Figure 1.
Flow diagram for the within-host model.
,
, and
are the variables describing uninfected cells, infected cells, and infectious virus. Uninfected cells become infected at rate
, infected cells produce virus at rate
and die at rate
. Virus decays at rate
. Solid lines indicate physical flows, dashed lines indicate interactions.
Table 1.
Initial conditions and parameter values for the within-host model.
Figure 2.
Flow diagram for the between-host model.
,
and
are the variables describing susceptible hosts, infected hosts, and pathogen (i.e. virus) in the environment. Transmission can occur directly between uninfected and infected hosts at rate
and through contact of uninfected hosts with virus in the environment at rate
. Infected hosts shed virus into the environment at rate
, and recover (and are assumed to become immune to re-infection) at rate
. Virus in the environment decays at rate
. Note that the parameters
,
and
, i.e. the rate of transmission between hosts, the rate of shedding and the rate of recovery all depend on the time since infection. Solid lines indicate physical flows, dashed lines indicate interactions.
Table 2.
Parameters for the between-host model.
Table 3.
Summary of quantities linking the within-host and between-host scales.
Figure 3.
Decay rate for 12 different influenza strains as function of temperature.
Symbols show data, lines show best fit of an exponential function. Virus decay for all strains was measured at the indicated temperature, a pH of 7.2, and salinity of 0. Decay for each strain was measured once for these specific conditions. See [33] for more experimental details.
Table 4.
Best fit values for the different influenza strains.
Figure 4.
Temperature trade-off between strains.
A) Decay rates for H8N4, H9N2 and H10N7, plotted on a log scale to illustrate the cross-over of decay rates. B) absolute values of and
for all strains, (note the log scale). C) Ranks of these parameters. Also plotted in each figure are regression lines.
Figure 5.
Best fit of within-host model to fecal virus load from influenza infections of mallards (Anas Platyrhynchos).
The limit of detection for the virus load was (
= 50% Egg Infectious Doses) and is indicated by the dashed horizontal line. See [69] for more details on the experiments and data. Fitting was done using a least squares approach for the logarithm of the virus load, corresponding to the assumption of log-normally distributed errors [89]. For data at the limit of detection (i.e. left-censored data), differences between model and data were accounted for if the model was above the data point, but not if the model took on any value below the limit of detection [75].
Figure 6.
Relative Fitness for the 12 influenza strains.
A) direct transmission (equation 19) and B) environmental transmission (equation 20) scenarios. We plot fitness for the three different link functions, , between within-host virus load and transmission/shedding described in the model section, i.e.
,
and
given by equations (12), (15) and (16). Strains are sorted according to within-host fitness, with H8N4 having the best within-host fitness (i.e. lowest value of
, see Table 4). We arbitrarily chose H1N1 as the reference strain, which therefore has a relative fitness of 1.
Figure 7.
Virus decay rate at different temperatures.
A) environmental, between-host temperature, , and B) within-host temperature,
, as a function of
(decay rate at 0 degrees Celsius). Solid lines are theoretical values obtained by choosing a value of
and computing the corresponding value for
from the regression equation
, where the values for
and
are the best-fit values obtained previously by fitting the decay data for the different strains. The dashed horizontal line indicates the level of
above which within-host fitness is so small that no infection takes place. H5N2 is highlighted as a strain with poor persistence at both low and high temperatures – see text.
Figure 8.
Normalized Fitness as measured by and
for direct transmission and environmental transmission.
The dashed vertical lines indicate the level of above which
becomes so large that no infection takes place (c.f. horizontal line in figure 7). Note that results for
and
are virtually indistinguishable and therefore the curves are on top of each other.