Robotics
Circuit Design
Medicine
Glass Melting

Equations

Let $e_i$ be the rate measuring the exposure to natural infection with disease $i$ (exposition factor),
$s_i$ the proportion of children previously with no detectable antibodies to disease$i$ who acquire
detectable antibodies to disease $i$ when vaccinated (seroconversion).

Then $q_i = e_i + (1-e_i)s_i$ measures the presence of antibodies to disease$i$ under vaccination
conditions.

From these data it is possible to obtain information about the expected antibody prevalence in general.

Let $p(\pm, \pm,\pm)$ with $+$ (resp. $-$) at the $i$-th place standing for the presence (resp. absence) of antibodies
to the $i$-th disease. Through laboratory testing the eight values $p(+,+.+), p(+,+,-), \dots, p(-,-,-)$ are
known. We obtain the following equations:

\begin{displaymath} \begin{array}{lclcl} p_7 = p(+,+,+) &=& \nu q_1q_2q_3 & + &... ...)(1-q_2)(1-q_3) & + & (1-\nu)(1-e_1)(1-e_2)(1-e_3). \end{array}\end{displaymath}

After the (technical) substitution $p_i = \frac{a_i}{n}$ using SINGULAR the variables $n, e_1, e_2, e_3, q_1, q_2, q_3$ could be
eliminated. The result is a quadratic equation:

\begin{displaymath} c_1(a_0, \dots, a_7) v^2 -c_1(a_0, \dots, a_7) v + c_0(a_0, \dots, a_7) = 0. \end{displaymath}

Using the factorization of SINGULAR we obtain $c_1 = f_1^2 f_4$ and $c_1 - 4c_0 = f_3^2$.

This leads to the following formula: $v_{1,2} = \frac{1}{2} \left(1 \pm \frac{f_3}{f_1\sqrt{f_4}}\right)$ with

\begin{eqnarray*} f_1 := & n=\bigl((a_0 + a_3 + a_5 + a_6) + (a_7 + a_4 + a_... ..._5a_2+a_0a_7a_6a_1+a_3a_4a_5a_2+a_3a_4a_6a_1+a_5a_2a_6a_1\bigr). \end{eqnarray*}

The solution is a closed formula for the proportion of children who have received the multivalent vaccine.

The Computation in SINGULAR:
Lille, 08-07-02 http://www.singular.uni-kl.de