Illustration of the “overshoot” effect.
Panel A shows the number of susceptibles and infecteds during an outbreak with R0 = 2. The initial growth phase of the epidemic is approximately characterized by an exponential increase in the number of infecteds, accompanied by a decline of susceptibles. The dashed horizontal line indicates the threshold level of susceptibles below which population immunity prevents further outbreaks for this given set of parameters. Once the number of susceptibles crosses a threshold level, the average number of new infections caused by an infected person falls below 1 and the epidemic wanes. The arrow indicates the difference between the number of susceptibles at the end of the outbreak and the threshold line. This difference was termed as the “overshoot” . Panel B shows how the magnitude of overshoot depends on R0. For the simple SIR model, the number of prevented infections is found to be highest for intermediate values of R0 ∼ 2. Panel C considers the optimal level of vaccination for the limiting case which is easy to analyze. We choose an infection with R0 = 2, high degree of dominance (second strain is introduced at day 150) and high cross-immunity which does not wane on this timescale. We plot the cumulative number of infections by either strain (as a fraction of the total population) as a function of time. The optimal level of vaccination (fraction vaccinated = 0.209, solid black line) against the first strain results in the fraction of susceptibles to the second strain declining to ∼1 − 1/R0 = 0.5. This level of herd immunity just prevents the second strain from spreading. At a lower level of vaccination (fraction vaccinated = 0.16, blue dotted line) leads to more infections with the first strain (and also prevents the second strain spreading). A higher level of vaccination (fraction vaccinated = 0.26, green dotted line) leads to fewer infections with the first strain but allows the second strain to spread and generate an overshoot.