Biomedical Engineering Reference
In-Depth Information
ogy these might be carnivores and herbivores, in immunology these might be
cytotoxic T cells and virus-infected cells. Immune effector cells proliferate in
response to antigen presented by infected cells, in which the rate of proliferation
is likely to be proportional to the number of infected cells presenting antigen.
The destruction of infected cells brings about a concomitant reduction in effec-
tor cell proliferation. We therefore expect oscillatory dynamics. In ecology one
of the principal measures of population stability is the variance in species abun-
dance. Large-amplitude oscillations are thought to make populations vulnerable,
whereas low-amplitude oscillations are a sign of robustness (38). In ecology
species extinction is at stake, in immunology a loss of effector cells and a loss of
regulatory control are at stake.
Jansen and de Roos (23) studied the following two-compartment model.
Consider two populations of predators,
P
1
and
P
2
, and prey populations,
N
1
and
N
2
. Predators are able to migrate from one compartment to another with prob-
ability
m
/2:
n nnp
=
,
[19]
1
1
1
1
m
p
=
np
N
p
+
(
p
p
)
,
[20]
1
1
1
1
2
1
2
n n np
=
,
[21]
2
2
2
m
p
=
np
N
p
+
(
p
p
)
.
[22]
1
2
2
2
1
2
2
The rate of predator proliferation is given by
rp
i
and the death rate N
p
i
. From an
immunological perspective we might think of two strains of infecting virus and
their corresponding T cell receptors.
Assuming equal densities of predators (
p
1
=
p
2
) and prey (
n
1
=
n
2
), the model
reduces to the non-spatial Lotka-Volterra model in which densities oscillate
permanently at an amplitude determined by the initial conditions. However, if
small differences in density are allowed between compartments, these tran-
siently increase with a correlated reduction in the amplitude of oscillations in the
average densities. This is because in the compartmental model large-amplitude
oscillations are diffusively unstable (statistical stabilization), whereas in the sin-
gle-population model oscillations of any amplitude can be maintained. Thus,
establishing compartments in which pathogens will be attacked (such as lymph
nodes), rather than fostering the likeness of a single population, should allow
organisms to limit variation in pathogen densities.
