4.4: Vaccination

Last updated
Save as PDF

Page ID: 93508

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Table \(4.1\) lists the diseases for which vaccines exist and are widely administered to children. Health care authorities must determine the fraction of a population that must be vaccinated to prevent epidemics.

We address this problem within the SIR epidemic disease model. Let \(p\) be the fraction of the population that is vaccinated and \(p_{*}\) the minimum fraction required to prevent an epidemic. When \(p>p_{*}\), an epidemic can not occur. Since even non-vaccinated people are protected by the absence of epidemics, we say that the population has acquired herd immunity.

We assume that individuals are susceptible unless vaccinated, and vaccinated individuals are in the removed class. The initial population is then modeled as \((\hat{S}, \hat{I}, \hat{R})=(1-p, 0, p)\). We have already determined the stability of this fixed point to perturbation by a small number of infectives. The condition for an epidemic to occur is given by (4.3.7), and with \(\hat{S}_{0}=1-p\), an epidemic occurs if

\[\mathcal{R}_{0}(1-p)>1 \nonumber \]

Therefore, the minimum fraction of the population that must be vaccinated to prevent an epidemic is

\[p_{*}=1-\frac{1}{\mathcal{R}_{0}} \nonumber \]

Diseases with smaller values of \(\mathcal{R}_{0}\) are easier to eradicate than diseases with larger values \(\mathcal{R}_{0}\) since a population can acquire herd immunity with a smaller fraction of the population vaccinated. For example, smallpox with \(\mathcal{R}_{0} \approx 4\) has been eradicated throughout the world whereas measles with \(\mathcal{R}_{0} \approx 17\) still has occasional outbreaks.