SEIRS system with continuous vaccination (dot indicates time derivative)

$$ \begin{align*} \frac{dS}{dt} = \dot{S} &= \underbrace{\mu N}_\textrm{birth} - \underbrace{\beta S I / N}_\textrm{infection} + \underbrace{\omega R}_\textrm{lost immunity} - \underbrace{ \mu S}_\text{death} - \underbrace{p S}_\text{vaccination} \\ \frac{dE}{dt} = \dot{E} &= \underbrace{\beta S I / N}_\textrm{infection} - \underbrace{\sigma E}_\textrm{latency} - \underbrace{\mu E}_\text{death} \\ \frac{dI}{dt} = \dot{I} &= \underbrace{\sigma E}_\textrm{latency} - \underbrace{\gamma I}_\text{recovery} - \underbrace{(\mu + \alpha)I}_\text{death} \\ \frac{dR}{dt} = \dot{R} &= \underbrace{\gamma I}_\textrm{recovery} - \underbrace{\omega R}_\textrm{lost immunity} - \underbrace{\mu R}_\textrm{death} + \underbrace{p S}_\textrm{vaccination} \\ \frac{dB}{dt} = \dot{B} &= \underbrace{\beta S I / N}_\textrm{infection} \end{align*} $$

SEIRS Parameters (mean values)

$$ \begin{align*} \textrm{contact rate} &= \beta \\ \textrm{latency period} &= 1/\sigma \\ \textrm{recovery period} &= 1/\gamma \\ \textrm{immunity duration} &= 1/\omega \\ \textrm{mean life expectancy} &= 1/\mu \\ \textrm{infection-induced death rate} &= \alpha \\ \textrm{annual vaccination rate} &= p \\ \textrm{mean infectious period} &= 1/(\gamma + \mu + \alpha) \\ \textrm{case fatality ratio} &= \alpha / (\sigma + \mu + \alpha) \end{align*} $$

Basic reproduction number

$$ R_0 = \frac{\sigma}{\sigma +\mu} \frac{\beta}{\gamma+\mu+\alpha} $$

Expected value of perfect information (EVPI) for the cumulative burden B_ij for model i and action j

$$ \text{EVPI} = \underbrace{\underset{j}{\text{opt}} \sum_i p_i B_{ij}}_\textrm{optimum of averages} - \underbrace{\sum_i p_i\,\underset{j}{\text{opt}}\,B_{ij}}_\textrm{average of optimums} $$

Infection trajectories show a numerical solution to the SEIRS equations with 1,000 time steps and initial parameters

$$S(0) = 0.999$$ $$E(0) = R(0) = 0$$ $$I(0) = B(0) = 0.001$$ $$S + E + I + R = N = 1$$

where B is the cumulative disease burden.