 0.15

2 0.1 1.5

1 percentage of total population 0.05 0.5

0 0 20 40 60 80 100 120 140 160 180 200 0 50 100 150 200 days days

Figure 3. Reproduction and replacement numbers (left) and sub-clinical cases (right) for the city of Rio de Janeiro for the City of Rio de Janeiro.

WHO states that 80% of infections are mild or asymptomatic4, in particular among children, as discussed by DeBiasi and Delaney (2020); Han et al. (2020). Recall that children are often not tested. Next we investigate numerically if this model can make reasonable predictions. To do so, after the day 200, we run the simulations based on an in-sample (days 1-200) calculated βw and µ, and compare with available data. We perform experiments considering the best and worst scenarios, i.e. lowest and highest number of cases and deaths. We also consider an intermediary case. The main idea shares similarities with some Value at Risk analyses used in risk management, where the past dynamics of investments help in predicting the future; see Jorion (2006). We make the predictions by going back j 10-day periods and performing simulations with the highest and lowest values of βw and µ in the period 200 − 10j and 200. Choosing j = 4, we show in Figure 4 the predicted number of new daily and accumulated cases and deaths with 50 days in advance. Note that, for most of the period of interest, the data is contained between the maximum and minimum value of predicted cases. We also plot the intermediary case, corresponding to the choice of βw and µ being the average of the extreme values, yielding a more accurate prediction. Regarding predictions for a long-term future (200 days), we plot in Figure 5 the accumulate cases for best, worst and average scenarios. One could wonder if the long term predictions of the number of deaths are pessimistic, since it is likely that the lethality will continue to

4https://www.who.int/emergencies/diseases/novel-coronavirus-2019/ question-and-answers-hub/q-a-detail/q-a-similarities-and-differences-covid-19-and-influenza 12 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES

New daily cases (SIR and data) ×10 -3 Daily deaths 0.018 1.8 SIR: Infected clinical 0.016 data 1.6

SIR 0.014 1.4 data

0.012 1.2

0.01 1

0.008 0.8

0.006 0.6

0.004 0.4 percentage of total population percentage of total population

0.002 0.2

0 0 200 210 220 230 240 250 260 270 280 200 210 220 230 240 250 260 270 280 days

Acumulated cases 1 0.09 SIR 0.9 0.08 dado real

0.8 0.07

0.7 0.06 0.6 0.05 0.5 0.04 0.4 0.03 0.3

0.02 0.2 percentage of total population percentage of total population

0.1 0.01

0 0 200 210 220 230 240 250 260 270 280 200 210 220 230 240 250 260 270 280 days days

Figure 4. The top ﬁgures display the number of daily new cases and deaths, while the ones on the bottom display cumulative results. In all cases, the red plots indicates the data and black plots represent the simulation results for

frozen βw and µ corresponding to the best/intermediary/worst case scenarios. decrease in the future. However, a sudden increase in the number of cases might induce a collapse of the health system, reversing the optimistic trend.

4. Vaccination

We explore in this section simple immunization policies, shedding some light on the following question: if an age-based vaccination strategy should be implemented, who should be protected ﬁrst? Certainly, real-life policies are never as simple since other aspects as economics, logistics, ethics, politics, etc, play signiﬁcant roles5. Not only that, but also it is

5For instance, free market policies are discussed in https://johnhcochrane.blogspot.com/2020/12/ free-market-vaccines.html. EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 13

Acumulated cases Acumulated deaths 2.5 0.25 SIR clinical data

2 0.2

SIR 1.5 0.15 dado real

1 0.1

0.5 0.05 percentage of total population percentage of total population

0 0 200 250 300 350 400 200 250 300 350 400 days days

Figure 5. Long term predictions for accumulated cases (left) and deaths (right) in the City of Rio de Janeiro. As before, the data is on red, and the SIR model results are in black. In both ﬁgures, the top/bottom graphs display the worst/best case scenario. The middle graph displays the results when we take the average of the best/worst parameters. not even clear that vaccinations schemes should be devised based on age groups; see Pastor- Satorras and Vespignani (2002) for instance. However, vaccination policies based on age are easy to implement in practice.

Based on the age-dependent reproduction and replacement numbers R0,` and Rt,`, we find out which age group contaminates most people. Consider for instance the dynamics of the disease in the City of Rio de Janeiro. In Figure 6 we display the age distribution of the population (left) and the respective age-dependent replacement numbers Rt,`. Note that although the group of those aged 25-30 is the largest one, the group that contribute the most for the dissemination of the disease is of those aged 15-19. And it is interesting to notice that those above 60, which are those most affected by the disease, are exactly the ones that contribute less for its spreading. Such heterogeneity leads to interesting immunization policy possibilities. The age group 15-19 (around 500 k people in Rio) has the most impact on dissemination, but few of those actually die. The group of those above 60 (around 900 k people) however is at risk, but they would have little impact on the dissemination of the disease. To offer a glimpse on these issues, we modify our effective contact matrix Ce, allowing for the possibility of immunization. Let us redefine (2) by e home work school other (12) C (t) = V (t) βh(t)C + βw(t)C + βs(t)C + βo(t)C V (t), 14 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES

×10 4 Population distribution 14 3 0-4 5-9 12 10-14 2.5 15-19 20-24 10 25-29 2 30-34 35-39 8 40-44 45-49 1.5 50-54 55-59 6 60-64 65-69 1 70-74 4 75-

0.5 2

0 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 0 50 100 150 200 age days

Figure 6. Population of the city of Rio de Janeiro (right), and age-dependent replacement numbers corresponding to the dynamics of the disease in Rio.

where the transient diagonal matrix V (·) models the vaccinated age groups. Its j-th diagonal element is zero if the corresponding age group is protected, and one otherwise. Of course, it is also possible to consider partial immunization by setting values between zero and one. Consider the hypothetical situation of having a region with the population distribution and

the same values for βh and βw (up to day 200) as in the City of Rio (see Figures 6 and 2, top-

right). Between the days 200-500, we assume that βw = βopt = 0.0063 or βw = βmax = 0.0098. Assume a constant lethality strength µ = 1. Suppose that there is a limited supply of vaccines made available only at a certain moment, in our case at day 100, 200 or 300 after the pandemic starts. Assume also that a single shot of the vaccine induces a perfect immunization and stop the virus spreading. The reader should be cautioned that these hypotheses constitute an important departure from reality, which we intend to address in future work. To compare experiments, we deﬁne an utility function that takes into account the number of lost lives. For Υ ∈ (0, 1), we deﬁne the Υ-discounted cost of death as

+∞ X (13) Υtd(t),

t=T0 where T0 is the day of vaccination and d(·) is the number of daily fatalities. We consider Υ = 0.9961/365 in the numerical tests. In our numerical experiments, we consider three age groups of similar size: those aged 15–34 (≈ 1, 584 people), 40–59 (≈ 1, 636 people) and those above 50 (≈ 1, 670 people). In all tested situations, the groups with best results were either the 15–34 or those above 50. EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 15

In Figure 7, we plot the accumulated deaths for the groups aged 15–34 and over 50, for smaller (left) and larger (right) rates of transmission after day 200. In solid blue we plot the total deaths if no vaccination takes place. The remaining plots correspond to the total deaths if vaccination occur on day 100 (black), 200 (red) or 300 (magenta). The solid lines correspond to vaccinating the young and the dashes lines corresponding to vaccination of the elderly. Basically, two different policies are possible: stop the spreading corresponds to vaccinating those in the 15–34 age group, and protect the vunerable corresponds to vaccinating those above 50. Note from the results that the impact of stop the spreading decreases as the vaccination date is delayed. That is because the percentage of young population already contaminated at latter dates is larger, and vaccinating the young group ends up not being so effective. Such effect is amplified for higher transmission rates, as one can see by comparing the figures on the right and left. Note also that for short terms, it is always preferable to protect the vunerable, and this indicates that vaccinating the elderly population is a wise option if a continuous supply of vaccines is available. Considering the results at day 500, for lower transmission rates (left figure), it is better to vaccinate the elders if vaccination takes place at days 200 or 300, but the results are reversed if vaccination takes place on day 100. The figure on the right indicates that under higher transmission rates, vaccinating the young group first might be appropriate. These results are confirmed in Table 1, where we consider the cost of death (13). For related results, see Giubilini et al. (2020); Zhao et al. (2020). We also remark that although most countries are pursuing policies that prioritize the elderly, there are exceptions; see, for instance, The Japan Times (2020).

5. Discussions and conclusion

Epidemics as the COVID-19 are hard to model, not only because of biological factors, but also due to the unpredictability of human responses. Political, economical, social and individual factors inﬂuence the behavior of people, leading to various degrees of risk behavior. Making long term predictions is riskier than usual in (non-human) biological systems, but they are crucial for planning non-pharmaceutical interventions, allocation of resources, economical decisions, etc. What we propose in the work is a general form to predict possible values of crucial parameters based on their past values. Our method has two steps. Based on the dynamics of the disease on a multi-generational SIR model, we ﬁrst determine the values of some time-dependent parameters, using a random optimization algorithm. Then, we feed the SIR 16 EDUARDO CAMPOS, RUBENS CYSNE, ALEXANDRE MADUREIRA, AND GÉLCIO MENDES

Acumulated deaths Acumulated deaths 0.16 0.3

0.14 0.25 0.12

0.2 0.1

0.08 0.15

0.06 0.1 0.04 percentage of total population percentage of total population 0.05 0.02

0 0 0 100 200 300 400 500 0 100 200 300 400 500 days days

Figure 7. Accumulated deaths for βw = 0.0063 (left ﬁgure) and βw = βmax = 0.0098 (right ﬁgure). In both cases, we assume no vaccination (solid blue), vaccination for group aged 15-34 (solid plots) and over 50 (dashed plots). The vaccination takes place on day 100 (black), 200 (red) or 300 (magenta). Note that vaccinating the elderly population is always optimal in the short term, but not necessarily in the long run.

βw = 0.0063 after day 200 βw = 0.0098 after day 200 Vaccination day Age group # vaccines 100 200 300 100 200 300 none 0 8.0 k 3.1 k 844 15.4 k 10.8 k 5.0 k 15-34 1,584 k 1.6 k 806 293 1.6 k 1.2 k 1.7 k 40-59 1,636 k 2.6 k 976 281 4.7 k 2.7 k 1.5 k 50- 1,670 k 2.1 k 759 206 3.7 k 2.3 k 1.1 k Table 1. Numerical investigation of immunization policies based on age. In each line we present the values of the death cost functional (13) assuming different age groups are vaccinated. The second column contains the population of the groups. In the three columns that follow, we present the results considering the vaccination takes place on the days 100, 200 or 300 and assuming

that βw = βopt. The following three columns display the results for βw = βmax and vaccination on the days 100, 200 or 300.

model using extreme values of such parameters, and predict best/worst case scenarios. We are also consider a prediction that is between the two extreme cases. EPIDEMIOLOGICAL FORECASTING WITH MULTI-GENERATIONAL SIR MODELING 17

We test the method for the City of Rio de Janeiro, and some other locations (in the Appendix), predicting the dynamics for 70 days. In all tests, the method performed well, and the accumulated cases and deaths stayed within or very close to the bounds. Our methodology is general and can be applied to other systems and circumstances, and might be useful to make long-term predictions in situations where unexpected occurrences might change the behavior of the system. We also explore age-based immunization strategies, under simplifying hypotheses. In particular we assume in our tests that vaccinated people no longer transmit the disease, an hypothesis that might not be correct with the available vaccines. Note that while we assume the behavior of the population as known, the very existence of vaccines changes people’s behavior. Moreover, several important aspects of vaccination are not considered here, as ethical problems, logistics, etc. We also point out that other policies might be better, as targeting certain classes of workers or individuals with comorbidities. In our limited setting, we find out that, optimal policies depend, among other things, on transmission rates and when vaccination takes place. Under most conditions, it pays off to vaccinate the elderly although they contribute the least for spreading the viruses. Earlier vaccination dates and higher transmission indices increase the impact of vaccinating the young. For short terms however, protecting the older population is always preferred. That seems to indicate that under a continuous supply of vaccines, protecting the elderly is a wise option. It is clear that the model provides different indications under different scenarios, and we shall explore more detailed policies in a future work. As we point out in our Disclaimer, our goal is not to predict actual number of cases and deaths or to propose general policies, but to investigate limited aspects of the disease dynamics and raise possibilities of immunization strategies.

Appendix A. Other locations

We next perform the same parameter estimation and prediction analysis for diﬀerent regions. We show the respective results for the city of Petrópolis (Figure 8), State of Rio de Janeiro (Figure 9), City of São Paulo (Figure 10), and State of São Paulo (Figure 11).

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ENCE - Instituto Brasileiro de Geografia e EstatÃstica, Rio de Janeiro - RJ, Brazil

EPGE - Fundação Getúlio Vargas, Rio de Janeiro - RJ, Brazil Email address: [email protected]

Laboratório Nacional de Computação Científica, Petrópolis - RJ, Brazil

EPGE - Fundação Getúlio Vargas, Rio de Janeiro - RJ, Brazil Email address: [email protected], [email protected]

INCA - Brazilian National Cancer Institute, Coordination of Assistance, Rio de Janeiro - RJ, Brazil Email address: [email protected]