Abstract
In this chapter, the authors present a simple model to determine the optimal choice of vaccination scheduling for a society composed of two groups of individuals in order to minimize the economic loss only, assuming herd immunity. First, a simple classical SIR model is presented to form the basis of the analysis; second, the model is revised to include the effects of vaccination which in turn will be extended to include two heterogeneous groups of individuals forming a society. The solutions of relevant differential equations will then be used to calculate the total economic cost of each scenario presented.
TopBackground: The Classical Sir-Model And Vaccination
The classical SIR model is proposed by Kermack and McKendrick (1927, 1932, 1933), which was one of the first formulations of a mathematical theory of epidemiology based on compartmental models. It is worth to note that the foundations of the entire such approach to epidemiology were laid, not by mathematicians, but by public health physicians such as Sir R.A. Ross, W.H. Hamer, A.G. McKendrick, and W.O. Kermack between 1900 and 1935 (Brauer, 2017).
In the canonical model (Murray, 1989; Anderson & May, 1991; Diekmann & Heesterbeek, 2000; Brauer & Castillo, 2000), a population of total size N is divided into three groups of individuals: (1) the susceptible S -- those who are not (yet) infected; (2) those who are infected and can spread the disease by contact with susceptibles; (3) the recovered R -- those who have been infected, then recovered and become immune for life, and do not spread infection. Their joint dynamics are modelled by the following system of ordinary differential equations:
,where 𝛽 is the transmission rate of the disease (or the probability of getting infected), 𝛾 is the rate at which the infected becomes recovered and immune for life. The initial population is given by
N(0)=
S(0)+
I(0)+
R(0), where
S(0),
I(0),
R(0) ≥0. Without loss of generality, the initial population may be normalized to
N(0)=1 by dividing
S, I, and
R by
N.
Key Terms in this Chapter
Vaccine: A vaccine is a substance that contain weakened or inactive parts of a particular organism (antigen) that triggers an immune response within the body.
Infected Population: Population group who has been already infected.
Age Group: Viruses may affect different age groups of a population differently. Thus, it is necessary to divide the population to age groups according to some criteria based on how virus affects ages groups.
Recovered Population: Population group who has been affected by the virus but recovered.
Vaccination Scheduling: The order in which the age groups are vaccinated.
Susceptible Population: Population groups whose members may be affected by virus.
SIR Model: A system of differential equations that describes how the Susceptible (S), Infected (I), and the Recovered (R) populations develops through time as related to a virus.