key: cord-0429772-c10agqnb authors: Miura, F.; Klinkenberg, D.; Wallinga, J. title: Dose-response modelling of endemic coronavirus and SARS-CoV-2: human challenge trials reveal the individual variation in susceptibility date: 2022-04-16 journal: nan DOI: 10.1101/2022.04.07.22273549 sha: 5f7e6c0fbb8d42d5a41712d9596cb7836457453c doc_id: 429772 cord_uid: c10agqnb We propose a mathematical framework to analyze and interpret the outcomes of human challenge trials. We present plausible infection risks with HCoV-229E and SARS-CoV-2 over a wide range of infectious dose, and suggest ways to improve the design of future trials and to translate its outcomes to the general population. Background 42 Quantifying the infectivity of pathogens is a crucial step towards the understanding of 43 infection risks. In human challenge trials the infection risk is observed as the proportion of 44 exposed participants that become infected. Dose-response models describe how this 45 proportion infected changes with an increase in the infectious dose used to expose the 46 participant [1, 2] . Such dose-response models can be used to improve trial designs [3] , to describe infectivity and immunogenicity in human hosts [4] , and to simulate the infection 48 risks via various transmission routes [5] . 49 Dose-response models can account for variation in host susceptibility, and most often such 50 variation has been modelled by a beta distribution [6, 7] . However, when a proportion of 51 individuals is completely immune, the variation is better captured by other distributions (e.g., 52 [8] ). 53 Here, we start by reformulating dose-response models with a flexible description of the 54 variation in host susceptibility that allows for an intuitive biological interpretation. We show 55 how variation in susceptibility determines the dose-response relationship for the endemic 56 human coronavirus HCoV-229E and we compute the plausible range of SARS-CoV-2 dose-57 response curves based on available outcomes of a challenge study. 58 Our approach suggests how the design of human challenge trials can be improved to better 59 capture the variation in susceptibility, and suggests how to translate the outcomes of human 60 challenge studies into infection risks for the general population. Methods 63 Human challenge studies 64 We conducted a literature search to collect available data from human challenge studies 65 with endemic coronaviruses and SARS-CoV-2. The collected data consists of 5 studies with 66 endemic coronaviruses HCoV 229E and one study with SARS-CoV-2. In all cases the study 67 population consisted of healthy adult volunteers, and the participants were intranasally 68 inoculated with certain doses in each trial. The challenge studies reported the challenge dose, 69 the number of challenged individuals, and the number of infected individuals as summarized 70 in Supplementary Material, Table S1 and Table S2 . 71 72 Dose-response models to analyze human challenge studies 73 The reported infectious doses are expectations of a Poisson distribution of the actual 74 infectious dose, with mean . If each host is equally susceptible, the probability of infection 75 ( ) given a challenge dose is ( ) = 1 − (− ). This assumes that each infectious 76 particle can independently establish an infection [2, 9] . 77 We capture the variation in susceptibility among study participants by assigning each 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 April 16, 2022 to reflect a situation where the level of susceptibility varies continuously; a bimodal 87 distribution with one fraction of the population almost immune, and the remaining fraction 88 with a single level of susceptibility; and a bimodal distribution with one fraction of the 89 population almost immune, and the remaining fraction with a gamma distribution for the level 90 of susceptibility to vary continuously. Detailed model descriptions and estimated parameters 91 are provided in Table S3 . Susceptibility distributions determine the shape of dose-response curves 95 We described the proportion of infections among individuals exposed to different doses of 96 the endemic coronavirus HCoV-229E by fitting dose-response models. Since the collected 97 trial data include participants who might have been exposed to viruses, we included bimodal 98 distributions of susceptibility. 99 The results reveal a strong statistical support for a distribution reflecting a situation where 100 a fraction of the population is almost immune whereas the remaining fraction of the population 101 has a single level of susceptibility. There is no statistical support for a homogeneous level of 102 susceptibility or continuous variation in susceptibility for all individuals (Table S4) . Plausible SARS-CoV-2 dose-response curves 105 In the available human challenge study with SARS-CoV-2 all participants were healthy 106 young adults with no evidence of previous SARS-CoV-2 infection or vaccination, and they 107 were all exposed to the same single dose [10] . Here we show how the variation in (Figure 1B and 1C) . 120 We compared the bootstrapped SARS-CoV-2 dose-response curves with the dose-response 121 curve from a SARS-CoV-1 mouse model obtained by Watanabe et al. [11] , which has been 122 widely used in risk assessments of SARS-CoV-2 (dotted lines in Figure 1 ). This reveals that . CC-BY-NC 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. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org/10.1101/2022.04.07.22273549 doi: medRxiv preprint In this study, we revealed the plausible range of infection risk over multiple orders of 132 magnitude of the infectious dose for the endemic coronavirus HCoV 229E and SARS-CoV- 133 2, based on human challenge trials. We presented how these dose-response relationships are 134 shaped by the underlying distribution of susceptibility to infection. 135 The range of SARS-CoV-2 dose-response curves arises from the unknown distribution of 136 background susceptibility in the population and the statistical uncertainty due to the limited 137 number of participants that have participated in the human challenge study. Our results 138 caution against assuming equal susceptibility in the population in risk assessments [5, 11] , as 139 this assumption results in a lower bound for infection risks at lower doses. 140 Our results provide implications for further research. We address three of them here. This would elucidate the unknown variation in level of susceptibility among individuals. 149 Secondly, the dose-response models proposed here, as many other dose-response models, 150 have underlying assumptions that the infectious particles are homogeneously mixed in the 151 inoculates and act independently in causing an infection [2, 9] . These assumptions suffice for 152 describing the outcome of human challenge studies, even though it might not hold, for 153 example, when virus particles aggregate. The dose-response model can be extended to allow 154 for variation in the per-particle probability, using methods explored previously [2, 14] , which 155 would allow for a built-in check of violating this assumption. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022 is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022 Figure S1 . Estimated dose-response models of endemic coronavirus. Figure S2 . Simulated trajectories of SARS-CoV-2 dose response curves. Table S1 . Human challenge data with endemic coronavirus. Table S2 . Human challenge data with SARS-CoV-2 Table S3 . Model description and estimated parameters. Table S4 . Model comparison of dose-response models employing different distributions. . CC-BY-NC 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. The copyright holder for this preprint this version posted April 16, 2022 The challenge data of endemic coronaviruses and SARS-CoV-2 were collected from published articles. We conducted literature search using PubMed and Google Scholar, and 13 human challenge studies were found in total. For further analysis, 7 studies [1] [2] [3] [4] [5] [6] [7] were excluded because the information on inoculated doses was unavailable. Thus, 5 studies of endemic coronaviruses [8] [9] [10] [11] [12] and 1 study of SARS-CoV-2 [13] were used in the dose-response analysis. The data consisted of the number of exposed doses, total participants, and infected individuals in each trial. The summary of analyzed data with details (i.e., inoculation methods and references) is shown in Table S1 . To synthesize the obtained data, we set two assumptions. First, the infection status was comparable across those studies. Several studies defined the infection status by antibody level, while others defined it by the presence of viruses. Second, there was negligible effect of aggregation of viruses. Since the detailed information of inoculated samples was not available, the unit of dose is defined as the reported unit (i.e., TCID50, Median tissue culture infectious dose). If there is data that quantify the level of aggregation, further extension of the dose-response analysis is also possible (see [14, 15] ). Here we denote the probability of infection in controlled infection experiments as ( ), a function of dose . In a host, it is reasonable to assume that all the particles are independently infectious and effective to establish an infection (i.e., single-hit theory [15, 16] ). The simplest doseresponse relationship is formulated by incorporating Poisson uncertainty in a microbial inoculum: where is the probability of establishing infection by a single-hit. While previous studies formulated the variation in (often with a beta distribution [15, [17] [18] [19] ), here we focus on the variation in a host. Suppose that the susceptibility to infection among individuals differs and is distributed as ( ) with a level of susceptibility . The interpretation of variable is that an individual with the level of susceptibility = ′ has ′ times higher probability of infection compared to an individual with = 1. By expanding Eq.1 and integrating the variation in susceptibility, the marginal probability of infection is written as where is a parameter vector of ( ). If a single-hit always results in infection, that is, = 1, Eq.2 can be further simplified where ℒ refers to the Laplace transform of ( ). As an illustrative example, we introduce the dose-response model where the level of susceptibility is distributed as a Gamma distribution, ∼ Gamma( , ). By solving Eq.3, the doseresponse model is derived as and this formula is the same as the so-called Beta-Poisson model [15, 19] . Note that we can derive Eq.4 without violating the single-hit principle, and the equation can be interpreted as the relationship between dose-dependent infection probability and the susceptibility distribution within a host. . CC-BY-NC 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. The copyright holder for this preprint this version posted April 16, 2022 Since the results of controlled infections are obtained as either infected or not in challenge experiments, such observation process leads to a binomial likelihood and thus the log-likelihood is where for each trial i we have a dose and a group of volunteers of which are infected. To estimate the set of parameters , maximum likelihood estimation (MLE) was performed. For this computation we used the optim() function in the R statistical programming environment version 3.5.1., and 95 % confidence intervals were computed from 1000 bootstrapped samples. Current risk assessments of SARS-CoV-2 infection risk among humans often refer to the animal dose-response model obtained by Watanabe et al. [20] . Their study used a Delta model (i.e., the first model in Table S3 ) and fitted it to available SARS-CoV-1 data based on mouse experiments. As a result, the estimated parameter was = 1 410 in Table S3 notation. For details, see the original article [20] . For comparison of dose-response curves, we converted the unit of inoculated doses using the ratio of PFU to TCID50 that is previously established as 0.7 [21] . . CC-BY-NC 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. Red plot and whiskers indicate the observed data and its 95% binomial confidence intervals. Each curve is obtained by bootstrapping with a gamma model. From panel A to F, the coefficient of variation is decreased from 10 2 to 10 -3 . . CC-BY-NC 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. The copyright holder for this preprint this version posted April 16, 2022 A population has two levels of susceptibility; a proportion ( 1 ) has a mass at level of susceptibility 1 and the other has another mass at level of susceptibility The level of susceptibility for a proportion 1 − 1 of population follows gamma distribution, and the other proportion 1 has a level of susceptibility 1 . . CC-BY-NC 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 April 16, 2022 2.0 *AIC: Akaike Information Criterion. The lowest value indicates the best model in terms of prediction. † Difference of >10 indicates strong evidence [22] . The values here suggest substantial support for the heterogeneity in susceptibility. . CC-BY-NC 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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