### Effect of vaccination on infection spread

Susceptible $$\frac{dS}{dt} = - \frac{\beta I S}{N}$$ Infectious $$\frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I$$ Recovered $$\frac{dR}{dt} = \gamma I$$ Recovery rate $$\gamma = \frac{1}{\text{infectious period}}$$ Basic reproduction number $$R_0 = \frac{\beta}{\gamma}$$

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

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

where p is the vaccination fraction.

### Points of Significance: Modelling infectious 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.

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

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

#### Version history

##### 9 April 2020 v1.0.0

Initial public release.

##### 17 April 2020 v1.0.1

Minor text changes. Fixed typos.

UI tweaks.