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While contact tracing and case isolation are considered as the first choice of interventions against a smallpox bioterrorist event, their effectiveness under vaccination is questioned, because not only susceptibility of host and infectiousness of case but also the risk of severe clinical manifestations among cases is known to be reduced by vaccine-induced immunity, thereby potentially delaying the diagnosis and increasing mobility among vaccinated cases. We employed a multi-type stochastic epidemic model, aiming to assess the feasibility of contact tracing and case isolation in a partially vaccinated population and identify data gaps. We computed four epidemiological outcome measures,

Contact tracing and case isolation constitute a crucial element of non-pharmaceutical interventions against directly transmitted infectious diseases including smallpox [

In the meantime, epidemiological study design and analysis of vaccine effects have greatly progressed during the past decades [

Because of independent progress of the abovementioned two subject areas, we have yet to understand how feasible the contact tracing and case isolation would be in the presence of vaccination in a population. This issue is motivated by previous epidemiological studies of smallpox [

We would like to assess if the contact tracing and case isolation, the first choice of interventions against smallpox, can remain feasible in a partially immune population. The present study is aimed to describe the transmission dynamics that involves contact tracing and case isolation under vaccination using a mathematical model, assessing the feasibility and identifying associated data gaps.

We start with considering the transmission dynamics in a randomly mixing population, although a heterogeneous model will be considered later. Let _{0} be the basic reproduction number of an infectious disease, representing the average number of secondary cases generated by a typical primary case throughout the course of its infectiousness in a fully susceptible population [_{0} may be decomposed as the product of the transmission rate _{0}.

We describe a population in which only a part of them are fully protected from infection due to vaccination that took place in advance of an epidemic. Due to the vaccination practice, a fraction _{f} is assumed as fully immune, while a fraction _{p} is partially immune (e.g., not protected from smallpox but protected from the severe illness). Partially immune individuals are assumed to have a reduced susceptibility by a factor _{s} ≤ 1. Even provided that the partially immune individuals are infected, their infectiousness is assumed to be _{i} times that among fully susceptible individuals (where _{i} ≤ 1). The relative reductions in susceptibility and infectiousness are common assumptions [_{m} ≥ 1. Lastly, the reduced severity may not only improve clinical outcomes but also delay diagnosis, and thus, we assume that the duration of infectious contact is lengthened by _{d} times (where _{d} ≥ 1). It should be noted that the last two effects, _{m} and _{d}, are vaccine-induced modifications in behavior, and thus, the estimates could vary with calendar time, geographic location and during the course of an epidemic (see Discussion). For simplicity and for the exposition of our modeling results, we focus on the early stage of an epidemic in a single hypothetical setting. In summary, we consider four individual effects of vaccination among which two act as protective (_{m} and _{d}, can increase the reproduction number of partially immune individuals relative to fully susceptible individuals.

Under the abovementioned scenario, we assess four epidemiological outcome measures, including: (i) the threshold of a major epidemic under the interventions (contact tracing/case isolation and vaccination); (ii) the probability of extinction given a single vaccinated or unvaccinated index case; (iii) the expected number of cases throughout the course of an epidemic, and (iv) the expected duration of a minor outbreak.

To describe the transmission dynamics under vaccination, we employ the next-generation matrix. We split the population into two sub-groups by vaccination history and the average number of secondary cases in sub-group _{ij} (where the subscripts

Namely, among unvaccinated contacts caused by unvaccinated primary cases, the proportion of those who do not possess residual immunity (1 − _{f} − _{p}) and contact tracing of primary cases (1 − _{0}. When the contacts are vaccinated, partially protected population _{p} has a reduced susceptibility by _{s} times as compared to unvaccinated. When the primary case is vaccinated, the transmissibility is multiplied by _{i}_{m}_{d} due to individual effects of vaccination on the primary case as mentioned above. Although the reproduction numbers (1) are heuristically described, _{ij} similar to Equation (1) can be derived from a variety of equation systems that adopt the abovementioned assumptions. The next-generation matrix,

The effective reproduction number, _{v}, of a population with vaccine-induced immunity is defined as the dominant eigenvalue of the next-generation matrix:

For the sake of clarity in our theoretical exposition (to apply model-based findings to the practical issue of smallpox control), we focus on the spread of infection that can be expected to be contained in the absence of partial immunity. Considering that 30% or more of the present day population has never been vaccinated, the assumption of “successful containment” indicates that the contact tracing is highly effective in preventing secondary transmission [_{v} < 1 for _{p} = 0, or equivalently, (1 − _{f})_{0} < 1.

Let _{u,i} and _{v,i} be the numbers of unvaccinated and vaccinated cases in generation _{u,0} unvaccinated and _{v,0} vaccinated index cases, the

The total number of unvaccinated and vaccinated cases, _{u} and _{v}, throughout the course of an epidemic is calculated as:

Combining Equations (4) and (5), we get:
_{v} < 1. A major epidemic, which does not decline to extinction without substantial depletion of susceptible individuals or concerted effort of control, occurs if and only if _{v} > 1. Solving the inequality with respect to vaccine effects, we obtain the following condition that allows a major epidemic to occur:

We assume that an outbreak starts with a single index case. In the sub-critical case (_{v} ≤ 1), the total number of cases converges to:

If an epidemic starts with a single infected individual who experiences infection at random, the probability that the index case is vaccinated is given by _{p}_{s}/(1 − _{f} − _{p} + _{p}_{s}). Otherwise, he/she is unvaccinated with the probability (1 − _{f} − _{p})/(1 − _{f} − _{p} + _{p}_{s}). Using this initial condition, the total number of cases who are capable of causing secondary transmissions is:

It should be noted that _{u} + _{v} increases if:

Caution must be exercised with respect to the total number of cases that is different from the calculation in Equation (9), because the abovementioned model has omitted the cases who were traced perfectly and isolated before developing infectiousness. That is, in addition to _{00} + _{10} in Equation (1), each unvaccinated cases have actually infected additional (1 − _{f} − _{p} + _{p}_{s})_{0} cases who were perfectly traced and were not involved in further transmission dynamics given their own infections. Similarly, in addition to _{01} + _{11} in Equation (1), each vaccinated cases have caused _{i}_{m}_{d}(1 − _{f} − _{p} + _{p}_{s})_{0} cases who were traced. Thus, the total number of cases

Whereas we consider a special case _{v} < 1 for _{p} = 0, a combined vaccine effect that satisfies in Equation (7) for _{p} ≥ 0 can lead to a major epidemic. We thus consider the relationship between the probability of extinction and vaccine effects in the following by continuing to employ the abovementioned multi-type branching process approximation.

As adopted above, we label unvaccinated as type 0 and vaccinated as type 1. To ease the computation of the probability of extinction, we assume that the infectious period is exponentially distributed so that the multi-type branching process model can be rewritten as a multivariate birth-and-death process [_{i} (_{ij} (0 ≤ _{ij} = _{ij}/_{j}.

Let _{j}(_{0} unvaccinated and _{1} vaccinated cases in the next generation. Following our foregoing study [_{j}(

Any introduction of initial cases decline to extinction with probability 1 as long as _{v} ≤ 1. In the case of _{v} > 1, the extinction probability by generation ^{t}(_{u,0}, _{v,0}) = (0,1) or (1,0). Since we consider a two-host population (^{t}

The probability of extinction, _{0} and _{1} are the probabilities of extinction given one unvaccinated and vaccinated case, respectively. As practiced with many other branching process models, each of the secondary cases of type _{0}, _{1}) is calculated as:

The expected duration of an epidemic, E(

Since the abovementioned descriptions rest on homogeneous mixing assumption between vaccinated and unvaccinated individuals, we additionally consider a heterogeneous pattern of transmission, at least by assuming that there are more frequent contacts within unvaccinated subpopulation as compared to between vaccinated and unvaccinated individuals. To address the so-called “assortative mixing” (_{i} measures the susceptibility of sub-group

Similarly, _{j} measures the infectiousness of primary cases in sub-group _{i} scales the population size of sub-group

Note that the parameterization in Equation (21) would result in (1) if

We consider a hypothetical scenario of biological warfare using variola virus. In advance of the bioterrorist event, we assume that no one possess full protection against smallpox any longer, and thus, _{f} = 0 [_{p} = 0.3 or 30% of the population still possesses partial protection (nevertheless, it should be noted that the fraction of susceptible individuals would continue to increase as time goes by). As mentioned above, our scenario is supposed to be a subcritical process in the absence of partial protection [_{v} < 1 for _{p} = 0. This leads to (1 − _{f})_{0} < 1 (or (1 − _{0} < 1 due to _{f} = 0). In the absence of intervention, _{0} is crudely assumed to be 5 which is in line with the goal of vaccination coverage during the Intensified Smallpox Eradication Programme without accounting for other interventions [_{v} < 1). One should remember that these arbitrarily allocated _{0} and _{s}_{i}_{m}_{d}, while we adopt a fixed value of _{s} at 0.8 given that a historical household data with probably limited vaccine potency indicates that susceptibility is reduced by a factor of 0.69 [

The effective reproduction number, _{v}, under vaccination and contact tracing/case isolation is computed using the assumed numerical values for smallpox and varying the uncertain product, _{s}_{i}_{m}_{d}, of combined effect of vaccination from 0 to 2 (_{v} > 1) as long as the inequality in Equation (7) is not satisfied. _{v}) in the expectations, the expected number of cases dramatically increases as _{s}_{i}_{m}_{d} becomes close to 1. When _{s}_{i}_{m}_{d} is greater than 1 in our hypothetical setting and Equation (7) is met, we would have _{v} > 1 and a major epidemic could occur. As long as the product _{s}_{i}_{m}_{d} is less than 0.9, the total number of cases is kept below 300, indicating the critical importance in quantitatively measuring _{s}_{i}_{m}_{d}.

Epidemic threshold and the total number of cases under vaccination and contact tracing. In both panels, the horizontal axis represents the product _{s}_{i}_{m}_{d} that measures the partial effects of vaccination. (

The probability of extinction is examined in _{s}_{i}_{m}_{d} satisfies inequality in Equation (7), we have _{v} ≤ 1, and thus, the probability of extinction given a single infected individual is always 1. Otherwise, the extinction probability lies between 0 and 1. Under the examined scenario with the relationship _{00} + _{10} < _{01} + _{11} for _{i}_{m}_{d} > 1, an introduction of single vaccinated individual is more risky to cause an epidemic than introducing an unvaccinated case into the population. For instance, when _{s}_{i}_{m}_{d} = 3, the probabilities of extinction given an unvaccinated and a vaccinated case are 78.8% and 49.8%, respectively. We also examined the impact of assortative (heterogeneous) mixing on the probability of extinction given a single unvaccinated index case (

_{v} ≤ 1 (_{s}_{i}_{m}_{d} ≤ 1, introducing an unvaccinated individual as index case would yield a longer duration of an outbreak as compared to introducing a vaccinated case. The expected duration of minor outbreak within the assumed parameter space was overall shorter than 10 generations (_{v} becomes closer to a critical level. The impact of heterogeneous mixing on the duration of an outbreak given an unvaccinated index case is examined in

Probability of extinction and combined effect of vaccination. In both panels, the horizontal axis represents the product _{s}_{i}_{m}_{d} that measures the partial effects of vaccination. (

Expected duration of minor outbreak under vaccination. In both panels, the horizontal axis represents the product _{s}_{i}_{m}_{d} that measures the partial effects of vaccination. (

For each combined effect of vaccination, the effective reproduction number with different assortativity coefficient was kept identical to allow comparisons. The basic reproduction number was assumed as 5. A fraction

Since an unvaccinated case is introduced, lower assortativity leads to observe longer durations of outbreak for _{s}_{i}_{m}_{d} > 0.5. However, the advantage of low assortativity is diminished as _{s}_{i}_{m}_{d} becomes smaller (

The present study modeled the transmission dynamics of an infectious disease, considering smallpox for the exposition and examining the contact tracing and case isolation in a partially immune population. Four different vaccine effects were considered, and especially, we took into account two specific effects that would adversely work for the epidemic control,

Our study is the first to identify the associated data gap that would influence the feasibility of implementing contact tracing and case isolation in the event of a bioterrorist attack or any other opportunities (e.g., containment phase of a pandemic influenza). Among four different effects, the delay in detecting cases and an increase in the frequency of movement are two critical variables that must be measured urgently and compared between vaccinated and unvaccinated cases. Nevertheless, at a clinical setting, the relative difficulty in clinical diagnosis of vaccinated cases as compared with unvaccinated has not been routinely measured as one of biological effects of vaccination. Moreover, although there have been published studies that showed the reduced frequency of contact during symptomatic period (as compared with during the incubation period) [_{m} and _{d}, are associated with behavioral aspect, and thus, it is likely that the estimates would not be regarded as biological constants, and rather, could vary with time and space. The estimates can also greatly vary with the recognition of a bioterrorist attack during the course of an epidemic. Although we have focused on the early stage of an epidemic in a single hypothetical setting and fixed these parameters as if these were constants, such assumptions may be subject to explicit evaluation.

To quantitatively measure these two effects (by assuming that these two are constants during the early stage of an epidemic), there would be two practical difficulties in the empirical observations. First, we do not have a widely accepted scale in measuring both vaccine effects in the way that directly influences the transmission dynamics (or the next-generation matrix). Second, smallpox has already been eradicated and our society does not have an opportunity to conduct any further empirical observations of naturally infected individuals. As a potential solution, one could consider gathering expert opinion [

Three technical limitations must be noted briefly. First, our model has been kept simple, and the realism has remained small. Our framework can be extended to realistic scenarios including the dynamics on a heterogeneous network [_{f} and _{p}, remain unknown in many epidemiological settings. Without understanding the background immunity levels, it is difficult to directly quantify the epidemic threshold using empirical data [

Although we have multiple tasks to be completed, we believe that the present study has contributed much to literature by identifying the associated data gaps in exploring the feasibility of contact tracing and case isolation under vaccination practice [

This work was suggested by Professor Martin Eichner. HN received funding support from the Japan Science and Technology Agency (JST) PRESTO program and St Luke’s Life Science Institute Research Grant for Clinical Epidemiology Research 2012. This work also received financial support from the Harvard Center for Communicable Disease Dynamics from the National Institute of General Medical Sciences (grant No. U54 GM088558). The funding bodies were not involved in the collection, analysis and interpretation of data, the writing of the manuscript or the decision to submit for publication.

The authors declare no conflict of interest.