Does susceptibility to novel coronavirus (COVID-19) infection differ by age?: Insights from mathematical modelling

Among Italy, Spain, and Japan, the age distributions of novel coronavirus (COVID-19) mortality show only small variation even though the number of deaths per country shows large variation. To understand the determinant for this situation, we constructed a mathematical model describing the transmission dynamics and natural history of COVID-19 and analyzed the dataset of fatal cases of COVID-19 in Italy, Spain, and Japan. We estimated the parameter which describes the age-dependency of susceptibility by fitting the model to reported data, taking into account the effect of change in contact patterns during the outbreak of COVID-19, and the fraction of symptomatic COVID-19 infections. Our modelling study revealed that if the mortality rate or the fraction of symptomatic infections among all COVID-19 cases does not depend on age, then unrealistically different age-dependencies of susceptibilities against COVID-19 infections between Italy, Japan, and Spain are required to explain the similar age distribution of mortality but different basic reproduction numbers (R0). Variation of susceptibility by age itself cannot explain the robust age distribution in mortality by COVID-19 in those three countries, however it does suggest that the age-dependencies of i) the mortality rate and ii) the fraction of symptomatic infections among all COVID-19 cases determine the age distribution of mortality by COVID-19.

The expected value of mortality (the number of deaths, hereafter referred to as 41 mortality) is calculated as the product of the number of cases and the mortality rate 42 among cases (hereafter referred to as morality rate). As the background mechanism of 43 the heterogeneity of mortality by age, the association of two epidemiological factors 44 with mortality can be considered: i) the age-dependency of susceptibility to infection, 45 which is related to the heterogeneity in the number of cases, and ii) the age-dependency 46 of severity, which is related to the heterogeneity in the mortality rate, e.g. the rate of Although not yet shown in relation to 58 severe acute respiratory syndrome corona virus 2 (SARS Cov-2), which is the causal 59 agent of COVID-19, the presence of age-dependent enhancement of severity has been 60 suggested in SARS coronavirus by the analysis of the innate immune responses in the 61 estimate of φ was 1.7 (95%CI = 1.4-1.9), 2.2 (95%CI = 1.9-2.5), and 2.5 (95%CI = 2.1-123 2.8) for 80%, 40%, and no reduction in contacts outside of the household. 124 125

126
In the present study, we explored the role of susceptibility to COVID-19 in 127 explaining the age distribution of mortality by COVID-19. Interestingly, the age 128 distributions of mortality from COVID-19 are quite similar between Italy, Japan, and 129 Spain (figure 1). When comparing the age distributions of mortality, only the 130 comparison between Italy and Spain is significant (p<0. 05  CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint required to explain their age distribution of mortality for both settings, i) age-141 independent mortality, and, ii) the fraction of infections that becomes symptomatic 142 among all COVID-19 cases, fs, does not depend on age. Although we cannot fully reject 143 the existence of age-dependency in susceptibility, our results suggest that it does not 144 largely depend on age, but rather that age-dependency in severity highly contributes to 145 the formation of the observed age distribution in mortality. 146 The estimates of φs assuming age independency in symptomatic infections were 147 smaller than those that assumed age independency in mortality. This suggests that the 148 age-dependency of the confirmed case fatality rate (cCFR), which can be biased by the 149 age-dependent difference of the fraction of symptomatic infections among all cases, 150 partially explains the age distribution in mortality. Indeed, when we assumed that the 151 fraction of symptomatic infections was not dependent on age, the estimate of φ in Japan 152 was close to zero in all scenarios regarding the fraction of symptomatic infections, 153 meaning that susceptibility is constant among age groups (figure 5). Although we 154 observed φs around 5 in Italy and 2 in Spain, this does not mean straightforwardly that 155 susceptibility is age dependent because there is room for an alternative explanation: not 156 susceptibility, but an age-dependent fraction of symptomatic infections can explain this 157 age-dependency. Unfortunately, as we do not yet have detailed data regarding the age-158 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint dependent fraction of symptomatic infections and the rate of diagnosis in COVID-19, 159 we cannot conclude which factors (i.e., susceptibility or the fraction of symptomatic 160 infection among all cases) contributed to the observed age-dependency. To understand the mechanism of age-dependency of mortality by COVID-19, an 167 accurate age-dependent mortality rate is required. To estimate the age-dependent 168 mortality rate, an accurate estimate of the case fatality rate is required. However, the 169 number of cases, which is the denominator of the case fatality rate, is difficult to 170 estimate for COVID-19 due to changes in the testing rate ( is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 9, 2020. . Based on our results and from the biological/epidemiological observations of past 186 SARS and MERS cases, the "increasing severity" assumption should be taken into 187 account when analyzing SARS Cov-2 epidemics. 188 In conclusion, the contribution of age-dependency to susceptibility is difficult to 189 use to explain the robust age distribution in mortalities by COVID-19, and it suggests 190 that the age-dependencies of the mortality rate and the fraction of symptomatic 191 infections among all COVID-19 cases determine the age distribution in mortality from 192 COVID-19. Further investigations regarding age-dependency on the fraction of 193 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted June 9, 2020. To understand the background of robust age distribution of mortality with varied 206 R0, we employed a mathematical model describing transmissions of COVID-19, an age-207 structured SEIR model, which can be written as; 208 (1) 209 210 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint where Sn, En, In, Rn and Dn represent the proportion of susceptible, latent, infectious, 214 recovered and dead among the entire population, and the subscript index n denotes age 215 group. We stratified the entire population by into eight groups, n = 1, 2, 3, 4, 5, 6, 7, and 216 8 for < 10 years old (yo), 10-19 yo, 20-29 yo, 30-39 yo, 40-49 yo, 50-59 yo, 60-69 yo, 217 and 70+ yo. β, kn,m, ε, γ and δn represent a transmission coefficient, an element of the 218 contact matrix between age group n and m, the progression rate from latent to 219 infectious, recovery rate and mortality rate among age group n, respectively. σn denotes 220 the susceptibility of age group n. For the sake of simplicity, births and deaths by other 221 than COVID-19 were ignored. To take into account the effect of behavioral changes 222 outside of the household during the outbreak, kn,m is decomposed by a matrix for 223 contacts within household kin,n,m and that for contacts outside the household kout,n,m; 224 where α denotes the reduced fraction of contacts outside of the household. We modelled 226 age specific susceptibility as 227 We parameterized ε and γ using the values from a previous modelling study of  19 (Prem et al., 2020). We referred to the contact matrices for Italy, Japan, and

Fitting 243
We calculated the proportions of deaths in the age group n among all deaths, dn 244 ), and fitted them to the observed data in each country. The 245 mortality rate among age group n, δn, is required to calculate dn, however, a reliable 246 estimate of δn for COVID-19 is difficult to obtain. Due to the uncertainty of the fraction 247 of symptomatic infections per age group, δn is difficult to estimate from observed data, 248 i.e., the confirmed case fatality rate among age group n (cCFRn). Since an estimate of δn 249 is difficult to obtain, we employed two different settings to calculate dn, i) δn is assumed 250 to be a constant among all age groups, or, ii) δn is calculated from cCFRn assuming that 251 the fraction of symptomatic infections among all COVID-19 cases ( fs) is not dependent 252 with age. 253 In setting i), the value of δn is not required to estimate dn once the value of R0 is 254 given. We calculated dn by calculating the proportions of recovered persons per age 255 group among all recovered persons " (∞)/ ∑ " (∞) . 256 In our model, shown in equation (1) is determined by the value 257 of R0 completely when all parameter values other than β and δn are fixed, and 258 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint if " ≠ 0. The proof can be found in 259 supplementary file 1. 260 The assumption in setting i), δn is constant among all age groups, may be too 261 strong for the COVID-19 epidemic. To take into account the age-dependency of 262 mortality by COVID-19, δn was calculated from the cCFRn assuming that fs is not 263 dependent with age. As for the setting ii), assuming three scenarios; fs = 0.05, 0.25, and 264 0.5, δn for each country were calculated using cCFRn in each country. We obtained δn 265 We solved the model shown in equations (1)-(5) numerically, and dn was 267 calculated after sufficient time was given to finish the epidemics. We estimated φ using 268 a log likelihood function describing the multinomial sampling process of deaths per age 269 group; 270 Competing interests 278 The authors declare no conflicts of interest. 279 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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(which was not certified by peer review)
The copyright holder for this preprint this version posted June 9, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted June 9, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted June 9, 2020.  . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint mortality when it was assumed that age-dependent mortality was proportional to cCFR 500 per age group. All parameters were fixed and parameterized as the setting for Spain 501 except the transmission coefficient β. Proportion . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint susceptibility among age groups assuming that mortality rate does not depend on age 521 and the fraction of infections that becomes symptomatic among all COVID-19 cases is 522 0.5. True and broken lines represent the maximum likelihood estimates and 95% 523 confidence intervals, respectively. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint susceptibility among age groups assuming that mortality rate does not depend on age 529 and the fraction of infections that becomes symptomatic among all COVID-19 cases is 530 0.05. True and broken lines represent the maximum likelihood estimates and 95% 531 confidence intervals, respectively. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 9, 2020. . https://doi.org/10.1101/2020.06.08.20126003 doi: medRxiv preprint