Modeling Influenza Viral Dynamics in Tissue - Artificial Immune Systems - page 28

where N nei is the number of neighbors a cell really has. Note that for N nei =6,
(4) reduces to (3).
The virion-related parameters D V , Δx , Δt in (4), and the release rate of
virions, g V , have been set as follows.

Diffusion Rate of Virions ( D V ): The diffusion rate or diffusion coecient for
virions, D V , measures how fast virions spread: the larger D V ,thefastervirions
will spread to neighboring sites and then to the entire grid. One way to deter-
mine D V from experimental results is to take a measure of the “patchiness” of
the infection, i.e. the tendency of infected cells to be found in batches. The au-
tocorrelation function offers a good measure of patchiness. Hence, we calibrated
D V by visually matching our simulation to the experimental autocorrelation.
We started with D V =3 . 18 10 − 12 m 2 / s which is the diffusion rate predicted

by the Stokes-Einstein relation for influenza virions diffusing in plasma at body
temperature. Ultimately, we found that D V =3 . 18

×

10 − 15 m 2 / s, a value 1,000-
fold greater than the Stokes-Einstein diffusion, yielded the best agreement to the
experimental autocorrelation. This is illustrated in Fig. 3 where the experimental
autocorrelation is plotted against simulation results for different values of D V .

×

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

01234 56 7 8 9 10 11 12 13 14

15

lag (number of sites away)

Fig. 3. Autocorrelation at 24 h post-harvest for the experiments (full line, full cir-
cles) compared against the autocorrelation produced by the simulation when using
a diffusion coecient of D V =3 . 18 × 10 − 12 m 2 / s (dotted line, empty squares), and
D V =3 . 18 × 10 − 15 m 2 / s (dashed line, empty triangles). All parameters are as in Ta-
ble 1 except for the D V =3 . 18 × 10 − 12 m 2 / s simulation where k was set to 4 per
virions per hour to preserve the same fraction of cells infected at 24 h post-harvest.
The autocorrelation have been “normalized” to be one for a lag of zero.

Grid Spacing or Diameter of Epithelial Cells ( Δx ): The diameter of
epithelial cells was estimated from “en face” and cross-section pictures of the
experimental monolayer. The average epithelial cell diameter was found to be
about 11

± 1 µ m. We use Δx =11 µ m.