John M. Drake

- Initial growth of the epidemic is
**exponential** **Susceptible depletion**results in departure from the**basic reproduction number**- Define
**effective reproduction number**and denote \( R_e \) - \( R_e \) scales with the proportion of susceptibles in the population (\( S=X/N \implies R_e = R_0 \times S \))
- This implies that there is a critical susceptible population size, \( S_c \) for the pathogen to invade

Recall that \( R_e = R_0 X/N \)

This implies that \( S_c = 1/R_0 \)

The complement of \( S_c \) is the **critical vaccination proportion**: \( p_c = 1-S_c = 1-\frac{1}{R_0} \)

A correction is needed for **imperfect vaccines** (e.g. SARS-CoV-2): \( p_c = \frac{1}{\epsilon}\left({1-\frac{1}{R_0}}\right) \) where \( \epsilon \) is the vaccine effectiveness at preventing transmissible infection.

We can obtain the same result by inspection of the equation for \( dY/dt \)

\[ \frac{dY}{dt} = \beta X \frac{Y}{N} - \gamma Y \]

Set \( dY/dt < 0 \) and solve

\[ \begin{align} \frac{dY}{dt} = \beta X \frac{Y}{N} - \gamma Y &< 0 \\ \beta \frac{X}{N} &< \gamma \\ \frac{X}{N} &< \frac{\gamma}{\beta} = \frac{1}{R_0} \end{align} \]

Due to the dependency upon social contact for transmission, an individual is afforded a degree of protection via the immunity of others in the population.

If contacts have been vaccinated, the probability of acquiring the disease is lower

**Implication:** Not everyone in a population needs to be immune to eradicate an infections disease

The extent of vaccination effort required is determined by \( R_0 \).

“Cocooning” strategies

**Assumption:** a fraction \( p \) of newborns are vaccinated/immunized at birth

\[ \begin{align} \frac{dS}{dt} &= \mu (1-p) - \beta SI - \mu S \\ \frac{dI}{dt} &= \beta SI - (\mu + \gamma) I \\ \frac{dR}{dt} &= \mu p + \gamma I - \mu R \end{align} \]

Set system to zero (equilibrium) and solving for \( I \) yields

\[ I^* = \frac{\mu}{\beta} (R_0 (1-p) -1) \]

Setting \( I^* \) to zero and solving for \( p \) yields

\[ p_c = 1-\frac{1}{R_0} \]

- \( p_c \) is the
**critical fraction**of newborns that must be immunized to achieve**eventual**eradication

\[ \begin{align} \frac{dS}{dt} &= \mu - \beta SI - \mu S - \rho S\\ \frac{dI}{dt} &= \beta SI - (\mu + \gamma I) \\ \frac{dR}{dt} &= \rho S + \gamma I - \mu R \end{align} \]

Set system to zero (equilibrium) and solving for \( I \) yields

\[ I^* = \frac{\mu}{\beta} (R_0-1-\frac{\rho}{\mu}) \]

Setting \( I^* \) to zero and solving for \( \rho \) yields

\[ \rho_c = \mu (R_0 - 1) \]

- \( \rho_c \) is the
**critical rate**at which susceptibles need to be immunized to achieve**eventual**eradication

**Discussion:** Provide an intuitive interpretation of \( \rho_c \)

Where infant vaccination and continuous vaccination are not feasible, a more economic and logistically efficient strategy may be **pulsed vaccination**

Pulsed vaccination seeks to vaccinate specific age cohorts at specified intervals