# Vaccination and Clinical Severity: Is the Effectiveness of Contact Tracing and Case Isolation Hampered by Past Vaccination?

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Mathematical Model

_{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 [8]. In a common transmission dynamics written by ordinary differential equations (e.g., the so-called “Susceptible-Infectious-Removed (SIR)” model), R

_{0}may be decomposed as the product of the transmission rate β and the average duration of infectiousness 1/γ. Under contact tracing practice which is accompanied by case isolation, we assume that the contact tracing takes place before the traced exposed individuals become infectious and that a fraction q of the total contacts of primary case can be traced and are effectively prevented. The average number of secondary cases generated by a primary case under the contact tracing is thus (1 − q)R

_{0}.

_{f}is assumed as fully immune, while a fraction v

_{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 α

_{s}≤ 1. Even provided that the partially immune individuals are infected, their infectiousness is assumed to be α

_{i}times that among fully susceptible individuals (where α

_{i}≤ 1). The relative reductions in susceptibility and infectiousness are common assumptions [4]. Moreover, we assume that vaccination elicits protection from severe disease. Because of the reduced severity, cases among partially immune individuals would be more mobile and have a greater contact rate than cases arising from susceptible individuals by a factor α

_{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 α

_{d}times (where α

_{d}≥ 1). It should be noted that the last two effects, i.e., α

_{m}and α

_{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 (i.e., reduce the average number of secondary cases per single primary case), while the remaining two factors, α

_{m}and α

_{d}, can increase the reproduction number of partially immune individuals relative to fully susceptible individuals.

_{ij}(where the subscripts i and j represent the vaccination history in which vaccinated individuals are denoted by 1 and otherwise 0). The average numbers of within- and between-group transmissions that are seen among those who are capable of causing further cases are parameterized as follows:

_{f}− v

_{p}) and contact tracing of primary cases (1 − q) are multiplied to R

_{0}. When the contacts are vaccinated, partially protected population v

_{p}has a reduced susceptibility by α

_{s}times as compared to unvaccinated. When the primary case is vaccinated, the transmissibility is multiplied by α

_{i}α

_{m}α

_{d}due to individual effects of vaccination on the primary case as mentioned above. Although the reproduction numbers (1) are heuristically described, R

_{ij}similar to Equation (1) can be derived from a variety of equation systems that adopt the abovementioned assumptions. The next-generation matrix,

**K**, is written as:

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

_{v}< 1 for v

_{p}= 0, or equivalently, (1 − q)(1 − v

_{f})R

_{0}< 1.

#### 2.2. Epidemic Threshold and Vaccine Effects

_{u,i}and n

_{v,i}be the numbers of unvaccinated and vaccinated cases in generation i, respectively. For mathematical convenience, here we focus on the exponential growth (linear phase) alone. Given the initial numbers of n

_{u,0}unvaccinated and n

_{v,0}vaccinated index cases, the i-th generation is written as:

_{u}and N

_{v}, throughout the course of an epidemic is calculated as:

_{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 R

_{v}> 1. Solving the inequality with respect to vaccine effects, we obtain the following condition that allows a major epidemic to occur:

_{v}≤ 1), the total number of cases converges to:

_{p}α

_{s}/(1 − v

_{f}− v

_{p}+ v

_{p}α

_{s}). Otherwise, he/she is unvaccinated with the probability (1 − v

_{f}− v

_{p})/(1 − v

_{f}− v

_{p}+ v

_{p}α

_{s}). Using this initial condition, the total number of cases who are capable of causing secondary transmissions is:

_{u}+ N

_{v}increases if:

_{00}+ R

_{10}in Equation (1), each unvaccinated cases have actually infected additional (1 − v

_{f}− v

_{p}+ v

_{p}α

_{s})qR

_{0}cases who were perfectly traced and were not involved in further transmission dynamics given their own infections. Similarly, in addition to R

_{01}+ R

_{11}in Equation (1), each vaccinated cases have caused α

_{i}α

_{m}α

_{d}(1 − v

_{f}− v

_{p}+ v

_{p}α

_{s})qR

_{0}cases who were traced. Thus, the total number of cases Z is written as:

#### 2.3. Probability of Extinction and Vaccine Effects

_{v}< 1 for v

_{p}= 0, a combined vaccine effect that satisfies in Equation (7) for v

_{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.

_{i}(i = 0 or 1) be the recovery rate of infectious individuals of type i and β

_{ij}(0 ≤ i, j ≤ 1) be the rate of increase in infectious individuals (i.e., the so-called “birth rate” of birth-and-death process) of type i generated by a primary case of type j. Considering a large population that consists of fully susceptible individuals, each element of the next-generation matrix is written as R

_{ij}= β

_{ij}/γ

_{j}.

_{j}(

**x**) is the probability that an individual of type j causes the x

_{0}unvaccinated and x

_{1}vaccinated cases in the next generation. Following our foregoing study [9], we have F

_{j}(

**s**) with an exponentially distributed infectious period that permits us to rewrite Equation (12) as:

_{v}≤ 1. In the case of R

_{v}> 1, the extinction probability by generation t,

**π**is described by using that at an earlier generation t − 1, i.e.,

^{t}(s)**u**represents the initial condition, (n

_{u,0}, n

_{v,0}) = (0,1) or (1,0). Since we consider a two-host population (i.e., unvaccinated and vaccinated cases), the probabilities of extinction at generation t given a single unvaccinated,

**π**(0,1) is given as a solution of the following simultaneous equations:

^{t}**π**can be obtained by taking the limit of t, which corresponding to:

_{0}and π

_{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 i generated by a primary case becomes an ancestor of an independent sub-processes (which restarts with a type i individual) behaving identically among the same type i [13]. Because of this multiplicative nature, the probability of extinction given multiple index cases (a

_{0}, a

_{1}) is calculated as:

_{i}measures the susceptibility of sub-group i, i.e.,

_{j}measures the infectiousness of primary cases in sub-group j, i.e.,

_{i}scales the population size of sub-group i:

#### 2.4. Numerical Illustration

_{f}= 0 [6,15]. On the other hand, we assume that v

_{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 [6]. Namely, R

_{v}< 1 for v

_{p}= 0. This leads to (1 − q)(1 − v

_{f})R

_{0}< 1 (or (1 − q)R

_{0}< 1 due to v

_{f}= 0). In the absence of intervention, R

_{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 [16] and also with the published estimate of 6.85 if accompanied by contact tracing [17,18]. The protective effect of contact tracing, q is arbitrarily assumed as 0.8 due to an assumption of sub-critical process (i.e., to adopt an assumption of R

_{v}< 1). One should remember that these arbitrarily allocated R

_{0}and q are very influential in discussing the feasibility of containment (using Equation (3)). Under these assumptions, we vary the combined effect of vaccination, denoted by α

_{s}α

_{i}α

_{m}α

_{d}, while we adopt a fixed value of α

_{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 [16]. Varying the combined effect of vaccination from 0 to 2, we calculate the estimate of the effective reproduction number, the expected total number of cases, the probability of extinction, and the expected duration of an outbreak. All statistical data were analyzed using a statistical software JMP version 9.0.0 (SAS Institute Inc., Cary, NC, USA).

## 3. Results

#### 3.1. Epidemic Threshold

_{v}, under vaccination and contact tracing/case isolation is computed using the assumed numerical values for smallpox and varying the uncertain product, α

_{s}α

_{i}α

_{m}α

_{d}, of combined effect of vaccination from 0 to 2 (Figure 1(A)). As can be seen from Equation (3), the reproduction number is a linear function of the combined vaccine effect. Also, the reproduction number leads to be supercritical (i.e., R

_{v}> 1) as long as the inequality in Equation (7) is not satisfied. Figure 1(B) shows the total number of cases based on Equations (9) and (11). Due to the use of geometric series 1/(1 − R

_{v}) in the expectations, the expected number of cases dramatically increases as α

_{s}α

_{i}α

_{m}α

_{d}becomes close to 1. When α

_{s}α

_{i}α

_{m}α

_{d}is greater than 1 in our hypothetical setting and Equation (7) is met, we would have R

_{v}> 1 and a major epidemic could occur. As long as the product α

_{s}α

_{i}α

_{m}α

_{d}is less than 0.9, the total number of cases is kept below 300, indicating the critical importance in quantitatively measuring α

_{s}α

_{i}α

_{m}α

_{d}.

**Figure 1.**Epidemic threshold and the total number of cases under vaccination and contact tracing. In both panels, the horizontal axis represents the product α

_{s}α

_{i}α

_{m}α

_{d}that measures the partial effects of vaccination. (

**A**) The effective reproduction number under vaccination and contact tracing. (

**B**) The expected total number of cases and the total number of cases who are capable of causing secondary transmissions. The basic reproduction number was assumed as 5. A fraction q = 0.8 of contacts was assumed to be protected by contact tracing. Vaccination was assumed to have conferred no full protection but partial protection among 30% of the population. Among partially protected individuals, susceptibility was assumed to be reduced by a factor of 0.8.

#### 3.2. Multi-Type Branching Process

_{s}α

_{i}α

_{m}α

_{d}satisfies inequality in Equation (7), we have R

_{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 R

_{00}+ R

_{10}< R

_{01}+ R

_{11}for α

_{i}α

_{m}α

_{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 α

_{s}α

_{i}α

_{m}α

_{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 (Figure 2(B)). As can be intuitively expected, the greater the assortativity coefficient is, the greater the probability of extinction would be.

_{v}≤ 1 (i.e., there would be only minor outbreaks) or α

_{s}α

_{i}α

_{m}α

_{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 (i.e., less than 150 days) and was not dramatically extended even when R

_{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 Figure 3(B).

**Figure 2.**Probability of extinction and combined effect of vaccination. In both panels, the horizontal axis represents the product α

_{s}α

_{i}α

_{m}α

_{d}that measures the partial effects of vaccination. (

**A**) Probability of extinction given a single infected individual is compared by vaccination history of the index case. Random mixing assumption was adopted. (

**B**) Probability of extinction given a single unvaccinated index case with different assortativity coefficient values.

**Figure 3.**Expected duration of minor outbreak under vaccination. In both panels, the horizontal axis represents the product α

_{s}α

_{i}α

_{m}α

_{d}that measures the partial effects of vaccination. (

**A**) Expected duration of outbreak given a single infected individual is compared by vaccination history of the index case. Random mixing assumption was adopted. (

**B**) Expected duration of outbreak given a single unvaccinated index case with different values of assortativity coefficient.

_{s}α

_{i}α

_{m}α

_{d}> 0.5. However, the advantage of low assortativity is diminished as α

_{s}α

_{i}α

_{m}α

_{d}becomes smaller (i.e., as the combined vaccine effect becomes larger).

## 4. Discussion

_{m}and α

_{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.

_{f}and v

_{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 [26].

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Mizumoto, K.; Ejima, K.; Yamamoto, T.; Nishiura, H. Vaccination and Clinical Severity: Is the Effectiveness of Contact Tracing and Case Isolation Hampered by Past Vaccination? *Int. J. Environ. Res. Public Health* **2013**, *10*, 816-829.
https://doi.org/10.3390/ijerph10030816

**AMA Style**

Mizumoto K, Ejima K, Yamamoto T, Nishiura H. Vaccination and Clinical Severity: Is the Effectiveness of Contact Tracing and Case Isolation Hampered by Past Vaccination? *International Journal of Environmental Research and Public Health*. 2013; 10(3):816-829.
https://doi.org/10.3390/ijerph10030816

**Chicago/Turabian Style**

Mizumoto, Kenji, Keisuke Ejima, Taro Yamamoto, and Hiroshi Nishiura. 2013. "Vaccination and Clinical Severity: Is the Effectiveness of Contact Tracing and Case Isolation Hampered by Past Vaccination?" *International Journal of Environmental Research and Public Health* 10, no. 3: 816-829.
https://doi.org/10.3390/ijerph10030816