On the study of full transmission dynamics of COVID-19 in Wuhan

Arising from Hao et al. Nature https://doi.org/10.1038/s41586-020-2554-8 (2020) Chong You1+, Xin Gai2+, Yuan Zhang3,4*, Xiao-Hua Zhou1,2,4* 1. Beijing International Center for Mathematical Research, Peking University, Beijing, China, 100871 2. Department of Biostatistics, School of Public Health, Peking University, Beijing, China, 100871 3. School of Mathematical Sciences, Peking University, Beijing, China, 100871 4. Center for Statistical Sciences, Peking University, Beijing, China, 100871 +Joint first author * Joint corresponding author email: azhou@math.pku.edu.cn, zhangyuan@math.pku.edu.cn

from . An individual in would be infected by individuals in , or with different transmissibility to get into and then after a latent period. At the time point of symptoms onset, an individual transited from to or A depending on whether they would be laboratory-confirmed in the future, and is the ratio that a patient would be laboratory-confirmed, namely, the ascertainment rate. Note that for a case to be laboratory-confirmed, the patient must be both symptomatic and tested positive by RT-PCR, which means individuals in must be symptomatic, while those who were in could be asymptomatic and their symptoms onset stage was just a hypothetical one which were included in the model for simplicity. The individuals in would then lose their transmissibility pathologically and got into .
In the meantime, individuals in would either lose their transmissibility pathologically ( ) before they got confirmed and isolated in hospital (which implies that a patient can be no longer infectious but still tested positive by RT-PCR), or got isolated in hospital ( , namely lost their transmissibility physically) and then lost their transmissibility pathologically ( ) eventually. The parameters (ascertainment rate) and (transmission rate) vary across five time periods on the basis of key events ("Chunyun") and containment interventions. It is worth noting that, different from most of other dynamic models fitting number of confirmed diagnosis at time , the numbers of individuals in all compartment in this model were not directly observable except in where ( ) is the number of laboratory-confirmed cases who reported their date of symptoms onset was on time .
Based on such interpretation of the SAPHIRE model, four major concerns are to be raised.
(1) The initial ascertainment rate was estimated based on the assumption of perfect ascertainments in Singapore ignoring asymptomatic individuals which certainly gave an over-conservative estimate of under the current model as mentioned in Hao et al (2020). In addition, should be a continuous function rather than a step function over the five time periods in Hao et al (2020) , see the justification in Appendix A.
(2) The individuals in can be very different including asymptomatic and mild cases, as well as severe cases as evidence by deaths of clinically confirmed cases reported in [2], it is hence not rational to assign a same transmission rate to all individual in (note that the proposed transmission rate in was identical to that of the presymptomatic infectious period , and was α = 55% of that in ). In fact, at the beginning of the pandemic the medical resources were overburdened, it was likely to have a larger fraction of patients with severe symptoms in and thus the transmission rate would be close to that of . On the other hand, when medical resources were replenished and strong screening and public awareness campaign were implemented, the remaining unascertained cases should be mostly asymptomatic or mildly symptomatic, and hence the transmission rate would be closer to that of . See why this issue can NOT be easily resolved in Appendix A.
(3) As mentioned in Hao et al (2020) the clinically diagnosed cases were excluded in the model, however, there was indeed a significant amount of cases in who were not laboratory-confirmed but clinically confirmed and isolated in (cabin) hospital in Wuhan during February 2020 and hence lost their transmissibility before they actually got into namely lost their transmissibility pathologically which implies that clinically confirmed cases in would have a faster rate to get into than other cases in [3]. Though only the data of laboratory-confirmed cases was used in Hao et al (2020), this does not mean the isolation due to clinical diagnosis can be simply ignored in the model.
(4) The pre-determined symptomatic infectious period = 2.9 Days is highly questionable. The symptomatic infectious period is the mean time from symptom onset to loss of transmissibility pathologically in our understanding, and the value was calculated based on the claim that 44% of secondary cases were infected during the index cases' presymptomatic stage by He et al (2020) [4]. Regardless of whether such claim is correct (a matters arising to that study was published), we have to notice that this 44% of presymptomatic spread was estimated based on the confirmed cases with isolation measure outside Wuhan, which is certainly not appropriate to be used to estimated mean time from symptom onset to loss of transmissibility pathologically. Furthermore, another defect in the calculation of is the inconsistency in the study of Hao et al (2020) where a constant infectiousness was assumed across the presymptomatic and symptomatic phases of ascertained cases in estimating while in the meantime α = 0.55 was used as the ratio of transmission rate of cases in (presymptomatic) to that of in (symptomatic). It is important to note that unlike other pre-determined parameters in the model, the value of is quite crucial to the model estimates of interest, see Table S1 in Appendix A for detail. Hence a more decent choice of is essential. Figure 2: Illustration of the modified SAPHIRE model. The modified model includes seven compartments: susceptible ( ), exposed ( ), presymptomatic infectious ( ), ascertainable infectious ( ), unascertainable infectious ( ) and removed ( ). A.
Relationship between different compartments. Two parameters of interest are (ascertainable rate) and (transmission rate). B, Schematic disease course of COVID-19. In this model, the unascertainable compartment includes asymptomatic and mild cases whose symptoms were not significant enough to be detected, while the ascertainable compartment includes symptomatic patients whose symptom were significant enough to be possibly ascertained.
To solve the aforementioned limitations in the current model, we propose a modified version of the SAPHIRE model. In our modified model, , and together with dynamics and parameters associated with them remain unchanged.
We redefine as "ascertainable infectious" than "ascertained infectious", that is, unlike in the SAPHIRE model, individuals are not guaranteed to be ascertained but are those ones with symptom significant enough that could be possibly ascertained, for example a severe symptomatic case might not get laboratory-confirmed if his/her RT-PCR test was negative due to the prolonged waiting time. In the meanwhile, is modified to include patients with no/mild symptoms exclusively who were NOT ascertainable. With such modification, individuals in or would be more homogeneous which is a required underlying assumption in any compartmental dynamic model. Furthermore, now stands for removed for any reason which is in turn a combination of and in the original model, see Figure 2 for illustration. Note that patients in can only transit to by losing transmissibility pathologically while patients in may reach by either losing their transmissibility pathologically, or isolation upon laboratory-confirmation (tested positive by RT-PCR), or isolation upon clinical diagnosis. Thus, the transition rate from to is given by are the period of the symptomatic infectious period and duration from illness onset to laboratory-confirmed diagnosis; and is the duration from illness onset to clinical diagnosis to be set as a step function which equals to infinity and 10 days before and after 2 February. Thus, the alternative model described above can be described by the following ODE system: where is the unknown transmission rate for ascertainable cases and varies over five time periods as in Hao et al (2020); is the ratio of the transmission rate of presymptomatic/unascertainable cases to that of ascertainable cases and is prefixed; is the unknown fraction of infections with significant symptoms; , , , and are the latent period, presymptomatic infectious period, symptomatic infectious period, duration from illness onset to isolation and duration from illness onset to clinical diagnosis respectively and are all predetermined. Under such setting, the transmission rate for is reasonable to be a constant over time and equal to the one for , and in addition, the ascertainment rate can be better presented as the function of the ratio between cases with insignificant (no/mild) and significant symptoms, and the time dependent ratio between the isolation/diagnosis and removal speed. We refer readers to Appendix B and C for estimation method, choices of initial values, parameter settings and sensitive analysis for our modified model. Note that all CI's without further specifications are 95% CI's throughout this paper. Compare with reproduction numbers in other published studies, our estimate in the first period is in the range but on the high side, it is possibly because most of the reproduction numbers were estimated for the period after 9 January namely after our first period, and in addition, the early data were that complete which might lead to an overestimation in reproduction number [5,6,7].  ⟶ with a rate of 1/7. Thus, the expectation of the symptomatic infectious period would be 3+7=10 days which is consistent with our choice of = 10 days.
(2) Patients in had a transmission rate of .
(3) Patients in had a transmission rate of β , where β ∈ [0,1] is the reduction factor of transmissibility in late stage, and is an unknown parameter to be inferred in the model.
Theoretically, this modification would grant us even better compartment homogeneity, and no additional technical difficulties were expected in inferring such a modified model with the same MCMC algorithm. Moreover, since the mean symptomatic infectious period remains the same more realistic value of 10 days, it is reasonable to expect the modified could still present the heavy tail phenomena as in the current study. However, in terms of coding, since such modification could cause substantial changes to the R-code of the original paper, we decide to first present theoretical argument here and postpone a full numerical report in the future work.        (1) Let (0) be (0) in Hao et al (2020) namely the number of symptoms onset cases during December 29 to 31 who would be lab-confirmed in the future; 0 = 0.23 be the initial ascertainment rate in Wuhan among symptomatic case which was calculated based on assuming complete ascertainment of early cases among symptomatic cases in Singapore; and 0 = 0.7 be the proportion of symptomatic patients [10,11,12].
(3) The ratio between A(0) and I(0), it should be roughly the same as the unknown ascertainable ratio in the  Table A2 with a sensitivity analysis on . Note that it is reasonable to believe that the mean duration from symptom onset to negative RT-PCR test result is less than 21 days, here we choose = 14 to be used in our main model (see upper left panel of Fig. 2 in [4] for reference). We can see that the estimates are relatively robust to the different choice of .  Table A2. Estimated transmission rates, overall ascertained rate and ascertainable ratio from the sensitivity analysis where = 14 is used in the main model.