The SEIRS model accounts for latency and loss of immunity
Scenario 1
Scenario 2
Basic reproduction number, R0
Infectious period, 1/γ (days)
Latent period, 1/σ (days)
Immunity duration 1/ω (years)
Life expectancy 1/μ (years)
Death onset 1/α (days)
Vaccination level, p (%)
Exploring the SEIRS phase plane
SEIRS system (dot indicates time derivative) $$ \begin{align*} \frac{dS}{dt} = \dot{S} &= \underbrace{(1-p)\mu N}_\textrm{unvaccinated birth} - \underbrace{\beta S I / N}_\textrm{infection} + \underbrace{\omega R}_\textrm{lost immunity} - \underbrace{ \mu S}_\text{death} \\ \frac{dE}{dt} = \dot{E} &= \underbrace{\beta S I / N}_\textrm{infection} - \underbrace{\sigma E}_\textrm{latency} - \underbrace{\mu E}_\text{death} \\ \frac{dI}{dt} = \dot{I} &= \underbrace{\sigma E}_\textrm{latency} - \underbrace{\gamma I}_\text{recovery} - \underbrace{(\mu + \alpha)I}_\text{death} \\ \frac{dR}{dt} = \dot{R} &= \underbrace{p \mu N}_\textrm{vaccinated birth} + \underbrace{\gamma I}_\textrm{recovery} - \underbrace{\omega R}_\textrm{lost immunity} - \underbrace{\mu R}_\textrm{death} \end{align*} $$ Parameters (mean values) $$ \begin{align*} \textrm{contact rate} &= \beta \\ \textrm{latency period} &= 1/\sigma \\ \textrm{recovery period} &= 1/\gamma \\ \textrm{immunity duration} &= 1/\omega \\ \textrm{mean life expectancy} &= 1/\mu \\ \textrm{infection-induced death rate} &= \alpha \\ \textrm{vaccination fraction} &= p \\ \textrm{mean infectious period} &= 1/(\gamma + \mu + \alpha) \\ \textrm{case fatality ratio} &= \alpha / (\sigma + \mu + \alpha) \end{align*} $$ Basic reproduction number $$ R_0 = \frac{\sigma}{\sigma +\mu} \frac{\beta}{\gamma+\mu+\alpha} $$ Endemic mean age of infection $$ A = \frac{\omega+\mu+\gamma}{(\omega+\mu)(\beta-\gamma-\mu)} $$ Endemic equilibrium $$ \begin{align*} S(\infty) &= 1/R_0 \\ E(\infty) &= \frac{(\gamma + \mu + \alpha)I(\infty)}{\sigma} \\ I(\infty) &= \frac{\mu (1-S(\infty))}{\beta S(\infty) -\frac{\omega\gamma}{\omega+\mu}} \\ R(\infty) &= \frac{\gamma I(\infty)}{\omega + \mu} \end{align*} $$

The inter-epidemic period is calculated from the largest imaginary part of eigenvalues of the Jacobian matrix evaluated at the endemic equilibrium.

Jacobian matrix $$ \mathbf{J} = \left[ {\begin{array}{ccccccccccccc} \frac{\partial\dot{S}}{\partial S} & \frac{\partial\dot{S}}{\partial E} & \frac{\partial\dot{S}}{\partial I} & \frac{\partial\dot{S}}{\partial R} \\ \frac{\partial\dot{E}}{\partial S} & \frac{\partial\dot{E}}{\partial E} & \frac{\partial\dot{E}}{\partial I} & \frac{\partial\dot{E}}{\partial R} \\ \frac{\partial\dot{I}}{\partial S} & \frac{\partial\dot{I}}{\partial E} & \frac{\partial\dot{I}}{\partial I} & \frac{\partial\dot{I}}{\partial R} \\ \frac{\partial\dot{R}}{\partial S} & \frac{\partial\dot{R}}{\partial E} & \frac{\partial\dot{R}}{\partial I} & \frac{\partial\dot{R}}{\partial R} \end{array}} \right] = \left[ {\begin{array}{ccccccccccccc} -(\beta I + \mu) & 0 & -\beta S & \omega \\ \beta I & -(\sigma + \mu) & \beta S & 0 \\ 0 & \sigma & -(\gamma + \mu + \alpha) & 0 \\ 0 & 0 & \gamma & -(\omega + \mu) \end{array}} \right] $$ Eigenvalue equation and inter-epidemic period $$ \begin{array}{cccc} |\mathbf{J}_{t = \infty} - \lambda \mathbf{I}| = 0 && T_{\mathrm{E}} = \dfrac{2 \pi}{\mathrm{Im} (\mathrm{max}_\textrm{Im} \lambda) } \end{array} $$ Approximation of inter-epidemic period $$ T_{\mathrm{E}} \approx 4 \pi \sqrt{4 \omega (R_0 - 1)(\gamma + \mu + \alpha)- \left(\omega-\frac{1}{A}\right)^2} $$

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

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

where p is the vaccination fraction.

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.

Download code

https://martinkrz.github.io/posepi2/

Citation

Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: The SEIRS model for infectious disease dynamics. (2020) Nature Methods 17:557–558.

Version history

23 May 2020 v1.0.0 — initial public release

22 June 2020 v1.0.1 — added links to first column

17 August 2020 v1.0.2 — added links to all columns column

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)

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