Mathematical Modeling of Two-Dose Vaccines

Abstract

Vaccines were invented through the advancements of technology and medicine to prevent premature death caused by infectious diseases. Although vaccines are a great way to inhibit the spread of disease, their disbursement to the population is a major component of their success. Both single and multi-dose vaccines require diligent planning to immunize the population. Even today, outbreaks of vaccine-preventable diseases such as pertussis and measles still occur in the United States. The purpose of this study is to examine the dynamics of multi-dose vaccines for diseases such as pertussis, with a particular goal of identifying circumstances when multiple doses are necessary for disease control. One of the basic models used in mathematical epidemiology is called the susceptible, infected, and recovered (SIR) model. In this modeling technique, the population dynamics are studied by dividing the population into susceptible, infected, and recovered individuals. In this project we plan to analyze the dynamics of a two-dose vaccination program by introducing a new class of vaccinated individuals to the basic SIR model. We explore if it is possible to have a single dose vaccine for an infectious disease such as pertussis. Further, the disease free equilibrium and its global stability are studied as is the endemic equilibrium and its local stability. We also calculate the critical vaccination rate. This is the minimum vaccination rate needed to eliminate disease from the population. Furthermore, this study looks into the number of secondary infections caused by an infected individual. This is called the basic reproduction number and it is specific to each type of model. These results characterize the epidemiological model. In conclusion, we find that a single dose vaccine for pertussis is only effective if there is a very low waning immunity from the first dose of vaccine. In any other instance, there must be a two-dose vaccination for the disease. The model can be further expanded to multiple doses, depending on the type of disease.