The SEIRS model of infection spread
##### Second trajectory
R0
Infectious period, 1/γ (days)
Latent period, 1/σ (days)
Immunity duration 1/ω (years)
Life expectancy 1/μ (years)
Disease 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: Adding realism to the SIR model for infectious disease epidemics

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.

4. Department of Statistics, The Pennsylvania State University, State College, PA, USA.

#### 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).