Sudden and unexpected deaths after the administration of hexavalent vaccines (diphtheria, tetanus, pertussis, poliomyelitis, hepatitis B, Haemophilius influenzae type b): is there a signal?

Abstract

Deaths in temporal association with vaccination of hexavalent vaccines have been recently reported. The objective of this paper is to assess whether these temporal associations can be attributed to chance. Standardised mortality ratios (SMR) for deaths within 1 to 28 days after administration of either of the two hexavalent vaccines in the 1st and 2nd year of life were determined using the respective annual rates for sudden unexpected deaths (SUDs) from the national vital statistics. The distribution of SUD cases and the vaccination uptake by month were estimated from surveys and sales figures for the individual vaccines. Sensitivity analyses were performed to account for limitations in the data sources. For one of the vaccines, Vaccine B, all SMRs were well below one. For the other, Vaccine A, SMRs exceeded one insignificantly on the 1st day after vaccination in the 1st year of life. In the 2nd year of life, however, the SMRs for SUD cases within 1 day of vaccination with vaccine A were 31.3 (95% CI 3.8–113.1; two cases observed; 0.06 cases expected) and 23.5 (95% CI 4.8–68,6) for within 2 days after vaccination (three cases observed; 0.13 cases expected). Extensive sensitivity analyses could not attribute these findings to limitations of the data sources. Conclusion: These findings based on spontaneous reporting do not prove a causal relationship between vaccination and sudden unexpected deaths. However, they constitute a signal for one of the two hexavalent vaccines which should prompt intensified surveillance for unexpected deaths after vaccination.

Appendix

Let T denote the time to death from SUD by months. To derive the expected deaths needed in calculating the denominator of SMRs, estimations of the following quantities are available:

$$ \begin{array}{*{20}l} {{c_{1} } \hfill} & { = \hfill} & {{{\text{cumulative}}\,{\text{incidence}}\,{\text{of}}\,{\text{SUDs}}\,{\text{in}}\,{\text{the}}\,1st\,{\text{year}}\,{\text{of}}\,{\text{life}}} \hfill} \\ {{} \hfill} & {{\text{ = }} \hfill} & {{P(0{\text{ }} \leqslant T{\text{ }} < 12)} \hfill} \\ {{c_{2} } \hfill} & { = \hfill} & {{{\text{cumulative}}\,{\text{incidence}}\,{\text{of}}\,{\text{SUDs}}\,{\text{in}}\,{\text{the}}\,2nd\,{\text{year}}\,{\text{of}}\,{\text{life}}} \hfill} \\ {{} \hfill} & {{\text{ = }} \hfill} & {{P(12{\text{ }} \leqslant T{\text{ }} < 24)} \hfill} \\ {{p^{{{\left( d \right)}}}_{{1,i}} } \hfill} & { = \hfill} & {\begin{aligned} & {\text{proportion}}\,{\text{of}}\,{\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{deaths}}\,{\text{from}}\,{\text{SUD}}\,{\text{occurring}}\,{\text{in}}\,{\text{the}}\,{\text{month }}[i,i{\text{ }} + {\text{ }}1){\text{ }} \\ & {\text{of}}\,{\text{the}}\,1st\,{\text{year}}\,{\text{of}}\,{\text{life}} \\ \end{aligned} \hfill} \\ {{} \hfill} & { = \hfill} & {{P{\left( {i \leqslant T < i + 1\left| {0 \leqslant T < 12} \right.} \right)} = \frac{1} {{c_{1} }}P{\left( {i \leqslant T < i + 1} \right)}\;,\;i = {\text{0}}{\text{,1}}{\text{,}} \cdots {\text{,11}}} \hfill} \\ {{p^{{{\left( d \right)}}}_{{2,i}} } \hfill} & { = \hfill} & {\begin{aligned} & {\text{proportion}}\,{\text{of}}\,{\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{deaths}}\,{\text{from}}\,{\text{SUD}}\,{\text{occurring}}\,{\text{in}}\,{\text{the}}\,{\text{month }}[i,i{\text{ }} + {\text{ }}1){\text{ }} \\ & {\text{of}}\,{\text{the}}\,2nd\,{\text{year}}\,{\text{of}}\,{\text{life}} \\ \end{aligned} \hfill} \\ {{} \hfill} & { = \hfill} & {{P{\left( {i \leqslant T < i + 1\left| {12 \leqslant T < 24} \right.} \right)} = \frac{1} {{c_{2} }}P{\left( {i \leqslant T < i + 1} \right)}\;,\;i = {\text{12}}{\text{,13}}{\text{,}} \cdots {\text{,23}}} \hfill} \\ {{p^{{{\left( v \right)}}}_{i} } \hfill} & { = \hfill} & {\begin{aligned} & {\text{proportion}}\,{\text{of}}\,{\text{children}}\,{\text{receiving}}\,{\text{a}}\,{\text{DTPa - Hib}}\,{\text{containing}}\,{\text{vaccine}}\,{\text{in}}\,{\text{month }}[i,i{\text{ }} + {\text{ }}1){\text{ }} \\ & {\text{to}}\,{\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{children}}\,{\text{in}}\,{\text{this}}\,{\text{age}}\,{\text{class}}\,{\text{for}}\,{\text{whom}}\,{\text{information}}\,{\text{on}}\,{\text{vaccination}}\,{\text{is}}\,{\text{available}}{\text{,}}\,i{\text{ }} = {\text{ }}0,1,... \\ \end{aligned} \hfill} \\ \end{array} $$

Let $\lambda _{i} = P{\left( {i \leqslant T < i + 1\left| {T \geqslant i} \right.} \right)}$ denote the mortality from SUD for month [ i,i + 1). From

$$\lambda _{i} \cdot P{\left( {T \geqslant i} \right)} = \left\{ {\begin{array}{*{20}c} {{p^{{{\left( d \right)}}}_{{1,i}} \cdot c_{1} \;for\;i = 0,1, \cdots ,11\;}} \\ {{p^{{{\left( d \right)}}}_{{2,i}} \cdot c_{2} \;for\;i = 12,13, \cdots ,23}} \\ \end{array} } \right.\;and\;P{\left( {T \geqslant i} \right)}\; = \;\exp {\left( { - {\sum\nolimits_{j = 0}^{i - 1} {\lambda _{j} } }} \right)}$$

we obtain the following recursive formula for the $\lambda _{i} {\text{'s}}$

$$\lambda _{0} = p^{{{\left( d \right)}}}_{{1,0}} \cdot c_{1} \;{\text{and}}\;\lambda _{i} = p^{{{\left( d \right)}}}_{{1,i}} \cdot c_{1} \cdot \exp {\left( {{\sum\nolimits_{j = 0}^{i - 1} {\lambda _{j} } }} \right)},\;i = 1,2, \cdots ,11$$

and

$$\lambda _{i} = p^{{{\left( d \right)}}}_{{2,i}} \cdot c_{2} \cdot \exp {\left( {{\sum\nolimits_{j = 0}^{i - 1} {\lambda _{j} } }} \right)},i = 12,14, \cdots ,23$$

Let S denote the set of all birth cohorts crossing the rectangle, and let N _k be the number of children from cohort $k \in S$ who contribute to the study period (being the rectangle in the Lexis diagram). Furthermore, let I _k denote the number of age classes of cohort $k \in S$ which fall into the study period. Because, children vaccinated in month [ i,i + 1) of age are at risk of sudden death in the following month with incidence

$$ \frac{{{\left( {\lambda _{i} + \lambda _{{i + 1}} } \right)}}} {2} $$

on average, concerning cohort $k \in S$, for the number of expected deaths among the vaccinated children in age class $i \in I_{k}$ contributing to the study period we obtain the expression $\frac{{\lambda _{i} + \lambda _{{i + 1}} }} {2}p^{{(v)}}_{i} N_{k} .$ Thus, the expected number from SUD cases in the 1st month after vaccination is

$$N_{{Exp}} = {\sum\nolimits_{k \in S} {N_{k} {\sum\nolimits_{i \in I_{k} } {\frac{{\lambda _{i} + \lambda _{{i + 1}} }} {2}} }} }p^{{(v)}}_{i} $$