Vaccination
Vaccination is included in an epidemic model by changing the way individuals move between compartments or by changing their risk of infection, infectiousness, or disease progression. The exact mathematical form depends on what biological effect the vaccine is intended to represent.
A simple vaccination model
Suppose susceptible individuals are vaccinated at per-capita rate \(\nu\). We introduce a vaccinated compartment \(V(t)\).
The total vaccination flow is \(\nu S\). It contributes \(-\nu S\) to the susceptible equation and \(+\nu S\) to the vaccinated equation.
Different ways to model vaccination
There is no single universal vaccination model. The compartment structure changes according to the biological effect being represented.
1. Perfect-protection model
Vaccination gives complete protection during the modelled period.
Vaccinated individuals no longer contribute to the susceptible pool. This is useful for understanding the basic population-level effect of vaccination, but it is an idealisation.
2. Leaky vaccine model
A leaky vaccine reduces each vaccinated person's susceptibility but does not eliminate it.
Let \(e_s\) be efficacy against infection. Breakthrough infections occur at rate
\[\boxed{(1-e_s)\beta\frac{VI}{N}}.\]For example, \(e_s=0.8\) leaves 20% of the original susceptibility.
3. All-or-nothing model
Vaccination completely protects some recipients but does not protect the remainder.
4. Waning-immunity model
If vaccine protection is temporary, vaccinated individuals eventually return to susceptibility.
Under a constant waning rate, the mean duration of protection is \(1/\omega_V\).
5. Vaccination in SIR and SEIR models
Vaccination can be added to an existing compartmental epidemic model.
For an SEIR model, an exposed compartment is inserted between susceptible and infectious: \(S\to E\to I\to R\), while vaccination provides an additional flow from \(S\) to \(V\). The vaccinated class may be fully protected or may still experience breakthrough infection.
6. Vaccines that reduce infectiousness
A vaccine may allow breakthrough infection but reduce onward transmission.
A simple force of infection is
\[\lambda=\beta\frac{I_U+(1-e_i)I_V}{N},\]where \(e_i\) represents efficacy against infectiousness.
7. Vaccines that reduce severe disease
When hospitalisation or severe illness is the outcome of interest, vaccinated and unvaccinated infections can have different progression rates.
This structure allows vaccination to strongly reduce severe outcomes even when protection against infection is incomplete.
| Model | Main assumption |
|---|---|
| Perfect protection | Vaccinated individuals cannot become infected during the modelled period. |
| Leaky | Each vaccinated individual has reduced susceptibility. |
| All-or-nothing | Some recipients are fully protected; others are not protected. |
| Waning immunity | Protection decreases and individuals return to susceptibility. |
| Reduced infectiousness | Breakthrough infections transmit less efficiently. |
| Reduced severity | Vaccination reduces progression to severe outcomes. |
How vaccination changes epidemic growth
In the simplest SIR setting, suppose a fraction \(p\) of the population is fully protected before an outbreak. Then
\[\frac{S(0)}{N}=1-p,\]and
\[\boxed{R_{\mathrm{eff}}(0)=R_0(1-p)}.\]To prevent initial epidemic growth in this idealised model,
\[R_0(1-p)<1,\]giving
\[\boxed{p_c=1-\frac{1}{R_0}}.\]Which vaccination model should be used?
The simplest model that captures the biological mechanism relevant to the question is usually preferable. Infection prevention requires susceptibility to be represented; hospital-demand questions require severe-disease progression; waning requires a return flow; and altered breakthrough transmission may require separate infectious classes.
These structures are not separate inventions belonging to a single person. They are extensions of compartmental epidemic modelling developed for different biological assumptions, so attaching one inventor to the general vaccination-model framework would be misleading.