SIRS Model with Waning Immunity and Inconsistent Vaccination

The SIR model of disease transmission models disease transmission as a mass-action system of ordinary differential equations, with pools for susceptibles (S), infected (I), and recovered (R)\(^5\). When immunity can decay, individuals return to the susceptible pool with decay rate \(\delta\).

\[\frac{dS}{dt} = \delta R -\beta I S\]

\[\frac{dI}{dt} = (\beta S - \gamma) I\]

\[\frac{dR}{dt} = \gamma I - \delta R\]

SIRS <- function(times, init, parms) {
  derivs <- function(t, init, parms) {
    with(as.list(c(init, parms)), {
      dS <- delta*R - beta*I*S
      dI <- (beta*S - gamma)*I
      dR <- gamma*I - delta*R

      list(c(dS, dI, dR))})}
  result <- ode(y=init, times = times, func = derivs, parms=parms)
  as.data.frame(result)}

Recovery rate \(\gamma\) = 0.5. Natural immunity decays at a rate of \(\delta = 0.05\). Eventually, a rate of endemic infection is reached.

We can further expand this model to include a vaccinated group, who also incur a decaying immunity level over time. Vaccination is assumed to occur with rate \(\theta\):

\[\frac{dS}{dt} = \delta_R R + \delta_V V -\theta S - \beta (1-\theta)S I\]

\[\frac{dI}{dt} = [\beta (1-\theta) S - \gamma] I\]

\[\frac{dR}{dt} = \gamma I - \delta_R R\]

\[\frac{dV}{dt} = \theta S - \delta_V V\]

VSIRS <- function(times, init, parms) {
  derivs <- function(t, init, parms) {
    with(as.list(c(init, parms)), {
      dS <- deltaR*R + deltaV*V - beta*(1-theta)*S*I - theta*S
      dI <- beta*(1-theta)*S*I - gamma*I
      dR <- gamma*I - deltaR*R
      dV <- theta*S-deltaV*V

      list(c(dS, dI, dR, dV))
      })
    }
  result <- ode(y=init, times = times, func = derivs, parms=parms)
  as.data.frame(result)
}

For measles, herd immunity is achieved when 90% of the population is vaccinated.

For n subpopulations who can cross-infect one another\(^6\),

\[\frac{dS_i}{dt} = \delta_R R_i + \delta_V V_i - \theta_i S_i - \beta (1-\theta_i) \left( \displaystyle\sum_{i=1}^n I_i \right) S_i\]

\[\frac{dI_i}{dt} = \beta \left( \displaystyle\sum_{i=1}^n I_i \right) S_i - \gamma I_i\]

\[\frac{dR_i}{dt} = \gamma I_i - \delta_R R_i\]

\[\frac{dV_i}{dt} = \theta_i S_i - \delta_V V_i\]

XVSIRS <- function(times, init, parms) {
  derivs <- function(t, init, parms) {
    with(as.list(c(init, parms)), {
      dS1 <- deltaR*R1 + deltaV*V1 - beta*(1-theta1)*S1*(I1 + I2) -theta1*S1
      dI1 <- beta*(1-theta1)*S1*(I1+I2) - gamma*I1
      dR1 <- gamma*I1 - deltaR*R1
      dV1 <- theta1*S1 - deltaV*V1

      dS2 <- deltaR*R2 + deltaV*V2 - beta*(1-theta2)*S2*(I1 + I2) -theta2*S2
      dI2 <- beta*(1-theta2)*S2*(I1+I2) - gamma*I2
      dR2 <- gamma*I2 - deltaR*R2
      dV2 <- theta2*S2 - deltaV*V2

      list(c(dS1,dI1,dR1,dV1, dS2,dI2,dR2,dV2))})}
  result <- ode(y=init, times = times, func = derivs, parms=parms)
  as.data.frame(result)}

• Herd immunity of the larger population is PERMANENTLY COMPROMISED by the low-vaccine rate of the subpopulation!!

• The overall population of 900 people following the CDC's recommendations should be well above the herd immunity threshold.

• Intermixed within the larger population are 10% of people with only a 70% vaccination rate.

• But introducing infection into this system causes ENDEMIC measles infection.

  • The larger population supports approximately 2.5 infected inviduals (a prevalence of 2.78 per 1,000 individuals)
  • The smaller population supports approximately 2 infected individuals (a prevalence of 20.67 per 1,000 individuals)
  • Overall, a prevalence of 4.552 infected people per 1,000 individuals.

• Prior to the introduction of the first measles vaccine in 1963, measles infection was considered a normal part of childhood. Infection rates peaked at 3-4 years, and over 90% of the population had already been infected by age 10.
  

• A measles vaccine was first licensed in 1963. Widespead vaccination began in 1969.

• Vaccine campaigns are generally mounted in response to outbreaks, but immunization rates wane as infections subside.

• One dose of MMR vaccine was recommended by a child's first birthday until a large outbreak in 1985 affected many school-age children who were appropriately vaccinated.

  • This demonstrated the vaccine's 92-95% failure rate. The CDC now recommends a child receive a second dose of MMR before entering Kindergarten.

• Re-exposure to measles leads to a rapid immune response and boosts antibody levels.

• Without re-exposure antibody levels, whether acquired naturally or artifically, decay over time.

• Prior to the widespread vaccination campaign against measles, children generally contracted clinical cases of measles in early childhood, and then adults were subsequently re-exposed when their own children contracted the illness.

• Indiginous measles virus eradication was declared in the United States in 2000\(^2\).

• With a population now largely protected only through childhood vaccinations or natural infection incurred over 45 years ago, and without widespread re-exposure, herd immunity is well below that required to suppress a measles outbreak.

• The groups now most susceptible to measles are surprising...

• Younger populations are smaller and very recently immunized. Herd immunity protects the few that are susceptible and decaying immunity does not occur fast enough to maintain an endemic infection.

• Older populations can hold a small-level endemic infection long enough for decaying immunity in their larger populations to provide enough susceptibles to start an epidemic.
  

• The current measles outbreak began in Disneyland Park over Christmas.

• To date, 142 cases in 7 states

• The median age is 22! That's in line with theoretical predictions, which show that mean age at first infection increases with vaccination.

• Recall the cross infection model - Most cases have been in unvaccinated children and adults. A small population of "anti-vaxxers" can have a major impact on the population as a whole.

• It's also important to consider waning immunity when endemic infection is no longer normal. Maybe something for us younger people to keep in mind regarding chickenpox.

  • The older you are when you get these diseases, the higher the risks.
    
  • The current outbreak has resulted in far higher hospitalization rates than were known historically.