Uncertainty in R0 and 1/ω will be modeled by simulating minimum and maximum values.

Remaining parameters are assumed to be accurately known and will remain fixed.

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 and will remain fixed.

Cumulative disease burden
Effect of vaccination
Expected value of perfect information
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{p S}_\textrm{vaccination} + \underbrace{\gamma I}_\textrm{recovery} - \underbrace{\omega R}_\textrm{lost immunity} - \underbrace{\mu R}_\textrm{death} \\ \frac{dB}{dt} = \dot{B} &= \underbrace{\beta S I / N}_\textrm{infection} \end{align*} 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 $$R_0 = \frac{\sigma}{\sigma +\mu} \frac{\beta}{\gamma+\mu+\alpha}$$

Infection trajectories show a numerical solution to the SEIRS equations with 500 time steps and initial parameters

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

where p is the annual vaccination rate and B is the cumulative disease burden.

### Points of Significance: The SEIRS model for infectious disease dynamics

Ottar Bjørnstad1,2, Katriona Shea1, 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.

#### Citation

Shea, K., Bjørnstad, O., Krzywinski, M. & Altman, N. Points of Significance: Uncertainty and the modelling and management of infectious epidemics. (2020) Nature Methods 17:in press.

#### Version history

27 July 2020 v1.0.0 — initial public release

#### Related columns

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).