Timing of epidemic peaks is impacted by uncertainty in R0 and immunity duration
Uncertainty in R0 and 1/ω will be modeled by simulating minimum and maximum values.
Remaining parameters are assumed to be accurately known.
Uncertainty in R0 and 1/ω will be modeled by simulating minimum and maximum values.
Remaining parameters are assumed to be accurately known.
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 Bij 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.
Katriona Shea1, Ottar Bjørnstad1,2, Martin Krzywinski3, Naomi Altman4
1. Department of Biology, The Pennsylvania State University, State College, PA, USA.
2. Department of Entomology, The Pennsylvania State University, State College, PA, USA.
3. Canada’s Michael Smith Genome Sciences Center, Vancouver, British Columbia, Canada.
4. Department of Statistics, The Pennsylvania State University, State College, PA, USA.
https://martinkrz.github.io/posepi3/
Shea, K., Bjørnstad, O., Krzywinski, M. & Altman, N. Points of Significance: Uncertainty and the management of epidemics. (2020) Nature Methods 17 (in press).
17 Aug 2020 v1.0.0 — initial public release
Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: Modelling infectious epidemics. (2020) Nature Methods 17:455–456. (interactive figures, download code)
Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: The SEIRS model for infectious disease dynamics. (2020) Nature Methods 17:557–558. (interactive figures, download code).