Biomedical Engineering Reference
In-Depth Information
Tabl e 1 Model ( 5 ) parameters and functions
Parameter
Description
γ
Survival “probability” of susceptibles and infectives
per generation
β t
T Per i od i c transmission constant (β t = β t + T )
φ
Frequency-dependent escape “probability” function
σ
Recovery “probability” of infective individuals
per generation
g
Density-dependent escape per-capita growth function
f
Recruitment function
Our assumptions and notation lead to the following frequency-dependent SIS
epidemic model:
β t I ( t )
N ( t )
S
(
+
)= γ
(
)+ σ
(
)+
(
(
))
,
S
t
1
φ
t
I
t
f
N
t
1
β t I ( t )
N ( t )
S
(5)
(
+
)= γ
(
σ )
(
)+
φ
(
)
,
I
t
1
1
I
t
t
where 0
. In Model ( 5 ), we assume that events
happen in the following order: disease transmission, recovery, reproduction, and
survival (death). However, in real biological systems, these three events may
happen in different orders. For example, when reproduction happens before disease
transmission and survival (death) happens after disease transmission, proceeding as
in the derivation of Model ( 5 ), we obtain the following system:
< γ <
1,
β t = β t + T and
σ (
0
,
1
)
β t I
(
t
)
S
(
t
+
1
)= γ
φ
(
S
(
t
)+
f
(
N
(
t
)))+ σ
I
(
t
)
,
N
(
t
)+
f
(
N
(
t
))
1
(6)
β t I
(
t
)
I
(
t
+
1
)= γ
(
1
σ )
I
(
t
)+
φ
(
S
(
t
)+
f
(
N
(
t
)))
.
N
(
t
)+
f
(
N
(
t
))
Clearly, Model ( 5 ) is different from Model ( 6 ). Cyclic permutations of the three
distinct temporal phases lead to models that are topologically conjugate to Model
( 5 ). However, noncyclic permutations of the three temporal phases may lead to
models that are not topologically conjugate to Model ( 5 ). For simplicity, we will
analyze Model ( 5 ). We summarize the model parameters and functions in Table 1 .
Below, we summarize some of the underlying assumptions in model ( 5 ).
(a) There is no acquired immunity.
(b) There is no latent period (or it is very short).
(c) There is no disease induced mortality.
(d) Transmission is frequency dependent rather than density dependent.
Theoretical and empirical investigations have been done on comparing these
assumptions. In both continuous-time and discrete-time epidemic models, it is
known that these assumptions are not universally applicable [ 4 , 12 , 16 - 18 , 29 , 30 ].
Model ( 5 ) is a deterministic frequency-dependent SIS epidemic model and has
no “probability” of transmission. The assumption of deterministic dynamics is valid
in a large population, where stochasticity is unimportant. This assumption places
 
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