The authors declare that one of coauthors Hiroshi Nishiura is a PLOS ONE Editorial Board member. This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials. Other than this point, the authors have no conflicts of interest to declare.
Conceived and designed the experiments: HN. Performed the experiments: KE HN. Analyzed the data: KE HN. Contributed reagents/materials/analysis tools: KE. Wrote the paper: KE KA HN.
The way we formulate a mathematical model of an infectious disease to capture symptomatic and asymptomatic transmission can greatly influence the likely effectiveness of vaccination in the presence of vaccine effect for preventing clinical illness. The present study aims to assess the impact of model building strategy on the epidemic threshold under vaccination.
We consider two different types of mathematical models, one based on observable variables including symptom onset and recovery from clinical illness (hereafter, the “observable model”) and the other based on unobservable information of infection event and infectiousness (the “unobservable model”). By imposing a number of modifying assumptions to the observable model, we let it mimic the unobservable model, identifying that the two models are fully consistent only when the incubation period is identical to the latent period and when there is no presymptomatic transmission. We also computed the reproduction numbers with and without vaccination, demonstrating that the data generating process of vaccineinduced reduction in symptomatic illness is consistent with the observable model only and examining how the effective reproduction number is differently calculated by two models.
To explicitly incorporate the vaccine effect in reducing the risk of symptomatic illness into the model, it is fruitful to employ a model that directly accounts for disease progression. More modeling studies based on observable epidemiological information are called for.
There are two intriguing characteristics in quantitatively modeling infectious disease data. First, the risk of infection to an individual is dependent on the risks of other individuals in the same population unit. Second, the infection event is seldom directly observable. Among these two, the dependence has been addressed during the process of model building, e.g., a heterogeneous contact structure has been explicitly considered in various types of models
Ignoring the unobservable nature during model formulation would complicate the model fitting to empirical data. In many instances, a temporal distribution of infected individuals (i.e. an epidemic curve) is analyzed, and most frequently, the best available dataset is the daily counts of cases. The data are usually collected based on observable information only, e.g. counts of cases according to the date of diagnosis of clinically apparent illness. Only in the better case, epidemiologists are granted an access to the daily frequency of illness onset. Nevertheless, the data generating process of the empirical information is rather different from assumed transition mechanism within the socalled SIR (susceptibleinfectiousremoved) model. The SIR model is considered as inconsistent with the data, because the transition from S to I state is determined by the event of infection (which is unobservable) and the other transition from I to R state is determined by the loss of infectiousness (which is even more difficult to observe)
Although a previous study recognized the importance of asymptomatic transmission in considering the feasibility of nonpharmaceutical public health interventions (e.g. contact tracing and case isolation)
Employing a mathematical modeling approach, the present study aims to assess the impact of model building strategy on the transmission dynamics of an infectious disease under vaccination practice. In particular, we investigate differential values of epidemic threshold between models that rest on observable and unobservable information.
We consider two different types of mathematical models, one based on observable variables including symptom onset and recovery from clinical illness (hereafter referred to as the “observable model”) and the other based on unobservable information including infection event and infectiousness (the “unobservable model”). Whereas the unobservable model in the following is a variant of the SEIR model
A. The compartment of an observable model. The model describes the transitions depending on illness onset and recovery from clinical symptoms. Once infected, all infected individuals experience asymptomatic period,
Here we briefly describe the timedependent growth of an epidemic based on the observable model, the compartments of which are drawn in
where
where
where
The basic reproduction number of this model is computed as follows (
where
which we will use in later discussion.
The other type of a model, i.e., the unobservable model, can be said to be the infectionage structured SEIR model that further classifies infectious individuals into symptomatic and asymptomatic cases
where
where the force of infection is
where
where
Using these two models under a homogeneously mixing assumption, we investigate the importance of appropriately capturing the observable natural course of infection in epidemiological models.
To explicitly account for the observable clinical course of infection, underlying assumptions of using a parameter
Subsequently, we investigate the differential impact of vaccination on the reproduction number (or, on the epidemic threshold) of the two models. In a published study, the nextgeneration matrix was employed to incorporate various different biological actions of vaccination into the transmission dynamics under vaccination
For numerical illustration, we examine the plausible parameter space for four different viral infectious diseases.
Description  Notation  Parameter values  References/Assumptions  
Smallpox  Influenza  HIV  Varicella  
The average number of secondary cases produced by an asymptomatic case 

0.69  0.60  3.67  3.24  
The average number of secondary cases produced by a symptomatic infection 

6.18  1.20  0.00  3.24  
The average number of secondary cases produced by a fully asymptomatic case 

1.37  0.96  6.12  6.47  
Probability of developing symptoms in the unobservable model  1.00  0.75  0.80  1.00  
Basic reproduction number of the observable model 

6.87  1.50  3.67  6.47  
Proportion of asymptomatic transmissions among all secondary transmissions 

0.10  0.40  1.00  0.50  
Proportion of presymptomatic transmissions among all asymptomatic infection 

1.00  0.60  0.67  1.00  
Vaccine efficacy of reducing infectiousness  VE_{I}  0.80  0.15  0.60  0.80  
Vaccine efficacy of reducing susceptibility  VE_{S}  0.95  0.41  0.40  0.50  
Vaccine efficacy of preventing progression to symptomatic illness  VE_{P}  0.87  0.67  0.60  0.50 
assumed.
To analytically describe the difference between two modeling approaches, we consider the unobservable model as a special case of the abovementioned observable model.
with the following boundary conditions:
where the force of infection,
where
(a)
(b)
and (c)
Writing in the way we computed the observable model in (4), the basic reproduction number is computed as
where
In summary, two models are rather different and can be consistent only in the case that the model could be written by ordinary differential equations and only when the incubation period can be equated to the latent period.
We continue to compare the special case of the observable model (
Let
The average number of secondary cases generated by a single fully asymptomatic case should be identical between (16) and (17), i.e.,
Similarly, the average number of secondary cases generated by a single presymptomatic case should also be identical between the two models as follows:
The impact of varying the proportion of presymptomatic transmissions among all asymptomatic transmissions (the horizontal axis; denoted by
or
In the following, a comparison of the reproduction numbers under vaccination is made between the observable model (
where the first row represents the exposure to unvaccinated susceptible individuals. It should be noted that
The nextgeneration matrix of the observable model under vaccination is given by the integral of
Let
The nextgeneration matrix of the unobservable model is obtained from (
where
where
To understand the extent of the different impact of
Effective reproduction numbers for the observable model and the unobservable model are compared as a function of vaccineinduced reduction in symptomatic illness. To permit comparison, in the absence of vaccination practice, the epidemic threshold values of the two models were assumed as identical. Vaccination coverage is fixed at 50%. The solid line shows the reproduction number of the unobservable model under vaccination. The dashed line shows the reproduction number of the observable model under vaccination. Except the vaccineinduced reduction in symptomatic illness, all parameters were fixed (see
The present study analyzed and compared observable and unobservable modeling approaches. Two major tasks have been completed. First, by rewriting the observable model as if it were an SEIRtype unobservable model, we aimed to clarify underlying assumptions of the unobservable model that involves asymptomatic transmission. For the two models to be identical, we have demonstrated that it is essential that the incubation period has to be identical to the latent period and also that no presymptomatic transmission occurs in both models. Only the observable model can directly incorporate vaccineinduced reduction in symptomatic illness (in the manner that the corresponding vaccine effect data is generated), and the probability of symptomatic infection in the unobservable model was shown to be multiplied to the transition rate from preinfectious to infectious state without phenomenological justification. Second, we numerically solved both models and examined the sensitivity of
The present study emphasizes that an appropriate model formulation would be essential to answer the corresponding scientific or public health question. As we have shown, an explicit formulation would also help clarify underlying assumptions that tend to be hidden in common model structures. Considering a practical example of vaccination that influences the symptom onset, we have shown that the modeling approach to tackle this issue requires a model building approach that can explicitly account for the natural course of infection including asymptomatic and symptomatic states. Since the use of SEIR structure with two or more types of Iclasses with different levels of symptom or clinical severity has also partially accounted for this matter of differential severity of symptom, and because the unobservable modeling approach to this issue has been proposed relatively early
Although our discussion might read as if we regard the observable model as always better than the unobservable one, this preference cannot always be true. In fact, the observable model is not perfect, largely missing the information of infectiousness in the model structure. However, if we handle the model fitting to the incidence of illness onset, the observable model must be most useful, because the renewal equation of only symptomatic cases can be computed and directly fitted to the data
Four limitations should be noted and described briefly. First, we conducted only univariate sensitivity analysis, ignoring any possible dependence between the frequency of presymptomatic transmissions among the total asymptomatic transmissions and other epidemiological variables. Ignoring such dependence structure could sometimes lead to overestimating the effectiveness of public health interventions
Considering that we were successful in gaining useful epidemiological insights into future quantitative modeling by formulating the vaccination issue using an observable model, it is suggested that more studies based on observable epidemiological variables are conducted. Future studies can also tackle the issue of abovementioned dependence between clinical illness and infectiousness based on an explicit model with both pieces of information as variables and analyzing individual datasets with multiple dimensions.