Adding realism to the SIR model for infectious disease epidemics

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: Adding realism to the SIR model for infectious disease epidemics

Ottar Bjørnstad^1,2, Katriona Shea¹, Martin Krzywinski^3*, 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.

* martink@bcgsc.ca

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https://github.com/martinkrz/posepi2

Citation

Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: Adding realism to the SIR model for infectious disease epidemics. (2020) Nature Methods 17 (in press).

Background reading

Bjørnstad, O., Shea, K., Krzywinski, M. & Altman, N. Points of Significance: Modelling infectious epidemics. (2020) Nature Methods 17:455–456.

First trajectory

second trajectory

Points of Significance: Adding realism to the SIR model for infectious disease epidemics

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Citation

Background reading

Version history

22 April 2020

5 May 2020