Epidemic interventions

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 \( \rho_c \)

Mathematical model:

\[ \begin{align} \frac{dS}{dt} &= \mu - \beta S I - \mu S - p \sum_n=0^\infty S(nT^{-1})\delta (t-nT) \\ \frac{dI}{dt} &= \beta SI - (\mu +\gamma)I \end{align} \]

Linear stability analysis yields an eradication criterion

\[ \frac{(\mu T - p)(e^{\mu T -1})+\mu pT}{\mu T(p-1+e^{\mu T})} < \frac{1}{R_0} \]

Shulgin et al. 1998. Pulse vaccination strategy in the SIR epidemic model Bulletin of Mathematical Biology 60:1123-1148.

Failures in

1. Degree (the level of protection, e.g. against disease but not infection)
2. Take (the fraction of the population which received protection, conditioned on vaccination)
3. Duration (how long protection lasts – waning immunity)

Mathematical model:

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

Yielding the following eradication criterion

\[ p_c = \left( 1-\frac{1}{R_0} \right) \left( 1+ \frac{\delta}{\mu} \right) \]

1. Physical distancing
2. Active case finding, isolation, contact-tracing, and quarantine
3. Infection barriers

Discussion: How would you incorporate these NPIs into a dynamical model?

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

Presentations and exercises draw significantly from materials developed with Pej Rohani, Ben Bolker, Matt Ferrari, Aaron King, and Dave Smith used during the 2009-2011 Ecology and Evolution of Infectious Diseases workshops and the 2009-2019 Summer Institutes in Statistics and Modeling of Infectious Diseases.

Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/bync/3.0/. Please share and remix noncommercially, mentioning its origin.