Biology Reference
In-Depth Information
the mass action principle, where the net rate of new infection is proportional
to the product of the
densities
of the susceptible and infectious populations,
and a homogenous mixing between the two populations is assumed. If we
let
S
and
I
to be the density of the susceptible and infectious populations,
respectively, and β be the probability that a contact results in a successful
transmission, then the infection rate is β
SI
. The transmission parameter, β
,
depends on a vast range of social, environmental, and epidemiological fac-
tors (Anderson and May 1991), and may be derived from epidemiological
data. Various alternatives for formulating disease transmission exist as well
(McCallum et al. 2001), with the most popular being:
β
S
(
I
/
N
), where
N
indi-
cates the total population density. With this formulation, the transmission
rate now takes into account the abundance of the infected per individual in
the total population. Naturally, the choice of transmission formula depends
on the characteristics of the disease under study.
With the state variables denoting density of the said population, the SEIR
model illustrated in Figure 3.1 can be written using mass-balance as
dS
dt
=−+−
αβ φµ
N
I
RS
(3.3)
dE
dt
=−−
βσµ
SI
EE
(3.4)
dI
dt
=−−
σγ µ
EI
I
(3.5)
dR
dt
=−−
γφµ
I
RR
(3.6)
Here, α and
µ
are birth and death rates, respectively. σ is the rate of exposed
individual moving into the infectious category, typically calculated as the
inverse of the latent period. γ
is the recovery rate, while φ is the rate for
the immunity to disappear.
When the disease transmission is indirect, another species serves as an
intermediary host for the pathogen. In this case, the model will include
the dynamics of the intermediate host population as shown in Figure 3.2.
Transmission occurs when the susceptible primary host comes in contact
with infected intermediary host (vector).
As we have previously mentioned, the transmission rates β depends on
many things, including environmental factors. One of the environmental
factors that have long been recognized to characterize infectious disease
pattern is seasonality (Fisman 2007). When the new infection case pattern
