Chicken Pox and Shingles - Dynamic Modeling of Diseases and Pests - page 130

8.1 Model Assumptions and Structure

For illustrative purposes, we initialize our model with population data from the 1990
U.S. Census and group them into six age cohorts. The first five cohorts span 10 years
(3,650 days) each, and the last comprises the population of 50 years and older. By
assumption, individuals in the last cohort remain a maximum of 30 years (10,950
days) in that cohort. The total initial population is 245,704, measured in thousands
of individuals. For simplicity, we assume that only 50 percent of individuals in the
10- to 19-year age cohort may reproduce, and that the total reproductive population
includes them as well as all individuals in the 20- to 39-year cohort. Further, we
assume a uniform and constant death rate of .000027 per day.
Following research on infections with chicken pox 1 , the average number of in-
fections with chicken pox by an infected individual is assumed to be

RsubZero = 10 (8.1)

and individuals are removed from the infective stock at a rate of

V = 1 / 7 . (8.2)

The general form for the transmission coefficient is 2

Beta = RsubZero ∗ V / Population . (8.3)

With an initial population of 245,704 and V = 1/7, Beta is .0000058.
A general representation of the transmission rate is

Transmission Rate = Beta ∗ Total Infective ∗ Susceptible (8.4)

Since transmission rates for chicken pox vary among age cohorts, we weight the
transmission coefficient for different age cohorts:

Transmission Rate = (Beta Weight ∗ Beta ) ∗ Total Infective ∗ Susceptible (8.5)

Individuals are most likely to contract chicken pox when they are young. This is
because children are typically kept in close contact with each other in places such
as school and daycare. We choose a Beta Weight of 1 for the first age cohort, thus
making the virus have its full effect in transmission. As people get older, however,
they are not as likely to come into contact with the virus. Therefore, lower weights
are used. The weight for the 10- to 19-year age cohort is set at .8. These individ-
uals continue to be in close contact with each other, but a large percentage of the

1 Hethcote, H.W. “Qualitative analyses of communicable disease models.” Mathematical Bio-
sciences from Mathematical Models in Biology .
May, Robert M. 1983.“Parasitic Infections as Regulator of Animal Populations.” American Sci-
entist. 71: 36-45 in Mathematical Models in Biology .
2

Edelstein-Keshet, Leah. 1988. Mathematical Models in Biology . Random House: New York.