20.2 Viruses as Pathogens

295

Fig. 20.1 Kinematic diagram of the SIR (susceptible–infected–recovered) model of infection.

rhoρ andbetaβ are the transition coefficients

There is no analytical solution to the model, but a numerical solution is straight-

forward. Typically, s 0s0 (the number of susceptible people at the beginning) is set to

1 Subscript i 01i0. i 0i0, the initial number of infected people, cannot be zero in the model, but it is

realistic to consider that it is a very small number.r 0r0, the initial number of recovered

people, would normally be expected to be zero. This model predicts a rapid initial

peak of infected people, which equally rapidly declines as the pool of susceptible

people declines, to be replaced by recovered (resistant) people. The immunization

criterion is obtained by setting the right-hand side of Eq. 20.2 to less than or equal

to zero:

StartFraction rho Over beta EndFraction greater than or equals s semicolon ^ρ

β ^{≥s ;}

(20.5)

at the very beginning of the infections almost equals 1s ≈1 (as noted above it is actually very slightly

less). The condition for population immunity (also called group of herd immunity) is

based on the assumption that all individuals who are not susceptible (i.e., the fraction

1 minus s1 −s) are immune, hence

1 minus s greater than or equals 1 minus rho divided by beta equals 1 minus 1 divided by upper R 01 −s ≥1 −ρ/β = 1 −1/R0

(20.6)

where the reproduction number upper R 0R0 is defined as

upper R 0 equals StartFraction beta Over rho EndFraction semicolonR0 = ^β

ρ ^;

(20.7)

it is the mean number of secondary infections created by a primary infected

individual.

The basic model can be usefully extended by including death as a possible outcome

or infection, and by allowing the immunity conferred by recovery to lapse (Fig. 20.2).