Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay

Each year, infectious diseases affect millions of lives, sometimes even claiming them. Therefore, controlling the spread of these diseases as well as the elimination of them has been, and is still intensely studied. One way that this has been studied is trying to figure out under what conditions a certain virus can penetrate a partially vaccinated population. That is, how large a fraction of the population has be vaccinated in order to keep the virus from starting an epidemic? Usually however, vaccines are only developed after the epidemic has struck. Thus, a more realistic approach is to study whether or not vaccination would be enough to eradicate the disease and if so, what kind of vaccination process should be used (i.e. how should the affected population be vaccinated)? The standard conventional approach has been constant vaccination, or a uniform and continuous effort of administering the vaccine to a population to try to stop the outbreak. Recently, another method—pulse vaccination—has been found to be more effective at eliminating epidemics. Pulse vaccination works by repeatedly applying a vaccine over defined age ranges, and at each vaccination time, “a constant fraction of the susceptible population is vaccinated”.

The above linked study, using mathematical models and quantitative means, analyzed whether or not pulse vaccination would be sufficient at eradicating SIR (Susceptible-Infectious-Removed) epidemics. To test this, the study varied variables such as proportion of those vaccinated successfully (i.e. pulse vaccination rate), period of pulsing, reproduction number, and maximum infectious period to see if the disease can be eradicated. The conclusion found was that a large pulse vaccination rate, 0.6 (or a short period of pulsing, 1) is sufficient at eliminating the disease.