Infectious disease modelling with SEIRV

Mathematical introduction

Infectious disease modelling informs decision making during disease outbreaks. I imagine this topic is now a canonical example in courses on mathematical modeling. The following set of ordinary differential equations describe an SEIRV model: susceptible, exposed, infected, recovered, vaccinated. $$ \begin{align*} \frac{dS}{dt} &= \mu N - \frac{\beta S I}{N} - \mu S - \kappa S &\quad \frac{dE}{dt}&=\frac{\beta S I }{N} - \sigma E - \mu E &\quad \frac{dI}{dt} &= \sigma E - \mu I - \gamma I \\ \frac{dR}{dt}&=\gamma I - \mu R &\quad \frac{dV}{dt} &= \kappa S - \mu V \end{align*} $$ These equations are more intuitively understood by considering the following diagram, showing the movement of people between groups:

This diagram shows the movement between groups. People start out as susceptible to the disease. This group increases through additional births $\mu N$, and decreases as people move to other groups. Being vaccinated or leaving this group via death depends on the population of this group and rates $\kappa$ and $\mu$, while becoming exposed depends on some rate $\beta$ and the proportion of the population who are infected $I /N$. Hence, the term $\beta S I / N$ is added to the exposed group. Leaving the exposure group occurs at rates $\sigma$ for infection and $\gamma$ for death. After exposure people become infectious at rate $\sigma$, and then there is a recovery rate $\gamma$.

Some simplifications in this model include: the rate of births is equal to deaths, the death rate is the same for each group ($\mu$), only people who are susceptible are vaccinated (at a rate of $\kappa$), and people who have recovered cannot be re-infected. Such a model will be appropriate for modelling some diseases but would need modification for others.