Question

Mumps is a contagious childhood disease. A vaccine has been widely available for several decades now, leading to a dramatic decline in the incidence of the disease. Data from the CDC show that vaccination coverage against the mumps virus is now about 90%.

Here are some estimates of the transmission parameters for the mumps virus:
gamma= 0.08
beta= 0.45    
R0 =5.6

a) What is the meaning of R0 in this disease model?

b) What fraction of the population should be vaccinated in order to prevent the spread of the disease? (explain)

c) Sometimes vaccines are not 100% effective. A single dose of the MMR vaccine is only about 90% effective. How does that change your estimate for the fraction of the population that should be vaccinated?

Answer 1

a) Ro in the provided disease refers to the measure of contagiousness (the measure of transmissibility) of the infection. This measure relates to the Herd immunity.

Herd immunity is an indirect form of protection in a population where most individuals have become immune to a particular infection by a previous occured infection or by vaccination. This will also protect the individuals in population who are non-immune.

b) The fraction of the population that has to be vaccinated in order to prevent the spread of disease depends on Ro.

The threshold level of Herd Immunity for the disease is calculated by taking into account the number of susceptible population (S) who are prone to infection multiplied to Ro and setting the product to 1.

Ro * S = 1

If the fraction of population that is immune to infection is considered p (that has been vaccinated) then,

The fraction of population susceptible can be written as (1 - p)

Therefore, Ro * (1 - p) = 1

1 - p = 1/Ro

p = 1 - 1/Ro

p is the fraction of population that has to be vaccinated to prevent the spread of the disease.

p = 1 - 1/5.6

p = 4.6/5.6 = 0.821

which means, about 82% of the population has to be vaccinated in order to prevent the spread of the disease.

c) In general, vaccines are considered to be 100% effective

In cases, where the vaccine is not 100% effective the Vaccination coverage (proportion of population need to be vaccinated) is calculated by the following equation:

Vc = Ic / E

where, Vc is the vaccine coverage,

Ic is the Herd immunity threshold (calculated as p in the previous case)

Ic = 1-1/Ro ==> 82%

E is the vaccine effectiveness given as 90% ===> 0.9

therefore, Vc = 0.82/0.90 = 0.91 ===>91%

Hence, 91% of the population has to be vaccinated in case of vaccine being 90% effective.