Appendix A - Derivation of Discrete-time Evolutionary
Model Parameters
Under the assumption that both natural mortality, µ, and average virulence, v̄ are constant
across all ages of infection, the quantities k and (s) in equations 1 and 2 of the main text
simplify to
k =
1
P
a ¯ (a) S
a
(1
v̄)a
µ
a=0
1
P
,
a (1
(A-1)
v̄)a
µ
a=0
(s) =
1
P
¯ (a) S
a
(1
µ
µ
v̄)s
v̄)a
a=s
s (1
,
(A-2)
where is the growth rate at the stable age distribution and (1 µ v̄)x gives the probability
of an infection lasting to age x. These are the discrete-time analogues of expressions presented
in the companion theory paper (Day et al., submitted ). For the first analysis focusing on
the evolutionary dynamics of transmission alone, we assume all genotype-specific virulences
are zero, thus v̄ is zero. For all analyses, µ is set to 0.005, which corresponds to a lifespan of
200 days (an underestimate for lab mice, but an over estimate for wild mice).
To predict the evolutionary trajectories in an endemic scenario, we need to define a situation
where the number of infected hosts is not growing in size. Mathematically, this occurs when
the leading eigenvalue of L is equal to 1 and there exists a unique value of the number of
susceptible individuals, S ⇤ , for which this occurs. This value is given by
1
S⇤ = P
n
¯ (a) (1
a=0
22
µ
v̄)a
.
(A-3)
Appendix B - Supplementary Results
Figure B-1 shows the stable age distribution in the epidemic and endemic scenarios. In an
expanding epidemic, most infections are very young, while in an endemic situation, older
infections become more frequent.
0
5
10
15
20
infection age
25
30
35
0.4
0.3
0.0
0.1
0.2
Proportion
0.3
0.0
0.1
0.2
Proportion
0.3
0.2
0.0
0.1
Proportion
(c)
0.4
(b)
0.4
(a)
6
8
10
12
infection age
14
16
6
8
10
12
14
16
infection age
Figure B-1: Average stable age distributions, q(s), in epidemic (red) and endemic (black scenarios). Shown are the average distributions across 10000 bootstrap replicates for infections
in (a) C57 mice, (b) C57 mice, truncated data set, (c) MF1 mice.
Figure B-2 shows how evolutionary trajectories can be qualitatively altered by changing
how host level measures are mapped to disease life history traits. We repeated all of the
analyses in the main text, but with a di↵erent mapping of within-host parasite densities to
transmission. In particular, we increased the linear coefficient of this mapping by an order
of magnitude. This increases estimates of genetic variance and covariance, while leaving the
qualitative patterns unaltered.
With this parameterization, the evolutionary predictions for the truncated C57 data set, in
the epidemic case, are altered. Unlike in the main text, selection for increasing transmission
rate can overcome selection for lower virulence, because the genetic variance in transmission rates has been increased. The predictions for the endemic case are, here, unchanged
presumably since the strength of selection for increasing transmission remains too low to
overcome the strength of selection for lower virulence. The other two data sets show quantitative changes in predictions (e.g., responses to selection for higher transmission rates are
now stronger in the epidemic cases since, again, genetic variance has been increased), but
hasn’t qualitatively altered evolutionary trajectories. Changing the mapping of within-host
densities to transmission in these cases also increases the magnitude of the covariances between transmission at di↵erent ages. Consequently, the evolutionary predictions are still
constrained by negative covariances (i.e., trade-o↵) between transmission rates, and the
qualitative patterns are unchanged.
23
20
25
30
6
8
10
15
20
infection age
25
30
14
5e-06
0e+00
16
6
8
35
rate of change of average virulence
4e-04
0e+00
6
8
10
12
infection age
10
12
14
16
14
16
infection age
-4e-04
rate of change of average virulence
0.0015
5
12
infection age
0.0005
-0.0005
rate of change of average virulence
0
10
-5e-06
rate of change of average transmission
1e-05
5e-06
-1e-05 -5e-06 0e+00
35
14
16
4e-05
15
infection age
0e+00
10
-4e-05
5
(c)
-8e-05
0
rate of change of average transmission
0.00010
0.00020
(b)
0.00000
rate of change of average transmission
(a)
6
8
10
12
infection age
Figure B-2: Evolutionary predictions for transmission rate (top panel) and virulence (bottom panel) when the relative cost of virulence is lower. As before, (a) C57 mice, (b) C57
mice, truncated data set, (c) MF1 mice. Red lines are predictions for epidemic conditions,
black lines for endemic. Solid lines denote mean predictions and dotted lines denote confidence intervals. Changing how within-host parasite densities map to transmission rates
qualitatively alters the evolutionary trajectories in (b) but not (a) or (c).
24